Insight Editions · No. 227
The Fabric of the Cosmos
Space, Time, and the Texture of Reality
Brian Greene
Brian Greene asks what space and time really are, from Newton’s bucket and Einstein’s spacetime to entanglement, the arrow of time, inflation, branes and holography.
A study guide to The Fabric of the Cosmos by Brian Greene. Shared free for personal reading; not for sale and not endorsed by the authors or publishers. Please read and buy the original.
The Book on One Page
Brian Greene's The Fabric of the Cosmos (2004) asks what space and time really are, and follows physics from Newton's spinning bucket of water to the possibility that spacetime itself is an approximation. Its two threads intertwine: is space a "something"? (Newton yes, Leibniz and Mach no, Einstein "spacetime is"), and why does time have an arrow? (because the universe began in an extraordinarily ordered state, which inflation may explain). The book ends by surveying how string theory, branes, experiments, time machines and holography may transform both.
The whole argument in ten moves
- Space is at least a something. Newton's spinning bucket shows acceleration is absolute: relative to what? Mach said the distant stars; general relativity says spacetime, a dynamic entity that is the benchmark for acceleration, though it bends to matter.
- Space and time are relative; spacetime is absolute. Light's constant speed forces moving clocks to slow and observers to slice spacetime into different nows; all agree on the four-dimensional whole.
- The universe is not local. Quantum mechanics lets entangled particles act as one system however far apart; Bell's theorem and experiments (Aspect 1982 to loophole-free 2015; Nobel 2022) rule out local hidden variables, yet no usable signal outruns light.
- Time may not flow. If reality is what is simultaneous with me, and a distant stroller's now is as valid as mine, every event is equally real: the block universe. The flow lives in memory, frame by frame.
- The arrow comes from the beginning. Laws are time-symmetric; entropy should rise toward the past as well as the future. Only a low-entropy big bang saves our memories from being fluctuations.
- Symmetry shapes laws, time and the cosmos. Change (broken time symmetry) makes time; uniformity (spatial symmetry) makes a cosmic clock; heating restores symmetries the Higgs ocean hides.
- Inflation supplies the bang. Negative pressure makes gravity repulsive; a field stuck on a plateau drove expansion by 1030 or more, explaining uniformity, flatness and, through stretched quantum jitters, the seeds of galaxies. It also leaves the low-entropy smoothness time's arrow needs.
- Quantum gravity needs strings (perhaps). Below the Planck length, quantum jitters wreck smooth geometry. Strings smear them out, contain the graviton, and require extra dimensions whose shape would fix particle properties.
- Branes, tests and time machines. M-theory adds branes; we might live on one, with large extra dimensions, TeV strings, micro black holes or a cyclic cosmos. Experiments were coming. Wormholes and teleportation show what physics permits in principle.
- Spacetime may be emergent. Quantum averaging, dualities (R ↔ 1/R), and black-hole entropy ∝ area suggest space is not fundamental; holography says a world with gravity can be equivalent to one with fewer dimensions and none.
Chapter map
| Part | Ch | Title | One-line payload |
|---|---|---|---|
| P | Preface | Space and time frame everything yet remain mysteries; the book explores them | |
| I. Reality's Arena | 1 | Roads to Reality | Classical, relativistic, quantum and cosmological realities; the arrow; unification |
| 2 | The Universe and the Bucket | Newton's absolute space vs Leibniz and Mach's relationalism | |
| 3 | Relativity and the Absolute | Special and general relativity: relative space and time, absolute spacetime | |
| 4 | Entangling Space | Double slit, EPR, Bell; the universe is non-local | |
| II. Time and Experience | 5 | The Frozen River | Now-slices, Chewie's stroll; the block universe; flow as memory |
| 6 | Chance and the Arrow | Entropy; the double-headed arrow; Boltzmann's quagmire; low-entropy big bang | |
| 7 | Time and the Quantum | Sum over histories, delayed choice, quantum eraser; measurement problem | |
| III. Spacetime and Cosmology | 8 | Of Snowflakes and Spacetime | Symmetry, cosmic time, expansion, shape of space |
| 9 | Vaporizing the Vacuum | Phase transitions, the Higgs ocean, electroweak and grand unification | |
| 10 | Deconstructing the Bang | Repulsive gravity, inflation, horizon and flatness, dark energy | |
| 11 | Quanta in the Sky with Diamonds | Quantum seeds of galaxies; 20-pound universe; inflation and the arrow | |
| IV. Origins and Unification | 12 | The World on a String | GR vs QM; superstrings; extra dimensions; Calabi-Yau shapes |
| 13 | The Universe on a Brane | M-theory, branes, braneworlds, large dimensions, cyclic cosmology | |
| V. Reality and Imagination | 14 | Up in the Heavens and Down in the Earth | Experiments; 2004 forecasts vs 2026 results |
| 15 | Teleporters and Time Machines | Quantum teleportation; time-travel paradoxes; wormholes | |
| 16 | The Future of an Allusion | Emergent spacetime, entropy bounds, holography, loops and strings | |
| S | Synthesis | 25 takeaways, frameworks, whole-book critique, who's who, connections |
How to read this edition. Each chapter opens with its payload in a lede, keeps the book's arguments, numbers, people and thought experiments, and closes with Going deeper (technical content from the book's endnotes), a Critical lens (errors, omissions and contested claims, flagged rather than silently corrected), a 2026 update (what twenty-two years of data have done to the argument), a So what box for practitioners, and a short Python experiment you can run. Book figures are reproduced where they carry information; all other visuals are new.
Then and now: the headline changes since 2004
- Found: the Higgs boson (2012), gravitational waves (2015), black-hole shadows (2019, 2022), loophole-free Bell violation (2015), a nanohertz gravitational-wave background (2023).
- Not found: supersymmetry, large extra dimensions, micro black holes, string harmonics, WIMP dark matter, primordial gravitational waves.
- Revised: universe age 13.8 billion years; recipe 4.9 / 26.4 / 68.7 percent; flatness to 0.2 percent; the Hubble tension; dark energy possibly weakening (DESI 2025).
- Transformed: entanglement as a technology (quantum networks, teleportation over 1,400 km) and as the possible origin of spacetime (holography, ER = EPR).
Preface: The Arena of Reality
Space and time are the arena of everything we do, think and experience, yet science still cannot say what they are. Greene's book follows three hundred years of investigation into a handful of deep questions: Are space and time real things or useful ideas? Are they fundamental or emergent? What does empty mean? Does time have a beginning and an arrow? Can we manipulate space and time? Running beneath all of them is a harder question: what is reality?
The questions
- Real or abstract? Is empty space, like a blank canvas, a real entity or a mere idea?
- Fundamental or emergent? Do space and time arise from more basic constituents?
- Empty? What does it mean for space to be empty at all?
- Beginning and direction? Does time start? Does it flow from past to future?
- Control? Can we bend, travel through or manipulate space and time?
Method over imagination. We only have access to perception and thought, so how do we know they reflect an external world? Philosophers and filmmakers (brains wired into artificial worlds) have long asked this. Physicists answer by choosing hard data and mathematics as guides, "not unrestrained imagination or unrelenting skepticism," and seeking the simplest theories that explain and predict experiments. That rules out brain-in-a-tank fantasies, yet the last century of physics has still produced revisions to everyday reality "as dramatic, as mind-bending, and as paradigm-shaking as the most imaginative science fiction."
The pendulum on empty space
Greene traces how questions are declared answered, overturned and reinterpreted across generations. The clearest example is empty space:
- 1600s, Newton: space is real, an absolute entity.
- 1800s, Ernst Mach: space is not real; only relations between material objects count.
- 1900s, Einstein: merges space and time into spacetime and largely refutes Mach, reformulating the question itself.
- Later: "empty" is redefined. Space is unavoidably filled with quantum fields and possibly a uniform cosmological constant: "modern echoes of the old and discredited notion of a space-filling aether."
- Coming experiments: space-based tests may confirm features of Mach's ideas that agree with general relativity (Gravity Probe B, Chapter 14).
What the book covers
In our own era: inflationary cosmology's insights into time's arrow, string theory's extra dimensions, M-theory's suggestion that our space may be a sliver floating in a grander cosmos, and the "wild speculation" that the universe is a cosmic hologram. Greene also covers Bell's theorem, delayed-choice experiments, quantum measurement, accelerated expansion, black holes made in particle accelerators and wormhole time machines.
- Audience: general readers with little formal training; as in The Elegant Universe, mathematics is replaced by metaphors, analogies, stories and illustrations, with warnings and summaries before the hardest sections.
- Unconventional angle: the early chapters cover standard relativity and quantum theory, but with a sustained focus on the corporeality of space and time: whether they are physical things.
- Controversy: unsettled issues get the leading views in the main text; points Greene considers closer to consensus have dissenting views in the notes.
- Manuscript readers: Raphael Kasper, Lubos Motl, David Steinhardt and Ken Vineberg read the whole manuscript, some repeatedly; also David Albert, Richard Easther, Shelley Goldstein, Andrei Linde, Maulik Parikh, Marlan Scully, John Stachel and others.
- Conversations: Andreas Albrecht, Sean Carroll, Alan Guth, Justin Khoury, Hiranya Peiris, Saul Perlmutter, Paul Steinhardt, Leonard Susskind, Neil Turok, Henry Tye, Erick Weinberg and others.
- Illustrations: Jason Severs. Funding: US Department of Energy, National Science Foundation, Alfred P. Sloan Foundation.
Roads to Reality: Space, Time, and Why Things Are as They Are
The book's thesis in one line: human experience is a misleading guide to the true nature of reality, and space and time are where that gap is widest. Greene sketches five successive pictures of reality (classical, relativistic, quantum, cosmological and unified), each overturning intuitions the last left intact, and introduces the thread that runs through the whole book: the arrow of time, an everyday feature that the laws of physics do not explain and that leads back to the highly ordered birth of the universe.
Camus and the case for physics
As a teenager Greene took a thin book down from the top shelf of his father's bookcase: Albert Camus's The Myth of Sisyphus, which opens by declaring that the only serious philosophical question is whether life is worth living, and that questions such as whether the world has three dimensions "come afterward." Greene agreed Camus chose the right ultimate question, but rejected his claim that physics is secondary to it.
Physical reality sets the arena for every judgment about life. If we lived in caverns and had never seen the surface, or had only the sense of touch, or our minds stopped developing at five, our appraisal of life would be thoroughly compromised, and would change radically once we gained sight, the surface or maturity. "Assessing life through the lens of everyday experience is like gazing at a van Gogh through an empty Coke bottle." The overarching lesson of the last century of science is that human experience is often a misleading guide to the true nature of reality.
Greene distinguishes this from the claims of occultists, astrologers and religious traditions that reality lies beyond experience: his claim rests on "ingenious innovators and tireless researchers" who have peeled back the cosmic onion "enigma by enigma." Space and time, from Aristotle to Einstein, from the astrolabe to the Hubble Space Telescope, top the list of ideas being "fantastically revised."
Isaac Newton: space and time simply are: an inert, universal stage on which events play out.
Gottfried Wilhelm Leibniz: "space" and "time" are only a vocabulary of relations between where objects are and when events happen.
Albert Einstein: space and time are raw material of reality: they warp, curve and take part in cosmic evolution.
Five pictures of reality
1. Classical reality
Galileo, Descartes and Newton launched the modern scientific age: the cosmos has an order accessible to reasoning and mathematics, so events are explicable and predictable. Galileo dropped weights from a leaning tower (or so legend has it); Newton studied falling apples (likewise) and the moon's orbit. Newton stole the show: a handful of equations synthesised everything known about motion on earth and in the heavens. Three centuries later they sit on introductory chalkboards, NASA flight plans and research calculations.
To make his equations meaningful Newton had to say what space and time are. In a few sentences of the Principia he declared them absolute and immutable, an invisible scaffolding. Critics objected that it made little sense to grant existence to something you can't feel, grasp or affect, but the theory's predictive power "quieted the critics": for two hundred years absolute space and time were dogma.
2. Relativistic reality
Classical physics matched intuition: push and it speeds up; throw harder and it hits harder; more mass, more gravitational pull. In the 1860s James Clerk Maxwell extended the framework to electricity and magnetism, with harder mathematics but equal success.
"Two clouds." In 1894 experimentalist Albert Michelson said "most of the grand underlying principles have been firmly established," quoting an "eminent scientist" (probably Lord Kelvin) that what remained was more decimal places. In 1900 Kelvin did note "two clouds" on the horizon: one concerning light's motion, the other the radiation emitted by heated objects. Both were thought minor. Within a decade each had ignited a revolution.
The first cloud led to relativity. In a few weeks in spring 1905 Einstein found that space and time are not independent and absolute but enmeshed and relative; in 1915 general relativity showed that space and time warp and curve and take part in cosmic evolution. Newtonian physics remains an excellent approximation except at extremes of speed and gravity, but "utility and reality are very different standards": features of space and time that feel like second nature are "figments of a false Newtonian perspective."
3. Quantum reality
The second cloud led to quantum mechanics (by the 1930s). Classical physics, and relativity too, says the past and future are etched into the present: know all positions and velocities now and you know them at every other time. Quantum mechanics says even perfect measurements yield only probabilities. "The universe ... Participates in a game of chance." Things can hover partly one way and partly another until an observation forces a definite outcome, which cannot be predicted, only its odds.
Quantum connections bypass space. A 1935 paper by Einstein, Boris Podolsky and Nathan Rosen, meant as an attack on quantum theory, can now be read as among the first to show that, taken at face value, quantum mechanics links something done here instantaneously to something happening there, at any distance. Einstein thought this ludicrous. By the 1980s experiments confirmed it. Normally, to affect the far side of a football field you must go there or send something across; quantum mechanics shows that, in some circumstances, two objects far apart behave as a single entity, and because space and time are linked, these connections have "temporal tentacles" too.
4. Cosmological reality: the arrow of time
Physics must also explain the reality we do experience, and here a gap remains: Arthur Eddington's arrow of time. Eggs break but don't unbreak; candles melt but don't unmelt; we remember the past, not the future; people age and never unage. If forward and backward in time were as symmetric as left and right, the world would be unrecognisable.
Yet the accepted laws of physics treat both directions of time identically. That mismatch is a huge puzzle. The most convincing resolution, rooted in Ludwig Boltzmann's nineteenth-century work and developed by Roger Penrose, invokes the beginning of the universe: special, highly ordered conditions at or just after the big bang may have imprinted a direction on time, as winding a clock spring sets it ticking. The breaking of an egg bears witness to conditions at the birth of the universe some 14 billion years ago.
That shifts the puzzle into cosmology. General relativity applied to the whole universe gave the big bang theory (Chapter 8); the mid-1960s discovery of the predicted microwave background and 1970s work on matter at extreme temperatures (Chapter 9) made it the leading theory. But it could not explain the shape of space, the uniform temperature of the microwave radiation, or why the early universe was so ordered. Inflationary cosmology (late 1970s, early 1980s; Chapter 10) inserted a burst in which the universe grew by a factor larger than a million trillion trillion (1030) in less than a millionth of a trillionth of a trillionth of a second (10−30 s), addressing all three (Chapter 11).
5. Unified reality
Inflation rests on general relativity, which fails when combined with quantum mechanics, exactly what the universe's first instants require. Combined equations predict infinite probabilities, which are not "a sure thing" but meaningless. For half a century most physicists kept the two theories apart: relativity for the big, quantum mechanics for the small. A few situations fall in the "demilitarised zone": the centre of a black hole (a star crushed to a point) and the big bang (the observable universe compressed smaller than an atom).
Einstein spent his last thirty years seeking a unified theory, which distanced him from mainstream physics; today unification ranks among the most important problems in theoretical physics. The leading contender is superstring theory (Chapter 12):
- Strings, not dots. Each particle is a tiny filament of energy, about a hundred billion billion (1020) times smaller than an atomic nucleus, whose vibration pattern sets its properties, as a violin string's patterns set its notes. Electron, quark and neutrino are different vibrations of the same entity.
- Consistency. It merges general relativity and quantum mechanics without infinite probabilities, and can embrace all forces and all matter.
- Extra dimensions. It requires nine spatial dimensions plus time; its fuller form, M-theory, requires ten plus time: eleven spacetime dimensions. We have "glimpsed but a meager slice of reality."
- Where they hide. Extra dimensions may be tightly crumpled (Chapter 12) or large but invisible to our probes (Chapter 13); their shapes may decide why the universe has stars and planets, and large ones could hide other nearby worlds.
- Testable soon? Next-generation accelerators might produce microscopic black holes or new particle species (Chapter 13).
If correct, "the reality we have known is but a delicate chiffon draped over a thick and richly textured cosmic fabric," and Camus's "three dimensions or more" would be anything but a detail.
Past and future reality
Superstring theory may one day let us calculate what happened at the very beginning. Superstring cosmology already offers new cosmological frameworks (Chapter 13), astrophysical tests (Chapter 14) and first insights into time's arrow. Letting imagination run, deeper understanding might even let us travel through spacetime (Chapter 15), "extremely unlikely," but understanding is its own empowerment.
Coming of age in space and time
"Even answers that are ultimately proven wrong represent the result of a deep engagement with the cosmos." Brian Greene
Camus found hope in Sisyphus, eternally pushing his rock uphill; Greene's heroes became Newton, Einstein, Niels Bohr and Richard Feynman. Feynman's description of a rose (he could enjoy its beauty as anyone, but physics added the wonder of its molecular, atomic and subatomic processes) hooked him for good. Professional physics is slower than teenage dreams (chalkboards, dead ends), but even stalled effort brings "a closer connection to the cosmos." Unlike Sisyphus, science's rock does not roll back down: each generation builds on the last, so the task is "the most exquisite and noble of tasks." Our species, "on the outskirts of the Milky Way," may be "finally reaching childhood's end." It all began with a simple question: What is space?
2026 update: where the five pictures stand
- Relativistic: confirmed in every new regime: gravitational waves (LIGO, 2015), black-hole images (EHT, 2019 and 2022), and gravitational time dilation measured with optical clocks over a millimetre height difference (JILA, 2022).
- Quantum: loophole-free Bell tests (2015) settled the EPR question experimentally; Aspect, Clauser and Zeilinger won the 2022 Nobel Prize.
- Cosmological: the universe's age is now 13.8 billion years (Planck 2018); inflation's predictions (flatness, nearly scale-invariant ripples) hold, but its gravitational-wave signature is still unseen.
- Unified: string theory remains untested; the LHC (from 2010) found the Higgs (2012) but no extra dimensions, mini black holes or superpartners through Run 3.
- Arrow of time: still explained by a low-entropy beginning (the "past hypothesis"); why the beginning had low entropy remains open.
So what. Greene's "utility versus reality" distinction is a useful discipline in enterprise modelling: Newtonian physics is useful even though it is not true, exactly like a credit scorecard or an LLM's apparent reasoning. Know which standard a model meets before relying on it outside the regime it was validated in. And the "two clouds" story is a classic warning about declaring a system finished: the anomalies everyone files as minor are often where the next architecture comes from.
The Universe and the Bucket: Is Space a Human Abstraction or a Physical Entity?
A spinning bucket of water poses the question of the whole first half of the book: is space a real thing or only a language for relations between objects? Newton argued the water's concave surface proves that acceleration is measured against absolute space, which must therefore be real. Leibniz held that space is like an alphabet without letters, meaningless apart from objects. Ernst Mach offered a third answer: acceleration is measured against the average matter of the universe, so in an empty universe spinning would feel like nothing. The bucket has resisted a final verdict for over three centuries; Einstein's relativity transforms the question (Chapter 3).
The experiment
Hang a bucket of water by a twisted rope and let it go.
| Stage | Bucket | Water | Relative motion water/bucket | Water surface |
|---|---|---|---|---|
| 1. Release | Spinning | Still (friction hasn't caught up) | Yes | Flat |
| 2. Catch-up | Spinning | Spinning with bucket | None | Concave |
| 3. Rope rewinds | Slows, stops | Still spinning | Yes (as in stage 1) | Concave |
The water's shape cannot be explained by its motion relative to the bucket: relative motion gives a flat surface at stage 1 and a concave one at stage 3. So what is the water spinning relative to? Greene calls coming to grips with this, still unfinished after three centuries, "among the most important steps toward grasping the structure of the universe."
Relativity before Einstein
- Velocity is relative. To the batter a fastball approaches at 100 mph; to the ball, the batter does. Sitting on a train beside another, you can't tell which is moving. Galileo: drop a coin on a smoothly sailing ship and it hits your foot exactly as on land; you may declare yourself at rest and the water rushing by.
- Acceleration seems absolute. Lurch, brake, round a bend or get caught in a whirlpool, and you feel it, eyes closed, with no external comparison. Constant-velocity motion can't be felt; changes in velocity can.
- The puzzle: if velocity only makes sense by comparison, why do changes in velocity have intrinsic meaning? Is there a hidden comparison at work whenever we accelerate?
- Descartes, wary after the Inquisition's treatment of Galileo, wrote in his Principia Philosophiae that bodies resist changes to rest or straight-line constant motion, but Newton asked: motionless or constant speed with respect to what?
Isaac Newton: so driven by truth that he once pushed a blunt needle between his eye and the socket bone to study vision, and later, as Master of the Mint, harshly punished counterfeiters ("sending more than a hundred to the gallows," in Greene's words). "Had no tolerance for false or incomplete reasoning," and so introduced the bucket.
Newton's answer: absolute space
Newton removes the bucket as the reference, then imagines the experiment in deep, empty space: a giant bucket like an amusement-park ride, with astronaut Homer strapped inside (Newton actually used two rocks tied by a rope). If it spins, Homer feels pressed to the wall, face taut, stomach compressed, "his hair (both strands) straining back." In totally empty space, what is it spinning relative to? Newton's answer: space itself, a real entity he called absolute space.
Newton's account:
- True rest, motion and acceleration are defined relative to absolute space.
- Stage 1: bucket spins relative to absolute space, water doesn't: flat.
- Stages 2 and 3: water spins relative to absolute space: concave.
- Everyday cases: you feel a car's turn, a plane's takeoff, your arms flung out while skating, because you accelerate relative to absolute space. Spin the whole ice arena around a stationary skater (frictionless skates) and the skater's arms would not splay.
- Absolute space has observable consequences, so it exists: "once one has any solid evidence for something's existence, no matter how indirect, that ends the discussion."
Newton defined it by fiat: "I do not define time, space, place, and motion, as [they] are well known to all," and "Absolute space, in its own nature, without reference to anything external, remains always similar and unmovable." Yet he admitted discomfort: it is "a matter of great difficulty" to distinguish true from apparent motions "because the parts of that immovable space ... Do by no means come under the observations of our senses." Newton considered absolute space his most important discovery: "For Newton, it was all about space."
Space jam: the long argument
Einstein: words like "red," "hard" or "disappointed" are clear, but "space," whose link to experience is less direct, carries "a far-reaching uncertainty of interpretation." The question goes back to Democritus, Epicurus, Lucretius, Pythagoras, Plato and Aristotle: Is space distinct from matter? Can it exist empty? Is it finite?
- Theology. Henry More (perhaps one of Newton's mentors) held that space is never truly empty because it is filled with spirit. Newton allowed "spiritual substance" that is no obstacle to matter, and called absolute space the sensorium of God.
- Leibniz's relationism. Space is only a convenient way of encoding where things are relative to one another. Like the alphabet, which orders 26 letters (a next to b; d six before j; x three after u) but means nothing without them, space without objects is meaningless.
- Leibniz's arguments. God, who never acts arbitrarily, could not have chosen where to place the universe in a uniform void. Stripped of theology: if the whole universe moved ten feet left, how would we know? What is the universe's speed through space? If space can't be detected, why say it exists?
Newton moved the debate from philosophy to data. The concave water is an observable consequence of absolute space, he argued. Leibniz conceded: "I grant there is a difference between absolute true motion of a body and a mere relative change of its situation with respect to another body." For two hundred years relationism "generated hardly an echo"; Newton's laws, built on absolute space, took centre stage.
Mach and the meaning of space
The Manhattan perspective game. Walking with his father, Greene would describe a scene from an odd viewpoint ("I'm walking on a dark, cylindrical surface surrounded by low, textured walls, and an unruly bunch of thick white tendrils is descending from the sky") and the other had to guess it: an ant on a hot dog being topped with sauerkraut. The game taught that every vantage point is equally valid, which made Newton distressing: to an ant on a skater's boot the arena spins; to a spectator the skater spins; Newton says only one is really spinning, the one whose arms splay.
Three nagging questions about absolute space:
- If it is real, why does it single out accelerated motion but give no benchmark for constant velocity?
- Why can't it tell us where we are absolutely?
- How can it affect us (splaying arms) while we cannot affect it?
In the 1870s the Austrian physicist and philosopher Ernst Mach (or, Greene notes, a prevailing modern interpretation of ideas Mach initiated) pointed out that Newton dismissed the bucket as a reference too quickly. We judge spinning against the laboratory, the ground, the trees, or the fixed stars. Perhaps relative motion with respect to them explains the water's shape.
Thought experiment 1, with stars: floating motionless, you see the stars stationary. Someone sets you spinning: your limbs feel pulled outward, and the stars sweep in great arcs. Force and motion relative to the stars go together.
Thought experiment 2, total emptiness: no stars, no galaxies, no air: "a real existential moment." Do you feel spinning? Experience says yes, but we have never been in such a universe. Mach: with no benchmarks, there is no distinction between spinning and not spinning. Newton's rocks' rope stays slack; your arms stay put; your inner-ear fluid is undisturbed.
Newton's hidden assumption: that laboratory results (concave water, taut rope) would hold in empty space. Mach challenged exactly that assumption.
Mach's was the first significant challenge to Newton in over two centuries and sent shock waves well beyond physics: in 1909, living in London, Vladimir Lenin wrote a philosophical pamphlet discussing Mach's work.
Mach, motion and the stars
Objection: in a universe with just a few distant stars, you could see yourself spin, so presumably feel it, and how can a few faint stars act as an on-off switch for the sensation? Mach's reply:
The force of acceleration is a collective effect of all the matter in the universe, proportional to its amount. Empty universe: no force. One star: a minuscule force. More stars: stronger, up to the full familiar force in our universe. Same for planes accelerating, cars braking, lifts climbing. "You feel acceleration only when you accelerate relative to the average distribution of other material inhabiting the cosmos." Space returns to Leibniz's role: a language of relations with no independent existence.
Mach versus Newton
As an undergraduate Greene found Mach "a godsend": a benchmark in the open for all to see, the distributed matter of the cosmos, rather than an invisible "something" bordering on the mystical. Einstein had the same reaction. But Mach's idea was not a complete theory: he never said how distant stars or the house next door produce the force you feel when you spin. Without a mechanism it couldn't be tested. Gravity was the natural candidate, and Einstein drew heavily on Mach while building general relativity, which ended up transforming the absolute-versus-relational question rather than simply answering it.
Critical lens.
- Date. Greene says Newton described the experiment "in 1689"; it appears in the Scholium of the Principia, published in 1687.
- The gallows. "More than a hundred" counterfeiters hanged overstates the record; historians document Newton's prosecution of a few dozen coiners (about 28 convictions are often cited) rather than a hundred executions.
- Mach's views are a reconstruction. Greene rightly warns that "Mach argued" should be read as "the prevailing interpretation of an approach initiated by Mach"; Mach never wrote a precise proportionality law.
- Inertia versus gravity. Treating Mach's effect as gravitational is natural but not forced; modern relational programmes (Julian Barbour's "shape dynamics") pursue Mach without gravity being the mechanism.
2026 update: Mach tested
- Gravity Probe B (launched 2004; final results 2011) measured the geodetic effect to about 0.3% and frame dragging (the Earth's rotation dragging spacetime, a partly Machian effect) to about 19%, both agreeing with general relativity. LAGEOS satellite tracking and LARES (2012) have measured frame dragging to a few per cent.
- Rotating black holes imaged by the EHT (M87*, 2019) and spins inferred from LIGO/Virgo mergers show frame dragging at its most extreme.
- The bucket in GR: relativity says the water is concave because it rotates relative to the local inertial frame defined by the gravitational field (spacetime), which is mostly but not entirely determined by distant matter. Chapter 3 tells this story.
So what. The bucket is a lesson in choosing the reference frame before interpreting a measurement. "Revenue is up" (relative to what: last quarter, plan, market?) and "the model drifted" (relative to which baseline?) are bucket questions. Newton's move, insisting that an observable effect (the curved surface) demands a real reference, is also good architecture practice: if a behaviour can't be explained by the components you've listed, there is an unlisted component, perhaps shared infrastructure acting like "absolute space."
Relativity and the Absolute: Is Spacetime an Einsteinian Abstraction or a Physical Entity?
Einstein did not make everything relative; he replaced absolute space and absolute time with something new and absolute: spacetime. Special relativity makes distances, durations and simultaneity depend on the observer, yet all observers agree on the whole spacetime "loaf" and on which paths through it are straight, so spacetime, not space, is the benchmark for acceleration, even in an empty universe. General relativity then makes spacetime dynamic: gravity is its curvature, and freely falling observers define "unaccelerated." The verdict on the bucket: spacetime is a something, but a changeable, field-like something that relationists can live with.
Is empty space empty? Fields and the aether
Some discoveries answer questions; others make them irrelevant, as learning the earth is round dissolves "what lies beyond its edge?" Einstein's two discoveries did that to Newton's space: the question became "Is spacetime a something?"
- Michael Faraday (early 1800s) ran tens of thousands of experiments and introduced the field. Iron filings around a bar magnet trace an invisible something that exerts force beyond the magnet's body, as "an army provides a dictator" influence. Fields explain airport metal detectors, MRI scans and compasses.
- Electric fields crackle in wool scarves, zap you at doorknobs and tingle your skin in mountain lightning storms. Hans Oersted and Faraday found that changing electric fields make magnetic fields and vice versa, like stocks moving bonds moving stocks.
- James Clerk Maxwell (mid-1800s) gave the four equations of electromagnetism. Our phones, radios and Wi-Fi ride on the electromagnetic fields "vibrating all around us, in fact right through us."
- The key result: electromagnetic disturbances travel as waves at 670 million mph, exactly the measured speed of light. Light is an electromagnetic wave.
But 670 million mph relative to what? Maxwell's equations give the number with no reference point, like directions to a party "22 miles north" of nowhere. Ocean waves move relative to water; sound (767 mph, "Mach 1," after the same Ernst Mach) relative to air. So physicists invented a medium for light, the luminiferous aether (Aristotle's name for the stuff of the heavens), and assumed Maxwell's equations hold for someone at rest in the aether. The aether was the electromagnetic twin of Newton's absolute space, both invented to supply a reference for motion.
Unlike absolute space, the aether was testable: swim toward a water wave and it reaches you faster. In 1887 Albert Michelson and Edward Morley found light's speed exactly 670 million mph regardless of their motion or the source's. Rescue attempts followed (experimenters dragging the aether along; equipment warped by motion) until Einstein.
Relative space, relative time
- As a teenager: Einstein asks what a light wave looks like if you chase it at light speed: it should look frozen, like a handful of snow. But Maxwell's equations forbid stationary light, and no one has ever caught any.
- April and May 1905: an intense five weeks of work.
- June 1905: "On the Electrodynamics of Moving Bodies": special relativity, the end of the aether.
Einstein's guiding principle was simplicity ("make everything as simple as possible, but no simpler"):
- Experiments can't find the aether because there is none. Maxwell's equations don't need a medium: light travels through empty space.
- Maxwell names no rest frame because none is needed. The speed of light is 670 million mph relative to anything and everything. Chase it or flee it, you always measure the same speed. That solves the teenage paradox: light is never stationary.
Bart races a light beam
Bart and Lisa. Bart's nuclear-powered skateboard tops out at 500 million mph. Lisa counts down from 11 (Schopenhauer's favourite number, she says) and fires a laser. Each hour she sees the light go 670 million miles and Bart 500 million, so the light pulls away from Bart at 170 million mph. Newton says Bart must agree, since distances and durations are absolute. Bart returns dejected: the light sped away from him at 670 million mph, whatever he did. A century of precision experiments backs Bart.
Speed is distance divided by time, so if Bart and Lisa disagree about the light's speed relative to Bart, their measurements of distance and time must differ. Newton's absolute space ("without reference to anything external") and absolute time ("flows equably without reference to anything external") are wrong. "Space and time are in the eye of the beholder": each of us carries an equally accurate clock and ruler, and when we move relative to each other they disagree, in exactly the compensating way that keeps light's speed fixed. The maths needs only high-school algebra; the difficulty is the strangeness.
The core insight: motion through time
Special relativity in one picture: the retrofitted skateboard. Bart's board is now limited to 65 mph. Heading north at full speed, all his speed is northward; veer onto a northeast highway and some speed goes east, so he moves north more slowly. He can't compensate because his total speed is capped.
- Everything moves through time. A parked car, the street and you all advance through time "second followed by second."
- The combined speed through space and time always equals the speed of light.
- Moving through space diverts motion from time, so moving clocks run slow. At Bart's 500 million mph, Lisa sees his watch run at about two-thirds her rate: 2 hours on Bart's watch for every 3 on hers.
- Maximum speed through space is light speed, when all motion through time has been diverted. That is why nothing outruns light.
- Light doesn't age. A photon's watch would not tick at all; light "realizes the dreams of Ponce de León and the cosmetics industry."
"This is not dexterous wordplay ... This is how the universe works." In 1971 Joseph Hafele and Richard Keating flew cesium atomic clocks around the world on commercial Pan Am jets; the flown clocks lost a few hundred billionths of a second relative to ground clocks, as relativity predicts.
When rumours spread that new experiments had found the aether, Einstein said: "Subtle is the Lord, malicious He is not." A beautiful theory made irrelevant by nature would be malicious. The experiments proved wrong, and the aether evaporated.
Not everything is relative: absolute spacetime
Special relativity dealt mainly with constant-velocity motion; accelerated motion waited until 1915. Yet Einstein and others concluded, like Newton and unlike Mach, that in an empty universe you would still feel the spinning bucket's pull. Why?
Relativity introduces a new absolute: spacetime. Velocities, distances and durations are relative, but the four-dimensional whole is not. Einstein disliked the name "relativity theory"; he and others suggested "invariance theory," stressing what everyone agrees on.
Carving spacetime: streets, flip books and loaves
Marge and Lisa redesign Springfield. Both centre their 100-metre street grid on the Soaring Nuclear Monument (5th Street, 5th Avenue). Marge puts the Kwik-E-Mart at 8th Street and 5th Avenue; Lisa puts it near 7th and 3rd. Neither is wrong; their grids are rotated. There are no absolute streets.
- Spacetime as a flip book. Record Itchy and Scratchy's duel page by page: each page is a time slice, space at one moment. Expand the binding and make the pages transparent: the whole thing is a block of spacetime, and the pages are just one way to organise it. (Hermann Minkowski, Einstein's maths professor, who once called him a "lazy dog," saw spacetime this way.)
- Different observers slice differently. Just as Lisa and Marge rotate street grids, observers in relative motion slice the block into "now" pages at different angles: the relativity of simultaneity.
The duel on the train. Itchy and Scratchy stand at opposite ends of a moving railway car and will draw when a gunpowder flare midway between them goes off. Referee Apu, on the train (sipping his Chutney Squishee), sees the light reach both at once: a fair draw. Referee Martin, on the platform, sees Itchy moving toward the flash and Scratchy away from it; since light's speed is fixed, the light reaches Itchy first: foul! Both are right. At train speeds the disagreement is under a trillionth of a second; near light speed it would be large. Synchronising clocks the same way, the platform sees Itchy's clock set ahead of Scratchy's (say 12:06 versus 12:04).
The angle is set by relative speed. Apu's and Martin's slices are rotated through space and time, and since the speed of light caps relative speed, the maximum tilt is 45°. Greene's new metaphor: spacetime as a loaf of bread that different observers slice at different angles.
The bucket according to special relativity
Why spacetime is the benchmark:
- Everyone reassembles the same loaf. Different slicings, one region of spacetime: absolute spacetime exists.
- Everyone agrees on shapes. Marge and Lisa disagree on which streets a painted trail from the nuclear plant to the Kwik-E-Mart crosses, but both see it is straight.
- A straight path through spacetime = constant velocity (straight in space, uniform in time). All observers agree whether a spacetime path is straight.
- Acceleration = a curved path through spacetime. A resting astronaut traces a straight line; one circling with a jetpack traces a spiral; one blasting off traces another curve.
- So the spinning bucket accelerates relative to spacetime, even in an empty universe. "Spacetime, not space alone, provides the benchmark." The pendulum swings back: Leibniz (relational), Newton (absolute), Mach (relational), Einstein (spacetime as a "something").
| Newton | Special relativity | |
|---|---|---|
| Space | Absolute | Relative (observer-dependent) |
| Time | Absolute, flows equably | Relative; moving clocks run slow |
| Simultaneity | Universal | Relative (tilted slices) |
| What is absolute | Space and time separately | Spacetime and the shapes of paths |
| Benchmark for acceleration | Absolute space | Absolute spacetime |
| Bucket in empty universe | Concave | Concave |
Gravity and the age-old question
Einstein still loved Mach's idea and suspected gravity might supply the missing mechanism. He had a second, sharper reason: Newton's gravity acts instantly, contradicting relativity's speed limit.
The vanishing moon. After a disappointing evening (team lost, birthday forgotten, the last chunk of Velveeta eaten) you go boating at midnight under a high tide. Aliens zap the moon to the far side of the galaxy. Newton says its pull vanishes instantly, so the tide starts receding about a second and a half before you see the moon disappear (light needs that long to cross the quarter million miles): "like a sprinter jumping the gun." Gravity would outrun light. Einstein was sure that was wrong.
From about 1907 Einstein was obsessed with a new theory of gravity. His notebooks show half-formed ideas, near misses and premature "eureka"s. By 1915, with help at key points from mathematician Marcel Grossmann, he had general relativity: "the rare heroic struggle of a single mind to master the universe ... The crowning jewel of pre-quantum physics." He was answering the question Newton dodged: how does gravity act across space? ("I leave this problem to the consideration of the reader.")
The equivalence of gravity and acceleration
Acceleration can be felt, and its forces resemble gravity's in one key way: you can't shield yourself from either. In 1907, in "one of those flashes of insight that scientists spend a lifetime longing for," Einstein saw that gravity and accelerated motion are two sides of one coin.
Barney's leap. Losing the Springfield Challenge (a belt-size weight-loss contest) after two weeks on a liquid diet of Duff Beer, Barney jumps out of the bathroom window with the scale stuck to his feet. On the way down the scale reads zero: it falls exactly as he does. In free fall he is weightless, like an astronaut. In an evacuated shaft every atom falls together, so all bodily stresses vanish too; two rocks tied by a rope would keep the rope slack. (NASA trains astronauts this way in a modified 707 nicknamed the "Vomit Comet.")
Reverse it. In a floating space capsule, fire the boosters with just the right thrust and the scale reads exactly Barney's bathroom weight. Stand on the inner wall of Homer's spinning bucket at the right spin rate: same reading.
The principle of equivalence: the force of gravity and the force of acceleration are the same. Hence the radical turnabout: only observers who feel no force at all, gravity included, are unaccelerated. Freely falling Barney is the true benchmark; you, reading calmly at home, are accelerating upward. "It was Newton's head that rushed up to meet the apple, not the other way around." You feel gravity only when you resist it. (Even Apu, Martin, Itchy and Scratchy were not truly unaccelerated; it didn't matter because the duel was horizontal.)
Warps, curves and gravity
Observers moving at constant relative velocity slice spacetime at different angles; an accelerating observer's angle changes moment to moment, and the slices merge into curved slices fitting together "like spoons in a silverware tray." Using the geometry of Carl Friedrich Gauss and Bernhard Riemann, Einstein concluded: since gravity and acceleration are equivalent, gravity itself is warping and curving of spacetime.
- The flood-warped floor. A marble rolls straight across a smooth floor but follows the bumps after a flood. Empty spacetime is flat; matter and energy warp it; objects follow the warps. The earth orbits because it follows the sun's curvature.
- Your chair. Right now your body would slide down an indentation in spacetime made by the earth; the floor's upward push stops you. Dive off a high board and you "give in" to a spacetime chute.
- Time warps too: clocks in stronger gravity tick slower, and for ordinary bodies like the earth and sun the warping of time is far larger than the warping of space.
- The Einstein field equations fix exactly how much given matter and energy warp spacetime, doing for gravity what Maxwell's equations did for electromagnetism. Head-to-head with Newton, Einstein's predictions match reality more closely.
- Gravity's speed: warps and ripples travel at exactly light speed. Zap the moon and the tides recede at the same moment you see it vanish.
In general relativity space and time "come alive": matter here warps space there, which moves matter over there, which warps space further. Spacetime is a player, not a stage.
General relativity and the bucket
Einstein named Mach's idea "Mach's principle," wrote Mach an enthusiastic letter in 1913 saying the new theory would vindicate his analysis of the bucket, and in 1918 listed Mach's principle as the third pillar of general relativity. As the theory's subtleties emerged, he found it "increasingly difficult" to include, and late in life renounced it.
Where GR is Machian: the benchmark for acceleration is the freely falling observer, and free-fall paths are set by the gravitational field produced by all matter and energy in the cosmos: earth, moon, planets, stars, gas clouds, quasars, galaxies and the house next door, nearer and heavier things mattering more.
Where GR is not Machian: in an empty, unchanging universe there is no gravity, spacetime is flat, and GR reduces to special relativity. The tiny bucket barely curves anything. You still feel the spin; the rope still goes taut. Empty spacetime still provides a benchmark.
Verdict: "Mach's principle is an example of a provocative idea that provided inspiration for a revolutionary discovery even though that discovery ultimately failed to fully embrace the idea that inspired it."
Spacetime in the third millennium
The most straightforward reading of general relativity: spacetime is a something. But relationists can celebrate too. Newton's space and special relativity's spacetime were fixed and unchangeable, so acceleration relative to them was absolute. In general relativity spacetime is dynamic, responsive and an embodiment of the gravitational field; as Einstein argued a few years before his death, acceleration relative to it is acceleration relative to a field: as real, tangible and changeable as stones or stars. (Even in special relativity, zero gravitational field is still a field that can be measured and changed.)
And the plot thickens: quantum mechanics redefines empty and nothing; aether-like entities (fields, a cosmological constant) return, though none sets a standard of rest; and the most basic role of space, separating one object from another, is about to be challenged by quantum connections (Chapter 4).
Going deeper: the technical notes
- Spacetime interval. All observers agree on s2 = c2t2 − x2 − y2 − z2 between two events, even though they disagree on t and x separately. This invariant is what "absolute spacetime" means mathematically, and straight spacetime paths are those that maximise elapsed proper time.
- Time dilation and length contraction. A clock moving at speed v ticks slower by γ = 1/√(1 − v2/c2); lengths along the motion shrink by the same factor. In Greene's "speed through spacetime" picture, (speed through space)2 + (speed through time)2 = c2 in suitable units.
- Slice tilt. Relative speed v tilts now-slices by an angle whose tangent is v/c, so the maximum is 45° at light speed.
- Flip-book continuity. The pages show sample moments; time is treated as infinitely divisible (whether it truly is returns later).
- Gravity's pull on time. Clocks deep in a gravitational potential (even inside a hollow at a star's centre where no force is felt) run slow; GPS corrects for ≈+45 μs/day from weaker gravity and ≈−7 μs/day from orbital speed.
- GR's equations. Gμν = 8πG Tμν/c4: curvature on the left, matter-energy on the right. In empty space, gravitational waves are solutions that propagate at c.
- Mach in GR, further. GR has partly Machian effects such as frame dragging: a rotating mass drags local inertial frames around with it (the Lense-Thirring effect). Solutions like the Gödel universe (rotating matter with no global rest frame) show GR does not enforce Mach's principle in general.
Critical lens.
- The "Subtle is the Lord" quote is usually dated to 1921, during Einstein's visit to Princeton, in reaction to Dayton Miller's reported aether-drift results, not to a 1908 rumour.
- Did Michelson-Morley drive Einstein? Greene says Einstein knew "at least some" null aether experiments; historians debate how much Michelson-Morley specifically mattered versus Einstein's reflection on electrodynamics and the moving-magnet-and-conductor problem.
- "Single mind" overstates it slightly. Beyond Grossmann, David Hilbert derived the field equations almost simultaneously (November 1915), and Michele Besso helped with the perihelion calculation.
- "Spacetime is a something" is a substantivalist reading; philosophers of physics still argue (the hole argument, raised by Einstein himself in 1913) over whether GR's diffeomorphism invariance favours relationism.
2026 update: relativity tested to the hilt
- Speed of gravity: the neutron-star merger GW170817 (August 2017) arrived 1.7 seconds before its gamma-ray flash after 130 million years of travel, showing gravitational waves move at light speed to about 1 part in 1015: Greene's vanishing-moon thought experiment, settled.
- Time dilation: optical atomic clocks detect gravitational slowing across a 1 mm height (JILA, 2022); aluminium-ion clocks show special-relativistic dilation at walking speeds.
- No aether: modern Michelson-Morley tests with optical resonators bound any light-speed anisotropy to about 10−18.
- Equivalence principle: the MICROSCOPE satellite (final results 2022) confirmed that different materials fall identically to ≈1 part in 1015.
- Frame dragging: Gravity Probe B (2011) confirmed it at about 19%; LAGEOS and LARES to a few per cent.
So what. Relativity's deep lesson for system design is "find the invariant." Observers disagree on timestamps (clock skew, event ordering across services), but they can agree on causal order within light cones, which is why distributed systems use vector clocks and causal consistency rather than global "now." Spacetime slicing is the physics of why there is no global clock in a distributed ledger: simultaneity is a choice of frame. Greene's equivalence-principle move, redefining "at rest" as "no forces felt," is also a model for redefining a baseline: often the "obviously stationary" reference is the one actually accelerating.
Entangling Space: What Does It Mean to Be Separate in a Quantum Universe?
Space is supposed to separate things: to affect something over there, something must travel from here. Quantum mechanics says otherwise. Greene builds the case from the double-slit experiment and Born's probability waves, through Heisenberg's uncertainty and the 1935 Einstein-Podolsky-Rosen argument that quantum mechanics is incomplete, to John Bell's 1964 insight that "hidden" definite properties have testable consequences. Experiments by Clauser, Aspect, Gisin and others show nature violates Bell's prediction. The universe is not local: two entangled particles, however far apart, behave as one system. No usable signal outruns light, but whether quantum non-locality truly squares with relativity remains, for some, an open question.
Relativity's weirdness versus quantum weirdness
You can train your mind to "feel" relativity: imagine your now drifting from others' as you drive; imagine time running faster for you on a mountaintop than in the valley. We don't feel it because evolution gave no survival advantage to relativistic intuition. Relativity's strangeness is "born of comparison": my view is one of infinitely many that fit into one spacetime. Quantum mechanics is different: its weirdness is evident without comparison; it "shatters our own personal, individual conception of reality." It began with papers on radiation by Max Planck and Einstein around 1900 and took three decades to mature.
The world according to the quantum
Each age has its cosmic metaphor: the Indian myth of the giant Purusa dismembered into sky, earth and wind; Aristotle's concentric crystalline spheres; Newton's clockwork. Relativity refined the clockwork (no universal clock, no universal now), but kept determinism: know every particle's position and velocity and you know past and future. Quantum mechanics breaks the clockwork: we can never know a particle's exact position and velocity, never predict even simple experiments with certainty, only probabilities.
Locality: Newton and Einstein agree that things separated by space are independent, and that influence must travel across the gap: a bird flying over to peck its companion, a slingshot pebble, sound waves jostling air molecules into an eardrum, a laser beam, or a gravitational ripple from a moon shaken by alien pranksters. Voodoo violates locality. Recent experiments show that measuring a particle here can be entwined with the outcome of measuring a distant particle, with nothing sent between them, even when there isn't time for light to cross. Einstein called it "spooky." Greene: of all experimentally verified assaults on our picture of reality, "I find none more mind-boggling than the recent realization that our universe is not local."
The red and the blue: Mulder and Scully
The alien spheres. On vacation in Provence, Scully gets a call from Mulder: they've each received 1,000 lightproof titanium boxes, numbered identically. Each holds a sphere that flashes red or blue at random when its door opens, but boxes with the same number always flash the same colour. Testing confirms it. Scully: obviously each pair was preprogrammed the same way. Mulder: the letter says the choice is random at opening, and that any tampering (examining the sphere's chemistry, say) also makes it flash, so the trick can't be checked. Scully: like claiming "I'm really a blonde, but I become a redhead whenever anyone looks." She seems to have science on her side, but quantum data side with Mulder.
Quantum mechanics says a particle can hang in limbo between properties until measured, then commit randomly, and that two particles can be entangled so their random choices match, however far apart. Example: a reflected photon has a 50-50 chance of passing through polarised sunglasses; its entangled partner, miles away, facing another polariser, does the same thing. Einstein, like Scully, argued the particles were "programmed" at emission. For five decades the dispute seemed as unresolvable as Mulder and Scully's, until John Bell in the 1960s showed it could be settled experimentally; by the 1980s it was.
Casting a wave
- Light is a wave. Shine a laser through two scratched lines on overexposed 35 mm film (or a coffee plunger's wire mesh): you see bright and dark bands. Where crest meets crest the wave is bigger; where crest meets trough they cancel. That's interference, the "smoking gun" of waves, settling a debate Newton opened by claiming light was particles.
- Electrons are waves too. In 1927 Clinton Davisson and Lester Germer fired electrons at nickel crystal (equivalent to two slits) and saw an interference pattern, not two stripes.
- Even one at a time. Fire one electron every ten seconds and record each impact: after an hour, half a day, a day, the dots build up the interference pattern. Individual spectators standing up at random are not doing "the wave," yet each electron somehow carries a wave.
Probability and the laws of physics
Schrödinger first guessed the wave was smeared-out electron "stuff," but we always find all of an electron's mass and charge at one point. Max Born proposed the decisive answer: the wave is a probability wave. Where it is large, the electron is likely to be found; where small, unlikely; where zero, never. (Precisely, the probability goes as the square of the wave's size.)
- No one has ever seen a probability wave, and orthodox theory says no one will. We compute it with the equations of Schrödinger, Bohr, Heisenberg, Dirac and others, then repeat identical experiments: identical starting conditions do not give identical results, but the frequencies match the predictions to "spectacular precision" over eight decades.
- Every probability wave extends through the whole universe, so there is a tiny but genuine chance an electron here could be found in Andromeda.
- Everything is wave and particle. Protons and neutrons too; and light comes in photons: a 100-watt bulb emits about 1020 photons per second.
- What the wave is (the electron itself, something associated with it, a calculating device, or a summary of our knowledge) is still debated.
Classical probability (weather, craps): reflects missing information. Know every molecule or every detail of the dice and table, and Newton says you could predict with certainty.
Quantum probability: fundamental. No amount of data or computing power can do better than odds. "Probability reigns supreme in the microcosmos."
Why don't we notice? The air molecules you breathe won't suddenly rematerialise on the dark side of the moon: the probability for one electron is tiny (making your chance of marrying Nicole Kidman or Antonio Banderas look enormous), and for all of them together you'd need to marry your heartthrob and win every state lottery every week for longer than the age of the universe.
Einstein's challenges
- 1927 Solvay Conference: an electron is always found at a definite spot X, so wasn't it at or near X just before? Then the probability wave is just a stand-in for a deeper, exact description. Bohr's reply: a definite path can't explain one-at-a-time interference; before measurement the electron simply has no position. Measurement creates the reality it records. Einstein: "Do you really believe that the moon is not there unless we are looking at it?"
- 1930 Solvay Conference: Einstein's photon box (a scale, a clock and a shutter) seemed to show particles have features quantum mechanics forbids. Bohr was "knocked back on his heels," then within days refuted it using general relativity: Einstein forgot that gravity warps time, so the clock's rate depends on the box's height.
- 1935: the EPR paper, Einstein's most potent attack.
Heisenberg and uncertainty
The uncertainty principle (Heisenberg, 1927) as a prix-fixe menu. Some Chinese restaurants list dishes in columns A and B: order the first from A and you can't have the first from B (Peking Duck or Lobster Cantonese). Nature splits features into such pairs: position and velocity, spin about different axes, and so on.
- Precise knowledge of one member of a pair degrades knowledge of its partner, and Heisenberg's formula fixes the minimum trade-off exactly.
- Everyday objects: a car passing a stop sign known to ±1 cm has speed uncertainty just shy of 10−36 mph: a trooper could say "between 89.999…9 and 90.000…1 mph." No uncertainty-principle defence.
- Electrons: position known to a billionth of a metre means speed uncertain by about 100,000 mph.
- More than clumsy measurement. Heisenberg's picture (light bouncing off an electron kicks it) helps but misleads: a wave with evenly spaced crests describes a particle with exact velocity and completely unknown position, with no one touching it. "In quantum mechanics, uncertainty just is."
EPR: is quantum mechanics complete?
Is uncertainty a limit on knowledge or on reality? For Bohr it was a Zen koan: physics addresses only what we can measure. In 1935 Einstein, Boris Podolsky and Nathan Rosen pressed the point.
Rod and Todd. The Flanders twins stand back to back in Springfield's Nuclear Desert and walk in opposite directions at the same speed. Nine hours later Ned finds Rod 45 miles east walking 5 mph east; without seeing Todd (90 miles away) he knows Todd is 45 miles west walking 5 mph west. Indirect measurement, no contact.
The EPR argument:
- Two particles fly apart from a decay with equal and opposite velocities and mirrored positions (quantum mechanics can't predict the values, only the relationship).
- Measure the right particle's position and you learn the left one's, without touching it. Since you did "absolutely nothing" to the left particle, it must already have had that position.
- You could instead have measured the right particle's velocity, learning the left one's velocity, again undisturbed.
- So the left particle has both a definite position and a definite velocity, which quantum mechanics can't describe.
- Therefore quantum mechanics is incomplete.
The key assumption: "an object over there does not care about what you do to another object over here." There's no clash with uncertainty itself (you still can't know both), but EPR say reality contains elements the theory omits.
Wolfgang Pauli answered for the pragmatists: worrying whether something unknowable exists is like "the ancient question of how many angels are able to sit on the point of a needle." EPR insisted the moon is there when no one looks. The standoff echoed Newton versus Leibniz on space, until, as Newton's bucket once did, John Bell (1964) turned philosophy into experiment: "the most profound discovery of science," as one commentator put it.
Bell and spin
Bell's key was a property with three or more mutually uncertain versions: spin.
- Quantum spin is always the same rate, clockwise or counterclockwise, about any axis you measure. Unlike a soccer ball spinning about a northeast axis (with spin shared between north and east), an electron never shows fractional spin: measuring along any axis forces all its spin to be clockwise or counterclockwise about that axis, and destroys information about other axes.
- David Bohm recast EPR for spin: two electrons emitted back to back with correlated spins; detectors set to the same axis always agree (Greene describes them as perfectly correlated; the usual convention is perfectly anti-correlated, an inessential relabelling). EPR logic then says each electron has a definite spin about every axis: so-called hidden variables.
Reality testing: Mulder's three doors
The new boxes have three doors (top, side, front). Opening any door makes the sphere flash red or blue at random; opening one door forever hides what the others would have shown. Same-numbered boxes, same door, same colour. Scully: each box carries a program such as red, blue, red (top, side, front), copied into its twin. Mulder: then let's choose doors randomly and independently and count agreements.
Mulder's (Bell's) argument:
- Take any program, say blue, blue, red. Label doors 1, 2, 3. There are 9 door combinations: (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3).
- Same color on 5 of 9: the three same-door pairs, plus (1,2) and (2,1), since doors 1 and 2 both give blue.
- Every program has at least two colours the same, giving at least 5/9; programs red-red-red or blue-blue-blue give 9/9.
- So if spheres (or electrons) carry programs, random door choices must agree more than 50% of the time.
- Translate: doors = three detector axes; colours = clockwise/counterclockwise. If EPR are right, spins agree more than half the time. Bell had found a way "to count Pauli's angels."
No smoke but fire: the experiments
- 1964: Bell's theorem; technology not yet ready.
- Early 1970s: Stuart Freedman and John Clauser (Berkeley) run the first tests.
- 1976: Edward Fry and Randall Thompson (Texas A&M).
- Early 1980s: Alain Aspect and collaborators (France): detectors 13 m apart; excited calcium atoms emit back-to-back photon pairs with perfectly correlated spins. Same settings: always agree. Random, independent settings: agreement is not more than 50%. It is 50%.
- 1997: Nicolas Gisin (Geneva) separates the detectors by 11 km: unchanged. "It might as well be 11 million kilometers, or 11 billion light-years."
Experiment, "not by theory, not by pondering, but by nature," proves EPR wrong. Their one assumption was locality: an object over there doesn't care what you do here. Since the measurements are simultaneous, no light-speed influence could connect them, so locality led to the >50% prediction, which fails. "An object over there does care about what you do to another object over here." Entangled particles are like magic dice, one thrown in Atlantic City and one in Las Vegas, each random, yet always agreeing. Entangled particles, though spatially separate, do not operate autonomously.
Entanglement and special relativity
The standard view: coexistence
If measuring one photon caused the other's spin, something would have to travel instantly, breaking relativity. The consensus: the conflict is illusory.
- One system. Their common origin binds the photons into one physical entity, so a measurement on it affects both parts at once.
- The information test. Relativity forbids faster-than-light matter, energy or information. Each detector's output is a random string of clockwise and counterclockwise results; you can't control any outcome, so no message can be sent. The lists are identical, but that can only be seen once they are compared by ordinary slower-than-light means (fax, email, phone). Correlation without cause and effect. No process has ever passed this test as faster than light.
The contrarian view: tension remains
Greene's gut sides with coexistence, but "two widely separated particles ... Somehow stay sufficiently 'in touch'." Some argue relativity's real core is that no observer's viewpoint is privileged, and the textbook account struggles with that:
- Collapse: in standard quantum mechanics, a measurement makes the probability wave collapse instantly across the whole universe: find a particle in New York and its wave in Andromeda drops to zero. Measuring the left photon clockwise zeroes the counterclockwise part everywhere, including at the right photon.
- This reproduces the data: with three axes chosen at random, it predicts agreement exactly 50% of the time, which is what Aspect found.
- But no one knows how, or whether, collapse happens. It doesn't follow from the equations; it is put in by hand.
- And "instantly" is frame-dependent. By the relativity of simultaneity, some observers say the left photon was measured first, others the right. A collapse simultaneous in one frame singles out that frame, clashing with relativity's egalitarianism. Proposals exist; the debate continues. Greene expects the majority view to prevail but notes that "subtle, foundational problems sometimes sow the seeds of future revolutions."
What are we to make of all this?
| Einstein's universe | Verdict |
|---|---|
| Local: what you do here matters only here | Ruled out by Bell and Aspect |
| Definite properties exist before measurement | Not ruled out by the data, though most physicists reject it |
David Bohm (1950s) built a version of quantum mechanics with both hidden variables and non-locality: particles always have definite positions and velocities, matching every quantum prediction, but the forces on a particle here depend instantly on conditions far away: "even more brazen" non-locality, little solace to Einstein.
Space no longer guarantees separateness: two things can be separated by enormous space yet lack independent existence. But Greene rejects "gushy" talk that everything is entangled with everything because all emerged from one place at the big bang. Entanglement is delicate: jostled by stray particles or equipment, it spreads into complex correlations with the environment and becomes virtually undetectable. That such connections exist at all, and can be seen across kilometres, shows that space is not what we thought. Next: what about time?
Going deeper: the technical notes
- Bell's 50% from quantum mechanics. For spin-½ particles prepared perfectly correlated, detectors at axes separated by angle θ agree with probability cos2(θ/2). With three axes 120° apart chosen at random: one-third of the time the axes match (agreement 1), two-thirds of the time they differ by 120° (agreement cos260° = ¼). Total: ⅓ + ⅔ × ¼ = ½ exactly, below the local-programs minimum of 5/9.
- CHSH form. Experiments usually test the Clauser-Horne-Shimony-Holt inequality: local hidden variables give |S| ≤ 2, quantum mechanics up to 2√2 ≈ 2.83 (Tsirelson's bound). Aspect measured S ≈ 2.70.
- Loopholes. Early tests left (a) the locality loophole (settings chosen slowly enough for a signal to pass), closed by Aspect's switching settings in flight and later by Weihs et al. (1998); and (b) the detection loophole (most photons undetected), closed with ions (2001). Both together were closed only in 2015.
- No-signalling theorem. Each side's statistics are unchanged by the other side's choice of setting: this is the precise statement that entanglement can't send information.
- Bohm's guidance equation. Particle velocity v = (ħ/m) Im(∇ψ/ψ), evaluated with the many-particle wavefunction, so each particle's motion depends instantly on all others' positions.
- Interpretation spectrum. Copenhagen (collapse), Many Worlds (no collapse; branches), Bohmian (hidden positions), GRW (spontaneous collapse), QBism and relational (probabilities are agent-relative). All reproduce Bell violations; they differ on what is real.
Critical lens.
- Aristotle and hell. Aristotle did use about 55 nested spheres, but the "seven circles of hell" belong to the medieval Christian (Dantean) cosmos, not to Aristotle.
- Born's date. Born proposed the probability interpretation in 1926 (Nobel 1954), not 1927.
- "Nature proved EPR wrong" is slightly too strong. Bell violations refute local hidden variables. Escape routes remain in principle: superdeterminism (detector settings correlated with the hidden variables), retrocausality, or Many Worlds (where measurements don't have single outcomes, so Bell's assumptions don't apply in the usual way). Greene notes the "most straightforward reading," which is fair.
- "Detectors agree exactly 50%" depends on Greene's chosen geometry (three axes at 120°); real experiments test CHSH inequalities with other angles.
- Collapse and relativity is presented as a live tension; relativistic collapse models (Tumulka's rGRWf, 2006) and Many Worlds both show non-locality can be made frame-independent, at some cost.
2026 update: entanglement goes global
- Loophole-free Bell tests (2015): Delft (Hensen et al., electron spins in diamonds 1.3 km apart), Vienna and NIST (photons) closed the locality and detection loopholes simultaneously.
- Freedom of choice: the BIG Bell Test (2016) used 100,000 volunteers' keystrokes to choose settings; cosmic Bell tests (2017 to 2018) used light from quasars billions of years old, pushing any superdeterministic conspiracy back nearly to the big bang.
- Space: China's Micius satellite distributed entangled photons over 1,200 km (2017) and enabled intercontinental quantum key distribution.
- Nobel Prize 2022: Alain Aspect, John Clauser and Anton Zeilinger "for experiments with entangled photons, establishing the violation of Bell inequalities."
- Technology: entanglement now underpins quantum computing (Google Willow, 2024, 105 qubits with below-threshold error correction), quantum sensing and quantum networks; Delft demonstrated a three-node quantum network (2021).
- Device-independent cryptography: keys whose security rests only on observed Bell violation were demonstrated in 2022.
So what. Mulder's argument is a template for testing a hidden-variable story in any system: if a simple "pre-programmed" explanation were true, what statistic would it force? Then measure that statistic. In fraud analytics or model monitoring, instead of arguing about whether correlations are "coincidence" or "a hidden common cause," derive the bound the simple story implies and test against it. And the "no-signalling" result is a sharp reminder for architects: correlation across systems is not communication; identical outputs at two endpoints prove nothing about a channel between them until the logs are compared through a real one.
# Mulder vs Scully: local "programs" vs quantum spin, three axes 120° apart
import random, math
def local(n=100000):
agree=0
for _ in range(n):
prog=[random.choice('RB') for _ in range(3)] # shared hidden program
a,b=random.randrange(3),random.randrange(3)
agree+= prog[a]==prog[b]
return agree/n
def quantum(n=100000):
ang=[0,120,240]; agree=0
for _ in range(n):
a,b=random.randrange(3),random.randrange(3)
p=math.cos(math.radians(ang[a]-ang[b])/2)**2 # spin-1/2 agreement
agree+= random.random()<p
return agree/n
print('local programs:',round(local(),3),'(always > 0.5)')
print('quantum: ',round(quantum(),3),'(exactly 0.5)')
The Frozen River: Does Time Flow?
Experience tells us two things about time: it seems to flow, and it seems to have an arrow. This chapter takes on the first. Using nothing but special relativity and a working definition of "what exists now," Greene argues that observers in gentle motion far away have now-slices that sweep through what we call the distant past and future. Grant their now the same standing as yours and every event in spacetime is equally real. The river of time is better pictured as a block of ice, every moment frozen in place; the "flow" is supplied by memory within each moment. Einstein accepted this with "painful but inevitable resignation." Greene sides with it by day and wavers at night.
Time and experience
Relativity already smashed Newton's universal time: each of us carries a personal shard of it, our own clock. This chapter asks a different question: not how my time compares with yours, but what time is for me, and whether experience reflects its true nature. Experience teaches two lessons:
- Time seems to flow. We stand on a riverbank as the future rushes toward us, becomes now, and recedes into the past. Or, inverting the metaphor, we ride the current from one now to the next.
- Time seems to have an arrow. Milk spills but never unspills. Cut a film of the spill into frames and anyone can reassemble the order without help. Eggs break, candles burn, people age, all aligned with one direction.
These raise sharp puzzles. If time flows, what flows, and how fast? Space has no inherent arrow (to an astronaut, left and right, up and down are equals), so where would time's arrow come from? Is it absolute, or can things evolve against it? This chapter treats flow; Chapter 6 treats the arrow. Both work in classical physics, ignoring quantum uncertainty until Chapter 7.
The core claim. Physicists have searched hard and found no evidence in the laws of physics for a flowing time. Nothing in the equations singles out one moment as "the present" and then moves that label along. Worse for intuition, a reframing of special relativity provides positive evidence that time does not flow.
The view from nowhere and nowhen
Return to the spacetime loaf of Chapter 3. Each slice is all of space at one moment for a given observer; stack every slice from the ancient past to the remote future and the loaf is the whole of spacetime. Every event, wherever and whenever, is a point in it. Zoom in on earth and you would find Alexander studying with Aristotle, Leonardo finishing the Mona Lisa, Washington crossing the Delaware, your grandmother as a girl, you reading this sentence, and the inauguration of your great-great-granddaughter. At coarser resolution you see the sun and earth condensing from a gas cloud and earth's demise when the sun swells into a red giant.
This perspective is fictitious: none of us can stand outside spacetime. Yet it is a powerful analytic tool. In it, the intuitive flow of time looks like a movie projector light that illuminates one slice, making it the momentary now, then lets it go dark as it moves to the next. No such light has ever been found in physics. No mechanism picks moment after moment to be momentarily real.
The persistent illusion of past, present and future
To argue that the whole loaf is real, Greene needs a working definition of reality. He uses the one most of us use implicitly:
The now-list. What exists at this moment is whatever appears in a mental freeze-frame of the entire universe right now. Greene's list as he types: midnight on his kitchen clock, his cat in mid-leap, first sunlight on Dublin, the Tokyo stock exchange, two hydrogen atoms fusing in the sun, a photon leaving the Orion nebula, a star's last moment before collapse. Charlemagne, Nero, Lincoln, Elvis: not on the list, so they do not exist now. Nor does anyone born in 2300 or 57000.
One subtlety: nothing you see belongs on your current now-list, because light takes time to arrive. You see this page as it was a billionth of a second ago, a room as it was 10 to 20 billionths of a second ago, the far rim of the Grand Canyon a ten-thousandth of a second ago, the moon 1.5 seconds ago, the sun 8 minutes ago, naked-eye stars years to 10,000 years ago. A now-list can only be compiled after the fact, by correcting for distance. Its entries are nonetheless the things we intuitively believe existed at that moment.
Newton versus Einstein on now-lists.
- Newton: absolute space and time give everyone the same freeze-frame. If something is on your now-list, it is on mine. Reality is one slice, shared.
- Einstein: observers in relative motion slice spacetime at different angles (Figure 3.4). Different nows mean different now-lists, so observers moving relative to each other have different conceptions of what exists.
Huge loaves magnify tiny angles
At everyday speeds the angle between two observers' slices is minuscule, which is why we never notice. Textbooks therefore crank speeds toward light. Greene uses the other lever: distance. Cut an ordinary loaf at two slightly different angles and the slices barely differ; cut a loaf billions of light-years long at the same angles and the slices separate enormously far from where they cross, just as a hair's opening between the blades of giant scissors puts the tips far apart.
Chewie's stroll. Chewie sits in his living room on a planet 10 billion light-years away, at rest relative to you (ignore planetary motion, cosmic expansion and gravity). Your now-lists coincide. Then he gets up and ambles away from earth at about 10 mph. His slice rotates slightly: no noticeable effect at home, but over 10 billion light-years it is amplified. The earth events on his new now-list happened about 150 years ago by your reckoning. In his equally valid now, you have not yet been born.
- Walk toward earth at 10 mph: his now includes earth 150 years in the future; you may no longer be alive.
- Fly the Millennium Falcon at 1,000 mph (slower than Concorde): 15,000 years past or future.
- Walk away at about 9.3 mph: John Wilkes Booth approaching the State Box at Ford's Theatre is on his now-list.
- Walk toward earth at about 6.4 mph: the election of the first president of the twenty-second century is on it. What is open for us has already happened for him.
No paradox follows. Chewie cannot save Lincoln: light from Ford's Theatre takes billions of years to reach him. Only his descendants, receiving the light, can place the assassination on the past now-list that also contains Chewie standing up. They will also find that a moment earlier his now-list held you, in the twenty-first century, reading. Now-lists encode no causal power: every observer agrees which events can affect which.
Two lessons. First, relativistic effects need not involve high speed: modest velocities produce huge effects when amplified by distance. Second, they bear on whether spacetime is a real entity or a mere abstraction stitched from "space now" plus history.
From many nows to one block
Chewie's freeze-frame is as real to him as ours is to us. It would be "stunningly narrow-minded" to exclude his. Since his location and velocity are arbitrary, we should include the now-slices of every possible observer, real or hypothetical, centred anywhere and tilted by any angle up to the 45-degree limit set by light speed. Their union fills a large region of the loaf; if space is infinite, it sweeps through every point.
"For we convinced physicists, the distinction between past, present, and future is only an illusion, however persistent." Albert Einstein, letter of condolence to the family of Michele Besso, 1955
The conclusion: if reality is your freeze-frame now, and your now has no special privilege over a distant mover's, then reality encompasses all events. The total loaf exists. We picture all of space as really out there; we should picture all of time as really out there too. The only thing that is real is the whole of spacetime.
Experience and the flow of time
In this picture events simply are. If you were having a great time at the stroke of midnight on New Year's Eve, 1999, you still are: that is an immutable location in spacetime. Against the cold facts of physics, the past-present-future scheme seems to find refuge only in the human mind. Our minds seem to supply the projector light; the flowing sensation comes from noticing change in thoughts, feelings and perceptions, which unfold as a coherent story.
Thought experiment: the faulty DVD player. Play Gone with the Wind on a player that jumps randomly between frames. You are baffled. Scarlett and Rhett are not. Stop on any frame and ask them about their memories: they give exactly the answers they would in a normal playback. Each frame contains memories rich enough to give the sensation of smoothly flowing time. Likewise, every slice of spacetime contains a you whose memories create the feeling of having lived up to that moment. The feeling of flow does not require previous frames to have been "sequentially illuminated."
Greene then shows the projector-light image is not only unnecessary but incoherent:
Why the moving spotlight fails. Suppose midnight, New Year's Eve 1999, is lit, then goes dark. Being lit would be a feature of that moment, as unchanging as anything else in it. To go from lit to dark is to change. But change has no meaning for a single moment: change happens through time, so we would need a second time in which the first time's moments change. A moment can no more change in time than a location can move in space; if it did, it would be a different moment. So every moment is illuminated, and remains so. The river becomes a giant block of ice.
Einstein's unease. Philosopher Rudolf Carnap recalled Einstein saying "the problem of the now worried him seriously": the experience of now is something special for humans, essentially different from past and future, yet "this important difference does not and cannot occur within physics." Einstein regarded this as a matter of "painful but inevitable resignation."
Greene's own verdict is honest ambivalence. By day he believes the mind imposes flow on a time that lacks it; by nightfall, ordinary life makes it hard to resist the idea that science misses something real. Someone may yet find a physical basis for flowing time. Meanwhile our language will keep lapsing into flow talk. "Don't confuse language with reality." Language is better at capturing human experience than deep physical laws.
| Situation | Distance | Speed | Shift in "now" on earth |
|---|---|---|---|
| Chewie strolls | 10 billion ly | 10 mph | about 150 years |
| Booth at Ford's Theatre | 10 billion ly | 9.3 mph (away) | about 140 years back |
| Election of 2100 | 10 billion ly | 6.4 mph (toward) | about 96 years ahead |
| Millennium Falcon | 10 billion ly | 1,000 mph | about 15,000 years |
| Andromeda observer | 2.5 million ly | 3 mph walk | about 4 days |
| Two drivers in Manhattan | 10 km | 60 mph | about 3 trillionths of a second |
Going deeper: the technical notes
- The formula. From the Lorentz transformation, t′ = γ(t − vx/c2). Events a moving observer calls simultaneous with his own now sit at earth time offset by Δt = vx/c2 = (v/c)(x/c). With v = 10 mph (1.5 × 10−8 c) and x/c = 1010 years, Δt ≈ 150 years. The shift is linear in both speed and distance, which is why distance can do the work speed usually does.
- Spacetime diagrams. The loaf is the standard spacetime diagram: horizontal lines are a resting observer's now-slices, the vertical axis is time. A moving observer's now-lines tilt, toward the light line but never past 45 degrees. Greene notes a curiosity: when Chewie switches frames, one moment for you appears on two of his now-lists, one before and one after he stood up. Now-lists jump because he jumps between frames; within any single coordinate system, causality is untouched and everyone agrees on what can affect what.
- Geometry assumption. The "union covers everything" step assumes flat Minkowski spacetime and infinite space. In other geometries the union need not cover the entire spacetime.
- Memory and identity. If your brain and body were placed in exactly their present state, you would feel you had lived the life your memories attest to, even if it never happened. Greene assumes all experience is grounded in the physical state of brain and body. He sets aside the fact that the brain needs processing time, so experience is never strictly instantaneous.
- "Moving through spacetime at light speed" without flow. Chapter 3's claim that everything travels through spacetime at speed c can be restated statically: an object's history is a curve (worldline). Its "speed" is the spacetime length of a segment divided by the proper time a carried clock records between its ends. Since ds2 = c2dt2 − dx2 and dτ2 = ds2/c2, ds/dτ = c for every worldline. No flow is needed, only readings at two points.
Critical lens.
- Uncredited lineage. The "distant observer swings your now" argument is the Rietdijk-Putnam argument (C. W. Rietdijk 1966; Hilary Putnam 1967), popularised by Roger Penrose as the "Andromeda paradox" (1989). Greene presents it without these names.
- The argument rests on a definition, not an experiment. Its first premise, that what exists is what is simultaneous with me, is exactly what a presentist denies once relativity is accepted. Its second premise, that existence is transitive across observers (real-for-Chewie implies real-for-me), was challenged by Howard Stein (1968, 1991). Stein argued the only frame-independent notion of "already determinate" is an event's past light cone. Relativity is compatible with eternalism; it does not force it.
- Rivals the chapter omits: the growing block (C. D. Broad; George Ellis's "evolving block universe"), in which the past and present exist and the future does not; causal set theory, in which spacetime "atoms" are born in sequence; and Lee Smolin's argument that time is fundamental and laws evolve. These are minority views but serious ones, so "the only thing that's real is the whole of spacetime" overstates the consensus.
- The Einstein quote comes from a letter of consolation to Besso's family (March 1955), a few weeks before Einstein's own death. It expressed comfort, not a technical result, and is often read as a stronger metaphysical claim than he defended in print.
- The incoherence argument against a moving now (change requires a second time) is McTaggart's (1908) regress in modern dress, again uncredited. Defenders of A-theories of time reply that "becoming" need not be change in a second time.
- The arrow is postponed. The block view says nothing about why memories point one way. Greene treats that in Chapter 6, but without that answer the DVD analogy borrows a direction it has not yet earned.
2026 update: what time is made of
- Time from entanglement. The Page-Wootters mechanism (1983) shows a static, "timeless" universe can contain subsystems that see evolution relative to a clock subsystem they are entangled with. Moreva and colleagues (2014) demonstrated it in a two-photon toy experiment: an outside observer sees nothing change, while an "internal" observer sees a photon evolve. This is the most concrete version of "flow is perspectival."
- Thermal time. Connes and Rovelli's thermal time hypothesis (1994) and Rovelli's The Order of Time (2017) argue that the flow we experience emerges from statistical coarse-graining, tying Chapter 5's question tightly to Chapter 6's arrow.
- The case for real becoming. Nicolas Gisin (Nature Physics, 2020) argues that if physics uses intuitionist mathematics, in which numbers have only finitely many determined digits at any moment, then the future is genuinely open and time passes. Smolin and Unger (2014) argue for the reality of time on cosmological grounds.
- Quantum causal order. Experiments since 2015 with the "quantum switch" put two operations into a superposition of orders, A before B and B before A. Definite temporal order itself appears not fundamental at the quantum level.
- The brain's now. Neuroscience (Dean Buonomano, Your Brain Is a Time Machine, 2017) finds the felt present is assembled over roughly 100 milliseconds to a few seconds, with postdiction effects. The experience of flow is constructed, consistent with Greene's DVD analogy.
- Clocks. Optical lattice clocks now resolve gravitational time dilation over a millimetre of height (JILA, 2022): every slice of the loaf has its own tick rate, measurably.
So what. The block universe is a lesson in separating the model from the user interface. In an event-sourced system the log of all events is the reality; "current state" is a projection computed at a reader's position, and different readers can legitimately project different "nows." Design accordingly: store immutable events and derive state, rather than treating any one snapshot as the truth. Personally, Greene's point that the feeling of flow lives inside each moment's memories suggests where experience is shaped: what a moment remembers is what that moment is like.
# How far does a distant observer's "now" swing across earth's history?
c = 299_792_458.0 # m/s
LY = 9.4607e15 # metres per light-year
mph = 0.44704 # m/s per mph
def shift_years(speed_mph, dist_ly):
v = speed_mph*mph
return (v/c)*dist_ly # (v/c)*(x/c) in years
for name,s,d in [('Chewie stroll',10,1e10),('Booth',9.3,1e10),
('2100 election',6.4,1e10),('Falcon',1000,1e10),
('Andromeda walker',3,2.5e6)]:
print(f'{name:17s} {shift_years(s,d):12,.1f} years')
# Speed needed to put a given past event on Chewie's now-list:
yrs = 2026-1865
print('Speed to see Lincoln now (mph):', round(yrs/shift_years(1,1e10),1))
Chance and the Arrow: Does Time Have a Direction?
Eggs splatter but never unsplatter, yet every known law of physics runs equally well backward. Greene shows that the second law of thermodynamics, which says entropy tends to increase, does not by itself explain time's arrow: because the laws are time-symmetric, the same statistics predict entropy should be higher in the past too. Followed faithfully, that logic says your memories are more likely a random fluctuation than a record, a conclusion that destroys the reasoning that produced it. The only escape is a special boundary condition: the universe began in an extraordinarily ordered, low-entropy state, and every splattering egg is the big bang's order still unwinding. Why it began that way is the question the arrow of time hands to cosmology.
The puzzle
Even if time does not flow, we can ask whether events are sprinkled along spacetime with a built-in order. A thousand times a day we see one-way sequences:
- Pizza cools on the way from the oven; it never arrives hotter.
- Cream stirred into coffee never unstirs.
- Eggs crack and splatter; splatter never reassembles into eggs.
- CO2 rushes out of an opened Coke; it never swooshes back.
- Ice melts in room-temperature water; it never re-forms.
- We remember the past, not the future.
Science codifies regularities, so you might expect a fundamental law behind this one. A "Law of Spilled Milk" would merely describe. We want a law of particles that shows why things go one way. No one has found one. From Newton through Maxwell to Einstein, the laws show complete symmetry between past and future. Worse, they positively permit coffee to unmix and eggs to unsplatter. Why don't we ever see it?
Time-reversal symmetry
Coolstroke's backhand. In the twenty-fifth-century interplanetary tennis league, your partner Coolstroke Williams hits a ball off Venus into deep space. A passing shuttle films it for the Celestial News Network. If CNN runs the film backward, can you tell from the footage alone? No. Left-to-right becomes right-to-left; both are allowed.
Now let Jupiter's gravity capture the ball, which arcs down and rightward to the surface. Run the film backward: the ball leaps up and leftward. Is the whole reversed path lawful, beyond its start? Yes. Launch it with the same speed in exactly the opposite direction, and gravity, which had mostly boosted its downward speed, now mostly saps its upward speed, shedding the extra rightward push too. The ball ends up drifting purely left at its original speed: an exact retrace.
Time-reversal symmetry, stated carefully. We are not reversing time. The claim is that reversing an object's velocity at any point makes it retrace its path, exactly as in a reverse-run film. Newton's mechanics, Maxwell's electromagnetism, and special and general relativity all have this property (quantum mechanics waits until Chapter 7). Whatever can happen forward can, in principle, happen in reverse.
Tennis balls and splattering eggs
Roll an egg off the counter: it falls, the shell cracks, yolk splatters, floorboards vibrate, air eddies swirl, friction warms the molecules. The same reasoning says that if we reversed the velocity of every atom involved, the whole thing would run backward: a converging wave of air molecules and floor vibrations would drive shell fragments and yolk droplets back to the impact site, fuse them into a smooth ovoid, and kick the egg up onto the counter, where it rolls a few inches and comes to rest. Greene stresses this is no hedge: shells crack because molecules are forced apart, and the reversed motions re-join them.
Principle and practice
Reversing the tennis ball was easy: grab it and throw it back. Reversing the egg needs every fragment sent back at once, beyond "all the King's horses and all the King's men." So is the answer just ease versus difficulty? It is part of the answer, but the full story is "far more subtle and surprising." The precise version of "easy versus hard" is entropy.
Entropy
Ludwig Boltzmann (1844 to 1906). His tombstone in Vienna's Zentralfriedhof, near Beethoven, Brahms, Schubert and Strauss, bears S = k log W. He developed statistical mechanics, linking the huge number of microscopic ingredients of a system to its overall properties, and was refined through debate with Maxwell, Kelvin, Loschmidt, Gibbs, Poincaré, Burbury and Zermelo. Boltzmann first thought he could prove entropy never decreases; their objections pushed him to the statistical view still used today. Ill and depressed, he took his own life in 1906; experiments soon confirmed the atomic picture he had defended.
Tossing War and Peace. Unbind Tolstoy's novel (693 double-sided sheets, 1,386 pages), throw the sheets in the air, gather them up. They are almost certainly out of order: there is one correct ordering and about 101878 wrong ones. "The more ways something can happen, the more likely it is." One lottery ticket gives one way to win; a million tickets give a million.
Entropy defined. Entropy counts the ways, consistent with physical law, that a situation can be realised. High entropy: many ways. Low entropy: few ways. Two features follow:
- Entropy measures disorder. High entropy means many rearrangements go unnoticed (shuffle an already jumbled stack and no one can tell). Low entropy means almost any rearrangement is detectable.
- Systems tend to evolve toward higher entropy, because disorder can be achieved in so many more ways than order.
The physics version counts rearrangements of atoms and particles that leave the gross, macroscopic properties unchanged. Because the numbers are vast, entropy is the logarithm of the count, which makes it additive when systems combine.
Why statistics, if Newton's laws are deterministic? In principle we could compute where each page lands. In practice, tracking 693 floppy sheets in air currents defeats any supercomputer, and the exact answer would be useless anyway: you care whether the book is in order, not which jumble you got. "If you've seen one disordered page arrangement, you've pretty much seen them all." The power of entropy is that it answers the question we actually ask. Intermediate classes exist (only Part One scrambled, say), but all out-of-order arrangements confined to Part One are just 10−178 of 1 percent of all disordered ones, so repeated tosses drift into full disorder.
The Coke bottle is the physics example. CO2 spread uniformly through a room is insensitive to huge numbers of molecular rearrangements (flail your arms; it still looks uniform): high entropy. Confined to the bottle or a corner, it has fewer places to be, so low entropy. Open the cap and the molecules explore the new territory, approaching uniformity, and stay there. A bottle holds about 1024 molecules; a list of their positions and velocities would tell you less than the single word "spread out."
Entropy, the second law and the arrow
The second law of thermodynamics: physical systems tend to evolve toward higher entropy. (The first law is conservation of energy.) It is statistical, not absolute: the pages can land in order; the CO2 can swoosh back into the bottle. These are merely so improbable that, "with more certainty than death and taxes," systems with many constituents evolve toward disorder.
The same holds for gas. Two molecules in an empty bottle wander back to within a millimetre of each other every few seconds; three take days; four take years; a blob of 1024 takes far longer than the age of the universe. Hence an arrow that emerges with numbers: a film of two molecules looks the same either way, but a film of 1024 molecules spreading out is unmistakably forward. Ice melting, eggs splattering: the forward direction is the one in which entropy rises. Entropy makes the "easy versus difficult" intuition precise: going from order to disorder is easy, so it happens all the time.
The double-headed arrow
Here Greene springs the chapter's trap. The second law modified nothing in Newton's laws; it is derivative, statistics applied to time-symmetric dynamics. So:
The astounding point. Since the laws have no built-in temporal orientation, the reasoning that says entropy will be higher in the future says, with exactly the same force, that it was higher in the past. A common misconception is that rising entropy toward the future implies falling entropy toward the past. The correct statement: if a system is not at maximum entropy now, it will subsequently have, and previously had, more entropy. The entropic arrow is double-headed.
Concretely: seeing half-melted ice on a warm day, you are sure the cubes will be more melted in half an hour. By the same statistics you should be equally sure they were more melted half an hour ago. Half of these conclusions seem flat-out wrong.
Following the math
Physics has a habit of believing equations over intuition (black holes, antimatter, entanglement were all predicted against common sense and later confirmed). So Greene follows the math under a kind of physicists' Hippocratic oath.
Thought experiment: a slow night at the bar. At 10:30 p.m. you have watched ice cubes melt for half an hour. Memory says the bartender added full cubes at 10:00; the guy across the bar and the surveillance video agree. Statistics says the opposite is likelier: at 10:00 the glass held plain room-temperature water, which fluctuated into partial ice by 10:30, while your neurons, his neurons and the camera's bits spontaneously arranged themselves to record a melting that never happened. Ridiculous, yet it is exactly where faithful entropic reasoning leads.
Pinpointing the disjuncture
Intuition and math agree that a spontaneous rise in order (water freezing itself, brains acquiring false memories) is very unlikely. They also agree that if full cubes existed at 10:00, melting is the likely sequel. They disagree on one thing: intuition ignores how unlikely full cubes at 10:00 are, given the one fact we accept, the partial cubes at 10:30.
The logic of what needs explaining. High entropy is the natural state; it never needs explanation. Low entropy is the aberration that does. Explaining today's low-entropy partial cubes by an even lower-entropy past (full cubes, a more pristine bar) is explaining the improbable with the more improbable. A fluctuation from ordinary water avoids that, and at every moment between 10:00 and 10:30 it passes through states of higher entropy, so it is hugely more likely.
Boltzmann took the small step to the whole universe. The order around us (biological, chemical, physical) is far more likely to be a rare fluctuation out of total chaos than the residue of an even more ordered past. Toss pennies long enough and they all land heads; wait long enough with flat Coke and the gas re-enters the bottle; wait nearly an eternity and the universe's high-entropy murk will coalesce into exactly what we see now, bodies and brains included, stocked with memories of a past that never happened.
Taking a step back: the quagmire
Greene admits that meeting this idea was a shock: textbooks only ever applied entropy forward. Two objections sink the fluctuation story:
- Too much order. A fluctuation should cut corners. If Thriller never was recorded and its millions of copies are a fluke, a fluke of a few copies would be far likelier. The same goes for an unnecessarily complete fossil record and over 100 billion galaxies when a handful, or one, would do. The universe is more ordered than it needs to be.
- Self-destruction. If memories and records can't be trusted, neither can the experiments that established the laws of physics, including the time-reversal symmetry that launched the argument. The reasoning saws off the branch it sits on.
That collapse is useful. It shows precisely why distrusting memory makes no sense, and it proves we left something out. What would let entropy be higher toward the future but lower toward the past? Only one thing: things must have been very special early on.
The egg, the chicken and the big bang
Following an argument emphasised by Roger Penrose, trust memory again and trace the egg's low entropy backward:
- Chicken. Living things take in about as much energy as they give out, but the quality differs: food is low-entropy, usable energy; body heat is high-entropy, diffuse. Every life form is a conduit: low entropy in, high entropy out.
- Plants. Following the food chain to vegetarians ("like me," Greene notes), plants use sunlight to split CO2, releasing oxygen and keeping carbon. Low entropy comes from the sun.
- Sun. Formed about 5 billion years ago (4.6 by current dating) when a diffuse gas cloud collapsed under gravity, heated, and ignited nuclear fusion, whose outward radiation halted the collapse.
- Gas cloud. Remains of earlier stars that went supernova, whose own gas came from the big bang: a few minutes after it, a nearly uniform hot gas of roughly 75 percent hydrogen and 23 to 25 percent helium, with traces of deuterium and lithium.
The current order is a cosmological relic. That primordial gas had extraordinarily low entropy, not because it was small, but because of gravity.
Entropy and gravity
Without gravity (the Coke bottle), uniform gas is high-entropy. With gravity, the reverse: every region pulls on every other and a large mass of gas fragments into clumps, like water beading on wax paper. When gravity matters, clumpiness is the norm.
Clumps look ordered; the books still balance.
- The playroom. Toys neatly packed in trunks are more ordered than toys strewn on the floor, but the fat burned and heat released by the parents who tidied more than compensate.
- The gas cloud. Clumping into stars lowers the gas's spatial entropy, but compression heat and then fusion's heat and light raise the total by far more.
The second law does not forbid order; it demands that order be paid for with more disorder elsewhere. Gravity exploits this "to the hilt."
Black holes are the limit: the densest clumps, they keep all their entropy (light can't escape; Hawking radiation only leaks a little), and because no rearrangement inside is detectable from outside, they hold more entropy than anything else (Chapter 16). So the buck stops here: the ultimate source of low entropy is the big bang itself, which produced not the black holes that probability favours but a hot, smooth, unclumped gas, astonishingly low in entropy once gravity is counted. Ever since, entropy has risen: gas clumped into stars and galaxies, a star fed a planet's life, a chicken laid an egg, and the egg carried forward the drive toward higher entropy by rolling off your counter.
The same story covers every example. The tossed War and Peace gains entropy only because it started ordered, thanks to Tolstoy, printer and binder, whose own order traces back to the big bang. The bartender's ice came from a freezer designed and built by highly ordered engineers. A splattering egg tells us something deep about the big bang.
The critical input and the remaining puzzle
Resolution. We can trust memories of a lower-entropy past only if the universe began in a highly ordered state. With that boundary condition, probabilistic reasoning runs freely toward the future (entropy rises) but is blocked toward the past, where it would contradict the low-entropy start. The arrow of time began its flight in the low-entropy state of the universe at its inception. Philosopher David Albert calls this assumption the "past hypothesis."
The puzzle that remains is enormous: why did the universe begin so ordered? Probabilistically, a fluctuation is "mind-bogglingly more likely" than the astoundingly low-entropy start the big bang requires. But the fluctuation route ends in the quagmire, so we "buck the bookies." The arrow of time points to cosmology (Chapters 8 to 11), after first asking what quantum mechanics adds (Chapter 7).
Going deeper: the technical notes
- Two kinds of asymmetry. The arrow concerns the order of events in time, not asymmetry of time itself (time may have a beginning but no end). Both, Greene concludes, lead back to cosmology.
- Formal time reversal. Laws are T-symmetric if, for evolution map U(t), there is an operation T with U(−t) = T−1U(t)T. For Newtonian particles T reverses all velocities. With electromagnetism, T must also flip magnetic fields so that the v × B force is unchanged; that is a detail, not a loophole.
- The real exception. Local, unitary, Lorentz-invariant quantum field theories always respect the combined symmetry CPT (charge conjugation, parity, time reversal). Cronin and Fitch (1964) found CP violation in K-mesons (Nobel 1980), implying T violation; CPLEAR (CERN) and KTeV (Fermilab) observed it directly. These exotic particles can tell a film's direction. Greene doubts this matters for the everyday arrow "but the truth is that no one knows for sure."
- Counting continuous states. Molecules don't come in countable arrangements, so physicists use phase space (6N dimensions for N particles), carve it into macrostate regions, and define entropy by region volume with a uniform measure over microstates. "Larger volume, more points, higher entropy."
- Poincaré recurrence. Via Liouville's theorem, any finite-energy system in a finite volume eventually returns arbitrarily close to its initial state: the CO2 will re-enter the bottle, after vastly more than the age of the universe.
- Why water freezes at all. Freezing lowers the water's entropy but releases heat that raises the surroundings' entropy. The real criterion is free energy F = U − TS: below 0 °C, the energy drop wins; above it, entropy wins. Freezers work the same way, pumping heat outside.
- Prior art. The unreliability-of-records argument appears in von Weizsäcker (1939) and David Albert's Time and Chance (2000); the low-entropy past goes back to Boltzmann and Reichenbach*, with Penrose's quantitative version in The Emperor's New Mind*** (1989).
- "Early on" without begging the question. Strictly, we need a low-entropy boundary condition at one end of time; "the past" is defined as the end where it holds. Could there be a future constraint too? Gell-Mann and Hartle explored it; we don't know.
- One arrow or many? The thermodynamic (heat flows hot to cold), electromagnetic (waves spread outward from sources), psychological (we remember the past) and cosmological (expansion) arrows might have separate explanations. Greene thinks all but the cosmological share one source: outgoing waves need ordered sources (stars, bulbs), whose order traces to the big bang. Computation's arrow is well understood thermodynamically (Bennett, Landauer). A universe that recollapsed would keep its thermodynamic arrow.
- Measures. "Fluctuation is more likely" assumes a measure uniform over microstates compatible with now. Albert's measure conditions on the past hypothesis too, so a low-entropy start has probability 1 by fiat, but why it holds is still unexplained.
- Small is not enough. The early universe was small, but small size alone doesn't mean low entropy: in a big crunch, entropy would keep rising as the universe shrank. Chapter 11 shows size does matter in the inflationary account.
Critical lens.
- The tombstone equation isn't Boltzmann's notation. S = k log W in that form, with the constant k, was written by Max Planck (1900 to 1901); Boltzmann expressed the relation differently and never used k.
- Boltzmann's death. He died in Duino, near Trieste, then part of Austria-Hungary (Italian only after 1918), not "in Italy." The confirming experiments, Jean Perrin's Brownian-motion measurements, came about two years later (1908 to 1909), not "a few months." Einstein's theory of Brownian motion (1905) actually preceded his death.
- "Disorder" is a leaky metaphor. Entropy depends on the choice of macrostates (coarse-graining), which Greene acknowledges only in passing. Many high-entropy states look more ordered to the eye: oil separating from vinegar, crystals forming from solution, the clumpy universe itself. Counting "ways" is the definition; "disorder" is a gloss.
- The fluctuation nightmare has a name. Greene describes what is now called the Boltzmann brain problem (Albrecht and Sorbo, 2004): a lone brain with false memories is a vastly smaller fluctuation than a whole ordered universe. His "too much order" objection is its seed but he doesn't push it to brains, where it bites hardest for modern cosmology.
- The arrows are less unified than he suggests. The psychological arrow ("we remember the past") is asserted to follow from thermodynamics via an analogy to computation; a detailed account of why records point only to the past (worked out by Albert, Loewer, Wallace and others) is more involved than the note implies.
- Alternatives omitted. Some physicists argue an arrow can arise without a special initial state: Barbour, Koslowski and Mercati's "Janus point" (2014) shows gravitating systems naturally develop arrows pointing away from a central point of minimal complexity; Carroll and Chen (2004) proposed baby universes budding from a high-entropy de Sitter space, giving a time-symmetric multiverse with local arrows.
- "100 billion galaxies" was the 2004 figure; deep-field counts later suggested up to 2 trillion (2016), revised downward to hundreds of billions by New Horizons background-light measurements (2021). The argument is unaffected.
2026 update: the arrow under the microscope
- Direct T violation, cleanly. BaBar at SLAC (2012) observed time-reversal violation directly in B-meson transitions, independent of CP arguments, at 14 standard deviations. It still has no known role in the everyday arrow.
- The second law broken, briefly and predictably. Fluctuation theorems (Evans and Searles 1994; Jarzynski 1997; Crooks 1999) give the exact odds of entropy-decreasing events. Optical-trap experiments (Wang et al. 2002) watched colloidal beads violate the second law over short times, at the predicted rate: Greene's "can happen" made quantitative.
- Information is physical. Landauer's principle (erasing one bit dissipates at least kT ln 2 of heat) was confirmed experimentally (Bérut et al., Nature 2012), supporting the notes' claim that computation's arrow is thermodynamic.
- Heat flowing backward. Micadei et al. (2019) showed that with initial quantum correlations, heat can flow from a cold spin to a hot one: the arrow depends on the absence of initial correlations, a microscopic version of the past hypothesis.
- The entropy budget. Egan and Lineweaver (2010) estimate the observable universe's entropy at about 3 × 10104 k, dominated by supermassive black holes, far below the ~10122 k maximum set by the cosmic horizon. Penrose's estimate of the initial state's improbability, 1 in 1010123, remains the standard dramatisation.
- Early clumping was faster than Greene's "billion years." JWST has confirmed galaxies within about 290 million years of the big bang (JADES-GS-z14-0, z = 14.3, 2024) and slightly earlier candidates (MoM-z14, 2025). Gravity started cashing in the low-entropy endowment remarkably early.
- Boltzmann brains now serve as a consistency test: cosmological models whose far future is an eternal de Sitter phase risk predicting that typical observers are fluctuations (Carroll, "Why Boltzmann Brains Are Bad," 2017).
So what. Greene's chapter is a masterclass in asking what actually needs explaining. High entropy is the default; order is the anomaly. In data work, a dashboard showing "normal noise" needs no story, but an unusually clean pattern does, and the tempting explanation (an even cleaner upstream state) may be more improbable than the thing explained. Second, check your reasoning for self-undermining: any conclusion that makes your own evidence unreliable must be rejected, or the premise that produced it fixed. Third, order is always paid for: every tidy system (a codebase, an organisation, a body) is a conduit taking in low entropy and exporting high entropy. Budget the energy to stay ordered.
# Ehrenfest urn: N molecules hop at random between two halves of a box.
# Entropy rises from an ordered start, then fluctuates near the maximum.
import random, math
def run(N, steps):
left = N # all molecules start on the left
S = []
for _ in range(steps):
if random.random() < left/N: left -= 1
else: left += 1
S.append(math.lgamma(N+1)-math.lgamma(left+1)-math.lgamma(N-left+1))
return S # S = log(number of arrangements)
for N in (4, 40, 4000):
S = run(N, 20*N); Smax = math.lgamma(N+1)-2*math.lgamma(N/2+1)
back = sum(1 for s in S[N:] if s < 0.5*Smax)/len(S[N:])
print(f'N={N:5d} final S/Smax={S[-1]/Smax:5.3f} '
f'fraction of later time below half-max: {back:.4f}')
# Small N: returns to order often (no arrow). Large N: never (arrow).
Time and the Quantum: Insights into Time's Nature from the Quantum Realm
Classical physics gives every present a single, definite past. Quantum mechanics does not. Greene shows how Feynman's sum over histories makes the observed present an average of all possible pasts, then escalates through Wheeler's delayed-choice experiment, the quantum eraser and the delayed-choice quantum eraser, in which measurements made years later and light-years away determine what story you may tell about photons detected today. The future never changes the past, but it can decide which kinds of history it is meaningful to recount. He then lays out the quantum measurement problem, four rival solutions and decoherence, and asks whether any of them builds an arrow into the laws. His answer: probably not, so the arrow still points back to the low-entropy big bang.
The past according to the quantum
In Chapter 6 probability was a convenience: track 1024 water molecules exactly and classical physics would tell you the past and future with confidence limited only by your data. In quantum mechanics probability is built in. That changes the future (only odds can be predicted) and, more surprisingly, the past.
Two kinds of history.
- Classical: a meteor's history is a unique succession of positions and velocities, described in the same language as the observation itself.
- Quantum: once observed, we know the outcome with certainty. But the unobserved past remains a matter of probabilities: a moment ago the electron had only chances of being here, there or way over there. And it is not that it was in one place and we don't know which: each possible history contributes to what we now see.
The double slit proves it. The classical story ("each electron went left or right") predicts no fringes; the fringes require something passing through both. Quantum mechanics says the electron's probability wave passes through both slits and interferes.
Feynman's sum over histories. Take it one step further: perhaps each electron itself travels through both slits, and the data reflect interference between classes of histories. Richard Feynman showed this is a fully viable formulation: if an outcome can be reached in several ways, all of them happen, each contributing to the outcome's probability; added correctly, they reproduce quantum mechanics exactly. A probability wave embodies all possible pasts that could have preceded a given observation.
To Oz: the beam splitter
Fire a laser at a half-silvered mirror (a beam splitter) that transmits half the light and reflects half; mirrors bring the two beams together at a detector screen, where they interfere. Now turn the laser down to one photon every few seconds. Classically each photon goes left or right and there is nothing to interfere with. Yet over time the dots build up fringes. Unlike Dorothy, baffled when the Scarecrow points both ways to Oz, each photon takes both routes.
Prochoice
With three slits there are three classes of history; with 300, three hundred; cut so many that the barrier disappears, and quantum physics says each electron traverses every possible path. "Quantum mechanics is the ultimate prochoice arena": every possible choice contributes to each outcome's probability.
Three questions Greene poses, and his answers.
- Is the sum over histories literally real? He can't say. He uses it so often in research "it certainly feels real," but the predictions, which match the data "spot on," don't depend on the story.
- Can we watch the histories merge? No. Probability waves are not directly observable, and neither are Feynman's histories. Observe the electron mid-flight and you see it in one place; you have merely pushed back what "history" means. "Quantum mechanics is starkly efficient: it explains what you see but prevents you from seeing the explanation."
- Why does classical physics work for baseballs? All paths contribute, but for large objects the Newtonian path dominates the average overwhelmingly. For electrons, quarks and photons, many paths contribute comparably.
Pruning history
Why not peek? Put detectors at each slit or on each path. You can, and two things happen: each particle is always found on one path only, and the interference pattern disappears, replaced by the classical result. The paradox ("how can it interfere with a path it demonstrably didn't take?") is averted because the new observation selects only histories through that path.
Niels Bohr's complementarity. Everything has wavelike and particlelike aspects. Pure particle thinking misses interference; pure wave thinking misses localised dots on a screen. How you interact decides which aspect shows. Leave electrons unobserved and the wave side emerges; watch them en route and your observation prunes the branches of quantum history. (Feynman's many histories belong to the wave side: "particle" here always means one trajectory.) "Nature does weird things. It lives on the edge. But it is careful to bob and weave from the fatal punch of logical paradox."
The contingency of history: delayed choice
Thought experiment: selling The Full Monty. Mr. Smithers of the Springfield Art and Beautification Society comes to see a painting left to you by great-uncle Monty Burns. You are torn. The conversation goes well (he turns out to have been your great-uncle's top aide) and you agree to sell. In retrospect it feels as if you had decided before he arrived. Had the visit gone badly, you would equally have "always known" you could never part with it. The past didn't change; a different present would have led you to describe a different history. Wheeler's experiment makes this psychological pattern precise and physical.
John Wheeler's delayed-choice experiment. Take the single-photon beam splitter and add a which-path detector on one route. Off: fringes. On: no fringes (no detection means the photon took the other path). Now move the detector far downstream, and set its switch after the photon has passed the splitter. The photon seemingly must "decide" at the splitter whether to travel both paths (wave) or one (particle), before it can know the detector's setting. Yet the photons always get it right: detector on, particle behaviour; detector off, fringes. "It's as if a consistent and definite history becomes manifest only after the future to which it leads has been fully settled."
In Wheeler's cosmic version, a galaxy's gravity acts as the beam splitter for light from a distant quasar. Switch on a path detector now and the photons act as if they went one way around the galaxy for billions of years; switch it off a few minutes later and new arrivals build fringes, as if they had travelled "in tandem with their ghostly partners."
What delayed choice does and doesn't mean. Turning the detector on today does not affect photons billions of years ago. "Quantum mechanics does not deny that the past has happened, and happened fully." The photon never decided one way, the other, or both: it was in the quantum norm, a hybrid of possibilities. Observation today makes one strand prominent in our recounting of the past. With detectors, which-path information is an essential detail of the story; without them it is fundamentally unavailable. "Both stories are valid." (Greene's note: don't explain this by saying time stops for photons; the same results hold for slower-than-light electrons.)
Erasing the past
If you can't change the past, can you erase its impact? A fielder who drops a routine fly can redeem himself with a diving catch, but nothing can restore the perfect game the error destroyed. The quantum eraser (Marlan Scully and Kai Drühl, 1982) seems to do just that.
The quantum eraser.
- Tag photons at each slit by twisting their spin (polarisation) in a slit-specific way. The screen can now tell which slit each came through: no interference.
- Just before the screen, add an eraser that sets every photon's spin to the same direction, destroying the which-slit mark.
- Result, confirmed by Raymond Chiao, Paul Kwiat and Aephraim Steinberg: the interference pattern returns.
Why it works. A detector at a slit records which-path information definitively; that cannot be undone. A tag provides only the potential for it, and "potentialities are just the kinds of things that can be erased." Tagging blurs each half of the wave relative to the other; both halves are still present. The eraser, "like a pair of glasses," refocuses them, and the pattern that "patiently lies in wait" reappears.
Shaping the past: the delayed-choice quantum eraser
The finale, also proposed by Scully and Drühl and carried out by Kim, Yu, Kulik, Shih and Scully (2000):
Step by step.
- Down-converters on each path of the beam-splitter setup turn one photon into two of half the energy: a signal photon continuing to the screen and an idler sent elsewhere. Which converter fired reveals which path the signal took, so no interference appears, even though we never touch the signal photons.
- Route the idlers through a maze of beam splitters to four detectors:
- Detector 1: only idlers from the left converter can reach it. Which-path known.
- Detector 4: only idlers from the right. Which-path known.
- Detectors 2 and 3: reachable equally from either converter (paths B-E or C-E; B-F or C-F). Which-path erased.
- Result: the full set of signal impacts shows no fringes. But pick out the subset whose idlers went to detector 2, and that subset does show fringes; likewise for detector 3. Subsets tied to detectors 1 and 4 do not. Colour detector-2 partners red and the rest green: a colour-blind viewer sees no pattern; everyone else sees red bands.
The ten-light-year version. Put the maze and idler detectors ten light-years away. Today you record signal photons: no fringes. You might conclude each went one way. Premature. Ten years later the idler results arrive; highlight the signal photons whose partners hit detector 2 (the first, seventh, eighth, twelfth...) and fringes emerge in data taken a decade earlier. And if a practical joker removes the erasing beam splitters one day before the idlers arrive, every idler lands in detector 1 or 4, and no subset of your old data shows fringes. "The story you'd tell to explain the signal photon data depends significantly on measurements conducted ten years after those data were collected."
"The future helps shape the story you tell of the past." Nothing in the data recorded today changes; what changes is which descriptions of today's photons are legitimate. Greene calls it "a magnificent affront to our conventional notions of space and time," but, having accepted EPR non-locality across space, entanglement across time is the natural next step.
Quantum mechanics and experience
Learning these experiments left Greene elated: everyday life looked like "an inverted magic act" hiding quantum reality. The task since has been to explain the trick: how quantum laws become classical laws for big objects. That bears directly on time's arrow.
Schrödinger's equation is time-symmetric too. Newton (1600s), Maxwell (1800s), Einstein (1905, 1915) and Schrödinger (1926) all treat past and future equally. Given a wavefunction at one moment, Schrödinger's equation gives it at any other, earlier or later; a reversed "movie" of a wavefunction is an equally valid solution (strictly, one must also take the complex conjugate).
But standard quantum mechanics has two stages:
- Stage one: the wavefunction evolves smoothly by Schrödinger's equation, like a water wave crossing a lake. Rigorous, unambiguous, universally accepted.
- Stage two: on measurement, the wavefunction collapses: zero everywhere except where the particle is found. This is an add-on: Schrödinger's equation never collapses anything. It was introduced because experimenters (and the rest of us) never see a chair in two places, a moon smeared across the sky, or a cat both dead and alive.
The quantum measurement problem
The open questions. How does measurement cause collapse? Does collapse really happen? Does every measurement cause it? When, and how fast? Which equation takes over from Schrödinger's, and how? And, for this book: does stage two introduce an arrow of time into the basic laws?
The problem is not a loose detail. It concerns the theory's universality: the experimenter and her apparatus are made of the same electrons and protons as what she studies, so why do different rules apply? Bohr answered that big things obey classical laws, but never said where the border lies or how the two sets of rules interface, and declared the questions out of bounds. Because predictions worked "astonishingly," the issue sat low on physicists' lists for decades. Many now see it as the single most important gap in quantum law.
Reality and the measurement problem: four proposals
All agree with experiment; they differ on what happens backstage. For Bohr there is no backstage; like a Spalding Gray monologue, the bare measurements are the show.
| Approach | Core idea | Collapse? | Main objection | Time-symmetric? |
|---|---|---|---|---|
| Wavefunction as knowledge (roots in Heisenberg) | Wavefunction encodes what we know; collapse is just learning | Yes, epistemic | Ties physics to awareness: what about mice, ants, computers, or the universe before minds? | Yes |
| Many Worlds (Hugh Everett, 1957, Wheeler's student) | Every outcome happens, each in its own branch; an infinity of spacetime blocks | No | Exorbitant proliferation; meaning of probability; preferred basis | Yes |
| Bohm (1950s; de Broglie earlier) | Particles have definite positions and velocities; a real pilot wave guides them | No | Lacks economy; explicitly non-local (Bell says that's unavoidable anyway) | Yes |
| GRW (Ghirardi, Rimini, Weber) | Wavefunctions collapse spontaneously and randomly, rare per particle, near-instant for big objects | Yes, physical | No experimental evidence yet; deviations too small to test (in 2004) | No |
How GRW works. Each particle's wavefunction collapses on its own about once in a very long time (Greene says roughly once every billion years), too rarely to disturb atomic physics. But a pointer has ~1023 particles, so some particle collapses within a tiny fraction of a second, and because they are entangled, the collapse cascades like dominoes. Big objects are always in one definite configuration: pointers point, the moon is in one place, cats are dead or alive.
Bohm's approach, in which the particle goes through one slit while its wave goes through both, never caught on, "perhaps unjustifiably." Greene, following John Bell, values it as a foil: it proves that particles can have definite positions and velocities at all times while matching every prediction of quantum mechanics.
Decoherence and quantum reality
The telltale difference. Classical probabilities (coins, roulette) describe separate, independent histories; the motions that produce heads don't interfere with those that produce tails. Quantum probabilities interfere: histories reinforce and cancel. Quantum probability is subject to interference; classical probability is not.
Decoherence, in its modern form from Dieter Zeh (1970) and developed by Erich Joos and Wojciech Zurek, applies Schrödinger's equation to real objects: big, complex and never isolated. Air molecules and photons are too feeble to move a book or a cat, but they constantly nudge its wavefunction, scrambling the orderly crest-trough sequence that interference requires, much as the eraser's tags blurred the photon waves. Once interference is gone, quantum probabilities behave like coin-toss probabilities.
Schrödinger's cat, decohered. A cat in a box faces a booby trap triggered by an electron with a 50 percent chance of hitting it. Decades of debate asked how a cat can be dead and alive, and how opening the box forces a choice. Decoherence says that long before you look, the environment has made "billions of observations" that turned quantum probabilities into classical ones. The weirdness leaks away, carried off by impinging particles. Observers become just another part of the environment; the stage one/stage two split dissolves.
Is that it? Some say yes: Robert Griffiths, Roland Omnès, Murray Gell-Mann and Jim Hartle have built decoherent (consistent) histories into what they consider a complete solution. Greene is "intrigued but not yet fully convinced." Decoherence removes Bohr's artificial border, which is real progress. But after decoherence, every outcome still vies for realisation: it explains why we get classical-looking odds, not why one outcome happens and where the others go. A coin's outcome is fixed by overlooked details; quantum mechanics offers no such details.
Two attitudes. Some, in Bohr's spirit, call the demand for more "unreasonable intellectual greediness." Greene disagrees: explaining data is science, but so is using confirmed theories for maximal insight into reality.
Quantum mechanics and the arrow of time
- Class one (knowledge, Many Worlds, decoherence) leaves Schrödinger's equation untouched: time symmetry is kept.
- Class two supplements or modifies it. Bohm's guiding equation is symmetric. GRW's collapse is asymmetric: an "uncollapsing" wavefunction violates the modified equations.
If symmetry survives (Greene's bet), Chapter 6 carries over intact: quantum entropy rises toward both future and past, and to avoid the quagmire we need a low-entropy big bang about 14 billion years ago. If symmetry fails, eggs might splatter but not unsplatter because unsplattering violates the true quantum equations. But that isn't enough either: an asymmetric law need not imply a low-entropy past, and GRW can't retrodict at all, since collapse destroys the information. Unless we find both a time-asymmetric measurement solution and proof that it makes entropy fall toward the past (Greene thinks both unlikely), the arrow points back to the origin of the universe.
Going deeper: the technical notes
- Classical limits. Even classical three-body problems lack exact solutions, but there is no fundamental limit to the approximation; better data and computers converge on the truth. Quantum uncertainty is a limit in principle.
- "No position before measurement" is too strong. Bell's results don't exclude definite positions; Bohm's theory has them. But it still needs a pilot wave spread over all locations, so the conclusion that a single-trajectory history is incomplete holds in every approach.
- The photon's-eye view. Since time stops at light speed, one might say the photon "knows" the detector setting. But electron versions behave identically, so that is not the explanation.
- The equation. HΨ = iħ ∂Ψ/∂t (H, the Hamiltonian). Heisenberg's matrix mechanics (1925) is equivalent. Time reversal maps Ψ(x,t) to Ψ*(x,−t).
- Smooth but jittery. Stage one is smooth; quantum "jitteriness" comes from the probabilistic reading of the wavefunction, not from the wavefunction itself.
- Bohm's non-locality is local in configuration space but non-local in real space. De Broglie had the idea first, hence de Broglie-Bohm. Bell noted neither man was fond of it; relativistic versions are harder but progressing (see Tim Maudlin, Quantum Non-locality and Relativity).
- Many Worlds' probability puzzle. If every outcome happens to some copy of you, in what sense do you have a 30 percent chance? And which basis defines the branches? Decoherence helps pick the basis.
- "Shut up and calculate." Greene is sympathetic to the idea that collapse will be dispensed with, and to environment-driven pseudo-collapse, but rejects giving up on what happens "when we are not looking." Measurement "is, or is akin to, or masquerades as" collapse, and that must be explained.
- Decoherence times for a dust grain: 10−36 s in air, 10−21 s in sunlight in vacuum, 10−6 s in deep space with only the cosmic microwave background. Bigger objects decohere faster still (Joos 2000; Zeh, The Physical Basis of the Direction of Time, 2001).
Critical lens.
- The eraser's decisive detail is missing. In the Kim et al. Experiment, the detector-2 subset shows fringes and the detector-3 subset shows anti-fringes, shifted by half a period. They sum to a featureless blob. That is exactly why the experiment cannot send signals to the past: nothing at the screen changes until the idler records arrive by ordinary means. Without this, readers often take "shaping the past" as retrocausation, which Greene disclaims but does not fully disarm.
- Anthropomorphic photons. Photons that "decide," "realize" and have "premonitions" make the paradox vivid but plant the very picture (a particle choosing a route at the splitter) that quantum mechanics rejects.
- Dates. Wheeler first proposed delayed choice in 1978, not 1980. The Kwiat-Steinberg-Chiao eraser was published in 1992. GRW's per-particle collapse rate (λ ≈ 10−16 per second) is about once per few hundred million years, not a billion.
- Measurement is more time-symmetric than "collapse" suggests. Aharonov, Bergmann and Lebowitz (1964) showed that if you condition on both initial and final measurements, quantum statistics are symmetric in time (the two-state-vector formalism). The apparent asymmetry of collapse comes from conditioning only on the past, much like the entropy story of Chapter 6. This strengthens Greene's conclusion but goes unmentioned.
- The interpretation menu is incomplete. Omitted: Penrose and Diósi's gravity-induced collapse, QBism, relational quantum mechanics (Rovelli), and retrocausal models that take delayed-choice experiments at face value.
- "Many worlds" as infinite spacetime blocks is one popular rendering; Everettians today usually describe a single universal wavefunction whose branches emerge through decoherence, not a literal multiplication of universes.
2026 update: the quantum past, tested
- Delayed choice done properly. Jacques et al. (Science, 2007) ran Wheeler's experiment with single photons and a random setting chosen after the photon entered the interferometer. Ma et al. (Nature Physics, 2012) performed delayed-choice entanglement swapping, deciding after the fact whether two photons had been entangled. Vedovato et al. (Science Advances, 2017) stretched delayed choice to a satellite link over 3,500 km. Wheeler's quasar version remains undone, but cosmic Bell tests (2017 to 2018) used quasar light to choose measurement settings.
- Bigger cats. Interference has been seen for molecules of about 2,000 atoms (25,000 daltons; Fein et al. 2019), and a 16-microgram crystal was placed in a Schrödinger-cat state (Bild et al., Science 2023). Decoherence, not size itself, is what hides quantumness.
- Watching decoherence. Serge Haroche (Nobel 2012, with David Wineland) observed photon "cat" states in a cavity decohering in real time, confirming decoherence quantitatively.
- Collapse models squeezed. Underground searches for the faint radiation that spontaneous collapse should produce (Gran Sasso, Donadi et al., Nature Physics 2021) ruled out the simplest parameter-free Diósi-Penrose model. GRW-type parameters are being cornered by LISA Pathfinder, cold-atom and levitated-particle experiments. No collapse has been seen.
- Wigner's friend. Theorems by Frauchiger and Renner (2018) and Bong et al.'s "local friendliness" experiment (Nature Physics, 2020) show that observers inside superpositions push interpretations into sharper corners: you cannot keep universality, single outcomes and observer-independent facts all at once.
- Indefinite causal order. The quantum switch puts operations into a superposition of orders; experiments since 2015 confirm it. Even the sequence of events can be quantum.
- Still no consensus. At quantum mechanics' centenary in 2025, a Nature survey of over a thousand physicists found the Copenhagen view still the most common (about a third), with many worlds, information-based views and Bohmian mechanics all holding significant minorities. Greene's 2004 guess, "no overwhelming favorite," still stands.
- Engineering decoherence. Quantum computers are machines for holding decoherence off; Google's Willow (2024) showed error rates falling as logical qubits grow, the first below-threshold quantum error correction.
So what. The delayed-choice eraser is the cleanest lesson in the book about conditioning. The raw screen data are featureless; structure appears only when you sort by information that arrives later. Data analysts meet this constantly: an A/B test that looks flat overall can hide opposite effects in segments that sum to zero (Simpson-like cancellation), and the segments can only be formed after joining a later dataset. Two rules follow. Don't conclude "nothing happened" from the aggregate, and don't believe a later join changed the past: it changed which story the data can support. Greene's "Full Monty" vignette is the human version: decide how you'll describe the past only after you know what you're conditioning on.
# Delayed-choice quantum eraser: sort signal photons by idler detector
import numpy as np
x = np.linspace(-3, 3, 13) # screen position (fringe periods)
env = np.exp(-x**2/4) # single-slit-like envelope
D1 = env/4 # which-path known (left)
D4 = env/4 # which-path known (right)
D2 = env*np.cos(np.pi*x)**2/2 # erased: fringes
D3 = env*np.sin(np.pi*x)**2/2 # erased: anti-fringes
total = D1+D2+D3+D4
for name, d in [('D1',D1),('D2',D2),('D3',D3),('ALL',total)]:
bar = ''.join(' .:-=+*#'[min(7,int(8*v/total.max()))] for v in d)
print(f'{name:3s} |{bar}|')
print('D2+D3 equals the envelope:', np.allclose(D2+D3, env/2))
Of Snowflakes and Spacetime: Symmetry and the Evolution of the Cosmos
If Feynman's one-sentence summary of science is "the world is made of atoms," the runner-up is "symmetry underlies the laws of the universe." Opening Part III, Greene shows that symmetry does three jobs. It generates the laws (translation, rotation, relativity's symmetries among observers). It makes cosmic time possible: time needs change, the absence of one symmetry, while a single cosmic clock needs spatial uniformity, which the microwave background confirms to one part in 100,000. And it narrows the shape of the universe to a handful of candidates. Along the way he explains expanding space with balloons, muffins and pennies. The history of the universe is, to a large extent, the history of symmetry.
Symmetry and the laws of physics
A symmetry is a manipulation that leaves no trace. Rotate a cue ball any way: unchanged. A plain dinner plate: any rotation about its centre. A snowflake: turns of 60 degrees. The letter A: a flip about its vertical axis. The more undetectable manipulations, the more symmetric the object: a sphere beats a cube, which survives only 90-degree turns about face axes. "Symmetries are like the deftest of prowlers."
Symmetries of laws, not of observations. The symmetries behind physics ask: what manipulations of you or your environment leave the laws unchanged? Your observations may change; the explanatory framework must not.
The gymnast. You train for four years in Connecticut until your body has an "innate sense of Newton's laws," then compete in New York. You bank on the same laws holding there, and they do: Newton's laws don't work one way in Connecticut and another in New York. On the moon, your trajectories would differ, but that difference is already in the law: gravity depends on mass. An environmental difference the law accommodates is not a change in the law. (Strictly, the trip from Connecticut to New York also rotates you, so it tests rotational symmetry too.)
| Symmetry | Manipulation that changes nothing in the laws | Who established it | What it buys |
|---|---|---|---|
| Translational | Moving to another place | Implicit since Newton | Laws learned on earth hold across the cosmos |
| Rotational | Turning to face another direction | Implicit since Newton | No preferred direction in deep space |
| Galilean / special relativity | Moving at any constant velocity, with light's speed also unchanged | Galileo; Einstein 1905 | All inertial observers equal |
| General relativity | Any motion, including acceleration, if gravity is included | Einstein 1915 | All observers equal; gravity = geometry |
| Gauge symmetries | Abstract internal "rotations" (Chapter 9) | Yang-Mills, 1950s onward | Electromagnetism, strong and weak forces |
Translational invariance need not have held. A universe could have laws as variable as local governments. We can't be absolutely sure laws are uniform everywhere, but if they change, it must be "way out there," beyond increasingly precise astronomical tests. That is symmetry's power: "because of translational symmetry, we can learn about fundamental laws at work in the entire universe without straying from home."
Rotational invariance looks false from earth (up and down differ) but that is the environment. In deep space, a prankster could re-orient your laboratory and no experiment would notice.
Relativity extends the march. Galileo saw that constant-velocity motion leaves the laws unchanged. Einstein added the speed of light to the list of invariant observations, elevating what had seemed an "environmental detail" into a law. General relativity goes further: you can feel acceleration, which seems to break the symmetry, but via the equivalence principle the force you feel is indistinguishable from a gravitational field, so every observer can claim to be at rest, attributing the forces felt to gravity. Newton's laws, by contrast, hold only for "inertial observers," a definition that is subtly circular (inertial observers are those for whom Newton's laws hold).
From consequence to foundation. Why these laws? Partly because experiments confirm them. But physicists also believe them because they feel right, and symmetry is the core of that feeling: no place, no constant velocity, and no vantage point should be special. General relativity is "the simplest theory incorporating this symmetry," and the theories of the other three forces rest on more abstract symmetries. Symmetries are not merely consequences of nature's laws; they are the foundation from which laws spring.
Symmetry and time
Time is, at least, "the bookkeeper of change": the hour hand moves, the sun shifts, the pages get more jumbled, the CO2 spreads. Paraphrasing Wheeler, "time is nature's way of keeping everything from happening all at once."
Time at the crossroads of symmetry.
- Time needs a broken symmetry. If the universe were perfectly symmetric along the time axis, with no change from moment to moment, the spacetime block could still exist, but time as we know it would be unrecognisable: no evolution, no ticking.
- Cosmic time needs an unbroken one. To say "the universe is 14 billion years old," clocks everywhere must have ticked alike. That requires symmetry in space: uniform conditions across the cosmos.
The microwave background: a fossil of uniformity
The universe is filled with microwave radiation, a cooled relic of the hot early cosmos, now about 2.7 degrees above absolute zero. (In analogue days it contributed a little to the snow on an untuned television.) It was discovered in 1964 by Arno Penzias and Robert Wilson at Bell Labs, who scraped out pigeon droppings before accepting that the noise was cosmic; Robert Dicke, Peter Roll, David Wilkinson and Jim Peebles interpreted it, building on earlier predictions by Gamow, Alpher and Herman.
How uniform? The temperature varies across the sky by less than a thousandth of a degree. On earth, the Weather Channel would be pointless: 85 degrees in Jakarta would mean 84.999 to 85.001 in Adelaide, Shanghai, Cleveland and Anchorage.
The uniformity yields two insights:
- Low entropy at the start. A young universe full of black holes and clumps would have left a clumpy imprint. Uniformity means the early universe was homogeneous, and with gravity, homogeneous means low entropy, exactly what Chapter 6 needs. Explaining that state is a goal of Part III.
- Uniform evolution everywhere. For temperatures in the Whirlpool galaxy and the Coma cluster to match ours to four decimal places, every region must have evolved the same way. Like water, lumpy at the molecular scale but smooth to the eye, the universe is homogeneous on scales of hundreds of millions of light-years.
Uniformity of change implies uniformity of elapsed time. Geologists on three continents agree on earth's age; physicists in the Milky Way, Andromeda and the Tadpole galaxy agree on the universe's. If the background were haphazard, "what were things like at 3 billion years?" would depend on whose clock you asked. The homogeneity of space provides a universal synchrony.
Stretching the fabric
How we learned the universe expands
- 1912 to 1925: Vesto Slipher measures galaxy redshifts; most are receding.
- 1922: Alexander Friedmann finds expanding and contracting solutions of Einstein's equations.
- 1927: Georges Lemaître independently derives expansion and a velocity-distance relation.
- 1929: Edwin Hubble, with the 100-inch Mount Wilson telescope, finds that two dozen galaxies recede faster the farther away they are.
- 1930: a Dutch newspaper cartoon of Willem de Sitter blowing up a ball: "A Lambda does the job!"
Modern versions of Hubble's law, as Greene quotes them: galaxies 100 million light-years away recede at about 5.5 million mph; at 200 million, twice that; at 300 million, three times.
Explosion in space versus expansion of space.
- Factory explosion: debris flies from a centre. Applied to galaxies, it would make our location the birthplace, a "pre-Copernican" asymmetry with no support in the data.
- General relativity: space is rubbery. Friedmann and Lemaître showed that the mutual gravity of cosmic matter means space must be stretching or contracting, never static, just as a ball popped above the catcher must be rising or falling. The analogy is mathematically exact: the ball's height obeys nearly the same equation as the universe's size. Galaxies are carried apart like poppy seeds in rising muffin dough. The outward motion is not an explosion within space; it is the swelling of space itself.
The balloon and the pennies
Glue evenly spaced pennies to a balloon and inflate it. Every Lincoln sees every other Lincoln rushing away, and no penny is the centre. Double the balloon: pennies 1, 2 and 3 inches apart become 2, 4 and 6 inches apart, so separation speed is proportional to distance. That is Hubble's law, explained with no special place: "almost too beautiful to be wrong."
Time in an expanding universe
Replace each penny with an identical clock. By symmetry, all experience the same conditions, so all tick alike. But aren't they moving apart at many speeds, which should desynchronise them?
Moving with space, not through it. Special relativity's time dilation concerns motion through space. A galaxy, like a glued penny, sits at rest relative to space; it separates from others only because space between them swells. So comoving clocks, those whose only motion is the cosmic flow, all record the same elapsed time. These are the clocks that define the age of the universe.
You may take your clock on a rocket, or hover just outside a black hole, and measure a different elapsed time since the bang. That is perfectly valid but individualistic. Cosmic age is the universal, agreed-upon measure.
A built-in speedometer. Move through space relative to the cosmic flow and the microwave background becomes warmer ahead (Doppler-compressed waves) and cooler behind. Earth sees exactly this dipole: from earth's orbit, the sun's galactic orbit, and the Milky Way's motion toward the constellation Hydra. Subtract it, and the exquisite uniformity appears.
Three subtleties of expansion
- The balloon's centre is meaningless. Only the surface represents space. A 3D version is a three-sphere: the 3D "surface" of a 4D ball. A solid bowling ball won't do, since it has a surface, an interior and a centre. Even professionals stick to the 2D analogy.
- Faster-than-light recession is fine. Distant galaxies do recede faster than light. Relativity limits motion through space, and galaxies barely move through it. Depending on whether expansion accelerates, light from some galaxies may stream toward us forever while the gap grows, a race "that would have made Zeno proud."
- You don't expand. Pennies don't grow as the balloon swells (a circle drawn in marker would). Nuclear, electromagnetic and gravitational forces hold atoms, bodies, planets, stars and galaxies together far more strongly than expansion pulls. Only on supergalactic scales does expansion win.
Cosmology, symmetry and the shape of space
Woken at night and asked the shape of the universe, you might despair: space is like Silly Putty. But symmetry rescues you. If every place and direction is equivalent on large scales, almost all shapes are excluded (a pear bulges at the bottom; an egg is pointy at the ends).
| Curvature | 2D picture | Triangle angles sum to | Walk straight ahead | Density vs critical (with no dark energy) |
|---|---|---|---|---|
| Positive | Sphere (3D: three-sphere) | More than 180° | Return home, like Magellan | Above critical |
| Zero (flat), infinite | Endless rubber sheet; infinite poppy seed muffin | Exactly 180° | Go forever, like the Energizer Bunny | Exactly critical |
| Zero (flat), finite | Video game screen that wraps (3D: three-torus) | Exactly 180° | Return home, no edge | Exactly critical |
| Negative | Pringles chip saddle (finite versions exist too) | Less than 180° | Go forever (if infinite) | Below critical |
Symmetry narrows infinity to a handful. Mathematicians have proven that uniform positive, zero or negative curvature exhausts the possibilities for a fully symmetric space. Several shapes (global topologies) can share one curvature: the infinite plane and the video screen are both flat. To choose among them we need Einstein's equations, which turn the average density into curvature. The dividing line is the critical density, about 10−23 grams per cubic metre, roughly five hydrogen atoms per cubic metre. In 2004 the data were "tipping the scales" toward flat; whether space is infinite or wraps around was "completely open."
A doughnut is a torus too, and it is flat in the curvature sense (its triangles sum to 180°); it only looks curved because of how it sits in our 3D world. Also note: even when space is flat, spacetime is curved, increasingly so toward the big bang, where Einstein's equations break down.
Cosmology and spacetime
Combine expanding space with the spacetime loaf. Comoving observers' watches all agree, so they slice the loaf identically. In a finite, flat, video-game universe, each slice is all of space at one cosmic time; earlier slices are smaller, with galaxies closer together.
The Coma astronomer. Light is a time capsule. We see Andromeda as it was about 3 million years ago (2.5 million by current measure) and the Coma cluster as it was 300 million years ago: if all its stars exploded now, we'd see nothing for 300 million years. An astronomer in Coma on our current now-slice, looking at earth, sees ferns, arthropods and early reptiles; she won't see the Great Wall or the Eiffel Tower for nearly 300 million years. Being well trained, she assigns what she sees to the right slices. If she moves through space relative to the cosmic flow, her slices tilt (as with Chewie in Chapter 5) and cut across many epochs, so cosmologists use the comoving slicing for convenience, though every view is equally valid.
Going back in time, the universe gets smaller, denser and hotter, like air in a bicycle pump. At about ten millionths of a second, ordinary matter dissolves into a primordial plasma; near time zero the known universe is smaller than a full stop. Our theories fail there, so Greene draws a fuzzy terra incognita at the start of the loaf.
If space is infinite
In a sphere or torus, the whole of space shrinks toward a dot. But infinite flat or saddle space stays infinite at every earlier time: halve infinity and it is still infinite. If space is infinite, the big bang did not happen at a point; it happened everywhere at once, with extreme density on an infinite expanse. Expansion then increases separations, not total size. Since data point to flatness and no wrap-around has been seen, infinite flat space is the front-runner.
The chapter closes with a hook: the universe's temperature has swept from ferocious to near absolute zero, and because of an interdependence of heat and symmetry, today's cosmos may be "a cool remnant of the far richer symmetry" of the early universe (Chapter 9).
Going deeper: the technical notes
- Redshift. Atoms emit at known wavelengths; light from receding galaxies is stretched, like a receding siren's pitch dropping, so measured wavelengths give speeds. Gravity also redshifts light climbing out of a well.
- Newton gives Friedmann. A test mass on a ball of radius R and density ρ has acceleration d2R/dt2 = −(4π/3)GρR. Identify R with the universe's scale and ρ with its density and you have Einstein's equation for a pressureless universe: the baseball analogy is literal.
- Cosmic horizons. Light follows ds = 0 in the metric ds2 = dt2 − a2(t)dx2. The comoving distance light covers from t to t0 is ∫dt/a(t). With accelerated expansion this integral stays bounded, so some regions can never exchange signals with us: they lie beyond our cosmic horizon.
- Curvature by triangles. Nineteenth-century differential geometry measures curvature intrinsically: angles summing to more than 180° (sphere), exactly 180° (plane, torus) or less (saddle).
- Curvature versus shape. Symmetry fixes three curvatures; each admits several global shapes (topologies).
- Penny caveat. Real galaxies differ; the claim is that on ~100-million-light-year scales differences average out.
- Clocks near black holes. Hovering near a horizon slows your clock severely: valid, but individualistic.
- No change, no time? If all change stopped, we'd experience no time passing; whether time "ends" is hard to answer and untestable. This differs from maximum entropy, where microscopic change continues.
Critical lens.
- Noether is missing. The deepest link between symmetry and law is Emmy Noether's theorem (1918): every continuous symmetry of the laws implies a conservation law. Translation gives momentum, rotation gives angular momentum, time translation gives energy. A chapter about symmetry generating laws omits it entirely. It also bears directly on cosmology: an expanding universe is not time-translation symmetric, so energy is not globally conserved (CMB photons lose energy as they redshift).
- "General relativity as symmetry" is contested. In 1917 Erich Kretschmann pointed out that almost any theory can be written in "generally covariant" form, so general covariance alone has little physical content. What is special about GR is the absence of fixed background structure, a subtler claim than "symmetry among all observers."
- Lemaître is under-credited. Lemaître derived the velocity-distance relation and estimated its slope in 1927, two years before Hubble; in 2018 the International Astronomical Union recommended calling it the Hubble-Lemaître law. Slipher's redshifts, which Hubble relied on, go unmentioned.
- Numbers that aged. Greene's speeds imply H0 ≈ 80 km/s/Mpc; modern values are 67 to 73. The universe is 13.8, not "about 14," billion years old. The background was released at about 380,000 years, not 300,000 (notes). Andromeda is 2.5 million light-years away, not 3 million.
- The torus needs more room. A wrapping universe would need sides far larger than "14 billion light-years from its midpoint": the observable universe's comoving radius is about 46 billion light-years, and Planck's search for matched circles in the microwave sky rules out a three-torus smaller than roughly the diameter of the surface of last scattering.
- Homogeneity is assumed, then confirmed only partly. The cosmological principle is an assumption tested mainly through the background's isotropy plus the Copernican assumption that we are not special. Isotropy around one observer does not by itself prove homogeneity.
2026 update: cosmology's symmetries under pressure
- Flatness nailed. Planck (2018) plus galaxy surveys give a curvature density Ωk = 0.001 ± 0.002: flat to better than half a percent. Whether space is infinite remains untestable directly; no matched-circle or topological signal has been found.
- Precise age. The universe is 13.79 ± 0.02 billion years old (Planck 2018), with ground-based ACT and SPT results (2023 to 2025) agreeing.
- The Hubble tension. Supernova distance ladders (SH0ES, H0 ≈ 73) disagree with the early-universe value (Planck, ≈ 67.4) at about 5 standard deviations. JWST Cepheid checks (2023 to 2024) confirmed the ladder is not obviously flawed; alternative ladders using tip-of-the-red-giant-branch stars land in between. The tension remains unexplained.
- Is dark energy constant? DESI's baryon acoustic oscillation maps (2024 and DR2, 2025) favour dark energy that weakens over time at 2.8 to 4.2 standard deviations, depending on supernova data. If confirmed, "a Lambda does the job" would need revising.
- Testing the cosmological principle. The quasar dipole (Secrest et al., 2021 and later) is about twice what our motion predicts, a roughly 5-sigma anomaly. Claimed giant structures (the Giant Arc, 2021; the Big Ring, 2024) approach or exceed the expected homogeneity scale. These are debated; most cosmologists still treat large-scale homogeneity as sound.
- Galaxy count. "More than 100 billion galaxies": deep-field counts rose to an estimated 2 trillion (2016), then fell to hundreds of billions after New Horizons measured the true darkness of the sky (2021).
So what. Greene's method generalises: impose the symmetry first, then let data choose among survivors. If you believe no location, direction or vantage point is privileged, the space of possible answers collapses from infinite to three. In engineering, the same move appears as invariance requirements: a pricing model that shouldn't care about currency units, a classifier that shouldn't care about image rotation. State those invariances explicitly and you prune hypotheses before fitting anything. And note the comoving-clock lesson: agree on a reference frame before comparing timestamps; the "age of the system" is meaningful only for clocks that ride with the system rather than through it.
# Age of the universe and the horizon from Friedmann's equation (flat ΛCDM)
import numpy as np
from scipy.integrate import quad
H0 = 67.4 # km/s/Mpc (Planck 2018)
Om, OL, Orad = 0.315, 0.685, 9e-5
H0_inv_Gyr = 977.8/H0 # 1/H0 in billions of years
E = lambda a: np.sqrt(Orad/a**4 + Om/a**3 + OL)
age = quad(lambda a: 1/(a*E(a)), 1e-8, 1)[0]*H0_inv_Gyr
Dh = quad(lambda a: 1/(a*a*E(a)), 1e-8, 1)[0]*H0_inv_Gyr # comoving, Gly
Devent = quad(lambda a: 1/(a*a*E(a)), 1, 1e4)[0]*H0_inv_Gyr
print(f'age of universe : {age:5.2f} billion years')
print(f'particle horizon today : {Dh:5.1f} billion light-years (comoving)')
print(f'event horizon (future) : {Devent:5.1f} billion light-years')
# Greene's Hubble-law speeds: 5.5 million mph per 100 million ly
v = 5.5e6*1.609/3600; d = 100/3.262
print(f'implied H0 from the book: {v/d:4.0f} km/s/Mpc')
Vaporizing the Vacuum: Heat, Nothingness and Unification
For 95 percent of cosmic history a reporter could fax in the same line: universe expands, cools, stays uniform. The first fraction of a second was different. Greene explains how the universe as a whole underwent phase transitions, like steam condensing to water, in which a Higgs field "condensed" into a non-zero value throughout space. That Higgs ocean gives fundamental particles their mass and hides a deep symmetry between the electromagnetic and weak forces, unified by Glashow, Weinberg and Salam. Grand unification may extend this to the strong force. The emptiest possible space is not nothing: it may be filled with a Higgs field, and only by heating the vacuum would nature's full symmetry reappear. In 2004 the Higgs particle was a prediction awaiting the Large Hadron Collider; in 2012 it was found.
Heat and symmetry
Heat ice and little happens until 0 °C, when it suddenly melts; heat water and little happens until 100 °C, when it boils. Without experience you'd never guess that a rock-hard solid, a viscous liquid and a hot gas are all H2O. These sudden changes are phase transitions, and most substances go through them. (Heat further and you reach a fourth state, plasma, where atoms come apart.)
Heating usually raises symmetry. Ice is a hexagonal lattice, unchanged only by special rotations (multiples of 60 degrees about particular axes). Liquid water is a jumbled clump unchanged by any rotation, and steam's freely flitting molecules are more symmetric still. Order is not symmetry: something is more symmetric if more transformations leave it looking the same. So melting and boiling increase symmetry; condensing and freezing reduce it. (Odd exceptions such as Rochelle salt become more ordered when heated.) Entropy still rises overall, because heat released to the surroundings pays for the drop in symmetry.
The 1970s insight: the cosmos as a whole can undergo phase transitions. As the universe expanded it cooled, like air escaping a bicycle tyre. Passing through critical temperatures, it may have experienced drastic reductions in symmetry. Many physicists think we live in a "condensed" or "frozen" phase. What condensed was not a liquid but a field: a Higgs field.
Force, matter and Higgs fields
| Field | Messenger / quantum | Range and strength | Status in 2004 | Status in 2026 |
|---|---|---|---|---|
| Electromagnetic | Photon | Long range; light, electricity, magnetism | Established | Established |
| Gravitational | Graviton (hypothetical) | Long range; by far the weakest (a fridge magnet beats the whole earth) | Graviton unseen | Gravitational waves detected (2015); graviton still unseen |
| Strong nuclear (Yang-Mills) | Gluons | Nuclear scales; binds quarks | Gluons seen (DESY, 1979) | Established |
| Weak nuclear (Yang-Mills) | W and Z | Nuclear scales; radioactive decay, solar fusion | W, Z seen (CERN, 1983) | Established |
| Matter fields | Electrons, quarks... | Particles are field excitations | Established | Established |
| Higgs | Higgs particle | Fills space with a non-zero value | Predicted | Discovered 2012 (125 GeV) |
Every particle is an excitation of a field: photons of the electromagnetic field, electrons of the electron field, up-quarks of the up-quark field. A difference: force particles can pile into the same state and build classical fields you can see when you switch on a light, while matter particles obey Pauli's exclusion principle and can't. Gravity is the one field that has resisted this quantum description (Chapter 12).
Fields in a cooling universe
The hotter it is, the more violently a field's value heaves up and down, like a boiling pot. At 10−43 seconds, the temperature was about 1032 K; today's deep space is 2.7 K. As things cooled, most fields' average values settled toward zero, what we intuitively call emptiness. Not the Higgs field.
Thought experiment: the frog in the hot bowl. Drop a frog into a hot metal bowl with a pile of worms at the centre. It leaps frantically, never near the worms. As the bowl cools it calms down and slides to the bottom, where dinner awaits. Now use a bowl with a bump in the middle and the worms on top of it. As it cools, the frog slides down the sides and settles in the circular valley, a non-zero distance from the worms, and stays there.
Translate: distance from the worms = the field's value; height = the field's energy. An ordinary field's energy bowl lets it settle at zero. A Higgs field's "Mexican hat" bowl makes it settle at a non-zero value, uniformly throughout space: a Higgs ocean (technically, a non-zero vacuum expectation value).
Why "empty" space contains a Higgs field. The vacuum is the lowest-energy state of a region. For ordinary fields, that means value zero. For a Higgs field, forcing the value to zero costs energy (the frog must jump up onto the central bump). So removing the Higgs field means adding energy; the emptiest possible space is full of Higgs field. Analogy: noise-cancelling headphones. You hear the least when they are actively producing sound, and more when you switch them off.
Spontaneous symmetry breaking. The valley is perfectly circular: every point is equivalent. But the field must land on one point, which then becomes special. The laws keep the symmetry; the state breaks it. (Points around the valley share a magnitude but differ in phase: the Higgs value is a complex number.)
The Higgs ocean and the origin of mass
Swing your arm, then swing a bowling ball: the bigger the mass, the harder your muscles work. Mass is resistance to acceleration (inertia). Newton, Mach and Einstein identified what accelerations are measured against (absolute space, the distant stars, spacetime), but none explained why objects resist acceleration at all. The Higgs field offers a mechanism.
Molasses, with three caveats. Quarks and electrons interact with the Higgs ocean, which resists their acceleration like molasses around a submerged Ping-Pong ball. That resistance is (part of) their mass. But:
- You can't take particles out. The ocean fills all space; masses are the same everywhere.
- It resists only acceleration. Constant-velocity motion feels no "friction," so the Higgs ocean doesn't pick out a rest frame.
- It isn't the main source of your mass. Protons and neutrons get much of their mass from the energy of gluons binding quarks (E = mc2). The Higgs gives mass to fundamental particles.
Inertial versus gravitational mass: the Higgs seems to explain only inertia, but the equivalence principle makes the two the same.
The paparazzi. Different particles couple to the Higgs ocean with different strengths. Photons pass through untouched (massless), like unknown people walking past photographers. The top quark, about 350,000 times heavier than the electron, interacts that much more strongly, like a movie star mobbed by the press. (This metaphor descends from David Miller's prize-winning answer to UK science minister William Waldegrave's 1993 challenge to explain the Higgs in a page.) Why each particle couples as it does remains unexplained; but without the Higgs ocean, all fundamental particles would be massless.
Unification in a cooling universe
The Higgs field condenses at about 1015 degrees, nearly 100 million times the sun's core temperature, reached about 10−11 seconds after the big bang. Before that, the field fluctuated around zero, there was no ocean, "the paparazzi vanished," and every known particle had the same mass: zero.
The electroweak transition: change plus lost symmetry. Like water freezing, the formation of the Higgs ocean brought (1) a qualitative change (massless particles acquired their present masses) and (2) a loss of symmetry (equal masses became unequal). The loss was bigger still. Before it, the force particles were massless too, and as Sheldon Glashow, Steven Weinberg and Abdus Salam found in the 1960s, photons and W and Z particles were interchangeable, a gauge symmetry, like a snowflake's indifference to rotation. Symmetry among messengers means symmetry among forces: at high enough temperature, the electromagnetic and weak forces are one electroweak force.
Today the symmetry is hidden because photons zip through the Higgs ocean while W and Z particles slog through it, acquiring masses 86 and 97 times the proton's (Greene's paparazzi for these are Bill Clinton and Madonna, "not to scale"). Maxwell unified electricity and magnetism in the 1800s; Glashow, Weinberg and Salam took the next chapter. They predicted W and Z particles before they were seen, trusting symmetry and theory; the particles turned up, and the three shared the 1979 Nobel Prize.
Who built the Higgs idea. Named for Peter Higgs, but Thomas Kibble, Philip Anderson, Robert Brout and François Englert (and Gerald Guralnik and Carl Hagen) were essential. Yang and Mills gave the strong and weak force fields their framework in the 1950s. Howard Georgi, Greene's freshman adviser, once filled a chalkboard for an hour explaining plans to test grand unification; the student "hardly understood a word."
Grand unification
If two forces merged in the early universe, might three or four merge earlier still, crystallising into today's forces through a series of phase transitions?
- 1974: Georgi and Glashow propose the first grand unified theory (based on the group SU(5)). With later work by Georgi, Helen Quinn and Weinberg, it says the strong, weak and electromagnetic forces were one above about 1028 degrees, before about 10−35 seconds. Photons, gluons, W and Z were all interchangeable.
- A grand unified Higgs condensed below that temperature, treating gluons differently and splitting off the strong force. Later, the electroweak Higgs split the weak and electromagnetic forces.
- Prediction: protons decay (into anti-electrons and pions, say), a faint relic of the early symmetry. Years of underground searches found none, ruling out the original model. Variants survive; none confirmed.
Greene's verdict: grand unification is "one of the great, as yet unrealized, ideas in particle physics." Superstring theory (Chapter 12) attempts to unify all four forces. Either way, today's universe shows only a remnant of the early universe's symmetry.
The return of the aether?
Higgs ocean versus nineteenth-century aether.
- Like the aether: it fills all space, permeates everything, can't be removed (unless you reheat the universe above 1015 degrees) and redefines nothingness.
- Unlike the aether: it has nothing to do with light: it doesn't carry light or change its speed, so the Michelson-Morley style experiments that killed the aether are irrelevant. And since it resists only acceleration, it picks out no preferred rest frame, so it is fully compatible with special relativity.
Greene's 2004 forecast: the Large Hadron Collider at CERN, then "slated to come online in 2007," should knock Higgs particles out of the ocean, as underwater collisions knock molecules out of the Atlantic. Finding it would confirm symmetry's power to guide theory, prove an ancient symmetric era, and show our notion of empty space has been naïve, oddly echoing Henry More's view (Chapter 2) that space is never truly empty.
Entropy and time
Through every phase transition symmetry was lost, but total entropy rose steadily. "Entropy can increase only if it starts out low": if War and Peace starts jumbled, tossing it changes nothing. The timeline is manifestly not eternal disorder, so the newborn universe must have been highly ordered and uniform. Explaining that requires going inside the fuzzy patch, the subject of Chapter 10.
Going deeper: the technical notes
- Value versus energy. In the frog picture the field's value is the distance from the centre and its energy is the height. In the valley the energy is minimal (zero by convention) but the value is non-zero.
- The groups. Electroweak theory has symmetry group SU(2) × U(1); the photon is a mixture of the U(1) boson and a U(1) inside SU(2), closely related to the W's and Z. Because the group has two factors, the unification is partial. Adding color SU(3) for the strong force gives SU(3) × SU(2) × U(1), three factors, which motivates a single simple group at high energy: SU(5) (Georgi-Glashow), SO(10) or E6.
- Mass ranges. The six quarks span about 0.0047 to 189 proton masses (Greene's figures); W and Z are 86 and 97 proton masses.
- Rochelle salts are the rare substances that get more ordered when heated.
Critical lens.
- The electroweak "phase transition" isn't one. Lattice simulations (Kajantie, Laine, Rummukainen and Shaposhnikov, 1996) showed that for a Higgs mass above about 70 GeV, the Standard Model's electroweak change is a smooth crossover, not a sharp transition like boiling. With the measured 125 GeV, there was no sudden condensation. A true first-order transition would need new physics, which is also what most theories of matter-antimatter asymmetry from electroweak physics require.
- "A significant fraction" undersells the gluons. The Higgs accounts for only about 1 to 2 percent of a proton's mass; the remaining ~98 percent is the energy of quarks and gluons confined by the strong force. Nearly all the mass of you and the bowling ball comes from QCD, not the Higgs.
- The molasses metaphor misleads. Physicists (including Higgs-mechanism textbook authors) object that drag implies dissipation and a preferred frame. Greene's caveat that the ocean resists only acceleration helps, but the mechanism is better described as particles' coupling to a constant background field, which shows up as a mass term in their equations.
- Dated numbers. The top quark is about 184 proton masses (172.6 GeV), not 189; the up quark's mass is about 0.0023 proton masses, not 0.0047. The ratio of top to electron masses (~340,000) holds.
- Gluon and W/Z discovery sites are right (DESY in Germany, 1979; CERN, 1983), but the W and Z were found by UA1 and UA2 at CERN's proton-antiproton collider, led by Carlo Rubbia and Simon van der Meer (Nobel 1984), unnamed here.
- "Unification" by phase transition is partly a picture. In electroweak theory the forces don't literally "merge into one": the SU(2) × U(1) structure is two factors even at high temperature, as Greene's own note concedes.
2026 update: the Higgs ocean, found and probed
- Discovery. On 4 July 2012, ATLAS and CMS at the LHC announced a new particle at 125 GeV. Higgs and Englert shared the 2013 Nobel Prize (Brout had died in 2011). The LHC's physics run began in 2010, three years after Greene's expected 2007, delayed by a 2008 magnet accident.
- It behaves as predicted. Couplings to the W, Z, top, bottom, tau and (since 2020 to 2021) muon have been measured, rising in proportion to mass, within about 5 to 10 percent for the best-measured ones. The paparazzi picture is quantitatively right.
- The shape of the bowl is next. The Higgs self-coupling, which fixes the Mexican hat's shape, is not yet measured; the High-Luminosity LHC (from about 2030) aims to see Higgs pair production.
- A metastable vacuum. With the measured Higgs and top masses, the Standard Model's Higgs potential turns over at very high field values: our vacuum appears metastable, with a lifetime vastly longer than the age of the universe. Empty space may not be in its lowest state after all.
- The W mass scare. CDF's 2022 W-mass measurement disagreed with the Standard Model at 7 sigma; ATLAS (2024) and CMS (2024) measured 80.36 GeV, in line with prediction.
- Proton decay, still absent. Super-Kamiokande puts the lifetime for p → e+π0 above 2.4 × 1034 years (2020), ruling out minimal SU(5) many times over. Hyper-Kamiokande (data from about 2027 to 2028) and DUNE will push an order of magnitude further.
- Unification without supersymmetry looks shaky. In the Standard Model the three force strengths nearly but not exactly meet at high energy; supersymmetry fixes this, but the LHC has found no superpartners up to roughly 1 to 2 TeV.
So what. Three habits worth borrowing. First, look for the "phase" behind sudden change: systems (markets, teams, codebases) often change little as a parameter drifts, then reorganise abruptly at a threshold, and the new phase usually has less symmetry (more structure, more specialisation). Second, the lowest-energy state need not be the "empty" one: the default state of an organisation may already contain a background field (norms, legacy dependencies) that is costly to remove. Third, the Higgs story is the gold standard of prediction: a 1964 symmetry argument predicted a particle found 48 years later. Trust an argument from symmetry, then build the collider to check it.
# Spontaneous symmetry breaking: a field cools in a Mexican-hat potential
# V(phi) = (|phi|^2 - 1)^2 ; noise ~ temperature, which falls over time
import numpy as np
rng = np.random.default_rng(1)
def cool(steps=4000, dt=0.01):
phi = np.zeros(2) # start at the symmetric point
for k in range(steps):
T = 2.0*(1 - k/steps) # cooling schedule
r2 = phi @ phi
grad = 4*(r2 - 1)*phi # dV/dphi
phi += -grad*dt + np.sqrt(2*T*dt)*rng.normal(size=2)
return phi
regions = np.array([cool() for _ in range(12)])
r = np.hypot(*regions.T); ang = np.degrees(np.arctan2(regions[:,1], regions[:,0]))
print('final |phi| (all ~1, the valley):', np.round(r, 2))
print('chosen angle (random, symmetry broken):', np.round(ang).astype(int))
Deconstructing the Bang: What Banged?
The big bang theory describes how the universe evolved from a split second after the beginning, but says nothing about the bang itself: what banged, why, or how. Gravity is attractive, so what pushed space apart? Greene's answer is gravity itself. In general relativity, pressure gravitates, and negative pressure produces repulsive gravity. Einstein's discarded cosmological constant had exactly this property; in 1979 Alan Guth realised a Higgs-like field stuck on a plateau behaves the same way, but enormously more strongly, driving a burst of inflation that expanded space by at least 1030 in about 10−35 seconds. Inflation explains the uniform microwave background (the horizon problem) and the flatness of space, and predicts the universe holds the critical density. The missing 70 percent turned up in 1998 as dark energy, accelerating the expansion: Einstein's "blunder" returned as the dominant ingredient of the cosmos.
The bang the big bang leaves out
A common misconception is that the big bang is a theory of origins. It isn't. It "says nothing at all about time zero." It also poses a puzzle: in the dense early universe gravity dominated, and gravity attracts. So what supplied the outward push? In the 1980s an old idea of Einstein's, resurrected, provided the answer: in the right environment gravity is repulsive.
Einstein and repulsive gravity
Einstein's cosmological rollercoaster
- 1915: General relativity explains Mercury's perihelion precession exactly; Einstein reports heart palpitations.
- 1917: Applying GR to the whole universe, he finds it cannot be static. Sharing everyone's belief in an eternal, unchanging cosmos, he adds the cosmological constant (Λ) to hold it still.
- 1919: Eclipse expeditions to Principe and Sobral, Brazil confirm light bending; Einstein becomes a celebrity overnight.
- 1929: Hubble finds the universe expanding. Einstein "rued the day," erased Λ, and everyone forgot it for decades.
- 1980s and 1998: Λ returns, first as the engine of inflation, then as dark energy.
Why a balancing term? A leaping dancer slows, stops and falls; to make her hover, a repulsive force must exactly cancel gravity. The same is true for the universe: to keep space at fixed size, something must push. Einstein found Λ supplies that push. The modern reading (going back to Lemaître): Λ is a uniform energy of space itself, transparent and dark, something like the aether or a Higgs ocean.
Pressure gravitates: the Hall of Wits
Thought experiment: two puzzles for parole. In the twenty-fifth century's Hall of Wits, white-collar convicts win release by solving puzzles. (The man next door must explain why Gilligan's Island reruns became the most popular show of the twenty-second century.)
- Two identical gold cubes must register different weights, with no change in their gold. Newton says impossible. You heat one: its atoms carry more kinetic energy, and in general relativity all energy gravitates, so it weighs more (by a minuscule amount: a 10 °C rise in a one-pound cube adds roughly 10−14 pound).
- Two identical Jack-in-the-boxes, same mass, same temperature. You close one, compressing its spring. The compressed spring stores energy and pushes outward: it has pressure. It weighs a touch more.
Pressure is a source of gravity. In Newton's theory only mass gravitates. In Einstein's, mass, energy and pressure all do. Positive pressure (outward, like a spring) adds to attraction. Negative pressure (inward suction) contributes repulsive gravity. This is separate from the ordinary force of a pressure difference (your eardrums underwater): a uniform pressure exerts no direct force, but it sources the gravitational field.
A cosmological constant fills space with positive energy and negative pressure, and the repulsion from the pressure wins: a positive cosmological constant exerts an overall repulsive gravitational force. Ordinary particles have positive pressure, which is why Λ can't be made of anything familiar. Its push is cumulative: negligible across the solar system, dominant across cosmic distances.
Had Einstein trusted his original equations, he would have predicted the expansion a decade before Hubble, perhaps "the greatest discovery of all time."
Of jumping frogs and supercooling
The big bang equations (Einstein, de Sitter, Friedmann, Lemaître) allow expansion but, like Newton's laws for a fly ball, say nothing about who threw it. Then, in December 1979, Alan Guth, a particle physicist and postdoc at SLAC, working with Henry Tye on Higgs fields in grand unified theories, asked: what if the frog of Chapter 9, in an early jump, landed on the central plateau and stayed there as the bowl cooled?
Supercooling. A Higgs field stuck on the plateau after the universe has cooled is supercooled, like ultra-pure water staying liquid below 0 °C because ice needs impurities to nucleate. Guth and Tye first cared because of the magnetic monopole problem (grand unified theories predict heavy single-pole magnets that would overfill the universe; none is seen). Then Guth's hunch: a field stuck on the plateau fills space with energy and negative pressure in exactly the proportions of a cosmological constant. (Others, including Martinus Veltman, had noticed the equivalence; Tye says a journal page limit cut their sentence about exponential expansion.)
Inflation
Guth saw two differences between a supercooled Higgs field and Einstein's Λ, "differences that make all the difference":
- It doesn't last. Like a restless frog, quantum jumps eventually knock the field off the plateau, perhaps within 10−35 seconds, and its energy and pressure relax to zero. Andrei Linde in Moscow and Paul Steinhardt with Andreas Albrecht at Penn found a smoother version: on a gently sloping hill the field simply rolls down, uniformly throughout space.
- It is enormous. Guth estimated the energy at more than 10100 times Einstein's value.
Brief plus monumental equals a phenomenal outward burst: "Guth provided the big bang with a bang."
The scale of the burst. The field driving it is called the inflaton. Depending on its potential, space expanded by 1030, 1050, 10100 or more. Even 1030 is like enlarging a DNA molecule to the size of the Milky Way in far less than a billionth of a billionth of a billionth of a blink, more than all expansion in the 14 billion years since. In many models, if the whole cosmos were shrunk to earth's size, the observable universe would be smaller than a grain of sand.
When the inflaton reached the valley, it handed its energy to ordinary particles and radiation, "like a foggy mist settling on the grass as morning dew," filling space uniformly. From then on the standard big bang takes over.
Old, new and chaotic inflation.
- Old inflation (Guth, 1980): the field tunnels off the plateau via bubbles of true vacuum. Flaw: the space between bubbles keeps inflating, so bubbles never merge into a smooth universe, and bubble-wall collisions leave matter lumpy (Guth and Erick Weinberg; Hawking, Moss and Stewart).
- New inflation (Linde; Albrecht and Steinhardt, 1982): the field rolls slowly down a gentle slope. One bubble grows to contain the whole observable universe; energy converts to particles uniformly, through "friction" with other fields.
- Chaotic inflation (Linde, 1983): no special potential or start needed. In a chaotic early universe, any tiny region (10−33 cm) where the field happens to be high on even a simple bowl rolls slowly, because high values create drag, and inflates. Many consider it the most convincing version.
- Precursors: Starobinsky (Moscow, 1979 to 1980) found inflation-like expansion but didn't stress its cosmological payoffs; Katsuhiko Sato (1981) proposed a version; Chibisov and Linde (1978) had the idea but didn't publish because of the bubble problem.
Two cautions. First, the inflationary bang is an event the pre-existing universe experienced, not necessarily the creation event; a fuzzy patch remains: why an inflaton, why that potential, why space and time, and, in Leibniz's phrase, why there is something rather than nothing. Second, inflation is a framework, not one theory: details (duration, push, expansion, potential shape, number of fields) are free. What transcends the details is the bang itself, plus solutions to two famous problems.
The horizon problem
The microwave background has the same temperature in every direction to better than a thousandth of a degree. How did distant regions coordinate? Entanglement dilutes too fast to help. Perhaps they were once in contact, like a hot kitchen and a cool living room sharing heat through an open door? In the standard big bang, no.
Secretariat's race, run backward. A region's horizon is the farthest distance light could have travelled since the bang. Run the cosmic film backward. In the standard big bang, gravity decelerates expansion: like Secretariat covering the first half of the course in 2 minutes and the second half in 3, you must rewind more than halfway to halve the distance between two regions. So the earlier you look, the less time they had to communicate relative to their separation, and the more puzzling their equal temperatures become.
With inflation, expansion accelerates: Secretariat runs the second half in 1 minute, so you rewind less than halfway. Going back, regions get easier to connect. With expansion of at least 1030, everything we see today was once as close as kitchen and living room: the universe "makes up for the sluggish start" by flinging the equalised regions apart faster than light, which relativity permits for the swelling of space.
The flatness problem
Density determines curvature: above the critical density (about five hydrogen atoms per cubic metre) space curves like a sphere; below, like a saddle; exactly at it, flat. If the density starts exactly critical it stays so, but any deviation grows: 99.99 percent of critical at one second would be 0.00000000001 of critical today.
The razor-thin ledge. By the 1980s, observations showed the density was not thousands of times off critical. For the standard big bang to fit, the density at one second had to be within a millionth of a millionth of a percent of critical: a mountaineer on a razor-thin ledge, or Greene's college shower, where the knob had to be set perfectly or the water ran scalding or freezing ("some students just stopped showering altogether"). A diehard can shrug "that's just how it was," but physicists find theories that hinge on unexplained fine-tuning grossly unnatural.
Inflation reverses the instability. Attractive gravity amplifies deviations from critical density; repulsive gravity shrinks them, like magnets in the climber's boots and the ledge pulling her foot into place. Geometrically: Nebraska looks curved draped on a sphere a few hundred miles wide, flat on earth, and flatter still on a sphere a billion times larger. Inflation stretched space so much that the observable universe is a tiny patch, and looks flat whatever the whole is like. Inflation predicts the observable universe should have very nearly 100 percent of the critical density.
Progress and prediction: where's the density?
For years, surveys of visible matter found only about 5 percent of critical: not wildly off (no fine-tuning disaster) but far short of inflation's 100 percent. The resolution came in two steps.
A prediction of darkness
The dark matter detectives.
- Fritz Zwicky (Caltech, early 1930s): galaxies in the Coma cluster move too fast for visible matter to hold them, like droplets that should fly off a spinning tyre. He posited unseen mass. (Zwicky, famously caustic, called colleagues "spherical bastards": bastards whichever way you looked at them.)
- Sinclair Smith (Mount Wilson, 1936): the same for the Virgo cluster.
- Vera Rubin and Kent Ford (Carnegie, 1970s): stars in spiral galaxies orbit too fast at large radii; galaxies must sit in giant halos of non-luminous matter. Like inferring a dark-robed mime from his white gloves.
Gravitational analyses put dark matter at about 25 percent of critical. Visible plus dark: 30 percent. What it is made of (exotic particles? Small black holes?) "no one knows." That left 70 percent missing.
The runaway universe
The independent check was the deceleration parameter: measure how fast expansion slowed in the past by comparing speeds (from redshift, easy) and distances (hard) of far-off objects. Distances need standard candles, objects of known intrinsic brightness, like 100-watt bulbs dotted across space.
Type Ia supernovae as standard candles. A white dwarf siphons material from a companion until it reaches about 1.4 solar masses, then explodes in a runaway nuclear reaction. Because every explosion happens at the same critical mass, their peak brightness is nearly uniform, and at a billion suns they're visible across the universe. They're rare (once every few hundred years per galaxy), so the teams of Saul Perlmutter (Berkeley) and Brian Schmidt (Australian National University) monitored thousands of galaxies with wide-field telescopes and found nearly four dozen.
1998: the expansion is speeding up. Both teams found that expansion slowed for roughly the first 7 billion years, then began to accelerate, like a car gliding through an EZ-Pass tollbooth and then hitting the gas. A cosmological constant explains it: its push stays fixed while matter's pull weakens as matter thins out, so eventually repulsion wins. The required dark energy: about 70 percent of the critical density.
The missing 70 percent
5 percent ordinary matter + 25 percent dark matter + 70 percent dark energy = 100 percent: exactly what inflation predicted. Two independent lines of evidence (inflation's flatness, supernova acceleration) give corroborating second opinions. And the Copernican revolution deepens: we are not at the centre, and the stuff we're made of is "flotsam on the cosmic ocean."
The far future. In about 100 billion years, all but the nearest galaxies will be carried beyond our view by faster-than-light expansion: a "vast, empty, and lonely" universe.
Puzzles remain. What came before inflation? What put the inflaton up the hill? Why the mishmash: 5, 25, 70? Many find the recipe "distinctly unattractive." Still, Greene calls inflation "far and away the front-running cosmological theory," with deeper support to come in Chapter 11.
Going deeper: the technical notes
- The equations. Einstein changed Gμν = 8πTμν to Gμν + Λgμν = 8πTμν. Positive Λ gives negative pressure and repulsion; negative Λ gives extra attraction.
- Acceleration equation. ä/a = −(4πG/3)(ρ + 3p). If p = −ρ (a cosmological constant or a field on a plateau), the right side is positive and expansion accelerates. Ordinary matter (p ≈ 0) and radiation (p = ρ/3) decelerate it.
- "Mass" for Newton means the summed rest masses of the constituents; Einstein adds every form of energy (E = mc2) and, separately, pressure. The Jack-in-the-box partly works through energy; the deeper point is that pressure itself gravitates. The repulsion is the internal gravitational field of a region filled with negative pressure.
- Quantum jumps of fields. For fields, the value and its rate of change play the roles of position and velocity, so the uncertainty principle makes field values jitter: the restless frog.
- Why not reuse the known Higgs? Models in which the electroweak or grand unified Higgs also drives inflation usually hit technical problems; most invoke a new field, the inflaton.
- Faster-than-light separation is essential. Two people who can exchange flashlight signals can never lose contact by running, since they can't outrun light. For regions once in contact to be out of contact now, they must have separated faster than light, which is exactly what inflation's swelling of space did.
- During inflation the horizon stays fixed while space swells exponentially, which is why our observable universe is a speck in a vastly larger whole.
- Critical density drops as the universe expands, but a universe that starts critical tracks it exactly.
- Caveats in the notes: not all type Ia supernovae may form by the same route, but their uniformity is well established. Peebles, Krauss, Turner and Steigman had argued for a small Λ before 1998. Some propose modified gravity on cosmic scales instead of dark energy.
Critical lens.
- Adam Riess is missing. The 2011 Nobel Prize for the accelerating universe went to Perlmutter, Schmidt and Adam Riess, who led the High-z team's decisive analysis. Greene names only the team leaders.
- Einstein's static universe was doomed anyway. Eddington (1930) showed it is unstable: a slight nudge makes it expand or collapse. Λ couldn't deliver the stillness Einstein wanted, a point that makes the episode more instructive than the "blunder" story.
- Inconsistent and dated numbers. The Coma cluster is "300 million light-years" away in Chapter 8 and "370 million" here (the modern value is about 320 million). The horizon "sphere" in the notes has a radius of "roughly 14 billion light-years"; the comoving radius of the observable universe is about 46 billion light-years. Acceleration began about 6 billion years ago (redshift about 0.6), when the universe was nearly 8 billion years old. The gold-cube weight gain is about ten times larger than Greene's figure.
- "Predicted 100 percent" is softer than it sounds. Before 1998, many inflationary cosmologists had resigned themselves to an open (sub-critical) universe, and open-inflation models were built to match it. Inflation can be made to produce curvature, so flatness is a natural prediction rather than a strict one.
- Inflation's critics are absent. Roger Penrose argues inflation can't explain low entropy, because it needs special initial conditions of its own. Ijjas, Steinhardt and Loeb (2017) argue that, with eternal inflation and a multiverse, "anything that can happen will happen," making inflation hard to falsify. Over thirty prominent cosmologists signed a reply defending it. The debate is live and goes unmentioned.
- The cosmological constant problem (quantum field theory naively predicts vacuum energy 10120 times larger than observed) is the elephant in this chapter; Greene defers it.
2026 update: inflation and the dark sector, tested
- Inflation's fingerprints confirmed, its simplest model ruled out. Planck (2018): space is flat to ~0.2 percent, and the spectral index ns = 0.965 ± 0.004, a slight tilt that slow-roll inflation predicted. But BICEP/Keck (2021) limits primordial gravitational waves to r < 0.036, excluding Linde's simplest chaotic models (φ2, φ4). The Starobinsky model (r ≈ 0.004) became the favourite; in 2025, ACT DR6 combined with DESI nudged ns to about 0.974, putting mild pressure even on it.
- A false alarm. BICEP2's 2014 announcement of primordial gravitational waves was traced to galactic dust within a year. The B-mode hunt continues with the Simons Observatory (operating since 2024 to 2025) and the planned LiteBIRD satellite.
- Dark matter still unidentified. The Bullet Cluster (2006) showed mass separated from gas, strong evidence for dark matter over modified gravity. But direct detection keeps coming up empty: LZ (2024 to 2025) set the strongest WIMP limits yet; the LHC found no candidates; axion searches (ADMX) continue.
- Dark energy may be evolving. DESI (2024; DR2, 2025) prefers dark energy that weakens over time at 2.8 to 4.2 sigma. If confirmed, Λ is not constant and the "lonely future" may differ.
- Better numbers. Planck 2018 composition: 4.9 percent ordinary matter, 26.4 percent dark matter, 68.7 percent dark energy; age 13.8 billion years. Greene's 5/25/70 holds up well.
- New eyes. Euclid (launched 2023) and the Vera C. Rubin Observatory (first images, 2025) will map billions of galaxies and find millions of supernovae, testing both dark components with unprecedented statistics.
So what. Three lessons generalise. Look for the hidden source term: Newton missed that energy and pressure gravitate; many models miss a contributor that seems negligible in everyday conditions but dominates at scale. Prefer mechanisms that make outcomes insensitive to unknown initial conditions: inflation's appeal is that it erases dependence on the details of the start, the same reason engineers prize self-correcting, convergent processes over ones that require perfect setup. And seek corroborating second opinions from independent methods: the 70 percent was believed because two unrelated measurements (geometry and supernovae) demanded the same number. Finally, Einstein's Λ is a caution: he bent a correct theory to fit a prejudice; the term returned, but for the opposite reason.
# When did the universe start accelerating? (flat ΛCDM, Planck 2018)
import numpy as np
from scipy.integrate import quad
from scipy.optimize import brentq
Om, OL, H0 = 0.315, 0.685, 67.4
tH = 977.8/H0 # Hubble time in Gyr
E = lambda a: np.sqrt(Om/a**3 + OL)
q = lambda a: (0.5*Om/a**3 - OL)/E(a)**2 # deceleration parameter
a_t = brentq(q, 0.3, 0.99); z_t = 1/a_t - 1 # q = 0: decel -> accel
age = lambda a: quad(lambda x: 1/(x*E(x)), 1e-8, a)[0]*tH
print(f'transition redshift z = {z_t:.2f}')
print(f'universe age then = {age(a_t):.1f} Gyr')
print(f'how long ago = {age(1)-age(a_t):.1f} Gyr')
# acceleration equation: a''/a ∝ -(rho + 3p); Λ has p = -rho
for name, w in [('matter',0),('radiation',1/3),('Λ',-1)]:
print(f'{name:9s} rho+3p sign: {"+" if 1+3*w>0 else "-"} ->',
'decelerates' if 1+3*w>0 else 'ACCELERATES')
Quanta in the Sky with Diamonds: Inflation, Quantum Jitters and the Arrow of Time
Inflation's early fame came from the horizon and flatness problems. This chapter covers three further payoffs that many think matter more. First, structure: quantum jitters in the inflaton field, stretched to astronomical size, seeded every galaxy, and their imprint matches the microwave background's temperature pattern with stunning precision. Second, energy: because the inflaton "mines" gravity, the entire contents of the observable universe could have grown from a twenty-pound nugget 10−26 centimetres across. Third, time's arrow: repulsive gravity smoothed space and left the universe with very low gravitational entropy, so a modest, Boltzmann-style fluctuation in a chaotic pre-inflationary era could launch a vast, ordered cosmos whose records we can trust. The 100 billion galaxies are quantum mechanics writ large across the sky. The catch: no one knows what the pre-inflationary era was like, so the odds of inflation starting can't yet be computed.
Quantum skywriting
If the universe is smooth on large scales, where do galaxies come from? The big bang loyalist shrugs: things were nearly uniform, with small lumps, and gravity grew them. "That's just how it was back then." Two things go unexplained: the overall smoothness and the tiny lumps. Inflation explained the first (Chapter 10). Now the second.
Uncertainty for fields. For a particle, position and velocity can't both be sharp. For a field, its value at a point and its rate of change there can't both be sharp. Greene's summary: quantum mechanics makes things jittery and turbulent. A field's value at every point undergoes a "frenzied, fuzzy, random jitter," normally confined to subatomic scales.
From jitter to galaxy, in four steps.
- Jitter. During inflation the inflaton field's value quivers randomly from place to place, so the energy in one spot differs slightly from another.
- Stretch. Space doubles in size about every 10−37 seconds, stretching microscopic differences to cosmic size, like Magic Marker wiggles on a balloon being blown up.
- Staggered finish. On average the inflaton rolls down its bowl, but jitters make it arrive at slightly different times in different places, so inflation ends a touch later here than there, leaving wrinkles, like a pizza maker stretching dough a bit more in one spot. (In many models the wrinkles come out too large, and the potential's shape must be adjusted.)
- Grow. After inflation, slightly denser regions pull in more matter. Over billions of years they become galaxies and stars.
The pioneers of this calculation in the early 1980s included Stephen Hawking, Alexei Starobinsky, Alan Guth, So-Young Pi, James Bardeen, Paul Steinhardt, Michael Turner, Viatcheslav Mukhanov and Gennady Chibisov. In a quantum world nothing is ever perfectly uniform; with inflation, that unavoidable non-uniformity is magnified into the seeds of cosmic structure. Greene calls it "one of the greatest wonders of the modern scientific age."
The golden age of cosmology
The background radiation is uniform to four decimal places, but at the fourth, it varies: 2.7249 K here, 2.7250 K there, 2.7251 K elsewhere. COBE first measured these variations in 1992; WMAP sharpened them. Denser regions are slightly cooler (photons lose energy climbing out of their deeper gravity), sparser ones slightly warmer.
Why "Wow." Satellites measured photons that had travelled toward us unimpeded for nearly 14 billion years. Their temperatures differ by only a few ten-thousandths of a degree, and those differences form a specific pattern across the sky. Inflationary calculations, built on quantum uncertainty, predict that pattern, and the data points fall on the predicted curve. Cosmology has matured "from a field based on speculation and conjecture to one firmly grounded in observation."
Creating a universe
Can inflation answer Leibniz: why is there something rather than nothing? No: any theory just moves the question back to why that theory. But it does recast a narrower question: where did all the mass and energy come from?
Thought experiment: the box of children. Thousands of children run and leap inside a huge, impermeable but flexible box. Their collisions push the walls outward, and energy goes into the walls' motion: the children's total energy drops, though nothing leaked out. In the expanding universe, fast-moving particles similarly do work against gravity, losing energy as space swells. (We see this directly: microwave background photons have been redshifted for nearly 14 billion years, their energy sapped by expansion. A particle's rest energy is unaffected; only its motion energy drains.)
Now pranksters stretch giant rubber bands between opposite walls. The bands pull inward (negative pressure), and as the box expands they get tauter, gaining energy. The inflaton is like the rubber bands: as the universe expands, matter and radiation lose energy to gravity, while an inflaton field gains energy from gravity. (The analogy has a flaw: the bands resist expansion, while the inflaton's negative pressure drives it, through gravity, not through direct force.)
Two bookkeeping problems.
- Standard big bang: matter and radiation lose energy as space expands, so the farther back you look, the more mass-energy you must explain: "an unending uphill battle."
- Inflation: the inflaton's energy density stays constant while space inflates, so its total energy grows with volume. Linear growth of at least 1030 means volume growth of at least 1090. The start needs very little: a nugget about 10−26 cm across, weighing about twenty pounds (10 kg at ~1082 g/cm3), "a whole lot less than my dog Rocky." In Linde's chaotic models, the seed is Planck-sized (10−33 cm) and weighs about 10−5 grams, like a speck of dust.
Gravity is a bank with unlimited credit. Gravitational energy can become arbitrarily negative, so the inflaton can borrow without limit. Could someone make a universe in the lab? Guth and Eddie Farhi studied the idea; the twenty pounds would need to be crammed into an impossibly small space.
Inflation, smoothness and the arrow of time
Chapter 6's conclusion: the arrow of time requires an early universe of extremely low entropy. With gravity, a uniform distribution of matter and a smooth spatial curvature are both highly ordered (there are many more ways for space to be warped and lumpy, like a population of black holes, than smooth). Why did the universe start that way?
Repulsive gravity irons space smooth. The usual reasoning (lumps grow, warps deepen) depends on gravity being attractive. During inflation gravity repelled: space swelled so fast that creases were stretched flat, like inflating a shrivelled balloon, and clumps were diluted to irrelevance, like the fish in your aquarium if the tank suddenly became an Olympic pool. At the end, the inflaton filled space almost uniformly with particles. The outcome of inflation is exactly the smooth, low-gravitational-entropy state that time's arrow requires. (Paul Davies and Don Page first debated this in Nature in 1983 to 1984.)
Entropy and inflation
Doesn't smoothing lower entropy, violating the second law? No.
The entropy gap. During inflation total entropy rises: the inflaton's decay produces about 1080 particles, a vast amount of entropy. But it rises far less than it could have, because the gravitational contribution (bumpiness of space, potential for clumping) is left near minimum. Greene's image: if entropy were property tax, inflation is New York City acquiring the Sahara: total taxes go up, but trivially compared with the acreage gained.
Ever since, gravity has been closing the gap: every galaxy, star, planet and black hole that clumps out of the near-uniformity (seeded by quantum jitters) raises entropy toward its potential. Inflation set the direction of time's arrow by creating a past with exceedingly low gravitational entropy; the future is the direction in which that entropy grows.
"Gravitational entropy" is a convenient fiction, made rigorous only for black holes. Greene's note concedes that splitting entropy into "gravity" and "everything else" is hard to define precisely. It can be rephrased: clumping converts gravitational potential energy into motion and radiation, which raises matter entropy. When clumping makes black holes, the horizon area gives an unambiguous gravitational entropy (Chapter 16).
But what started inflation?
If igniting inflation needs its own highly special, low-entropy conditions, we have just traded one mystery for another. So: how likely was it?
Boltzmann redux
The cosmic slot machine. Suppose the pre-inflationary realm was ordinary and high-entropy: space riddled with warps, the inflaton jumping wildly like the frog in the hot bowl. Play a fair slot machine long enough and you hit triple diamonds. Sooner or later a fluctuation sets the inflaton to the right, uniform value in some 10−26 cm nugget, and inflation takes off. Most fluctuations fail (wrong value, not uniform enough), but we see only one universe, so the machine need pay out only once. (Greene's note adds a subtlety: the jackpot must also include uncorrelated quantum fluctuations, or the seeds could conspire to un-grow.)
Boltzmann's fluctuation versus the inflationary fluctuation.
- Boltzmann (Chapter 6): a fluctuation straight into today's universe. Two fatal flaws: it should cut corners (why billions of galaxies when one would do?), and it most likely happened just now, so memories and records are fake, and the reasoning self-destructs.
- Inflationary version: a small fluctuation in a tiny nugget, long ago. Inflation then inexorably amplifies it into a vast, smooth universe, so there is no corner-cutting puzzle; and it yields a specific large universe: ours, with the right shape, uniformity and seeds for galaxies and background temperature variations. The past really happened. "Boltzmann may well have been right."
Many jackpots: the multiverse. In a chaotic realm, nothing limits the winning fluctuation to one nugget. As Andrei Linde proposed, many regions may inflate, sprouting universes from universes, perhaps forever. Then inflation is no longer a "front end" to the big bang; rather the big bang is incorporated into inflation, one bubble at a time. Whether such separate universes can ever be detected is doubtful.
Inflation and the egg
Greene's complete chain:
The smooth, low-entropy spatial fabric inflation produced is the counterpart of War and Peace with its pages in order. The absence of severe warps and gargantuan black holes primed the universe for its long climb toward higher entropy, giving the arrow we all experience. (Dark energy's return to repulsion about 7 billion years after the bang affects only the largest scales; within galaxies and solar systems attractive gravity still rules.)
The fly in the ointment
The unjustified assumption. To compute the odds of inflation starting, we must specify the pre-inflationary era. "Wild, chaotic, energetic" sounds reasonable but is only a guess and hard to make mathematically precise. Different assumptions give different odds. Linde argues the uniform Planck-sized nugget isn't special, since in so tiny a region its entropy is about the same as any other configuration's. Greene is "partial to this optimistic vision" but cautions that it rests on unjustified assumptions about trans-Planckian modes, the field behaviour at scales below the Planck length.
Inflation bundles the horizon, flatness, structure and low-entropy problems and solves them with one mechanism. That "feels right." Going further requires a theory of the fuzzy patch, with its extreme heat and density, which means healing the rift between general relativity and quantum mechanics (Chapter 12).
Going deeper: the technical notes
- Energy can't be "created," but gravity's ledger can go negative. As inflaton energy rises, gravitational energy falls by the same amount; total energy can stay near zero.
- Nugget numbers depend on the model. Greene's twenty pounds assumes an energy density of ~1082 g/cm3 in a volume of 10−78 cm3. Chaotic inflation uses the Planck length and Planck mass.
- Entropy accounting. A classical smooth spacetime is like a coherent state of gravitons, essentially a unique state with low entropy; assigning entropy to it needs coarse-graining. Only black hole horizons give an unambiguous gravitational entropy.
- Correlated fluctuations. Just as shell fragments could reassemble, sufficiently correlated fluctuations could conspire to suppress growth. Inflation's role in the arrow also needs uncorrelated initial fluctuations.
- Rubber-band caveat. Uniform pressure exerts no force; negative pressure drives expansion gravitationally.
Critical lens.
- Page's side of the debate is muted. Greene cites Davies and Page together, but Don Page's 1983 paper was titled "Inflation does not explain time asymmetry." Page, and later Roger Penrose, argued that inflation itself needs a special, low-entropy initial patch: if you pick a generic state, it is overwhelmingly more likely to be a lumpy, black-hole-filled one than a smooth inflating patch. Penrose estimates the needed specialness at 1 in 1010123, a number inflation does not reduce.
- Boltzmann brains come back. In eternal inflation, infinitely many universes and observers arise, and "which observers are typical" depends on an undefined measure. Some measures predict that lone fluctuated brains vastly outnumber evolved observers (the problem Greene's Boltzmann redux was meant to cure), others predict a "youngness paradox." This measure problem has no accepted solution.
- How quantum jitters became classical lumps is itself an open question: a homogeneous quantum state evolving by symmetric laws cannot by itself pick out where lumps are. Decoherence is usually invoked, but some (Perez, Sahlmann and Sudarsky, 2006) argue an extra ingredient such as collapse is needed. Greene's "quantum mechanics writ large" quietly assumes a resolution of the measurement problem he left open in Chapter 7.
- Energy "mined from gravity" is a heuristic. In general relativity there is generally no well-defined global energy in an expanding universe (no time-translation symmetry, so no Noether conservation law). The "bank with unlimited credit" is a useful picture, not a theorem.
- Uncredited prize-winners. John Mather and George Smoot won the 2006 Nobel Prize for COBE's discovery of the anisotropies; Jim Peebles shared the 2019 prize largely for the theory of the acoustic pattern that Greene's Figure 11.1 shows.
- The particle count. The observable universe holds about 1080 baryons but about 1089 photons (and as many neutrinos); the photons carry far more entropy than Greene's "1080 particles" suggests.
2026 update: the quantum seeds, measured
- Planck nails the pattern. Planck's final release (2018) measured the spectrum through seven acoustic peaks, fitting the six-parameter ΛCDM model with inflationary initial conditions. The seeds are adiabatic, Gaussian (non-Gaussianity fNL = −0.9 ± 5.1, as single-field slow-roll inflation predicts) and nearly scale-invariant with a red tilt (ns ≈ 0.965; ACT DR6 in 2025 suggested ≈ 0.974).
- The smoking gun is still missing. Inflation also predicts primordial gravitational waves that would twist the background's polarisation into B-modes. None found: BICEP/Keck (2021) bounds r < 0.036. The Simons Observatory, South Pole Observatory and Japan's LiteBIRD (planned for the 2030s) will push toward r ≈ 0.001, covering the Starobinsky-type models still favoured.
- A gravitational-wave background, but probably not from inflation. Pulsar timing arrays (NANOGrav 15-year, plus EPTA, PPTA and the Chinese PTA, June 2023) detected a nanohertz gravitational-wave hum. Most likely it comes from supermassive black-hole binaries; some cosmological explanations were proposed.
- Persistent anomalies. Large-angle oddities (a low quadrupole, a hemispherical power asymmetry, the Cold Spot) remain at roughly 2 to 3 sigma in both WMAP and Planck. Most attribute them to chance; a few see hints of pre-inflationary physics.
- The critics persist. Ijjas and Steinhardt develop bouncing cosmologies that produce the same smooth, flat, near-scale-invariant universe without inflation. Penrose's conformal cyclic cosmology (claimed "Hawking points," 2018) has not been confirmed. Inflation remains the mainstream framework, but the "initial conditions for inflation" problem Greene flagged is as open as in 2004.
- Early galaxies grew fast. JWST finds bright galaxies within ~300 million years of the big bang, more than many models expected. So far these tensions concern galaxy formation physics, not the inflationary seeds themselves.
So what. Inflation's strongest move is use: a modest, unremarkable event, amplified by a powerful process, explains a vast, highly ordered outcome, and avoids the absurdity of needing the outcome itself to be the fluke. When you find a surprisingly ordered system, ask what amplifier could have scaled up a small, likely seed: an early design decision copied across a codebase, a founding norm replicated through hiring. The entropy-gap idea is just as practical: a smooth, under-exploited state (a new market, a fresh platform) is valuable precisely because of how far it is from its maximum. And the fly in the ointment is the honest scientist's habit: Greene stops exactly where the assumptions run out, and says so.
# Quantum jitters, stretched: a nearly scale-invariant random field
import numpy as np
rng = np.random.default_rng(7)
N, ns = 2**16, 0.965 # grid size; Planck spectral tilt
k = np.fft.rfftfreq(N)[1:]
P = k**(ns - 2) # 1D slice of a 3D spectrum k^(ns-4)
amp = np.sqrt(P/2)*(rng.normal(size=k.size) + 1j*rng.normal(size=k.size))
field = np.fft.irfft(np.concatenate([[0], amp]), n=N)
field /= field.std()
# Scale invariance: equal variance per logarithmic band of wavelengths
for lo, hi in [(1e-4,1e-3),(1e-3,1e-2),(1e-2,1e-1)]:
m = (k>=lo)&(k<hi)
print(f'band {lo:g}-{hi:g}: k*P(k) avg = {np.mean(k[m]*P[m]):.2f}')
print('overdense fraction:', np.mean(field>0).round(3), '(seeds of galaxies)')
print('regions > 2 sigma (first to collapse):', np.mean(field>2).round(4))
The World on a String: The Fabric According to String Theory
Physics has advanced by unifying: falling apples with orbiting planets, electricity with magnetism, space with time, the weak force with electromagnetism. One rift remains: general relativity is classical and smooth, quantum mechanics is probabilistic and jittery, and below the Planck length (10−33 cm) their union yields only infinities. That rift is why the earliest moments of the universe, and the centres of black holes, are a fuzzy patch. Greene tells the accidental history of superstring theory, from a 1968 formula for the strong force to the 1984 anomaly cancellation that converted a thousand physicists, and explains its claims: every particle is a vibrating string; one vibration is the graviton; strings' finite size smooths out the quantum frenzy; and consistency demands nine dimensions of space, six of them curled into tiny Calabi-Yau shapes whose geometry would fix every particle property. If string theory is right, the code of the cosmos may be written in the geometry of hidden dimensions. No string has ever been seen, and the theory has not yet explained a single measured particle mass.
Piecemeal progress and the pull of unity
Imagine a universe where understanding anything required understanding everything. Science would have stalled. Ours lets us work piecemeal: Newton needed no atoms, Maxwell no electrons, Einstein no primordial spacetime. New theories don't so much overthrow old ones as refine them: Newton isn't wrong, just limited to slow speeds and weak gravity, though Einstein's refinement needed a whole new conceptual scheme.
From diversity to unity. The explanatory arrows "seem to be converging on a powerful, yet-to-be discovered framework." Einstein spent more than 30 lonely years seeking a unified theory, alienated from mainstream physics. Since the 1980s his dream has driven a generation. But the focus shifted: three forces now share one quantum language; general relativity stands outside it. The goal is to merge general relativity and quantum mechanics.
Quantum jitters and empty space
The uncertainty principle forbids a field from having both a definite value and a definite rate of change. A field that sat exactly at zero would violate it. So even in perfectly empty space, fields jitter around zero: vacuum fluctuations. On large scales they average out, like a marble surface that looks smooth to the eye but is jagged under a microscope.
The Casimir effect: nothing pushes. In 1948 Dutch physicist Hendrik Casimir realised that two uncharged metal plates facing each other in vacuum would restrict the electromagnetic jitters between them (only waves that vanish at both plates fit). Fewer jitters inside than outside creates a pressure imbalance, like thinner air on one side of your eardrum, and pushes the plates together. Marcus Sparnaay made rough tests about a decade later; Steve Lamoreaux (1997) confirmed the prediction to 5 percent (using a lens and a plate; the Padova group later managed true parallel plates to 15 percent). For playing-card-sized plates a ten-thousandth of a centimetre apart, the force equals the weight of a teardrop.
The mathematics of quantum fields (quantum electrodynamics and its relatives) matches experiment spectacularly: the effect of vacuum fluctuations on the electron's magnetism agrees to one part in a billion. Greene's note adds a sting: quantum jitters fill space with energy, effectively a cosmological constant, and naive estimates come out "wildly" larger than observed. Explaining the vacuum energy is "one of the most important open problems in theoretical physics."
Jitters and their discontent
Why gravity and quantum mechanics clash. The gravitational field is the shape of space itself. Like every field, it must jitter, so the geometry of space fluctuates, more violently the smaller the scale. Magnify space step by step: calm and flat at everyday scales, increasingly frenetic below, and at the Planck length (1.6 × 10−33 cm) "a seething, boiling cauldron" in which left and right, up and down, even before and after (below the Planck time, 10−43 s) lose meaning. General relativity's core principle (gently curving spacetime) collides head-on with quantum mechanics' core principle (uncertainty).
Greene's analogy: water is a smooth fluid at everyday scales and a sparse crowd of molecules at small scales. Smooth spacetime is likely the same kind of approximation. What its "molecules" are is unknown.
Does it matter?
In practice the clash shows up as infinity: combine the equations and calculations "almost always" blow up. Contrast special relativity and quantum mechanics, whose tension over entanglement (Chapter 4) is philosophical, while their combination yields the most precise predictions in science. So why not keep general relativity for big things and quantum mechanics for small ones?
Three reasons to care.
- Gut-level unity. The universe doesn't come with a line in the sand. "We have one universe and therefore, many strongly believe, we should have one theory."
- Black holes are both massive (need GR) and tiny at their centres (need QM). There, the laws break down. (Greene's note: inside the horizon space and time swap roles, so the "centre" is better thought of as a moment in time, possibly where time ends.)
- The beginning. Rewind the film: the observable universe shrinks to the size of the sun, earth, a bowling ball, a pea, a grain of sand, and finally the Planck length, all its mass-energy in a speck far smaller than an atom. "The projector jams, the cosmic film burns up." Without a merger, the origin of space and time stays in the fuzzy patch.
A further puzzle in the notes: quantising general relativity the standard way gives the Wheeler-DeWitt equation, in which time does not appear at all. Is time emergent? No one knows.
The unlikely road to a solution
The right solution for the wrong problem
- 1968: Gabriele Veneziano at CERN notices that Euler's beta function, a 200-year-old formula, fits strong-force collision data. The equation came before any picture.
- 1970: Leonard Susskind, Holger Nielsen and Yoichiro Nambu explain it: particles connected by tiny elastic strings. Susskind's paper is returned as of minimal interest: "I was stunned... So I went home and got drunk."
- Early 1970s: finer data contradict strings; quantum chromodynamics (particles and fields, no strings) explains the strong force. By 1974 strings seem knocked out.
- 1971: Pierre Ramond, then André Neveu and John Schwarz, add half-integer spins; with Gliozzi, Scherk and Olive this reveals supersymmetry: superstring theory.
- 1974: Schwarz and Joël Scherk note that the theory's unwanted massless spin-2 particle has exactly the properties of the graviton. String theory is a quantum theory of gravity, if strings shrink to Planck size. The community ignores them.
- Summer 1984: working late on a stormy night at the Aspen Center for Physics, Schwarz and Michael Green show that all anomalies (quantum inconsistencies) cancel. The number of string theorists jumps "from two to over a thousand": the first superstring revolution.
Greene began graduate school at Oxford in autumn 1984, as rumours of breakthroughs ran daily through the corridors and there was "serious talk that the end of theoretical physics was within reach."
What the theory says
The Greeks asked what you'd reach by cutting matter in half again and again. Atoms split; nuclei hold protons and neutrons; those hold quarks (found at SLAC in the late 1960s). Standard theory treats electrons and quarks as dots, the last Russian doll.
Everything is a string. String theory says each "point" particle is a tiny, one-dimensional vibrating filament of energy, about 10−33 cm long, a hundred billion billion times smaller than a nucleus, so small it looks like a point to any accelerator. As one cello string plays many notes, one kind of string produces every particle: each vibrational pattern is a particle species, with its mass, charge and spin set by how the string vibrates. There is no "electron string": the same string, vibrating differently, is an electron or a quark or a photon or, in one special pattern, a graviton.
| Family | Charged lepton (mass, GeV) | Neutrino | Up-type quark | Down-type quark |
|---|---|---|---|---|
| 1 | Electron (0.000511) | Electron-neutrino (< 0.45 eV) | Up (0.0022) | Down (0.0047) |
| 2 | Muon (0.106) | Muon-neutrino | Charm (1.27) | Strange (0.093) |
| 3 | Tau (1.78) | Tau-neutrino | Top (172.6) | Bottom (4.18) |
| Messengers | Photon (0), gluons (0) | W (80.4), Z (91.2) | Graviton (0, unseen) | Higgs (125.1, found 2012) |
Each family repeats the same pattern with heavier masses; each particle has an antiparticle. The standard model takes all of this as input and predicts interactions brilliantly, but can't say why three families, why the electron has three times the down quark's charge, why the top is ~340,000 times heavier than the electron. Those numbers matter: small changes would switch off stellar fusion. String theory promises to calculate them.
Why string theory works: a minimum size
Pixels of space. The frenzy below the Planck length comes from applying uncertainty to arbitrarily short distances. Describe the gravitational field by its gravitons: if gravitons are points, nothing limits how finely the field can be probed, and the frenzy is unbounded. In string theory gravitons are Planck-sized strings, so the field can't be resolved below the string size, just as a TV image can't be resolved below its pixels. At the Planck scale the jitters are still there but mild enough for general relativity's mathematics to be modified sensibly. By limiting how small you can get, string theory limits how violent the jitters become, and the clash disappears.
What replaces continuous space? Perhaps a grid, with motion as leaps from strand to strand, like an ant hopping threads of fabric, and no distance smaller than the Planck length (as no American coin is smaller than a penny). Or, Greene's preference, space and time morph into other concepts for which "shrinking smaller" is as meaningless as asking whether the number nine is happy, as temperature and viscosity mean nothing for a single water molecule.
The finer points: matching the real world
No one has seen a string, and seeing one directly would be like reading this page from 100 light-years, a resolving power a billion billion times beyond today's. Some call such a theory "philosophy or theology." Greene calls that view premature: atoms and black holes were accepted through indirect evidence. The real test is detail: string vibrations must reproduce Tables 12.1 and 12.2.
Mass is vibrational energy. By E = mc2, a string's vibrational energy is its mass: heavier particles are strings vibrating more furiously. (The sun converts 4.3 million tons of mass to energy each second; strings use the equation in reverse.) Massless particles are the most placid patterns, whose small positive energy is exactly cancelled by negative-energy quantum jitters.
Too many vibrations
At first blush, string theory fails:
- There are infinitely many vibrational patterns, but only a finite list of particles.
- To match gravity's weakness, strings must be tiny and therefore stiff: a tension of about 1039 tons, 1041 times a piano string's. Any wiggle costs enormous energy, so masses come in multiples of the Planck mass, about 1019 proton masses (the mass of a dust mote or bacterium). More precisely, mass squared comes in integer multiples.
The way out. Heavy patterns would decay quickly (the top quark lasts 10−24 s) and are far beyond any collider, which in 2004 reached about 1,000 proton masses. Every known particle weighs less than 10−17 Planck masses: to better than one part in 1017, they are all zero times the Planck mass, "just as predicted," much as most people's wealth is zero times the Sultan of Brunei's. The task is to explain the tiny departures from zero.
But even the massless list is wrong: too many spin-1 and spin-½ patterns, and no family structure. And then the biggest shock: the equations work only with nine dimensions of space. "That's not a fine point; that's a problem. Or is it?"
Unification in higher dimensions
Theodor Kaluza and Oskar Klein. In April 1919 Einstein received a paper from a little-known German mathematician, Kaluza (who had learned to swim by reading a treatise and diving into the sea). Kaluza wrote general relativity in four space dimensions. The extra equations that appeared were Maxwell's equations of electromagnetism: gravity and electromagnetism as warps in a five-dimensional spacetime. In 1926 Swedish physicist Klein explained why we don't see the extra dimension: it is curled up, with a radius near the Planck length. Einstein pursued Kaluza-Klein theory into the 1940s, but it couldn't accommodate the electron's charge and tiny mass, and interest faded.
Thought experiment: the worm on the tightrope. Philippe Petit walks a rubber-coated tightrope between Everest and Lhotse. From miles away the rope is a line: one dimension. A worm ahead of him seems doomed. But with a telescope you see a second, curled-up dimension, the circumference, and the worm can crawl around and let Petit pass. Dimensions come in two kinds: large and extended, or small and curled. Klein's proposal: space is like the rope, with a tiny circle attached at every point of our three big dimensions, too small for any instrument (2004's best resolve about a thousandth of a nucleus, short of the Planck length by a million billion).
Why string theory changes the game. Kaluza and Klein assumed an extra dimension: why five dimensions rather than six or 7,000? "Why not?" was the only answer. String theory demands nine space dimensions. "A calculation, not an assumption, not a hypothesis, not an inspired guess, determines the number of space dimensions," the first time in history a theory has done so.
Why nine? A string on a tabletop can wiggle in two directions; let it rise off the table and new vibrational patterns open up. More dimensions, more patterns. A precise equation in string theory requires the number of independent vibrational patterns to meet a constraint, or the mathematics falls apart. With 3, 4, ... 8 space dimensions there are too few; with 9, the constraint is met exactly. (The original bosonic string needs 25.)
The shape of hidden dimensions
Generalise Klein's circle: a hollow sphere at every point (two extra dimensions, five numbers to locate a dinner party, six with the time), a solid ball (three extra), and, impossible to draw, six extra. String theory's equations pick a special class of six-dimensional shapes: Calabi-Yau spaces, discovered by mathematicians Eugenio Calabi and Shing-Tung Yau long before physicists needed them. You and everyone else would be surrounded and filled by them; walking across a room, you circumnavigate them countless times.
Geometry determines physics. Strings are so small they vibrate into the curled dimensions. Like air in a tuba, whose notes depend on the instrument's twists and chambers, a string's vibrations, and hence the masses and charges of particles, depend on the size and shape of the extra dimensions. The early calculations that found too many massless patterns assumed flat, unfurled dimensions; curling them changes the answer.
- Candelas, Horowitz, Strominger and Witten (mid-1980s) showed each "hole" in a Calabi-Yau shape yields one family of particles. Three holes, three families. Shapes with three holes exist, some with the right messenger particles, charges and nuclear force properties.
- But which shape? There are "at least hundreds of thousands" of candidates, and the known (approximate) equations treat all as equally valid; they don't even fix the size. Greene thinks sharper equations will decide; so far they haven't.
- Masses probably come from a stringy Higgs ocean of coordinated strings, with drag forces that, unlike the standard model's inputs, would be calculable.
The fabric according to string theory
In mending the rift, string theory suggests space has more dimensions than we see and that space and time may not extend below the Planck scale. Early on, all nine space dimensions were presumably small; for reasons unknown, three were singled out to expand for 14 billion years. The fuzzy patch becomes the domain "where familiar space and time have yet to emerge" from something more fundamental. Progress on that requires better tools, and with M-theory (Chapter 13) that progress came faster than optimists expected.
Going deeper: the technical notes
- Converging forces. The strengths of the three non-gravitational forces change with energy and nearly converge at very high energy, circumstantial evidence for unification.
- Energy-time uncertainty. Energy fluctuations are inversely proportional to the time resolution: the finer you look, the wilder the jitter.
- Vacuum energy as Λ. Had Einstein not introduced the cosmological constant, quantum theorists would have: the jitters fill space with energy. Summing them gives a value vastly larger than the supernova data allow.
- Planck units. (ħG/c3)1/2 ≈ 1.616 × 10−33 cm, built from light speed, gravity's strength and Planck's constant: the natural size for curled dimensions unless a large number emerges from the theory.
- Mass formula. Squared masses are integer multiples of the string scale squared; usually close to the Planck mass, but Chapter 13 considers larger strings.
- Extra time dimensions? Itzhak Bars has built consistent two-time theories, but the second time is generally seen as a device. The extra space dimensions of string theory are meant to be fully real.
- Symmetry doesn't constrain hidden shapes. Chapter 8's homogeneity applied to the three large dimensions only.
Critical lens.
- "Hundreds of thousands" became a landscape. There are about half a billion reflexive polytopes generating Calabi-Yau shapes (Kreuzer and Skarke, 2000), and once fluxes are included the number of stable vacua was estimated at around 10500 (Bousso and Polchinski, 2000; KKLT, 2003), later far more. Greene's hope that sharper equations would single out one shape has largely given way to the landscape and anthropic reasoning (he discusses this in The Hidden Reality). The claim that string theory calculates particle properties is, in practice, still unrealised.
- "Predicts nine dimensions" needs fine print. Ten dimensions are required for the critical perturbative superstring; M-theory (Chapter 13) has eleven; the bosonic string 26. Non-critical strings exist in other dimensions. The prediction is real but conditional on the framework.
- "Almost always infinity" overstates the clash. General relativity works perfectly well as an effective quantum field theory at low energies (Donoghue, 1994), giving calculable, finite quantum corrections. The breakdown is specifically at the Planck scale, where it is non-renormalisable.
- Rival approaches get little room. Loop quantum gravity appears only in a note; asymptotic safety, causal dynamical triangulations and causal sets are absent. None is confirmed either.
- Uncredited co-discoverers. Tamiaki Yoneya found independently in 1974 that strings contain gravity. Gunnar Nordström proposed a five-dimensional unification in 1914, before Kaluza. Supersymmetry in field theory dates to Golfand and Likhtman (1971) and Wess and Zumino (1974). (Also: Kaluza's paper reached Einstein in 1919 but was published in 1921; Sparnaay's name is misspelled in the text.)
- The Casimir effect may not prove vacuum energy is "real." Robert Jaffe (2005) showed it can be computed as a relativistic van der Waals force between the plates' charges, with no reference to vacuum fluctuations. It demonstrates quantum field theory, not necessarily energy in empty space.
2026 update: twenty years of looking for strings
- No supersymmetry, no extra dimensions at the LHC. Running at 13.6 TeV since 2022, the LHC has found no superpartners (gluino and squark limits around 2 TeV), no Kaluza-Klein particles and no microscopic black holes. Supersymmetry may still exist at higher energy, but the hope that it would explain the Higgs mass "naturally" has faded.
- Gravity tested at short range. Torsion-balance experiments (Eöt-Wash, 2020) confirm Newton's inverse-square law down to about 50 micrometres, ruling out large extra dimensions bigger than that.
- Precision QED triumphs. The electron's magnetic moment was measured to 0.13 parts per trillion (Northwestern-Harvard, 2023), still matching theory. The muon's g-2 final result (June 2025) agrees with the updated lattice-QCD prediction, dissolving a long-running anomaly.
- The Casimir effect, refined and set in motion. Measurements now reach about 1 percent; the dynamical Casimir effect, real photons created from vacuum by a rapidly moving mirror, was observed in a superconducting circuit (Wilson et al., Nature 2011).
- Neutrino masses bounded. KATRIN (2025) caps the electron-neutrino mass at 0.45 eV, the first direct laboratory limit below half an electronvolt. Oscillations prove the masses aren't zero.
- String theory's own turn. The swampland programme (from Vafa, 2005; the de Sitter conjecture, 2018) asks which low-energy theories can arise from strings at all; some versions suggest a truly constant positive Λ is impossible, making DESI's hint of evolving dark energy (2024 to 2025) intriguing to string theorists. Explicit heterotic models reproducing the exact standard-model particle content (2005 onward) exist; machine-learning scans of Calabi-Yau data began in 2017.
- Tabletop quantum gravity. Proposals to test whether gravity can entangle two masses (Bose et al.; Marletto and Vedral, 2017) are driving experiments with ever larger quantum superpositions, but none has yet reached the needed sensitivity.
So what. String theory's history is a case study in how ideas survive: a formula found by pattern-matching (Veneziano), a picture that explained it (strings), a failure in its first job (the strong force), and a re-purposing that turned its embarrassing extra particle into its central virtue (the graviton). Keep "failed" ideas on a back burner and ask what problem their bugs might solve. The pixel argument, a minimum resolution that tames runaway behaviour, appears everywhere from numerical simulation (grid regularisation) to image processing. And the critical lens offers the opposite lesson: a theory that can accommodate 10500 worlds has traded predictive power for flexibility. Judge frameworks by what they forbid.
# Why 26 and 10? The zero-point energy of a vibrating string.
# Sum over harmonics n of n/2 per transverse direction, regulated by
# zeta(-1) = -1/12. Consistency (a massless photon) fixes the dimension D.
from fractions import Fraction as F
zeta_m1 = F(-1, 12)
# Bosonic string: D-2 transverse directions, each contributes zeta(-1)/2
# Lorentz invariance needs the ground-state shift a = 1.
a = lambda D: -F(D-2)*zeta_m1/2
print('bosonic: D =', [D for D in range(4, 40) if a(D) == 1])
# Superstring (NS sector): fermions with half-integer modes add
# (D-2)*(1/48); need total shift a = 1/2.
a_ns = lambda D: F(D-2)*(F(1,24) + F(1,48))
print('superstring: D =', [D for D in range(4, 40) if a_ns(D) == F(1,2)])
# Planck units from c, G, hbar
c, G, hbar = 2.998e8, 6.674e-11, 1.055e-34
print(f'Planck length {(hbar*G/c**3)**.5:.2e} m, '
f'time {(hbar*G/c**5)**.5:.1e} s, mass {(hbar*c/G)**.5*1e3:.1e} g')
The Universe on a Brane: Speculations on Space and Time in M-Theory
By the early 1990s there were five string theories, an embarrassment for a candidate "theory of everything." In 1995 Edward Witten showed they are five translations of one master theory, M-theory, which lives in eleven spacetime dimensions and contains not only strings but membranes and higher-dimensional "branes." Joe Polchinski then showed that branes are sticky: open strings' ends cannot leave them. That makes possible the braneworld: our entire universe might be a three-brane, with light and matter trapped on it and only gravity leaking into extra dimensions that could be as large as a tenth of a millimetre. Strings might be big enough, and gravity strong enough at short range, for the Large Hadron Collider to see string harmonics or even microscopic black holes. A braneworld also inspires the cyclic model, in which our brane collides with a neighbour every trillion years or so, each "splat" a new big bang. If we live on a brane, the old question of whether space is a something gets its firmest answer: space would be a physical object. All of this is speculative, and two decades of experiments have so far found none of it.
A theory without a core principle
Special relativity has the constancy of light's speed; general relativity, the equivalence principle; quantum mechanics, uncertainty. String theory has no such core principle. It grew piecemeal, like a primitive tribe excavating a buried spacecraft and learning, toggle by toggle, that the controls work together. The second superstring revolution strengthened that sense of coherence and led to extra dimensions, experimental hopes, neighbouring universes and a cyclic cosmos.
The second superstring revolution
Five theories, one master text. By the 1990s there were five consistent superstring theories: Type I (with open strings, which have loose ends, as well as closed loops), Type IIA, Type IIB, Heterotic-O and Heterotic-E. Unification was meant to lead to one theory; five raised the question of why the other four existed. In spring 1995, drawing on work by Chris Hull, Paul Townsend, Ashoke Sen, Michael Duff, John Schwarz and others, Edward Witten showed that the five are different mathematical descriptions of a single theory, like five translations of one book. He named it M-theory: Master? Majestic? Mother? Magic? Mystery? Matrix? Membrane? String theory is a single theory: a meta-unification.
The power of translation
Why was this revolutionary rather than a mere correction? Because the exact equations of each theory are intractable, and researchers relied on approximations. Witten's dictionary means every hard question has four reformulations, and sometimes one of them is easy.
The untranslatable manuscript. Imagine a master text so full of puns, rhymes and culture-bound jokes that no single translation captures it: some passages read easily in Swahili but not in Inuit; others are clear in the Inuit version; Sanskrit catches yet others; and a few are opaque in all five, so only the original will do. The five string theories are such translations. A question impossible in one may be simple in another; for the hardest, we need M-theory itself, which nobody yet fully understands. Witten's discovery gave string theorists "five times the firepower."
Eleven dimensions
Witten showed the approximate equations had missed a dimension. M-theory has ten space dimensions, eleven spacetime dimensions. The approximations had implicitly assumed the tenth spatial dimension was tiny, too small for them to resolve. Like five villages that look separate from the ground but are linked by roads visible from a mountaintop, the five theories connect through the extra dimension. When Witten announced this at the 1995 Strings conference, it "shook the foundations of the field." Earlier work survives if the tenth dimension is small; for most readers, picturing seven curled dimensions is about as hard as picturing six.
Branes
Why strings, and not blobs?
- Before 1995: attempts by Heisenberg and Dirac, among others, to build theories from 2D or 3D constituents failed basic tests (probabilities between 0 and 1, no faster-than-light signals). One-dimensional strings work because their equations have an infinite set of symmetries (conformal symmetry: changes of resolution, zooming in or out on the surface a string sweeps through spacetime); for higher-dimensional objects the symmetry drops off steeply.
- After 1995: Witten's work and an avalanche of follow-ups showed that M-theory does contain higher-dimensional objects: membranes or two-branes, three-branes, and in general p-branes for any whole number p below ten. Strings are one ingredient, not the ingredient.
Why hadn't anyone seen them? In the regimes studied, branes are much heavier than strings, and the approximate equations fail at high energies, which is where branes live. They don't invalidate older work when they're heavy, but they can be light. Greene's own work with Andrew Strominger and David Morrison showed that a brane wrapped around a shrinking sphere inside a Calabi-Yau shape becomes lighter as the sphere shrinks, letting space tear without catastrophe (a theme for Chapter 15). Greene's note adds: as the tenth dimension grows in Type IIA, strings swell into inner-tube-shaped membranes.
Braneworlds
Only low-energy strings are tied to Planck size by gravity's weakness. Pump in energy and a string grows, without limit. In the big bang's aftermath, long strings stretching across the sky might have formed and could leave detectable imprints.
Branes need not be small either, and higher dimensions open a new possibility. A long one-brane is like a power line within our three dimensions; a huge two-brane is like an impossibly large drive-in movie screen. But a huge three-brane would fill all three dimensions we know.
Snow White's screen. Snow White lives on a two-dimensional screen inside a three-dimensional theatre. Might we live inside a three-brane within the higher-dimensional space of M-theory? Then what Newton, Leibniz, Mach and Einstein called space would be a specific physical object, and four-dimensional spacetime would be the wake of a three-brane moving through time. (We could even live on a four- or five-brane, with some dimensions filling small extra directions.)
Sticky branes and vibrating strings
Polchinski's insight (October 1995). Years earlier, with Jin Dai and Robert Leigh (and in related work by Petr Hořava and Michael Green), Joe Polchinski had found that open strings' endpoints can be confined to particular regions, like beads on a wire or pinballs on a table, while the string itself vibrates freely. What holds them? After Witten, the answer was obvious: the region is occupied by a p-brane (a "D-brane," for the Dirichlet boundary conditions involved). Branes are "the stickiest things imaginable." And because a brane's properties are captured by the open strings whose ends it holds, like learning about a carpet by running a hand through its pile, decades of string mathematics could now be used to study branes.
Our universe as a brane
If we live inside a three-brane, why can't we see it? Because in the braneworld scenario photons are open strings: their ends move freely within the brane, so the brane is perfectly transparent, and they can't leave it.
Why big extra dimensions could stay hidden. We detect things with the electromagnetic force: eyes, microscopes, accelerators. If photons can't leave our brane, they can't probe extra dimensions of any size. The same holds for gluons, W and Z particles, and all matter particles (electrons, quarks): all are open strings. You and I and everything we've ever seen are permanently imprisoned within our three-brane, like an ant on a lily pad unaware of the deep water below. Except for gravity: gravitons are closed loops with no ends, free to leave. Gravity is our only window onto the extra dimensions.
Gravity and large extra dimensions
Why does gravity fall off as the square of distance? Picture the sun's gravitational field lines spreading out evenly. At distance d they cross a sphere of area proportional to d2, so their density, and gravity's strength, falls as 1/d2. The inverse-square law is geometry of spheres in three dimensions.
Gravity counts dimensions.
| Space dimensions | Field lines spread over | Force falls as | Double the distance |
|---|---|---|---|
| 1 | Nothing (no room) | constant | no change |
| 2 | Circle (∝ d) | 1/d | ÷ 2 |
| 3 | Sphere (∝ d2) | 1/d2 | ÷ 4 |
| 4 | 3-sphere (∝ d3) | 1/d3 | ÷ 8 |
| 3 + n | (∝ d2+n) | 1/d2+n | ÷ 22+n |
Thought experiment: the one-dimensional lab. Physicists in a universe that seems to have one dimension have found for centuries that gravity doesn't vary with distance, but they've only tested it down to a tenth of a millimetre. Secretly, space has a second, curled dimension, like Petit's tightrope. Bring two objects much closer than its circumference and the field lines spread around it: gravity starts to fall as 1/d. Beyond the circumference, the lines have saturated the curled dimension and gravity goes back to constant. Like water pressure in an old house: each new open faucet lowers the shower pressure, but once every faucet is open, it stops dropping. From the crossover distance they'd learn both that there are two dimensions and how big the hidden one is.
1998: Arkani-Hamed, Dimopoulos and Dvali pointed out that in 1998 no one had tested gravity below a millimetre, so in a braneworld extra dimensions could be that large and still undetected. Experiments soon pushed to a tenth of a millimetre (2004), with no deviation. The non-gravitational forces probe down to 10−18 m without finding extra dimensions, but in a braneworld they're blind to them. "Right now, right next to you... There could be another spatial dimension," large enough to swallow this page.
Large extra dimensions and large strings
Gravity may only look weak. The gravity we feel combines its intrinsic strength and the volume of the extra dimensions it spreads into, like water pressure diluted by wide pipes. Old calculations assumed Planck-sized extra dimensions, so gravity looked weak because it is weak. With large extra dimensions, gravity could be intrinsically strong, diluted by the extra room, and then strings could be far longer. Building on Witten, Joe Lykken and Constantin Bachas, Ignatios Antoniadis with Arkani-Hamed, Dimopoulos and Dvali showed strings could be as large as about 10−18 m, some 1017 times the Planck length.
String theory confronts experiment?
How likely? Greene's intuition says "unlikely," though he concedes it was formed by fifteen years of Planck-sized thinking. If strings are that big:
- String harmonics would weigh only a thousand to a few thousand proton masses, within reach of the LHC. Detectors would "light up like the Times Square crystal ball on New Year's Eve," with a tower of particles whose masses relate like a cello's harmonics.
- Microscopic black holes. If gravity is intrinsically strong at short distances, much less squeezing is needed to make a black hole. The LHC might become "a factory for producing microscopic black holes" (as proposed by Giddings and Thomas and by Dimopoulos and Landsberg). They'd evaporate almost instantly via Hawking radiation, posing no danger.
Braneworld cosmology
Two research paths:
- Conservative: string/M-theory will provide a front end to inflation, unfuzzing the fuzzy patch by introducing a minimum size that tames the singularity, much as strings tame quantum gravity. Progress, but "the fuzzy patch still looks fuzzy."
- Radical: the cyclic model.
Cyclic cosmology
Cycles in cosmology
- Ancient cosmologies: many saw the world as cyclic, like days, seasons and moon phases.
- 1930s, Richard Tolman (Caltech): the universe expands, stops, contracts, bounces, repeats. But entropy rises each cycle, more entropy means a longer cycle, so earlier cycles were shorter and shorter: the cycles can't stretch infinitely into the past. Tolman's model also needed a spherical universe, now ruled out.
- 1996, Hořava and Witten: one of M-theory's seven extra dimensions could be a line segment bounded by two "end-of-the-world" branes, linking Heterotic-E to the other theories.
- 2001 to 2002, Paul Steinhardt and Neil Turok (with Burt Ovrut, Nathan Seiberg and Justin Khoury): the ekpyrotic then cyclic model, built on that setup.
The "big splat," stage by stage. Two parallel three-branes, separated by a sliver (a fraction of a millimetre) in a fourth spatial dimension, attract each other "as though connected by tiny rubber bands." The other six dimensions are a Calabi-Yau shape.
- Splat. The branes collide and rebound. The collision fills each with hot radiation and matter whose properties, Steinhardt and Turok argue, match what inflation would leave.
- Standard cosmology. Our brane expands and cools; stars and galaxies form.
- Dark energy. About 7 billion years in, dark energy dominates and expansion accelerates (we are here, about 14 billion years in).
- Emptiness. For roughly a trillion years, acceleration dilutes matter and radiation to almost nothing; the brane becomes flat and uniform.
- Approach. The branes draw together; quantum jitters ripple our brane, and the ripples grow.
- Next splat. The ripples imprint the non-uniformities that seed the next cycle's galaxies.
| Feature | Inflation | Cyclic (big splat) |
|---|---|---|
| Smoothing and flattening | A burst of exponential expansion in ~10−35 s | A trillion years of gentle acceleration at the end of the previous cycle |
| Seeds of structure | Quantum jitters of the inflaton | Quantum ripples of the approaching brane |
| Role of dark energy | An unexplained add-on | Essential: it resets each cycle |
| The bang | Singular start (time zero) | Brane collision; only the gap dimension shrinks, perhaps finite temperature |
| How it began | Needs conditions to start inflation | Needs a first cycle (entropy and quantum effects forbid infinite past) |
| Distinguishing test | Primordial gravitational waves (B-modes) | No detectable primordial gravitational waves |
A brief assessment
Neither model is complete. Inflation assumes the conditions needed to start it and hasn't been embedded smoothly in string theory. The cyclic model can't run forever: entropy buildup and quantum fluctuations (one brane might twist relative to the other and stop the engine) mean cycles had a beginning. Why six dimensions curl into a Calabi-Yau shape while one becomes a segment, why the branes line up and attract just so, and above all what happens at the collision, are open; existing analyses suggest the mathematics breaks down there too.
The cyclic model's best feature is dark energy: inflation treats the observed acceleration as a "clumsy add-on," while the cyclic model needs it. If acceleration had failed to hold up, inflation would have survived but the cyclic model would not.
New visions of spacetime
Both ideas are "highly speculative." Greene includes them to show how M-theory reframes old questions. If space is a three-brane, it is definitely a something, perhaps one among many branes floating in a higher-dimensional expanse; and time as we know it may span only one of many cycles. "What we consider to be 'everything' may be but a small constituent of a far richer reality."
Going deeper: the technical notes
- Conformal symmetry. A string sweeps out a 2D surface in spacetime, and its equations are invariant under the 2D conformal group, which is infinite-dimensional; for higher-dimensional objects it is finite. That is the "peak" in symmetry that made strings special.
- T-duality. If a dimension is a circle of radius R, closed-string physics is identical for radius R and 1/R: strings moving around the circle and strings wound around it swap roles. Open strings seem to break this, until Dirichlet boundary conditions (D-branes) are included; this was the puzzle Polchinski, Dai, Leigh, Hořava and Green resolved.
- Dimension counting for the cyclic model: two three-branes plus the gap = 4 space dimensions; plus time = 5; the remaining 6 form the Calabi-Yau.
- Contraction isn't reversal. In a bouncing universe, eggs still splatter forward during contraction; entropy rises in both phases.
- Effective description. The cyclic model can be written as four-dimensional field theory on one brane, resembling scalar-field inflation models; its novelty is conceptual (colliding branes).
- Modified gravity. Some propose changing gravity on large scales instead of invoking dark matter and dark energy; Greene calls these "highly speculative" with little support.
Critical lens.
- Randall and Sundrum are missing. Lisa Randall and Raman Sundrum (1999) showed that a warped extra dimension, even an infinite one, can explain gravity's weakness and localise gravity on a brane. Their models drove much of the braneworld phenomenology Greene describes and became among the most cited papers in physics; neither name appears.
- "Nearly identical predictions" was disputed at the time. Andrei Linde, Renata Kallosh and others argued that the ekpyrotic/cyclic models needed fine-tuning and that their fluctuation spectra were not straightforwardly scale-invariant. Later analysis showed simple ekpyrotic models predict large non-Gaussianity, which Planck (2013 to 2018) did not see; proponents had to build more elaborate versions. Steinhardt and collaborators have since moved to bouncing models without branes (Ijjas and Steinhardt, 2019).
- The "front end to inflation" path quietly became the landscape. The best-known string-inflation constructions (KKLMMT, 2003) arrived alongside the 10500 vacua problem, which made string cosmology less predictive, not more.
- Large extra dimensions were always a long shot, and astrophysics cut deep. Even in 2004, cooling of supernova SN 1987A and heating of neutron stars by escaping gravitons pushed two large extra dimensions well below a millimetre. The LHC's prospects were real but optimistic.
- M-theory remains undefined. Nearly three decades after 1995 there is still no complete formulation; matrix theory (Banks, Fischler, Shenker and Susskind, 1996) and AdS/CFT (Maldacena, 1997, discussed in Chapter 16) give non-perturbative definitions only in special settings.
2026 update: what the experiments said
- No string harmonics, no black holes at the LHC. Through Run 3 (13.6 TeV), ATLAS and CMS found no microscopic black holes (minimum masses excluded around 9 to 10 TeV), no Kaluza-Klein gravitons (Randall-Sundrum gravitons excluded up to roughly 4 to 5 TeV) and no string resonances (excluded below about 8 TeV). The cello did not play.
- Gravity's inverse-square law holds down to 52 micrometres (Eöt-Wash, 2020), shrinking the room for large extra dimensions by half since Greene wrote. For two large extra dimensions, astrophysical bounds now require the fundamental gravity scale to sit well above 100 TeV.
- Gravitational waves count dimensions. The neutron-star merger GW170817 (2017) produced gravitational waves and light that arrived 1.7 seconds apart after 130 million years; comparing distances from both signals showed no leakage of gravity into extra dimensions on cosmic scales (consistent with four spacetime dimensions).
- Cosmic strings hunted. Planck limits cosmic-string tension to Gμ below about 10−7; LIGO-Virgo-KAGRA searches for string "cusps" and the 2023 pulsar-timing background (NANOGrav and others) tightened or tested cosmic-superstring models. The pulsar signal is most likely from merging supermassive black holes.
- Cyclic versus inflation. Planck's tiny non-Gaussianity hurt simple ekpyrotic models. If B-mode primordial gravitational waves are detected by the Simons Observatory or LiteBIRD, cyclic and bouncing models in their standard forms would be ruled out; if r stays below about 0.001, inflation loses its cleanest confirmation. The question Greene posed is now an experimental race.
- M-theory's mathematics flourishes. Branes became central tools in pure mathematics (mirror symmetry, knot invariants) and in holography (Chapter 16), even as direct tests remain out of reach.
So what. Witten's dictionary is the chapter's most transferable idea: when a problem is intractable in one representation, translate it. Engineers do this constantly (time domain to frequency domain, primal to dual, SQL to graph) and the move often converts the impossible into the routine. The braneworld adds a lesson about instruments: we may be blind to a dimension not because it is small but because every tool we use shares the same constraint; ask what your measurement channel cannot, by construction, see. And the cyclic-versus-inflation table shows good scientific practice: when two stories fit the same facts, find the one observation they disagree on (here, primordial gravitational waves) and go measure it.
# Large extra dimensions (ADD): how big must they be if the true
# strength of gravity is felt at the TeV scale? M_Pl^2 ~ M_D^(n+2) R^n
hbar_c = 1.97e-19 # metres x TeV (so 1/TeV = 1.97e-19 m)
M_Pl = 1.22e16 # Planck mass in TeV
for M_D in (1, 10):
for n in range(1, 7):
R = hbar_c/M_D * (M_Pl/M_D)**(2/n) # size of each extra dim
print(f'M_D={M_D:>2} TeV, n={n}: R ~ {R:9.2e} m')
# Gravity between two masses at distance d with n extra dims of size R:
def F_ratio(d, R, n):
"""force relative to Newton's law (=1 for d >> R)"""
return (R/d)**n if d < R else 1.0
for d in (1e-3, 1e-4, 5e-5, 1e-5):
print(f'd={d:.0e} m, n=2, R=1e-4 m: F/F_Newton = {F_ratio(d,1e-4,2):g}')
Up in the Heavens and Down in the Earth: Experimenting with Space and Time
Since the mid-1980s theoretical physics has been "the victim of its own success," racing into regimes beyond experiment. This chapter is Greene's 2004 forecast of the experiments that might catch up: Gravity Probe B measuring frame dragging, LIGO hearing gravitational waves, gravity tests and cosmic rays hunting extra dimensions, Fermilab and the LHC chasing the Higgs and supersymmetry, WMAP, Planck and CMBPol reading primordial gravitational waves in the microwave sky to decide between inflation and the cyclic model, and searches for dark matter and the nature of dark energy. Twenty-two years later most of these experiments have reported. The scorecard is striking: every prediction of established theory (frame dragging, gravitational waves, the Higgs) came true; every speculative signal (supersymmetry, extra dimensions, micro black holes, dark matter particles) has so far failed to appear. This edition's main addition is that scorecard.
Einstein in drag: frame dragging and Mach
General relativity grew from nineteenth-century geometry (Gauss, Bolyai, Lobachevsky, Riemann) and from Mach. Einstein found that his theory wasn't fully Machian: Newton's bucket spinning in an empty universe still goes concave, implying absolute acceleration. But it does include Machian features.
Frame dragging. In 1918 Josef Lense and Hans Thirring showed that a rotating mass drags space and time around with it, like a spinning stone in a bucket of syrup. An asteroid falling toward a spinning neutron star gets whipped around; from its own point of view it falls "straight down," but the spatial grid itself is swirling.
Mach's bucket inside a spinning shell. Calculations begun by Einstein (1912), extended by Dieter Brill and Jeffrey Cohen (1965 to 1966) and completed by Herbert Pfister and K. Braun (1985) show that a massive rotating hollow sphere drags the space inside it. A bucket of water at rest (seen from far away) inside the spinning shell goes concave. If the shell has about the mass of the universe, it makes no difference whether the shell spins around the bucket or the bucket spins inside the shell: only relative rotation matters, just as Mach argued, and the shell completely screens the influence of space beyond it. Still, unlike Mach, GR says a bucket in a truly empty infinite universe would go concave.
Gravity Probe B. In 1960 Leonard Schiff and George Pugh independently proposed testing frame dragging with an orbiting gyroscope. Earth's rotation should twist its axis by about a hundred-thousandth of a degree per year, the angle a clock's second hand sweeps in about two millionths of a second. After four decades of development and nearly a hundred PhD theses, Francis Everitt's Stanford team, with NASA, launched four of the most stable gyroscopes ever built 400 miles up (April 2004). Greene hoped for the first direct evidence of a Machian effect, or a crack in general relativity.
Catching the wave
General relativity predicts that moving masses send ripples of curvature through space, as charges racing along an antenna send out radio waves. Einstein found this in 1916 to 1918. Unlike light, sound or water waves, which travel through space, gravitational waves travel within it: they are distortions of geometry itself. Machian scruples (how can empty space wiggle?) kept the subject controversial for decades, but the theory converged: space can ripple.
How feeble? A passing wave stretches space one way and squeezes it the other. The Trinity atomic bomb (July 16, 1945; 20,000 tons of TNT) would have stretched someone standing beneath the tower by a minuscule fraction of an atomic diameter. A supernova 10,000 light-years away would stretch a one-metre rod by about 10−17 cm, a hundredth of a nucleus. (Thinking of a wave as coordinated gravitons shows why detecting one graviton is hopeless.)
LIGO's design (as Greene described it in 2004).
- Two L-shaped detectors, at Livingston, Louisiana and Hanford, Washington, about 2,000 miles apart, so a real wave appears in both while trucks, chainsaws and falling trees don't.
- Arms 4 km long, because stretching is cumulative: 10−20 m over 4 m becomes 10−17 m over 4 km. Light bounces over a hundred times, making each beam's path about 800 km.
- Sensitivity: changes smaller than a trillionth of a human hair's width, a hundred-millionth of an atom.
- Sounds: many signals are at a few thousand cycles per second, audible frequencies: coalescing neutron stars chirp with rising pitch; colliding black holes sound like "the trill of a sparrow that's received a sharp blow to the chest."
- Indirect evidence already existed: the Hulse-Taylor binary pulsar (discovered 1974) spirals inward exactly as GR predicts from energy lost to gravitational waves.
Why it matters. Gravity's weakness lets gravitational waves pass through matter opaque to light, and its universality means everything emits them. From naked eyes to Lippershey and Galileo's telescopes to radio, infrared, X-ray and gamma rays, each new window transformed astronomy. "For millennia we have looked into the cosmos; now it's as if, for the first time in human history, we will listen to it."
The hunt for extra dimensions
Before 1996, extra dimensions were assumed Planckian, seventeen powers of ten beyond reach. If they are "large" (above about 10−20 m), there is hope:
- Very large (near a millimetre): precision gravity tests should show the inverse-square law failing. None found yet, as of 2004.
- Large: gravity's intrinsic strength would be higher, so microscopic black holes could form at the LHC (then "slated for completion by 2007") or, as Alfred Shapere and Jonathan Feng noted, in collisions of ultra-high-energy cosmic rays with the upper atmosphere.
- Missing energy: gravitons escaping into extra dimensions would make collisions appear to lose energy, like coins slipping between the sofa cushions.
Nature's accelerator. Cosmic rays, discovered by Victor Hess in 1912, rain particles through your body every second. On October 15, 1991, Utah's Fly's Eye detector recorded a particle carrying the energy of 30 billion proton masses, roughly a Mariano Rivera fastball in one subatomic particle, about 100 million times the LHC's design collision energy. No known process explains such energies. The Pierre Auger Observatory in western Argentina, with a collecting area the size of Rhode Island, could see about a dozen tiny black holes a year if extra dimensions were as large as 10−14 m. Drawbacks: we can't control what hits what; the most energetic particles strike a given square kilometre about once a century; and they hit slow atmospheric targets rather than head-on beams.
Greene confesses his bias: after fifteen years working on extra dimensions, he can't imagine a discovery that "would more thoroughly shake the foundation of physics."
The Higgs, supersymmetry and string theory
- Higgs: expected at 100 to 1,000 proton masses. Fermilab's upgraded Tevatron had a chance if light; the LHC should make them "galore by the end of the decade."
- Supersymmetry: every particle should have a partner differing by half a unit of spin: selectrons, squarks, sneutrinos, gluinos, photinos, winos and zinos ("Yes, physicists get carried away"). Expected around 1,000 proton masses, so the Tevatron had a shot and the LHC "should produce them with ease." Confirmation would be "the most important development in elementary particle physics in more than two decades," circumstantial support for strings, but not proof, since supersymmetry fits point-particle theories too.
- String harmonics: in a braneworld with extra dimensions around 10−16 cm, strings would be long and floppy, with vibrations at about 1,000 proton masses. LHC collisions would be like "a well-hit golf ball ricocheting around the inside of a piano," exciting many octaves: "the most likely scenario for a direct confirmation of string theory."
Cosmic origins
The microwave background was released about 300,000 (now 380,000) years after the bang, when electrons and protons combined into neutral atoms; about 400 million of its photons fill every cubic metre. COBE (1992) found its splotches; WMAP (launched 2001) mapped them with about forty times better resolution and sensitivity; Planck (then due in 2007) would improve another tenfold. COBE marked a shift: before it, cosmological theories needed to match only broad features; afterwards, precision data winnowed them.
The decisive test between inflation and the cyclic model.
- Inflation (in many flavours: old, new, warm, hybrid, hyper, assisted, eternal, extended, chaotic, double, weak-scale, hypernatural...) violently stretches space, generating primordial gravitational waves along with density ripples.
- The cyclic model builds its ripples gently over a trillion years as two branes approach, like coarse sandpaper slapped together, so it predicts no primordial gravitational waves.
- The signature: primordial gravitational waves would leave a distinctive twist, B-mode polarisation, in the microwave background. Planck and the proposed CMBPol might see it. Detection would rule out the cyclic model and crown inflation. (Greene's note: some complicated inflation models also suppress gravitational waves.)
Dark matter, dark energy and the future
Not ordinary matter. The abundances of hydrogen, helium, deuterium and lithium match big bang nucleosynthesis only if the dark matter is not made of protons and neutrons. (Electrons are under 0.05 percent of ordinary matter's mass.)
Candidates in 2004.
- Neutrinos: about 55 million per cubic metre of each type survive from the big bang; they'd supply the dark matter if one weighed about 10−8 proton masses. They have mass, but too little, short by a factor of more than a hundred.
- Supersymmetric particles: the lightest of the photino, zino or higgsino (it must be stable) could pass through the whole earth untouched. To be the dark matter they'd need 100 to 1,000 proton masses, the same range supersymmetry and string theory suggest on their own, a "puzzling and completely unexplained confluence" unless the idea is right. (This is the "WIMP miracle.")
- Direct searches: of a million dark matter particles crossing a quarter-sized area each second, at most one per day would register. DAMA (Gran Sasso) claimed a detection; CDMS appeared to rule it out.
Dark energy: constant or quintessence? The background's splotch sizes show space is flat (bigger splotches would mean a sphere, smaller a saddle), and 30 percent matter plus 70 percent dark energy makes exactly the critical density. The proposed SNAP satellite would measure twenty times more supernovae and test whether dark energy is a true cosmological constant (eternal acceleration, a "barren" future) or quintessence, a slowly rolling field, "an extremely gentle version of the inflationary burst," whose push eventually ends.
The scorecard: 2004 forecasts, 2026 results
| Greene's 2004 expectation | What happened | Verdict |
|---|---|---|
| Gravity Probe B measures frame dragging; first direct Machian effect | Results (2011): geodetic effect to 0.3 percent; frame dragging 37 ± 7 milliarcsec/yr vs 39 predicted. Confirmed, but only to ~19 percent (unexpected electrostatic patch effects). Laser-ranged satellites (LAGEOS, LARES) reached a few percent | Confirmed |
| LIGO makes the first direct detection of gravitational waves | 14 September 2015 (GW150914): two black holes, 29 and 36 suns, 1.3 billion light-years away. Nobel 2017 (Weiss, Barish, Thorne). Neutron-star merger GW170817 seen in light too. By 2025, 300+ events; GW250114 tested Hawking's area theorem | Confirmed, spectacularly |
| Inverse-square law fails at small distances (large extra dimensions) | Holds to 52 μm (2020) | Not seen |
| Micro black holes at the LHC or in cosmic rays | LHC (from 2010): none, excluded to ~9 to 10 TeV. Auger: none; showed the highest-energy cosmic rays come from outside the galaxy (2017) | Not seen |
| Missing energy into extra dimensions | No excess in mono-jet or mono-photon searches | Not seen |
| Higgs at 100 to 1,000 proton masses, Tevatron or LHC | Found at the LHC, 4 July 2012, 133 proton masses (125 GeV). The Tevatron (closed 2011) saw a hint | Confirmed |
| Supersymmetry at ~1,000 proton masses, "with ease" at the LHC | None; gluinos and squarks excluded to ~2 TeV | Not seen |
| String harmonics at the LHC (braneworld) | None; string resonances excluded below ~8 TeV | Not seen |
| Planck: finer CMB maps winnow inflation models | Launched 2009; final 2018 data: flat, ns = 0.965, Gaussian, adiabatic. Simplest chaotic models excluded | Confirmed (framework) |
| Primordial gravitational waves (B-modes) decide inflation vs cyclic | BICEP2 claim (2014) was dust. R < 0.036 (2021). Undecided | Pending |
| Dark matter identified "within the next few years" | No particle found. LZ, XENONnT, PandaX: WIMP limits ~100 times stronger; DAMA's signal refuted by COSINE-100 and ANAIS-112 (2025) | Not seen |
| SNAP: is dark energy constant? | SNAP folded into Nancy Grace Roman Space Telescope (launch due by May 2027). DESI (2024 to 2025) hints dark energy weakens: quintessence-like | Open, intriguing |
Reading the scorecard. General relativity and the standard model went three for three (frame dragging, gravitational waves, Higgs). Every prediction that depended on new physics just beyond reach (supersymmetry, large extra dimensions, TeV-scale strings, WIMP dark matter) came back empty. This is the "nightmare scenario" many feared: a Higgs and nothing else at the LHC. It does not refute string theory, which never required TeV-scale signals, but it removed every near-term route to testing it that Greene listed.
Space, time and speculation
Understanding has often preceded control: electromagnetism gave the telegraph, radio and television; quantum mechanics gave computers and lasers; nuclear physics gave weapons and perhaps, one day, energy from "vats of salt water." Might understanding space and time make us masters of them? Chapter 15 speculates.
Going deeper: the technical notes
- Warped time matters more than warped space. For ordinary objects like the sun, the warping of time (clocks run slower nearer the sun) dominates the warping of space in producing orbits; rubber-sheet pictures show only space because time is hard to draw.
- Earlier frame-dragging evidence came from lunar and satellite orbits (Bertotti, Ciufolini, Bender; Shapiro and colleagues); Gravity Probe B's advantage was being a fully controlled experiment.
- Hulse-Taylor pulsar. Two neutron stars orbiting closely; the orbital period shrinks exactly as GR predicts for energy carried off by gravitational waves, to about one part in a thousand.
- Energy conservation in GR is subtle. Matter's stress tensor is covariantly conserved, but there is no general global energy conservation, because gravitational energy has no good local definition. Locally (in accelerator experiments) energy conservation holds.
- Cosmic-ray energetics. About 10 billion cosmic rays with ~1 GeV energy hit each square kilometre each second; the most energetic, about 100 billion proton masses, roughly once per square kilometre per century.
Critical lens.
- Optimism bias in forecasts. Greene wrote that the LHC would produce superpartners "with ease" and that dark matter might be identified "within the next few years." These reflected the community's expectation in 2004, driven by naturalness arguments (the Higgs mass should not be fine-tuned). The null results have made naturalness itself a contested principle, a shift the book could not foresee but which its confident phrasing makes conspicuous.
- The WIMP "confluence" was weaker than presented. The right relic abundance arises for weak-scale masses over a broad range, and many non-WIMP candidates (axions, sterile neutrinos, primordial black holes) were already viable. The "miracle" was suggestive, not "puzzling and completely unexplained" otherwise.
- The cyclic-versus-inflation test is asymmetric. Detection of primordial B-modes would kill the cyclic model, but non-detection does not kill inflation: many inflation models predict tiny r. Greene's note concedes this, but the main text implies a clean decision.
- The Fly's Eye "30 billion proton masses." The 1991 "Oh-My-God particle" carried about 3.2 × 1020 eV, about 340 billion proton masses. Greene's figure appears to be an order of magnitude low (his own note cites 100 billion for the most energetic class).
- Name spelling: Josef Lense; the Brill-Cohen paper is 1966.
2026 update: new windows opened
- Gravitational-wave astronomy is routine. LIGO-Virgo-KAGRA's O4 run (2023 to 2025) added hundreds of candidate events, including black holes in the supposed "mass gap" and the record-loud GW250114, which confirmed Hawking's area theorem at high confidence. LISA was formally adopted by ESA in January 2024, for launch around 2035.
- Pulsar timing arrays (NANOGrav, EPTA, PPTA, CPTA; June 2023) detected a nanohertz background, probably from supermassive black-hole binaries: a fourth band of the gravitational-wave spectrum.
- Black holes imaged. The Event Horizon Telescope photographed the shadows of M87* (2019) and Sagittarius A* (2022), testing GR in the strong-field regime; polarisation maps (2021 to 2024) traced magnetic fields at the horizon.
- Ultra-high-energy cosmic rays. The Amaterasu particle (Telescope Array, reported 2023) carried about 2.4 × 1020 eV from an apparently empty region of sky. Auger confirmed an extragalactic origin (2017). Neither found micro black holes.
- Dark sector surveys under way. Euclid (2023), DESI (from 2021), the Vera C. Rubin Observatory's ten-year survey (from 2025) and Roman (due by 2027) aim to pin dark energy's equation of state to the percent level. DESI's 2025 data already prefer weakening dark energy at up to ~4 sigma.
- Neutrino mass from cosmology. DESI plus CMB data limit the sum of neutrino masses to below about 0.07 eV, close to the minimum oscillations allow, a tension being watched closely.
So what. Keep a scorecard for your own forecasts. Greene's chapter is a rare, honest record of what a leading theorist expected from the next two decades, and the pattern it reveals is general: predictions grounded in well-tested frameworks (GR, the standard model) came true; predictions grounded in aesthetic or naturalness arguments did not. When evaluating a roadmap, separate claims that follow from established models from claims that rely on "it would be elegant if." And note the asymmetry of tests: design experiments whose failure would be as informative as their success.
# A toy gravitational-wave chirp (leading-order inspiral) you can listen to
import numpy as np, wave
G, c, Msun = 6.674e-11, 2.998e8, 1.989e30
m1, m2 = 36*Msun, 29*Msun # GW150914-like
Mc = (m1*m2)**0.6/(m1+m2)**0.2 # chirp mass
tc, fs = 0.4, 8192 # time to merger, sample rate
t = np.arange(0, tc-1e-3, 1/fs); tau = tc - t
f = (1/(8*np.pi))*(5/tau)**(3/8)*(G*Mc/c**3)**(-5/8) # frequency (Hz)
phase = 2*np.pi*np.cumsum(f)/fs
h = (f/f.max())**(2/3)*np.cos(phase) # amplitude grows as f^(2/3)
print(f'frequency sweeps {f[0]:.0f} -> {f[-1]:.0f} Hz in {tc} s')
# Speed it up x1 (already audible) and save a WAV file
audio = np.int16(h/np.abs(h).max()*30000)
with wave.open('chirp.wav','w') as w:
w.setnchannels(1); w.setsampwidth(2); w.setframerate(fs)
w.writeframes(audio.tobytes())
print('saved chirp.wav: the sparrow-like trill of two merging black holes')
Teleporters and Time Machines: Traveling Through Space and Time
As a boy, Greene found Star Trek's all-knowing computer harder to believe than warp drive; within half a century his laptop did most of it. So what of teleporters and time machines? Quantum mechanics seems to forbid copying an object, since measuring it changes it, yet entanglement lets a particle's exact quantum state be teleported, as first done with photons in 1997. Teleporting a DeLorean or a person is "eons, if not an eternity" away, and raises the old puzzle of personal identity. Travel to the future is easy in principle (move fast and come back). Travel to the past threatens paradox, but the block universe of Chapter 5 dissolves the paradoxes: you can't change the past, only be part of it, or, in Many Worlds, arrive in a different branch. No law of physics yet known forbids a time machine; Kip Thorne's wormhole design shows what one would require, and every requirement presses against the edge of known physics. Greene, like most physicists, expects that past-directed time travel will one day be ruled out, but says no proof exists.
Teleportation in a quantum world
A science-fiction transporter scans an object, sends the blueprint, and rebuilds it far away, either from the original atoms or from local ones. Real teleportation resembles the second kind, and raises two questions:
- Is an exact replica the original? A philosophical puzzle.
- Can an object be scanned perfectly? Classically yes, apart from complexity. Quantum mechanically, measurement changes what it measures: a "quantum Catch-22." To replicate you must observe; observing destroys what you wanted to copy.
Quantum physics answers the identity question cleanly for particles. All electrons are exactly identical in mass, charge, nuclear charges and total spin, and those exhaust their properties; likewise every up-quark, every photon (smallest packets of the same field, or identical vibrations of the same string). Two particles of one species can differ only in their quantum state (probabilities of position, spin, energy). Put a second particle into exactly the state of the first, far away, and it is indistinguishable in principle. That is quantum teleportation. Since the original's state is unavoidably destroyed, there is never a copy, so "teleportation" rather than "faxing" is the right word.
Thought experiment: United Quantum Van Lines. You ask movers to ship your DeLorean from New York to London; secretly they teleport it, particle by particle, every quantum state and relationship reproduced. Bumps, nicks, the squeaky left wing-door, the dog's musty smell, the handling on a sharp turn: all identical. You could never know, even in principle. Now do it to your cat, or to yourself. Greene: "a living being whose constituent atoms and molecules are in exactly the same quantum state as mine is me." If the original survived, there would be two of me, neither with priority. He goes further: we lose and gain atoms constantly (ice cream, MRIs, heart transplants; about a trillion atoms replaced every millionth of a second), so even an imperfect copy might be him. Those who believe a person is more than physical structure will demand more; "no one can claim to have the definitive answer."
How teleportation works
From theory to the lab
- 1993: Charles Bennett (IBM), Gilles Brassard, Claude Crépeau, Richard Jozsa (Montreal), Asher Peres and William Wootters show that entanglement makes teleportation possible.
- 1997: Anton Zeilinger's group (Innsbruck) and Francesco De Martini's (Rome) teleport single photons across a lab. The hard part was inventing a way to perform the joint (Bell-state) measurement on two photons.
Teleporting Photon A from New York to Nicholas in London.
- Share entanglement. Greene holds Photon B, Nicholas holds Photon C; B and C are entangled (measured along any axis, they agree).
- Joint measurement (the quantum stage). Greene does not measure A. He measures a joint property of A and B, for example whether they have the same spin about a vertical axis, without learning either spin. This disturbs A and B, but through entanglement the disturbance is imprinted on C in London.
- Phone call (the classical stage). Greene tells Nicholas the result: how A relates to B after the measurement. This travels at or below light speed.
- Correction. Nicholas applies a simple operation (at most a rotation) to C, "subtracting out" the disruption. C is now in A's original state, the state A had before the measurement.
Key features. A's original state no longer exists in New York, so there's no copy. Nobody ever learns A's state: "we don't need to know the photon's quantum state in order to teleport it." And nothing travels faster than light: without the phone call, C is random.
Realistic teleportation?
Zeilinger's fanciful recipe for the DeLorean: two chambers of particles, New York's entangled with London's, enough for a car; a device to make joint measurements between every particle of the car and the New York chamber; a phone call carrying about 1030 results; and Nicholas's manipulations. "Every step... Is fantasy." A DeLorean has more than 1027 particles; entangling even a few was hard in 2004, and joint measurement of two photons was a feat. Still: forty years ago, the Enterprise's computer looked unlikely too.
The puzzles of time travel
Travel to the future is settled physics. Build a ship reaching 99.9999999996 percent of light speed, fly out for a day, ten days or 27 years of ship time, and turn back: you return 1,000, 10,000 or about 10 million years into earth's future. Time dilation has been confirmed with airliners and with muons, which live about two millionths of a second at rest but much longer at high speed. "Technology, not physics, keeps each of us tethered to this epoch."
Travel to the past threatens paradoxes:
Two paradoxes.
- The grandfather paradox: go back and stop your parents meeting. If you're never born, who went back?
- The knowledge paradox (Michael Dummett, highlighted by David Deutsch): Greene travels ten years ahead, lunches at Tofu-4-U (which displaced McDonald's after the "great mad-cow pandemic"), and reads that string theory has been completely solved in a paper by his mother, Rita Greene, who thanks him for years of tutoring. He returns, tutors her, fails, and finally dictates the paper from memory. Who discovered the results? Not him (he read them), not her (he dictated them). Knowledge from nowhere.
Rethinking the puzzles
The block universe dissolves the grandfather paradox. In Chapter 5's frozen river, moments don't change; they just are. The intuitive story ("midnight, New Year's Eve 1965, was initially when your parents met; then you changed it") assumes a moment can be one way and later another. If you travel to 11:50 p.m., December 31, 1965, you were always there. Whatever you do there is part of the one unchanging history that led to your trip. If you travel to the past, you can't change it any more than you can change the value of pi. "Applying the concept of change to a moment makes as much sense as subjecting a rock to psychoanalysis."
The man from the future. You remember your father once telling you he hadn't planned to propose. But ten minutes before midnight at that New Year's party, a stranger popped out of nowhere claiming to be from the future, and your father was so shaken he proposed to your mother on the spot. You were the stranger. Your trip is not a threat to the past but a cause of it: the events "interlock with the rigid economy of a cosmic crossword puzzle."
Free will, Many Worlds and time travel
What stops you from succeeding? Here free will enters, unresolved even without time machines.
- If free will is an illusion (classical determinism, or a purely Schrödinger-equation quantum world), the laws simply prevent you: the time machine lands you on the wrong side of town, the gun jams, you shoot your father's rival instead, or you no longer want to.
- If free will is real (perhaps through the unresolved measurement problem; some have even proposed conscious observation as the trigger of collapse, which Greene finds "extremely unlikely"), Deutsch's Many Worlds resolution applies: the gun works, your father dies, but you have arrived in a different branch, one where your parents never met. Your parents live in the branch you left. In the knowledge paradox, the string theory paper was written by the version of Greene's mother in a branch where she is a physics whiz; time travel just carried her discovery to another branch.
None of this proves past travel possible; it shows that paradoxes alone can't rule it out.
Is time travel to the past possible?
"Most sober physicists would answer no. I would say no." But it is a qualified no: unlike the light barrier, no law yet known forbids it, and physicists have published hypothetical blueprints for civilisations with unlimited technology. Einstein thought about it while developing general relativity but "never made much progress."
| Proposal | Year | Mechanism | Catch |
|---|---|---|---|
| W. J. Van Stockum | 1937 | Dense infinite cylinder spinning about its axis drags time "on its side"; circling it returns you before you left | Infinite cylinder; refined by Frank Tipler (1974), who found finite versions need singularities or exotic conditions |
| Kurt Gödel | 1949 | The whole universe rotates (meaningless to Mach, meaningful in GR); some orbits return before departure | Our universe isn't rotating (light doesn't spiral) |
| Kip Thorne and students (Morris, Yurtsever) | 1988 | Wormhole with one mouth moved at high speed | Needs exotic, negative-energy matter; Hawking's feedback objection |
| J. Richard Gott | 1991 | Two cosmic strings passing at near light speed | Shown to require unphysical total momentum or a closed universe (Deser, Jackiw and 't Hooft; Carroll, Farhi and Guth, 1992) |
Blueprint for a wormhole time machine
What a wormhole is. A tunnel through space that differs from a mountain tunnel: remove the mountain tunnel and its space remains; remove a wormhole and its space vanishes. It touches ordinary space only at its mouths. Standing in Springfield, you'd never see a wormhole between the Kwik-E-Mart and the nuclear power plant strung across the sky; you'd see a hole in the Kwik-E-Mart through which the plant's interior is visible. General relativity allows wormholes; John Wheeler studied them in the 1950s.
Bart, Lisa and the Andromedan fish fingers. Bart, at the power plant mouth, loads it onto his ship to fetch Homer's birthday fish fingers from Andromeda, at 99.999999999999999999 percent of light speed. Through the wormhole, whose length doesn't change however far the ship flies, he chats with Lisa at the Kwik-E-Mart the whole way: four hours out, four hours back. He lands on the lawn of Springfield High, and the scoreboard over the rollerball stadium reads 6 million years later. Lisa, looking out the Kwik-E-Mart window, sees no ship; looking through the wormhole, she does. "You're looking at the right place, but you're not looking at the right time." Bart crawls through the wormhole, buys a Butterfinger from Apu, and goes home for dinner: a moment's crawl carried him 6 million years into the past. The trip moved the mouth not just through space but through time.
Two steps, in principle: (1) find or make a wormhole wide enough to use; (2) create a time difference between its mouths.
Building one: the obstacles
- Getting a wormhole. Perhaps tiny ones appear in the quantum foam and could be enlarged, or one could be engineered by shaping matter and energy, which requires tearing space. (Greene's string theory work shows some tears are possible, but not whether they help.) "A very long way from being realised."
- Keeping it open. Wheeler and Robert Fuller (1960s) showed wormholes collapse in a fraction of a second. Morris and Thorne and Matt Visser found that exotic matter with negative energy could prop them open. Quantum mechanics allows negative energy in special conditions (the Casimir effect is one), but Visser estimated a one-metre wormhole needs negative energy equal in magnitude to the sun's total output over 10 billion years.
- Surviving the switch-on. Stephen Hawking and others suggested that the moment the wormhole becomes a time machine, vacuum fluctuations could cycle through it endlessly, future to past to future, building up energy like a screeching microphone and destroying it. Plausible, but unproven: the calculations strain quantum theory in curved spacetime.
Cosmic rubbernecking
Why no tourists from the future? If time machines will ever exist, historians would surely visit the first atomic bomb or the moon landing. Greene's answer: the proposed machines can't reach times before the first machine was built. A wormhole time machine connects only times after its mouths' time difference was established. If the first is built 10,000 years from now, that moment will be a tourist magnet, but our era stays out of reach. "Every moment that goes by without our having succeeded in building a time machine is a moment that will be forever beyond our reach."
Going deeper: the technical notes
- Measurement disrupts. Except when a particle is already in a definite state of the measured property, observation yields a result that doesn't reflect its prior fuzzy state.
- The four outcomes. The Bell-state measurement has four possible results (two classical bits); for each, Nicholas applies one of four operations, at most a simple rotation.
- Exotic matter versus Λ. A cosmological constant repels through negative pressure with positive energy; wormhole support needs negative energy itself, a stronger violation.
- Where's the wormhole? Even its mouths are hard to depict: the mouths in Figure 15.3 are really inside the Kwik-E-Mart and the power plant.
Critical lens.
- The self-consistency principle has an author. Greene's block-universe resolution is the Novikov self-consistency principle (Igor Novikov, 1980s), tested in the "billiard ball" paradox studies of Echeverria, Klinkhammer and Thorne (1991), who found that every initial condition they tried admitted at least one consistent history (sometimes many, which raises its own puzzle). Neither is named. Deutsch's quantum treatment of closed timelike curves (1991) is referred to only informally.
- Teletransportation and identity: Parfit. Greene's position (a perfect physical copy is you; two copies are both you) is close to Derek Parfit's reductionist view in Reasons and Persons (1984), built on the same teletransporter thought experiment. Parfit concluded that identity is not what matters; survival is a matter of psychological continuity. The literature is richer than "some disagree."
- The no-cloning theorem (Wootters and Zurek; Dieks, 1982), which explains why the original's state must be destroyed, deserves a mention: it is the deep law behind the "no faxing" point.
- Gott's cosmic-string machine was rebutted soon after: building it requires a total momentum that is "tachyonic" (Deser, Jackiw and 't Hooft, 1992), and in an open universe there's never enough energy (Carroll, Farhi and Guth, 1992).
- Negative energy is more constrained than "highly specialised conditions" suggests. Quantum inequalities (Ford and Roman, 1990s) limit how much negative energy can exist and for how long; the averaged null energy condition has since been proved for broad classes of quantum field theories (Faulkner and colleagues, 2016; Hartman and colleagues, 2017). Macroscopic traversable wormholes look harder now than in 2004.
- The 1997 experiments succeeded only part of the time: Innsbruck's linear-optics Bell measurement distinguished one of the four outcomes (about 25 percent), a limitation of the method, not of the physics.
2026 update: teleportation became engineering
- Distance records. Teleportation over 143 km of open air between Canary Islands (2012); from ground to the Micius satellite over 1,400 km (2017); over metropolitan fibre networks (2016 onward), including fibre simultaneously carrying ordinary internet traffic (2024).
- Matter and memories. Quantum states have been teleported between trapped ions, between atomic memories, and between nodes of the Delft three-node quantum network (2022), including teleportation between non-neighbouring nodes.
- Teleporting operations. In 2025 Oxford researchers teleported a logic gate between two separate ion-trap processors, a step toward distributed quantum computing.
- Bigger entangled systems. Entanglement now spans thousands of atoms in spin-squeezed clocks and macroscopic mechanical oscillators (2021), still some 20 orders of magnitude short of a DeLorean.
- A "wormhole" on a chip? In 2022 a team (Jafferis et al., Nature) ran a tiny holographic model on Google's Sycamore processor and reported dynamics resembling traversable-wormhole teleportation. Critics showed the learned model was too simple to behave holographically in general; it was a quantum simulation of a toy model, not a wormhole. Theoretical work by Gao, Jafferis and Wall (2017) does allow traversable wormholes in certain settings, but they are no faster than going around and cannot become time machines.
- Post-selected "time travel." Experiments simulating closed timelike curves with photons and post-selection (Lloyd and colleagues, 2011) illustrated how a self-consistent quantum time loop would behave, without sending anything into the past.
- Time dilation, tabletop scale. Optical clocks now detect relativistic slowing from walking-pace motion and from height changes of a millimetre (JILA, 2022): Greene's "technology, not physics" for future travel is vividly confirmed, if only by nanoseconds.
So what. Quantum teleportation is a perfect design pattern: split a task into a fast but meaningless channel (entanglement, instant but random) and a slow but meaningful one (the phone call), and neither alone leaks or transmits anything. Security engineers use the same idea in secret sharing and one-time pads. Greene's treatment of time paradoxes offers a second lesson: many apparent contradictions come from smuggling in a hidden assumption (here, that moments can change); name the assumption and the paradox evaporates. And the "no tourists" argument models good inference: absence of evidence is evidence only if your hypothesis predicted the evidence.
# Quantum teleportation of a random qubit, simulated exactly with numpy
import numpy as np
rng = np.random.default_rng(3)
psi = rng.normal(size=2) + 1j*rng.normal(size=2); psi /= np.linalg.norm(psi)
bell = np.array([1,0,0,1])/np.sqrt(2) # B,C entangled: |00>+|11>
state = np.kron(psi, bell) # order: A, B, C
I, X, Z = np.eye(2), np.array([[0,1],[1,0]]), np.diag([1,-1])
B4 = {'00':np.array([1,0,0,1])/2**.5, '01':np.array([0,1,1,0])/2**.5,
'10':np.array([1,0,0,-1])/2**.5,'11':np.array([0,1,-1,0])/2**.5}
fix = {'00':I, '01':X, '10':Z, '11':Z@X} # Nicholas's corrections
for bits, b in B4.items():
C = b.conj() @ state.reshape(4,2) # project A,B onto Bell state
p = np.vdot(C, C).real; C /= np.sqrt(p)
out = fix[bits] @ C
print(f'outcome {bits} (p={p:.2f}): fidelity = {abs(np.vdot(psi,out))**2:.6f}')
The Future of an Allusion: Prospects for Space and Time
Kant thought we could not think without space and time. Many physicists now suspect space and time are not fundamental but emergent, like the hardness of a cannonball or the scent of a rose. Greene's final chapter gathers the clues. Quantum averaging: smooth spacetime is an average over Planck-scale jitters, as a TV image averages pixels. Geometric duality: equivalent string theories disagree about the size, shape and even the topology of space (a circle of radius R is indistinguishable from one of radius 1/R). Black-hole entropy: the maximum entropy in any region grows with its surface area, not its volume, as if the true degrees of freedom live on a boundary, which suggests the holographic principle, made concrete by Maldacena's 1997 correspondence, in which a five-dimensional theory with gravity is equivalent to a four-dimensional theory without it. The form of spacetime, its shape, size and even number of dimensions, changes in translation, like the letters of "cat" and "gato"; it may be an adorning detail rather than a foundation. String theory and loop quantum gravity each offer candidates for spacetime's "atoms"; Greene hopes their merger will yield a background-independent theory in which Leibniz, Newton, Mach and Einstein can all claim a share of victory.
Are space and time fundamental?
Physicists live "in a state of confusion"; it's an occupational hazard. Kant argued we cannot do without space and time, and Greene admits that when he closes his eyes, "space, through context, or time, through change, always manages to seep in." Only deep in a calculation does he come close to escaping them. "Truly eluding space and time makes escaping your shadow a cakewalk."
Emergence, not illusion. A cannonball's hardness, a rose's scent and a cheetah's speed emerge from atoms, molecules and muscle; they are real, but we wouldn't build a theory of the universe from them. Likewise, if spacetime is composite, Newton's bucket and Einstein's gravity stay real at our scales. The "illusion" would be ours: the belief that the deepest description must bring space and time into sharpest focus. Space and time may dissolve when nature's laws are written at their most fundamental level.
Clue 1: quantum averaging
Chapter 12's jitters of geometry are tamed by strings but not eliminated. We see smooth spacetime because our instruments average over the undulations, as your eyes average a television's pixels at a comfortable distance. Averages are useful but can describe nothing real in detail: the average American family has 2.2 children, and the average gallon of milk cost $2.783 (in 2004); you won't find either.
The jitters also suggest that dividing distance and duration stops near the Planck length (10−33 cm) and Planck time (10−43 s). Whatever lies beneath (an "ur-ingredient") is likely so altered by averaging that, like Beethoven's Ninth heard note by note or a Monet haystack seen brushstroke by brushstroke, "nothing resembling it exists at the most fundamental level."
Clue 2: geometry in translation
Geometric duality. Witten's dictionary between string theories (Chapter 13) translates not only questions but geometry. Type IIA string theory with a curled dimension of radius R (in Planck units) is exactly equivalent to Type IIB with radius 1/R: big and small circles are physically indistinguishable.
Why: strings on a circle come in two kinds. Winding strings, wrapped like a rubber band around a tin can, have energy proportional to R (bigger circle, more stretch). Momentum strings, sitting on part of the circle, have energy proportional to 1/R (smaller circle, more hemmed in, more quantum jitter). Swap R for 1/R and swap the two kinds, and every energy, and all physics, is unchanged.
Mirror symmetry does the same for Calabi-Yau shapes: a shape with, say, 2 holes of one kind and 5 of another gives the same physics as one with 5 and 2, since strings respond only to the difference between the two counts. Size, shape and the number of holes can all differ.
Two observers using equivalent theories would disagree about the size and shape of space, and both would be right. This isn't relativity's different slicings of one spacetime; they disagree about spacetime itself. If spacetime were fundamental, most physicists would expect everyone to agree on its geometry.
Clue 3: the entropy of black holes
Black holes: no hair, maximum entropy. From outside a black hole shows only mass, charge and spin: "black holes have no hair." Whatever its interior constituents, rearranging them changes nothing visible, so black holes have the highest entropy of anything their size. Try to beat it by pumping gas into a hollow sphere of the same size: before the gas's entropy catches up, its mass makes the sphere collapse into a black hole. Pump more in and the hole grows. So: the maximum entropy that can fit in any region of space equals the entropy of a black hole that just fills it.
Surprise: area, not volume. Join two Tupperware boxes of air and the entropy doubles: ordinary entropy scales with volume. But in the 1970s Jacob Bekenstein and Stephen Hawking found that a black hole's entropy is proportional to its horizon's area. Double the radius: volume ×8, area ×4. Multiply it by 100: volume ×1,000,000, area ×10,000. The Tupperware reasoning ignores gravity, which at high entropy is decisive.
Two consequences.
- Atoms of space. Tile a horizon with Planck squares (each about 10−66 cm2): the entropy equals their number (strictly, one quarter of it). Each square seems to carry one unit of entropy, as if nothing can happen inside a Planck square, another hint of a minimal spatial unit.
- The bookkeeping test. A fundamental theory must be able to track everything that could happen in a region, no more, no less. Imagine recording a group of hyperactive children: the more bedlam possible, the more you must track. Since gravity caps a region's disorder at its area, a theory with gravity has fewer degrees of freedom than one without. And perhaps those degrees of freedom live on the surface.
Clue 4: is the universe a hologram?
The holographic principle. A hologram is a flat sheet of etched plastic that projects a 3D image (Dennis Gabor, Nobel 1971, recorded light's phase as well as its intensity; phase differences encode depth). In the early 1990s, Gerard 't Hooft and Leonard Susskind proposed that our 3D world might be a projection of processes on a distant 2D surface. It inverts Plato's cave: the flattened "shadows" are the real thing, and the richer world is the projection.
Hard questions follow. Where is the boundary of a universe that is infinite or wraps around? We seem to control events here, in the "bulk," not on some faraway surface. Is holography a duality, where you choose bulk or boundary description by taste?
Maldacena's breakthrough (1997). Juan Maldacena found a concrete, mathematically tractable (but unrealistic) example. Take a universe with four space dimensions and one time, uniformly negatively curved (a higher-dimensional Pringles chip: anti-de Sitter space) plus curled dimensions (AdS5 × S5). Its boundary has one dimension fewer, like the can around tomato soup. A string theory with gravity in the soup is exactly equivalent to a quantum theory without gravity on the can. Neither is primary; they're translations, and every calculation so far agrees. Not only shape and size but the number of dimensions changes in translation.
"Much as the number of letters, syllables, and vowels in the word cat differ from those in gato, its Spanish translation, the form of spacetime... Also changes in translation." Of all the clues, Greene bets on holography to dominate future research: it rests on black-hole entropy, which any sensible theory of gravity must reproduce.
The constituents of spacetime
A century ago, the atomic hypothesis was itself controversial (Mach rejected it). Serious thinking about spacetime's microscopic makeup only became possible with general relativity and quantum mechanics, so lacking its "atoms" is "not a black mark." Two candidates:
Strings as the threads of spacetime. Perhaps spacetime is stitched from strings, as a shirt from thread, with matter as embroidery on the fabric. Attractive, but no one has made it precise. Unravel a shirt and you get a pile of thread, but a "pile" of strings that haven't yet woven spacetime would have no space or time to vibrate in. String theory needs a background-independent formulation, "the single greatest unsolved problem facing string theory," which might also choose the extra dimensions' shape.
Matrix theory. M-theory also contains zero-branes, point-like but with strings attached, so they share strings' gentle quantum behaviour. Tom Banks, Willy Fischler, Stephen Shenker and Leonard Susskind formulated Matrix theory, in which zero-branes are the fundamental ingredient from which strings, branes and perhaps space and time are built.
| String theory | Loop quantum gravity | |
|---|---|---|
| Tradition | Particle physics: starts small (quantum) and reaches for gravity | General relativity: starts large (gravity) and reaches for quantum |
| Origin | A theory of the strong force; gravity appeared by accident | Since the mid-1980s (Ashtekar, Rovelli, Smolin and others) |
| Unifies all forces? | Yes: every particle is a string vibration | No: gravity only |
| Background independent? | No: strings move in a spacetime put in by hand | Yes: no spacetime assumed |
| Recovers general relativity at large scales? | Yes, readily | Hard; still being worked out |
| Spacetime's atoms | Strings, branes, zero-branes (Matrix theory) | Loops woven like crocheted wool into a "spin network" |
| Discreteness | Hinted: minimal length, Planck squares | Explicit: areas come in discrete units (no fractions) |
| Black-hole entropy | Reproduced | Reproduced |
One theory from two ends? Both involve loops, both find atomised space, and both get black-hole entropy right. Greene's forecast: loop-gravity techniques will give string theory a background-independent formulation and spark a third superstring revolution.
The relationist debate, resolved. In such a theory the ingredients stand only in relation to one another, with no pre-existing arena, as Leibniz and Mach wanted. When they coalesce into large-scale spacetime, that spacetime is a something, as Newton and Einstein found (a bucket spun in empty flat spacetime still goes concave). The line between spacetime and matter "would largely evaporate," both emerging from more basic ingredients in a theory that's "fundamentally relational, spaceless, and timeless." Leibniz, Newton, Mach and Einstein could all claim a share of the victory.
Inner and outer space
Space and time will keep framing experience; what changes is our understanding of them. The twentieth century revealed time's slowing, relative simultaneity, sliced spacetime, gravity as curvature, quantum probability and entanglement. Our age adds dark matter and dark energy, gravitational waves, a Higgs ocean, inflation, strings, extra dimensions, braneworlds, and perhaps emergent spacetime.
Greene hoped accelerators would test many of these in the next decade. But his deepest hope is astronomical: just as inflation stretched quantum jitters across the sky (Chapter 11), perhaps it stretched imprints of strings, quantum gravity, or the atomic structure of spacetime itself. "There would be nothing more poetic... Than for us to confirm our theories of the ultrasmall... By turning our most powerful telescopes skyward and gazing silently at the stars."
Going deeper: the technical notes
- Why entropy doubles. The number of arrangements of two joined boxes is the product of each box's count; entropy, a logarithm, therefore adds.
- Area versus volume compares growth rates, not like quantities.
- The precise bound. Raphael Bousso's covariant entropy bound (1999): the entropy passing through a light-sheet (a null surface that isn't expanding) is at most A/4 in Planck units, A being the area of a cross-section.
- The exact formula. A black hole's entropy is its horizon area in Planck units divided by 4, times Boltzmann's constant: S = kA/4ℓP2. Cosmic horizons are also thought to carry entropy proportional to area.
- Anti-de Sitter boundary. In AdS space, light can travel from the interior to spatial infinity and back in finite time, which is what makes its boundary a sensible place to put a theory.
- Sociology, not physics. The "small versus large" contrast describes traditions; only string theory currently convincingly reduces to general relativity on large scales.
Critical lens.
- The information paradox is the missing elephant. Hawking's 1976 argument that evaporating black holes destroy information (violating quantum mechanics) drove most of the work Greene summarises, including holography and AdS/CFT; he doesn't present it. Hawking publicly conceded in July 2004, the year this book appeared.
- Strominger and Vafa (1996) first counted a black hole's microstates in string theory and reproduced S = A/4 exactly, for special (extremal, supersymmetric) black holes. That is the strongest evidence for "black-hole entropy is reproduced," and it is not named.
- Loop gravity's area spectrum is less tidy than "1, 2, 202 square Planck lengths." LQG areas are sums of terms like √(j(j+1)) times a constant involving the Barbero-Immirzi parameter; they are discrete but not integer multiples. And loop gravity reproduces S = A/4 only after fixing that parameter to match, which critics call a fit rather than a derivation.
- AdS is not our universe. Our universe has a positive cosmological constant (de Sitter-like), and no comparably precise holographic duality for de Sitter space exists. Greene notes Maldacena's model is unrealistic, but the gap remains the central open problem of holography.
- "Every calculation done to date supports this assertion" was true but selective: AdS/CFT is a conjecture, supported overwhelmingly by tests in regimes where both sides can be computed. No proof exists.
- The forecast of a third revolution via background independence has not materialised; instead, the field moved toward entanglement and quantum information as the source of spacetime (see the update).
2026 update: spacetime from entanglement
- Entropy equals area, inside the bulk too. Ryu and Takayanagi (2006) showed that the entanglement entropy of a region of the boundary theory equals the area of a minimal surface in the bulk. Mark Van Raamsdonk (2010) argued that removing entanglement tears spacetime apart: geometry is built from entanglement.
- ER = EPR. Maldacena and Susskind (2013) conjectured that entangled particles (EPR, Chapter 4) are connected by microscopic wormholes (Einstein-Rosen bridges), uniting this book's two strangest themes.
- Holography as error correction. Almheiri, Dong and Harlow (2015) showed the bulk is encoded in the boundary like a quantum error-correcting code, explaining how local bulk physics survives the loss of parts of the boundary.
- The information paradox (largely) resolved in principle. The firewall argument (AMPS, 2012) sharpened the crisis. Then in 2019, Penington and Almheiri, Engelhardt, Marolf and Maxfield computed the Page curve: radiation entropy rises then falls as information escapes, via "islands" and quantum extremal surfaces. How the information gets out in detail is still debated.
- Spacetime-free formulations. The amplituhedron (Arkani-Hamed and Trnka, 2013) computes certain particle-scattering amplitudes from geometry with no reference to spacetime or locality, which emerge as derived properties.
- Testing Planck-scale structure. Gamma-ray bursts observed by Fermi (GRB 090510, 2009) and LHAASO (2022 to 2024) show photons of very different energies arriving together after billions of years, ruling out the simplest linear Planck-scale violations of Lorentz symmetry. Fermilab's Holometer (2015) excluded one model of "holographic noise." Black-hole area-theorem tests with gravitational waves (GW250114, 2025) confirm horizon behaviour expected from the entropy story.
- Background independence is still unsolved. Neither string theory nor loop gravity has the fully spaceless, timeless formulation Greene hoped for, but most researchers now expect that, if it comes, it will be quantum-informational: spacetime as an emergent property of entanglement.
So what. The chapter's deepest move is to distrust the coordinates you think in. Duality shows that a "big" and a "small" description can be the same system; holography shows that a description in three dimensions can be fully equivalent to one in two. In modelling, this means asking which features of your representation are invariant (the physics) and which are conventions (the language). The entropy bound is also a practical heuristic: the information capacity of a system is set by its interface, not its interior. Teams, APIs and organisations often behave the same way; their effective complexity is bounded by what crosses their boundary.
# Black holes as the universe's hard drives: Bekenstein-Hawking entropy
import numpy as np
G, c, hbar, kB = 6.674e-11, 2.998e8, 1.055e-34, 1.381e-23
lP2 = hbar*G/c**3 # Planck length squared
Msun = 1.989e30
def bh(M):
r = 2*G*M/c**2 # horizon radius
S = 4*np.pi*r**2/(4*lP2) # entropy in units of k
T = hbar*c**3/(8*np.pi*G*M*kB) # Hawking temperature (K)
t = 5120*np.pi*G**2*M**3/(hbar*c**4)/3.156e7 # evaporation time (yr)
return r, S, T, t
for name, M in [('1 solar mass',Msun),('Sgr A* (4.3e6)',4.3e6*Msun),
('M87* (6.5e9)',6.5e9*Msun)]:
r,S,T,t = bh(M)
print(f'{name:16s} r={r:9.2e} m S={S:8.1e} k T={T:8.1e} K '
f'evaporates in {t:8.1e} yr')
print('Entropy of the sun itself ~ 1e58 k: a solar-mass black hole holds')
print(f'{bh(Msun)[1]/1e58:.0e} times more, all on its surface.')
Master Takeaways
Strip away the analogies and The Fabric of the Cosmos makes a handful of strong claims about space and time, and a larger set about how physics reasons when its best theories outrun its instruments. The first six below are the ones to remember; they also appear on the back cover.
The ideas that matter most
- Spacetime is a "something," but not Newton's. The concave water in a spinning bucket proves acceleration is absolute; in general relativity it is measured against spacetime, a dynamic entity shaped by matter. Absolutists and relationists were each partly right.
- Space and time are relative; spacetime is not. Because light's speed is the same for all, moving clocks slow, simultaneity depends on motion, and everything always moves through spacetime at light speed; observers slice one shared four-dimensional reality differently.
- The universe is non-local. Entangled particles are correlated beyond anything locally pre-programmed (Bell), confirmed by every test since 1982 and loophole-free since 2015; yet entanglement cannot send signals.
- Time's flow may be a feature of memory, not physics. Grant every observer's "now" equal reality and the whole of spacetime exists: a frozen river. Each moment contains memories that create the sensation of flow.
- Time's arrow points back to the big bang. Every known law runs equally well backward; entropy explains the arrow only if the universe began in an extraordinarily low-entropy, smooth state. "A splattering egg tells us something deep about the big bang."
- Inflation turned quantum jitters into galaxies. Repulsive gravity from negative pressure blew up space by ≥1030, explaining its uniformity and flatness, seeding structure with stretched quantum fluctuations whose pattern the microwave background confirms, and leaving the smooth, low-gravitational-entropy start the arrow requires.
- Pressure gravitates. In Einstein's theory, energy and pressure create gravity; negative pressure repels. That single fact underlies inflation, dark energy and Einstein's cosmological constant.
- The universe is 5 percent familiar. Ordinary matter is about 5 percent of the cosmic energy budget; dark matter ~25 percent, dark energy ~70 percent, the latter accelerating the expansion since about 6 billion years ago.
- Symmetry is the source of law. Translation, rotation and relativity's equivalences generate the laws; time requires broken symmetry (change); cosmic time requires spatial uniformity.
- Empty space is full. The vacuum hosts quantum jitters (Casimir effect), probably a Higgs ocean (found 2012), and dark energy. Nothingness is the lowest-energy state, which need not be "nothing."
- Hot universes are more symmetric. Cooling triggers phase transitions that hide symmetries: the Higgs ocean distinguishes electromagnetism from the weak force and gives particles mass.
- Quantum mechanics enlarges the past. The observed present is an average over all possible histories (Feynman); delayed-choice and eraser experiments show that later measurements decide which kinds of history may be told, without changing what happened.
- The measurement problem is unsolved. Decoherence explains why we see classical outcomes, not why one occurs; the laws remain time-symmetric in most interpretations, so quantum mechanics does not supply the arrow.
- Entropy is counting, not mess. High entropy means many microscopic arrangements look the same; it rises because disorder has overwhelmingly more ways to happen, and it is statistical, not absolute.
- With gravity, smooth means ordered. Gravity makes clumping the high-entropy direction; black holes are the maximum-entropy objects. The early universe's smoothness was its most improbable feature.
- General relativity and quantum mechanics clash at the Planck scale. Below 10−33 cm, geometry fluctuates wildly; the equations give infinities. Black-hole centres and the big bang sit on both sides of the divide.
- String theory proposes a minimum size. Particles as vibrating strings smooth the jitters, include the graviton, and require extra dimensions whose geometry would set particle properties; the theory's mathematics is rich, its predictions so far elusive.
- Branes reframe what "space" is. If our universe is a three-brane, space is literally an object, light is trapped on it, and only gravity probes extra dimensions, which could have been large; experiments say they are smaller than about 50 micrometres.
- Translate hard problems. Dualities make the five string theories one, make big and small circles equivalent, and make a gravitational world equal to a gravity-free one in fewer dimensions.
- Information lives on boundaries. The maximum entropy in a region scales with its surface area; the holographic principle suggests reality's degrees of freedom live on a boundary.
- Spacetime may be emergent. Quantum averaging, duality and holography all point to space and time arising from more basic, spaceless and timeless ingredients, a resolution in which Leibniz, Newton, Mach and Einstein all share the victory.
- Travel to the future is easy; to the past, not ruled out. Special relativity sends travellers forward; wormholes with one moving mouth would allow travel back, but need negative energy and may be destroyed by quantum feedback.
- Paradoxes come from treating moments as changeable. In a block universe you can't alter the past, only be part of it; in Many Worlds you'd arrive in a different branch.
- Teleportation is real, for quantum states. Entanglement plus a classical message rebuilds a particle's exact state far away, destroying the original and transmitting nothing faster than light.
- Established theory delivered; speculation has not (yet). Of the experiments Greene anticipated, those testing general relativity and the standard model (frame dragging, gravitational waves, the Higgs) all succeeded; those seeking new physics just beyond reach (supersymmetry, large extra dimensions, WIMPs) found nothing.
Frameworks and mental models
| Name | What it says | Use it when |
|---|---|---|
| The bucket test (Ch 2) | Find an effect that only makes sense if some background is real | Deciding whether an abstraction (a "market," a "culture") does causal work |
| Find the invariant (Ch 3) | Observers disagree on parts but agree on the whole | Reconciling conflicting metrics or timestamps across systems |
| Speed budget (Ch 3) | Motion through space is diverted from motion through time | Trade-offs under a fixed total resource |
| Bell bound (Ch 4) | Derive what a simple hidden story would force; measure it | Testing "it's just a common cause" explanations |
| Correlation is not communication (Ch 4, 15) | Matching outputs prove no channel | Security, distributed systems, attribution |
| Now-list reasoning (Ch 5) | Small angles × huge distances = big disagreements | Understanding why small policy differences diverge at scale |
| What needs explaining (Ch 6) | High entropy is the default; order needs a story | Anomaly detection; avoiding "explaining" noise |
| Self-undermining check (Ch 6) | Reject conclusions that make your evidence untrustworthy | Auditing arguments and models |
| Condition on later data (Ch 7) | Structure can appear only after sorting by later information | Segment analysis, Simpson-like cancellation |
| Invariance first (Ch 8) | Impose symmetries, then fit | Model design, feature engineering |
| Comoving clocks (Ch 8) | Agree on a reference frame before comparing times | Distributed logging, cross-team metrics |
| Phase transitions (Ch 9) | Little change, then sudden reorganisation and lost symmetry | Organisational change, market regimes |
| Insensitivity to initial conditions (Ch 10) | Prefer mechanisms that erase dependence on unknown starts | Robust design, self-correcting processes |
| Corroborating second opinions (Ch 10) | Trust results that independent methods agree on | Validating models, audits |
| Use of a small seed (Ch 11) | A modest event amplified explains a vast ordered outcome | Root-cause analysis of large patterns |
| Entropy gap (Ch 11) | Value lies in distance from the maximum | Spotting under-exploited opportunities |
| Pixel limit (Ch 12) | A minimum resolution tames runaway behaviour | Regularisation, discretisation |
| Translate the problem (Ch 13, 16) | Hard in one description, easy in another | Primal/dual, time/frequency, SQL/graph |
| Know what your instruments can't see (Ch 13) | Every probe shares the same blind spots | Monitoring and observability |
| Forecast scorecard (Ch 14) | Check predictions grounded in tested theory vs aesthetics | Evaluating roadmaps |
| Two-channel design (Ch 15) | A fast meaningless channel plus a slow meaningful one | Secret sharing, key exchange |
| Boundary capacity (Ch 16) | Capacity is set by the interface, not the interior | Throughput in APIs, teams, platforms |
Critical lens on the whole book.
- Uncredited lineages. The block-universe argument is Rietdijk-Putnam (and Penrose's Andromeda paradox); the "no moving now" argument is McTaggart's; the time-travel consistency resolution is Novikov's; the teleporter identity puzzle is Parfit's. Greene's syntheses are excellent but often present inherited arguments without their names.
- Omissions that matter. Noether's theorem is missing from a chapter about symmetry generating laws; the black-hole information paradox, central to holography, is absent; Randall-Sundrum warped braneworlds and Strominger-Vafa microstate counting are unnamed; inflation's critics (Penrose, Ijjas, Steinhardt and Loeb) are muted.
- Optimism about near-term tests. Chapters 13 and 14 expected TeV-scale strings, supersymmetry "with ease," micro black holes and dark matter identification "within the next few years." None appeared. This reflects the 2004 consensus, but the book's tone sometimes outruns the evidence it reports.
- Settled versus speculative. Parts I to III are physics on firm ground (with the measurement problem honestly flagged). Parts IV and V are speculative, as Greene says, but readers can lose track of the boundary; the editions' 2026 boxes mark it explicitly.
- Errors and dated figures flagged in this edition: Newton's bucket (1687, not 1689); "Subtle is the Lord" (1921); Born's interpretation (1926); Boltzmann's death (Duino, Austria-Hungary); the tombstone formula's notation (Planck's); Wheeler's delayed choice (1978); GRW collapse rate; top and up quark masses; H0 implied by Greene's speeds (≈80); CMB release (380,000 years); Coma's distance (inconsistent between chapters); Andromeda (2.5 million light-years); the horizon radius (46, not 14, billion light-years); acceleration onset (~6 billion years ago); the Fly's Eye particle (≈340, not 30, billion proton masses); Riess omitted from the supernova discovery.
- What the book does superbly. It makes the arrow of time genuinely comprehensible; its explanations of Bell's theorem, delayed-choice erasers and the double-headed entropic arrow remain among the clearest anywhere; and it consistently separates what physics knows from what physicists hope.
Who's who
| Person | Role | Why they matter |
|---|---|---|
| Isaac Newton | Absolute space; the bucket (1687) | Showed acceleration needs a reference; laws of motion and gravity |
| Gottfried Leibniz | Relationalism | Space is only relations; sufficient reason |
| Ernst Mach | Mach's principle (1880s) | Inertia from the distant stars; inspired Einstein |
| Albert Einstein | Special (1905) and general (1915) relativity; Λ (1917); EPR (1935) | Spacetime as entity; questioned quantum completeness |
| Hermann Minkowski | Spacetime (1908) | Unified space and time geometrically |
| Niels Bohr, Werner Heisenberg, Max Born, Erwin Schrödinger | Quantum mechanics (1920s) | Probability waves, uncertainty, the measurement framework |
| John Bell | Bell's theorem (1964) | Made non-locality testable |
| Alain Aspect, John Clauser, Anton Zeilinger, Nicolas Gisin | Bell tests; teleportation (1997) | Confirmed non-locality; Nobel 2022 (Aspect, Clauser, Zeilinger) |
| David Bohm, Hugh Everett, Ghirardi-Rimini-Weber | Pilot wave; Many Worlds; spontaneous collapse | Rival answers to the measurement problem |
| Richard Feynman | Sum over histories | All pasts contribute to the present |
| John Wheeler | Delayed choice; wormholes; "time keeps everything from happening at once" | Shaped quantum-gravity thinking |
| Marlan Scully, Kai Drühl | Quantum eraser (1982) | Later measurements decide which histories are tellable |
| Dieter Zeh, Wojciech Zurek | Decoherence | Environment hides quantum interference |
| Ludwig Boltzmann | Statistical entropy | Entropy as counting; the fluctuation puzzle |
| Roger Penrose, David Albert | Low-entropy past | The past hypothesis; gravitational entropy |
| Alexander Friedmann, Georges Lemaître, Edwin Hubble | Expanding universe (1922 to 1929) | Space stretches |
| Penzias and Wilson; COBE, WMAP, Planck teams | Microwave background (1964 onward) | Uniformity and its ripples |
| Peter Higgs and colleagues | Higgs mechanism (1964); found 2012 | Mass from a field filling space |
| Glashow, Weinberg, Salam; Georgi | Electroweak (1960s); grand unification (1974) | Symmetry hidden by cooling |
| Alan Guth, Andrei Linde, Albrecht and Steinhardt, Starobinsky | Inflation (1979 to 1983) | Repulsive gravity, structure from jitters |
| Fritz Zwicky, Vera Rubin | Dark matter (1930s, 1970s) | Most matter is unseen |
| Perlmutter, Schmidt, Riess | Accelerating expansion (1998) | Dark energy; Nobel 2011 |
| Veneziano, Nambu, Nielsen, Susskind, Schwarz, Scherk, Green | String theory (1968 to 1984) | Graviton; anomaly cancellation |
| Kaluza, Klein | Extra dimensions (1919 to 1926) | Geometry unifies forces |
| Edward Witten, Joe Polchinski | M-theory; D-branes (1995) | Second superstring revolution |
| Arkani-Hamed, Dimopoulos, Dvali | Large extra dimensions (1998) | Braneworld phenomenology |
| Steinhardt, Turok | Cyclic model (2001 to 2002) | Colliding branes as big bangs |
| Bekenstein, Hawking | Black-hole entropy and radiation (1970s) | Entropy ∝ area |
| 't Hooft, Susskind, Maldacena | Holographic principle (1993); AdS/CFT (1997) | Spacetime with gravity equals a boundary theory without |
| Kip Thorne | Wormhole time machines (1988); LIGO (Nobel 2017) | Physics of time travel; gravitational waves |
| Bennett and colleagues | Quantum teleportation (1993) | Entanglement as a resource |
| Ashtekar, Rovelli, Smolin | Loop quantum gravity | Background-independent quantum geometry |
Connect it
- Distributed systems are relativity in miniature. There is no global "now" across servers; vector clocks and causal consistency are the engineering form of light cones and frame-independent causal order.
- Event sourcing is the block universe. Store immutable events and derive state at the reader's position; different readers' "current state" can legitimately differ, as different observers' now-lists do.
- Entropy and data quality. "Disorder" in data is the default; well-ordered pipelines need continual low-entropy input (validation, curation) and export disorder elsewhere. Budget for it.
- Conditioning traps in analytics. The delayed-choice eraser is Simpson's paradox with photons: flat aggregates can hide opposite effects in segments formed only by later joins.
- Model risk and the measurement problem. A model's output is a probability distribution; any decision collapses it. Separate the model's maths from the narrative of "what it decided," as Greene separates Schrödinger's equation from collapse stories.
- AI evaluation and the arrow of time. Benchmarks degrade as models train on their test sets: information leaks one way. Keep evaluation data "low-entropy" (unseen) to preserve meaning.
- Platform capacity and holography. Throughput in microservices is bounded by interfaces (APIs, queues), not internal compute, just as a region's information is bounded by its surface.
- Roadmaps and the Chapter 14 scorecard. Separate forecasts that follow from proven capabilities from those resting on "it would be elegant if"; track both, and expect the second category to slip.
About this book
The Fabric of the Cosmos: Space, Time, and the Texture of Reality by Brian Greene, 2004. This is a chapter-by-chapter study of the book, with summaries, maps, tables and diagrams; short quotations keep their page references. It is shared free for personal study, is not for sale, and is not endorsed by the author or publisher. Please read and buy the original.
Insight Editions · rajeshrmahapatra.com/insight-editions