Insight Editions · No. 249
Time Travel Physics
Understanding the Possibilities of Time Travel
Lisa Visintainer
Relativity, black holes, wormholes and paradoxes: travel into the future is measured every day, while a trip into the past needs geometries that nature may forbid.
A study guide to Time Travel Physics by Lisa Visintainer. Shared free for personal reading; not for sale and not endorsed by the authors or publishers. Please read and buy the original.
Travel into the future is real and measured; travel into the past is allowed by general relativity only in exotic geometries (wormholes, rotating universes, cosmic strings) that need negative energy or fail under quantum effects. Only a theory of quantum gravity can give the final answer.
Special and general relativity with worked numbers, black holes, the arrows of time and spacetime, every major time-machine design from Gödel (1949) to the 2024 warp drives, Feynman's sum over histories and quantum CTC models, and the paradoxes; updated to 2026.
Einstein, Minkowski, Schwarzschild, Kerr, Gödel, Wheeler, Feynman, Hawking, Thorne, Novikov, Morris, Yurtsever, Tipler, Gott, Alcubierre, Deutsch, Lloyd, Everett, Lewis.
Beginners who want the real physics of time travel, with the formulas, the experiments and an honest map of what is proven, what is permitted and what is speculation.
The Book on One Page
Lisa Visintainer asks whether time travel is possible or pure science fiction, and answers by walking through Einstein's relativity, black holes, the nature of time, proposed time machines and the paradoxes. The answer splits in two: every clock that moves or sits lower in gravity already travels into the future, but a return trip to the past needs a closed timelike curve, and nature may forbid them.
- Speed slows time (ch 2). Einstein's postulates give the Lorentz factor γ = 1/√(1 − v²/c²). Moving clocks run slow by γ: muons survive the atmosphere, Hafele–Keating's airborne clocks lost and gained nanoseconds, GPS corrects −7.2 µs/day. Faster-than-light signals would be backward-in-time signals.
- Gravity slows time (ch 3). The equivalence principle implies that clocks lower in a gravitational field tick slower, by √(1 − 2GM/rc²); measured from Pound–Rebka (1959) to optical clocks a millimetre apart (2022). GPS gains +45.9 µs/day from it.
- Black holes are the extreme (ch 4). At the Schwarzschild radius clocks stop for distant observers and light cones tip inward; Kerr's spinning holes even contain closed timelike curves near a ring singularity, though their interiors are unstable.
- Time has an arrow and a shape (ch 5). The one-way direction comes from entropy and a low-entropy beginning; Minkowski fused space and time into spacetime, and the block universe view is the one in which the past is a place one could revisit.
- Time machines (ch 6). Fast capsules and gravity wells move travellers forward only. Wormholes with time-shifted mouths, Gödel's rotating universe, Tipler cylinders, Gott's cosmic strings and superluminal warp drives reach the past, but need exotic matter, infinite structures or unrealistic conditions.
- Quantum views (ch 7). Feynman's sum over histories includes time loops; Deutsch (1991) and Lloyd (2011) give rival quantum models of CTCs, simulated with photons and now used in metrology.
- Paradoxes and escapes (ch 8). The grandfather paradox is met by chronology protection (Hawking 1992), self-consistency (Novikov; Polchinski's billiard ball has consistent solutions) or many worlds.
- Verdict (ch 9). Future travel: proven. Past travel: unresolved, pending quantum gravity.
| Ch | Title | One-line payload |
|---|---|---|
| 1 | Introduction | Separate forward travel (proven) from backward travel (open); a time machine is a spacetime with a CTC. |
| 2 | Special Relativity | γ controls time dilation and length contraction; light cones fix causality; FTL means backward in time. |
| 3 | General Relativity | Equivalence principle; gravity slows clocks; GPS budget +38.6 µs/day; curvature can tilt light cones. |
| 4 | Black Holes | Schwarzschild radius, horizons, Hawking radiation, EHT image; Kerr ring singularity hosts CTCs. |
| 5 | Time | Entropy and the arrows of time; Minkowski spacetime; A-theory vs block universe. |
| 6 | Time Machines | Turbo and parking capsules go forward; wormholes, Gödel, Tipler, Gott, warp drives go back at a price. |
| 7 | Sum Over All Histories | Feynman paths, antiparticles as backward particles, Deutsch vs Lloyd quantum CTCs. |
| 8 | Time Paradoxes | Grandfather, bootstrap, twins; chronology protection, Novikov consistency, many worlds. |
| 9 | Conclusion | Scoreboard: future travel measured; past travel awaits quantum gravity. |
Introduction: Can We Travel Through Time?
Visintainer sets out to test a single question against real physics: is time travel possible, or does it belong to science fiction? Her route runs through Einstein's two theories of relativity, the nature of time, proposed time machines and the paradoxes they raise. The question splits at once into two very different problems: travel into the future is a measured fact of nature, while travel into the past is an open problem at the edge of quantum gravity.
Why the question is serious
The book opens with a collection of aphorisms about time and space, and one of them drives the whole inquiry. Stephen Hawking observed that nobody has ever met a visitor from the future: if time travel were possible and became cheap, some traveller should have strayed into our present. Hawking put it more sharply elsewhere: the best evidence that time travel is impossible is that we have not been "invaded by hordes of tourists from the future". Visintainer takes the remark as a challenge rather than a verdict.
Her motives are candid. A childhood astronomy book started a lifelong interest in the universe; the Back to the Future trilogy (1985–1990) supplied the specific question, because it combined humour with a physics premise; Interstellar (2014) showed that a film could be built on real general relativity. Kip Thorne, the Caltech theorist who later shared the 2017 Nobel Prize for detecting gravitational waves, was executive producer and science adviser on Interstellar, and its time-dilation planet and black hole "Gargantua" were computed from Einstein's equations rather than invented.
Hawking's party for time travellers. On 28 June 2009 Hawking hosted a champagne reception at Gonville and Caius College, Cambridge, with balloons and canapés laid out for guests. He sent the invitations, giving the date, time and exact coordinates, only after the party was over. Nobody came. Hawking called it "experimental evidence that time travel is not possible", and filmed the empty room for his 2010 documentary series. The joke carries a real point: a time traveller would need only the invitation, which still exists, to attend.
The stakes are practical as well as philosophical. If anyone could move freely through time, history could be edited, causes could follow effects, and the order of events that law, memory and science depend on would be at risk. Visintainer argues that we have never had so many technical means at our disposal, so settling whether time travel is possible matters now.
The two kinds of time travel
The single most useful step for a beginner is to separate two questions that popular culture runs together. Forward time travel means arriving in the future having aged less than the people who stayed at home. Backward time travel means arriving at an event in your own past. Physics treats them completely differently.
| Into the future | Into the past | |
|---|---|---|
| Status | Proven, measured daily | Unproven, possibly forbidden |
| Mechanism | Speed (special relativity) or gravity (general relativity) | Closed timelike curves: wormholes, rotating spacetimes, cosmic strings |
| Paradoxes | None: no one can revisit their past | Grandfather, bootstrap, causality loops |
| Record so far | Gennady Padalka, 879 days in orbit: about 22 milliseconds younger | Nothing above the scale of quantum simulation |
| Main obstacle | Energy and shielding | Exotic matter and quantum instability |
Forward time travel is not speculation. Every astronaut, every GPS satellite and every muon arriving from the upper atmosphere is a small time traveller. The hard question is the return ticket: whether the laws of physics allow a path through spacetime that loops back to its own past.
Scope and method of the book
Visintainer works as a careful reader of physicists who have written for general audiences: Hawking, Michio Kaku, Andreas Müller, Rüdiger Vaas and textbook authors on relativity. She compares their definitions and opinions to reach her own conclusion. The plan follows the logic of the question:
The route through the book
- Special relativity: speed slows clocks; time dilation, length contraction, mass–energy.
- General relativity: gravity slows clocks; the equivalence principle; curved spacetime.
- Black holes: the most extreme clocks in nature and the raw material for several time machines.
- Time: the arrows of time, entropy and Minkowski's spacetime.
- Time machines: the high-speed capsule, the parking capsule near a massive star, wormholes and Gödel's rotating universe.
- Sum over histories: Feynman's quantum picture, in which microscopic time loops appear naturally.
- Paradoxes: grandfather, twins, self-meeting, causality, and the escapes (consistency, chronology protection, many worlds).
- Conclusion: what is possible now, what is possible in principle, and what needs a theory of quantum gravity.
This edition follows that route and deepens every step: the formulas are given with worked numbers, the experiments that tested them are named, and the major time-machine designs proposed since Gödel's 1949 solution are covered, from Morris, Thorne and Yurtsever's wormholes to Gott's cosmic strings, the Alcubierre warp drive and the quantum models of Deutsch and Lloyd.
The tourist argument, tested. Suppose a future civilisation builds a wormhole time machine in the year 3000. Every known design shares one feature: you cannot travel back to before the machine was switched on, because the time shift is set by the moment the two mouths are separated. So the absence of tourists today does not rule time travel out; it rules out only machines that already existed in our past. Hawking's empty party is consistent with time machines that will be built later. The argument still bites against cheap, universal time travel, which is the version science fiction usually imagines.
What a "time machine" means in physics
A physicist's time machine is not a vehicle; it is a region of spacetime whose geometry lets an ordinary object, always moving slower than light in its own neighbourhood, return to an earlier event. The technical name for such a path is a closed timelike curve (CTC): "closed" because it returns to its starting event, "timelike" because every step along it is one a massive body could take. Every serious design in this book is an attempt to build a CTC without breaking one of three constraints:
- Einstein's field equations. The geometry must be a solution of general relativity, which ties curvature to the matter and energy present.
- Energy conditions. The matter required must be physically reasonable; most designs need negative energy density, which quantum theory allows only in small, brief doses.
- Stability. The geometry must survive the effects of quantum fields, which in most calculations pile up and destroy the time machine as it forms.
The introduction promises to decide whether time travel "is possible". No book can yet do that for the past: the answer depends on a theory of quantum gravity that does not exist in finished form. Visintainer concedes this in her conclusion. The honest framing, which this edition uses throughout, is: forward travel is established; backward travel is allowed by classical general relativity in exotic geometries, and probably blocked by quantum effects that nobody can yet compute with confidence.
The Special Theory of Relativity
Einstein's 1905 special theory of relativity ended the idea that space and time are absolute. From two postulates follow time dilation, length contraction and E = mc², all measured to high precision. A moving clock runs slow by the Lorentz factor γ, so anyone who moves fast enough and comes back has travelled into the future of those who stayed home. The same theory sets the light cone, the boundary that any time machine to the past has to break.
Before 1905: absolute space and time
Until 1905 physics ran on Newton's picture: one universal time ticking identically for everyone, and one fixed space in which things move. Two observers moving relative to each other would always agree on the duration of an event and the length of a rod. That picture works perfectly at everyday speeds, which is why relativistic effects are invisible in daily life: at highway speed the correction to time is about one part in 10¹⁵.
Visintainer separates two kinds of motion. Uniform motion is in a straight line at constant speed; accelerated motion changes speed or direction. Special relativity (STR) deals with observers in uniform motion, called inertial frames: frames in which Newton's first law (the law of inertia) holds, so a body left alone stays at rest or keeps moving in a straight line.
Einstein's two postulates
- Principle of relativity. No inertial frame is preferred. The laws of physics take the same form in all inertial frames, so no experiment done inside a closed, uniformly moving laboratory can reveal its speed.
- Constancy of the speed of light. Light in vacuum travels at c = 299,792,458 m/s for every inertial observer, whether the source, the observer or both are moving.
The second postulate is the strange one. If you chase a light beam at half the speed of light, you still measure it receding at c. For both postulates to hold, something else must give way, and what gives is the universal agreement on time intervals, lengths and simultaneity.
The Lorentz factor
Every effect in special relativity is controlled by one number, the Lorentz factor:
γ = 1 / √(1 − v²/c²)
At everyday speeds γ is indistinguishable from 1. It reaches 1.15 at half the speed of light, 2.29 at 90%, 7.09 at 99%, 22.4 at 99.9% and grows without limit as v approaches c. Because γ diverges at v = c, no massive object can reach the speed of light: the energy required becomes infinite.
- v = 0.1c (30,000 km/s): γ = 1.005, a clock loses 0.5%.
- v = 0.866c: γ = 2, one hour on board for every two at home.
- v = 0.995c: γ = 10, the factor Visintainer uses for her turbo capsule.
- v = 0.99999999c (LHC protons in Run 3 at 6.8 TeV): γ ≈ 7,250.
- Parker Solar Probe, fastest human-made object (192 km/s, 2024): γ − 1 ≈ 2 × 10⁻⁷, about 18 milliseconds lost per day.
Time dilation
The first consequence Visintainer draws is time dilation: from the point of view of an observer at rest, time passes more slowly for an observer in motion. In her notation, with t_b the time elapsed for the moving observer and t_r the time elapsed for the observer at rest:
t_b = t_r · √(1 − v²/c²) = t_r / γ
The time kept by a clock that travels with an object is its proper time (τ). Proper time between two events is always shortest along the path that moves relative to the observer, and longest for the clock that stays inertial between the same two events.
The light clock. Picture a clock made of two mirrors 1 metre apart with a photon bouncing between them; each round trip is one tick. Put the clock on a fast train. To you on the platform, the photon must travel a diagonal zigzag, because the mirrors move sideways while it is in flight. The zigzag is longer than the straight up-and-down path, and light travels at the same speed for you as for the passenger. So each tick takes longer by your watch. Pythagoras on the triangle gives exactly the factor γ. Nothing is wrong with the clock: time itself runs slower along the moving path, and every process on the train (heartbeats, chemistry, ageing) slows in step.
Muons: time dilation you can count
Cosmic-ray muons are created about 15 km up when protons from space strike the atmosphere. A muon at rest decays with a mean lifetime of 2.2 microseconds. Even at nearly light speed, 2.2 µs carries it only about 660 metres on average, so almost none should reach the ground. In fact large numbers do.
Frisch and Smith, 1963. David Frisch and James Smith counted muons at the top of Mount Washington (1,917 m) and at sea level in Cambridge, Massachusetts. Of about 563 muons per hour at the summit, roughly 27 should have survived the descent without time dilation. They counted about 408. The surplus matches a Lorentz factor of about 8.8: in the muons' own frame the descent took only about 0.7 µs. In 1977 the CERN muon storage ring circulated muons at γ ≈ 29.3 and measured their lifetime stretched to 64.4 µs, matching Einstein's prediction to about 0.1%.
From the muon's point of view the same result appears as length contraction: the atmosphere rushing towards it is shortened by γ, so it has less distance to cover. Both descriptions are correct; they are the same physics described from two frames.
Atomic clocks on aircraft and satellites
Hafele and Keating, October 1971. Joseph Hafele and Richard Keating flew four caesium-beam clocks around the world on commercial airliners, once eastward and once westward, and compared them with clocks at the US Naval Observatory. Both speed and altitude matter (the altitude part is general relativity, Chapter 3). Prediction: eastward −40 ± 23 ns, westward +275 ± 21 ns. Measured: eastward −59 ± 10 ns, westward +273 ± 7 ns. The eastward clocks lost time because they moved faster than the ground clock relative to Earth's centre; the westward clocks gained.
The Global Positioning System depends on the same correction. GPS satellites orbit at about 20,200 km altitude at 3.87 km/s. Their speed alone makes their clocks lose about 7.2 microseconds per day relative to the ground. That sounds tiny, but light travels about 2.2 km in 7.2 µs. Combined with the larger gravitational effect (Chapter 3), the uncorrected error would build up at around 10 km per day.
Every smartphone that finds its position uses special relativity. The satellites' clocks are deliberately set to tick at 10.22999999543 MHz instead of 10.23 MHz before launch, so that once in orbit they tick at the right rate as seen from the ground. Engineering accepted relativity long before most of the public did.
Length contraction
Objects moving relative to an observer are measured as shortened along the direction of motion. Visintainer's formula, with l_b the length measured by the observer for whom the object moves and l_r the object's proper length (measured at rest with it):
l_b = l_r · √(1 − v²/c²) = l_r / γ
Only the dimension along the motion contracts; widths and heights are unchanged. A 100-metre spaceship passing at 0.866c is measured as 50 metres long. The ship's crew see nothing odd about their ship; they see the outside world contracted instead. This is what makes long voyages possible within a human lifetime: at γ = 10, a star 100 light years away is only 10 light years away in the traveller's frame.
Relativistic mass is an outdated concept. Visintainer presents a "dynamic mass" that grows with speed, m_D = m₀ / √(1 − v²/c²), and explains that it means an increase in inertia, not size. The formula is arithmetically correct but most physicists stopped using it decades ago (Lev Okun campaigned against it from 1989). Modern practice keeps one invariant mass m and lets energy E = γmc² and momentum p = γmv carry the factor γ. The physical point is unchanged: pushing a body towards c requires ever more energy for ever smaller gains in speed. The phrase "mass would be infinitely large", repeated in Chapter 6, should be read as "energy would be infinite".
Mass and energy: E = mc²
Visintainer writes total energy as rest energy plus kinetic energy and arrives at E = mc², the equivalence of mass and energy. The full relations are:
- Total energy: E = γmc², which at rest reduces to E₀ = mc².
- Kinetic energy: E_kin = (γ − 1)mc², which at low speed reduces to Newton's ½mv².
- Energy–momentum relation: E² = (pc)² + (mc²)², which also covers massless light (E = pc).
The size of c² makes rest energy enormous: 1 gram of mass corresponds to 9 × 10¹³ joules, about one and a half times the energy released by the Hiroshima bomb. This number sets the price of relativistic travel in Chapter 6.
Relativity of simultaneity
Visintainer's chapter stops at E = mc², but the effect that matters most for time travel is the one she does not discuss: events that are simultaneous in one frame are not simultaneous in another. The Lorentz transformation for time makes this explicit:
t′ = γ (t − vx/c²)
The term vx/c² means that two events at different places (different x) that share the same t do not share the same t′. Which of two distant events happened "first" can depend on who is asking.
Einstein's train. Lightning strikes both ends of a moving train at the moment, by the platform's clocks, that the train's midpoint passes a woman standing on the platform. The two flashes reach her together, so for her the strikes were simultaneous. A man sitting at the train's midpoint is moving towards the front flash and away from the rear one, so the front flash reaches him first. Light moves at c in his frame too, and both flashes covered the same half-train distance, so he concludes the front strike happened first. Neither is wrong. "Now" at a distance is a convention that depends on motion.
Spacetime intervals and the light cone
Observers disagree about time and distance separately, but they all agree on one combination, the spacetime interval:
s² = c²Δt² − Δx² − Δy² − Δz²
The sign of s² sorts every pair of events into one of three classes, and this classification is the foundation of all talk about causality and time travel.
| Interval | Condition | Meaning |
|---|---|---|
| Timelike | s² > 0 | A signal slower than light can link them; all observers agree on their order. One can cause the other. |
| Lightlike (null) | s² = 0 | Linked exactly by a light ray. |
| Spacelike | s² < 0 | Too far apart for any signal; their time order depends on the observer. Neither can cause the other. |
At every event, the light rays leaving and arriving form a double cone in spacetime: the light cone. The upper half holds the event's absolute future (everything it can influence), the lower half its absolute past (everything that can influence it), and the region outside is elsewhere, causally disconnected. A body's history, its world line, always stays inside the light cone at each of its points.
Travel into the past means leaving your own light cone, or bending spacetime so that the cones themselves tip over and a world line can return to its starting event. Special relativity on flat spacetime forbids both. Every time machine in Chapter 6 is a way to tilt light cones with gravity, which only general relativity allows.
Why faster than light means backward in time
Combine the relativity of simultaneity with a signal faster than light and causality breaks. If a signal could cover a spacelike separation, there is always some inertial frame in which it arrives before it was sent. Send such a signal to a fast-moving friend who relays it back the same way, and the reply can reach you before your original message left. Richard Tolman pointed this out in 1917; it is now called the tachyonic antitelephone.
The antitelephone. Alice on Earth fires a hypothetical signal at 10c to Bob's ship receding at 0.8c. In Bob's frame the signal's emission and arrival are spacelike-separated, and the order can be reversed. Bob immediately answers at 10c in his own frame. Run the Lorentz transformation and Bob's reply reaches Alice before she sent the original. If her message was "do not send this message", there is a contradiction. The lesson: in relativity, faster than light and backward in time are the same problem. That is why no particle or signal has ever been confirmed to beat c, and why warp drives (Chapter 6) are time machines in disguise.
Velocities also do not add in the familiar way. Two speeds u and v combine as (u + v) / (1 + uv/c²), so two rockets each moving at 0.9c relative to a station, approaching each other, see each other approach at 0.994c, not 1.8c.
Special relativity has been tested at ever higher precision since Visintainer's sources were written. Lorentz invariance has survived searches for any preferred frame at levels near 10⁻¹⁸–10⁻²⁰ in some sectors (optical cavities, atomic clocks, gamma-ray burst timing from the Fermi telescope). The 2011 OPERA claim of faster-than-light neutrinos was traced to a loose fibre-optic connector and withdrawn in 2012. Time dilation itself was measured at walking-pace speeds of under 10 m/s with aluminium-ion clocks at NIST in 2010. Nothing in physics has broken the light cone.
The General Theory of Relativity
Special relativity covers only uniform motion, so Einstein spent 1907–1915 building a theory of acceleration and gravity. Its starting point, the equivalence principle, implies that light bends and that clocks run slower deeper in a gravitational field. Gravity is not a force but the curvature of spacetime, and curvature that is strong enough can tilt light cones: the only route physics offers towards a time machine to the past. Every one of these predictions, down to clocks a millimetre apart, has now been measured.
From uniform motion to gravity
STR holds only in inertial frames, yet nothing in the real universe moves without acceleration, and gravity acts everywhere. Einstein wanted a theory valid for every observer, accelerated or not. He called the insight that unlocked it, in 1907, "the happiest thought of my life": a person falling freely from a roof does not feel their own weight.
The equivalence principle in three strengths
- Weak form (Galileo, Newton). Inertial mass equals gravitational mass: the m that resists acceleration in F = ma is the same m that gravity pulls on. So all bodies fall with the same acceleration, whatever they are made of.
- Einstein's form. In a small enough region, no experiment of any kind (mechanical, optical, electrical) can distinguish a uniform gravitational field from an accelerated frame, or free fall from floating in empty space.
- Consequence. Gravity can be "switched off" locally by falling freely, so it is not a force acting in spacetime but a property of spacetime: its curvature.
Visintainer illustrates with an astronaut. A rocket accelerating at 9.81 m/s² in deep space feels exactly like standing on Earth at mid-latitude; a rocket coasting far from any star feels exactly like a rocket in free fall around one. In her source's phrase: by no experiment whatsoever can one distinguish between inertia and gravity.
The weak equivalence principle is one of the best-tested laws in physics. Loránd Eötvös confirmed it to about one part in 10⁹ around 1908. The French MICROSCOPE satellite (2016–2018, final results 2022) compared free-falling cylinders of platinum and titanium and found any difference in their accelerations to be smaller than about one part in 10¹⁵. Atom-interferometer tests now compare individual isotopes in free fall at the 10⁻¹² level.
Light bends in gravity
If gravity and acceleration are indistinguishable, light must bend in a gravitational field. Visintainer's argument: a photon enters an accelerating rocket through a side window and, seen from outside, flies in a straight horizontal line. While it crosses, the floor rushes upward, so the photon strikes the far wall lower than where it entered. Inside the rocket, the photon's path is a downward parabola. By equivalence, light must curve the same way in a gravitational field.
Full general relativity predicts that starlight grazing the Sun is deflected by 1.75 arcseconds, twice the value a Newtonian calculation gives. Arthur Eddington's eclipse expedition of 29 May 1919 confirmed it and made Einstein a world celebrity. The same bending produces gravitational lenses and Einstein rings, which reappear in Chapter 6 as a way to hunt for wormholes.
Gravity slows clocks
The prediction that matters most for time travel is gravitational time dilation. Visintainer derives it from energy. A photon sent upward from a planet's surface loses energy climbing out of the field: its wavelength stretches and its frequency drops (gravitational redshift). A photon sent downward gains energy and arrives blueshifted. Interpret each photon's crests as the ticks of the sender's clock:
- Observer A, high up, receives B's signals at a lower rate than B sends them, and concludes that B's clock, deeper in the field, runs slow.
- Observer B, low down, receives A's signals at a higher rate, and concludes that A's clock runs fast.
Both agree: the clock closer to the mass ticks more slowly. This is not an illusion of signalling; when the two clocks are brought together later, the lower one has recorded less elapsed time.
The exact relation, for a clock at rest at distance r from the centre of a non-rotating mass M, compared with a clock far away:
Δτ_near = Δt_far · √(1 − 2GM/(rc²))
For weak fields this reduces to the approximation the book prints, Δτ_near ≈ Δt_far · (1 − GM/(rc²)). For two clocks separated by a small height h near Earth's surface, the fractional rate difference is simply Δf/f ≈ gh/c² ≈ 1.1 × 10⁻¹⁶ per metre.
The book's formula has its labels reversed. It writes T_A = T_B · (1 − GM/(c²r)) while defining T_A as the time for the distant observer and T_B for the observer near the mass. Read literally, that makes the distant clock slower, the opposite of the effect described in the same paragraph. The factor (1 − GM/rc²) multiplies the far time to give the near time. The companion formula for lengths has its labels the other way round, which adds to the confusion. Use the version above.
- Earth's surface vs deep space: GM/(Rc²) ≈ 7 × 10⁻¹⁰, so a ground clock loses about 22 milliseconds per year.
- Top of Everest vs sea level (8,848 m): about 30 microseconds per year more for the climber.
- Sun's surface vs deep space: about 2 × 10⁻⁶, roughly 67 seconds per year.
- Neutron star surface: about 36% slower, a minute outside is 38 seconds on the surface (Chapter 6 table).
- Black hole event horizon: the factor reaches zero for a distant observer.
The penthouse and the basement. Twins are born on the same day. One lives her whole life in a penthouse 100 m up, the other in the basement. Gravity makes the basement clock run slow by 100 × 1.1 × 10⁻¹⁶ ≈ 10⁻¹⁴. Over 80 years (2.5 × 10⁹ seconds) the basement twin ends up about 25 microseconds younger. The effect is far too small to matter biologically and far too large to ignore for modern clocks, which can see it across a single floor of a building. Time is not one river; every height has its own rate.
How gravitational time dilation was measured
- 1959 Robert Pound and Glen Rebka fire gamma rays (iron-57, using the Mössbauer effect) up and down the 22.5 m Jefferson tower at Harvard; they measure the predicted shift of 2.5 × 10⁻¹⁵ to about 10% (published 1960; Pound and Snider reach 1% in 1964).
- 1971 Hafele–Keating flights: the altitude term contributes about +144 ns eastward and +179 ns westward, in agreement with prediction.
- 1976 Gravity Probe A: a hydrogen maser launched to 10,000 km on a rocket confirms the redshift to about 1.4 × 10⁻⁴.
- 1978 onwards GPS corrects for both effects in every satellite.
- 2010 Chou, Hume, Rosenband and Wineland at NIST compare two aluminium-ion clocks with one raised by 33 cm and see the higher one run fast by about 4 × 10⁻¹⁷.
- 2018 Two Galileo satellites stranded in eccentric orbits test the redshift to about 2.5 × 10⁻⁵, five times better than Gravity Probe A.
- 2020 Hidetoshi Katori's team compares strontium lattice clocks at the base and the 450 m observation deck of Tokyo Skytree.
- 2022 Jun Ye's group at JILA resolves the redshift across a single cloud of atoms about 1 mm tall, a fractional shift near 10⁻¹⁹.
GPS: both relativities in one budget
A GPS satellite sits higher in Earth's gravity well than a receiver, so gravity makes its clock run fast; it moves at 3.87 km/s, so special relativity makes it run slow. The gravitational effect wins.
A net drift of 38.6 µs/day multiplied by the speed of light is about 11.6 km of ranging error per day. Without Einstein's two theories, satellite navigation would be useless within hours. GPS is a working, continuously operating demonstration that time runs at different rates in different places.
Curved spacetime and Einstein's field equations
The equivalence principle shows that gravity is geometry. Einstein's task was to find the equation linking the geometry to its source. He presented it to the Prussian Academy on 25 November 1915:
G_μν + Λg_μν = (8πG/c⁴) T_μν
The left side describes the curvature of spacetime (the Einstein tensor G_μν, plus an optional cosmological-constant term Λ); the right side, the stress–energy tensor T_μν, describes the density and flow of energy, momentum and pressure. John Wheeler's summary is the one to remember: spacetime tells matter how to move; matter tells spacetime how to curve. Free bodies follow geodesics, the straightest possible paths through curved spacetime.
The equations were first confirmed by the anomalous precession of Mercury's orbit, 43 arcseconds per century that Newton could not explain. Because the equations are nonlinear, they have strange exact solutions: black holes (Chapter 4), expanding universes (Chapter 5) and spacetimes that contain closed timelike curves (Chapter 6). General relativity does not by itself forbid time travel to the past; it relates geometry to matter, and the question becomes whether any physically acceptable matter can produce the needed geometry.
"Scales shrink in a gravitational field" is a coordinate statement, not an observable one. The book argues that because light speed is constant and clocks slow down, rulers must shrink by the same factor, and gives L_A = L_B · (1 − GM/(c²r)). A local observer never sees her own ruler shrink, just as she never sees her own clock slow. What is real is that space is curved: near a mass the measured radial distance between two spheres is larger than the naive difference of their radii, by the factor 1/√(1 − 2GM/(rc²)). The comparison between "near" and "far" lengths depends on coordinates; the curvature does not.
Frame dragging: why rotation matters
A rotating mass does more than curve spacetime: it drags it around with it, the Lense–Thirring effect (1918). Near a rotating body, local inertial frames are swept in the direction of rotation. Gravity Probe B (results 2011) measured Earth's frame dragging at 37 milliarcseconds per year in the orientation of orbiting gyroscopes, with about 19% uncertainty; laser-ranged satellites such as LAGEOS and LARES confirm it at a few per cent.
Frame dragging is the hidden thread connecting several time machines: Gödel's rotating universe, van Stockum and Tipler's spinning cylinders, the interior of a spinning Kerr black hole and Ronald Mallett's circulating laser beams all try to drag spacetime so hard that light cones tip over and a circle in space becomes a loop in time.
General relativity has passed every test since the book was written. LIGO detected gravitational waves from merging black holes in 2015 (Nobel Prize 2017 to Weiss, Barish and Thorne); the Event Horizon Telescope imaged black-hole shadows in 2019 and 2022; pulsar-timing arrays reported evidence for a low-frequency gravitational-wave background in 2023. In 2025 optical clocks reached systematic uncertainties below 10⁻¹⁸ (NIST's aluminium-ion clock about 5.5 × 10⁻¹⁹; Finland's VTT MIKES strontium-ion clock 7.9 × 10⁻¹⁹). At that level, a clock senses a height change of under a centimetre, and metrologists plan to redefine the SI second around 2030, which will make gravitational time dilation a routine tool of geodesy.
Black Holes
Black holes are the extreme laboratory for time: at their boundary, the event horizon, a distant observer sees clocks stop. Visintainer explains how they form from dying stars, how they are detected, how Hawking radiation lets them evaporate, and how the Event Horizon Telescope photographed one in 2017. Inside a black hole the roles of space and time swap, and inside a spinning one, general relativity even predicts regions where time loops close on themselves.
How black holes form
A star lives by balancing gravity against the pressure of nuclear fusion in its core. When fuel runs out, stars swell into red giants and then collapse; what remains depends on the mass of the collapsing core.
- White dwarfs. Remnant cores below the Chandrasekhar limit of about 1.4 solar masses (Subrahmanyan Chandrasekhar, 1930) are held up by electron degeneracy pressure. Sirius B packs a solar mass into an Earth-sized sphere.
- Neutron stars. Heavier cores crush electrons into protons to form a ball of neutrons about 20 km across, held up by neutron degeneracy and nuclear forces. The maximum mass, the Tolman–Oppenheimer–Volkoff limit, is now estimated at about 2.2–2.3 solar masses.
- Black holes. Above that, no known pressure can stop the collapse. Typically this needs a star born with more than about 20–25 solar masses. Robert Oppenheimer and Hartland Snyder described the collapse in 1939; John Wheeler popularised the name "black hole" in 1967.
The Schwarzschild solution and the event horizon
Karl Schwarzschild found the first exact solution of Einstein's equations in December 1915, while serving as an artillery officer on the Russian front, and died of illness in May 1916. His solution describes spacetime outside any spherical, non-rotating, uncharged mass:
ds² = −(1 − r_s/r) c²dt² + dr² / (1 − r_s/r) + r²(dθ² + sin²θ dφ²)
The critical radius is the Schwarzschild radius, r_s = 2GM/c². At r = r_s the clock factor √(1 − r_s/r) reaches zero: this is the event horizon, the surface from inside which nothing, not even light, can escape.
- The Sun: r_s ≈ 2.95 km (actual radius 696,000 km).
- The Earth: r_s ≈ 8.9 mm, the size of a marble.
- A 5-solar-mass stellar black hole: r_s ≈ 15 km.
- Sagittarius A* (Milky Way centre, 4.3 million solar masses): r_s ≈ 12.7 million km, about 18 solar radii.
- M87* (6.5 billion solar masses): r_s ≈ 128 astronomical units, bigger than the orbit of Pluto.
Visintainer's image of a black hole as a funnel with vertical walls is the standard embedding diagram: it shows how the geometry of space (not spacetime) is stretched around the mass, drawn as a rubber sheet with one dimension suppressed.
At the horizon the light cones tip over. Outside, they point "up" in time with some room to move outward; at r_s the outward edge of each cone runs exactly along the horizon; inside, every future-directed path points towards smaller r. The radial coordinate becomes timelike: the singularity is not a place you can avoid but a moment in your future, as unavoidable as tomorrow. This tilting of light cones by gravity is the same move every time machine tries to make in a controlled way.
Falling in: two views of the same journey
- Seen from far away, an infalling clock ticks ever more slowly and its light reddens; it appears to freeze at the horizon and fade from view, never quite crossing.
- For the infaller, nothing special happens at the horizon. She crosses it in finite proper time and reaches the singularity soon after: for a 10-solar-mass black hole, in about a tenth of a millisecond; for Sgr A, in about a minute*.
- Tidal forces differ by size. Near a stellar black hole, the difference in pull between head and feet stretches a body apart ("spaghettification") well outside the horizon. At a supermassive hole the tides at the horizon are gentle; the danger comes later.
"Infinite density at a point" is what classical general relativity predicts, not something known. Roger Penrose's 1965 singularity theorem (Nobel Prize 2020) shows that once a horizon forms, classical general relativity breaks down inside. Most physicists expect quantum gravity to replace the singularity with something finite, but there is no agreed description. Visintainer's phrasing should be read as "general relativity stops working here".
How black holes are detected
The book lists three routes; the update since 2015 adds a fourth.
| Method | What is seen | Landmark case |
|---|---|---|
| Kinematic | Visible stars or gas orbiting an invisible mass | Star S2 orbits Sgr A* every 16 years; Genzel and Ghez, Nobel 2020 |
| Accretion disks | X-rays from gas heated to millions of degrees as it spirals in | Cygnus X-1 (discovered 1964), about 21 solar masses |
| Imaging the shadow | The dark silhouette inside a ring of lensed light | M87* (2019), Sgr A* (2022) |
| Gravitational waves | Ripples from merging black holes | GW150914 (2015), GW250114 (2025) |
The first image of a black hole
In April 2017, the Event Horizon Telescope linked radio dishes at eight observatories on six sites, from Hawaii to the South Pole, into a virtual telescope the size of Earth, observing at a wavelength of 1.3 mm. The image of M87*, 55 million light years away, was released on 10 April 2019. It shows a ring of emission about 42 microarcseconds across around a dark centre.
As Visintainer notes, the image does not show the black hole itself, but its surroundings. The bright side is brighter because gas moving towards us is boosted by the relativistic beaming effect described in Chapter 6.
The EHT imaged Sagittarius A*, our own galaxy's black hole, in May 2022. Repeat observations of M87* in 2018 and 2021 show a ring of the same size, as expected for a fixed mass, while the brightest region shifts; images released in 2025 show the ring's magnetic polarisation pattern reversing direction between years. On 14 January 2025 LIGO recorded GW250114, the clearest gravitational-wave signal yet (signal-to-noise ratio about 80), from two black holes of about 33 and 32 solar masses merging into one of about 63. It confirmed Hawking's 1971 area theorem (the total horizon area increased) and identified the ringdown "overtone" predicted for a Kerr black hole at 4.1σ significance.
Hawking radiation
In 1974 Stephen Hawking showed that quantum effects make black holes radiate. Visintainer gives the popular picture: quantum fluctuations create particle–antiparticle pairs near the horizon; occasionally one member falls in while the other escapes, carrying away energy. To conserve energy, the black hole's mass decreases, and over immense times it can evaporate completely.
The radiation has a temperature inversely proportional to the mass:
T_H = ħc³ / (8πGMk_B) ≈ 6 × 10⁻⁸ K × (M_☉ / M)
- A solar-mass black hole: about 60 nanokelvin, far colder than the 2.7 K cosmic microwave background, so it currently absorbs more than it emits.
- Evaporation time for a solar-mass hole: about 10⁶⁷ years, vastly older than the universe (1.38 × 10¹⁰ years).
- M87*: a temperature near 10⁻¹⁷ K, lifetime around 10⁹⁶ years.
- A primordial black hole of about 5 × 10¹¹ kg (a mountain's mass), smaller than a proton, formed in the big bang, would be finishing its evaporation now in a final burst of gamma rays; none has been seen.
Two points need correction. First, Visintainer lists Hawking radiation as a way to detect black holes; for any black hole formed by a star it is far too faint ever to observe, and it has never been seen from a real black hole. Analogue versions have been measured: Jeff Steinhauer observed spontaneous "Hawking" phonons at a sonic horizon in a Bose–Einstein condensate (2016, 2019). Second, the electron–positron pair is the wrong example: a stellar-mass hole is far too cold to emit massive particles and radiates almost entirely photons and gravitons. Hawking himself called the pair story a heuristic; the actual calculation concerns how quantum fields are defined on either side of the horizon.
Spinning black holes: Kerr and the ring singularity
Real black holes rotate, because the stars that formed them did. In 1963 the New Zealand mathematician Roy Kerr found the exact solution for a rotating black hole, specified by its mass M and its spin parameter a = J/(Mc), where J is the angular momentum. The structure is richer than Schwarzschild's. The table compares the four classic black-hole solutions; the list below it gives Kerr's distinctive features.
| Black hole | Parameters | Singularity | Time-travel relevance |
|---|---|---|---|
| Schwarzschild (1916) | Mass | Point (spacelike moment) | None; only forward travel by hovering |
| Reissner–Nordström (1916–18) | Mass, charge | Timelike point | Inner horizon leads to other universes on paper |
| Kerr (1963) | Mass, spin | Ring | CTCs near the ring (Carter 1968) |
| Kerr–Newman (1965) | Mass, spin, charge | Ring | Same as Kerr |
- Two horizons. An outer event horizon and an inner Cauchy horizon, at r± = GM/c² ± √((GM/c²)² − a²). They merge when the spin reaches its maximum, a = GM/c² (an "extremal" black hole).
- Ergosphere. Outside the outer horizon is a region where frame dragging is so strong that nothing can stay still; everything must co-rotate. Penrose (1969) showed energy can be extracted from it.
- Ring singularity. The singularity is not a point but a ring of radius a in the equatorial plane. A path through the middle of the ring avoids it.
- Closed timelike curves. Brandon Carter (1968) showed that just inside the ring, in a region where the radial coordinate is negative, the rotational direction becomes timelike: circles around the axis are closed timelike curves. A traveller there could in principle return to her own past.
The Kerr interior is almost certainly not usable. The exact solution describes an eternal black hole in an empty universe. In a real collapse, radiation falling in piles up at the inner horizon and is infinitely blueshifted (mass inflation, Poisson and Israel, 1990), turning the Cauchy horizon into a singular barrier. The passage through the ring and the CTC region beyond are features of an idealised mathematical solution, and no one expects them to survive in nature.
Miller's planet. In Interstellar the crew land on a planet orbiting Gargantua, a black hole of 100 million solar masses spinning within about one part in 10¹⁴ of the maximum. Each hour on the planet costs seven years on the mother ship, a time-dilation factor of about 61,000. Kip Thorne checked that this is only possible because extreme spin lets a stable orbit sit almost on the horizon. The crew get exactly what general relativity promises: a one-way journey into the future, paid for by being stuck deep in the well. No amount of gravity sends anyone back; it only slows the clock of the one who stays down.
Time
Before asking whether we can travel through time, Visintainer asks what time is. Everyday time seems absolute, flowing uniformly and beyond our control; relativity shows it is woven with space into a single spacetime, and thermodynamics shows its one-way character comes from entropy. The laws of motion run equally well forwards and backwards; the arrow of time comes from the universe having started in an extraordinarily ordered state. Whether the past still "exists" to be visited turns out to be a question about which picture of time is right.
Absolute time and clock time
In daily life time seems to have no influence of its own: we cannot stop it, speed it up or slow it down, and it seems to pass at the same rate for everyone. This is Newton's absolute time, defined in the Principia (1687) as time that flows equably "without relation to anything external". Visintainer adds a distinction that matters later: time is not a substance, but it is measured by clocks, which are material objects. Space and time also feel utterly different: we move freely in three spatial directions, but only one way in time.
Relativity destroyed the first half of Newton's picture. Chapters 2 and 3 showed that clocks moving relative to each other, or at different heights, accumulate different amounts of time between the same two events. There is no single cosmic clock, only the proper time along each world line. What survives is the second half: time still has a direction, and that direction needs explaining.
The arrows of time
Space has no preferred direction: left and right are interchangeable. Time does: we remember the past and not the future, eggs break but do not unbreak, and the universe expands. Visintainer calls this one-way property the arrow of time and explains it through the second law of thermodynamics.
Entropy and the thermodynamic arrow
Entropy measures disorder, or more precisely, how many microscopic arrangements are consistent with what we see at large scale. Ludwig Boltzmann's formula, carved on his tombstone in Vienna, defines it:
S = k_B ln W
where W is the number of microstates and k_B = 1.38 × 10⁻²³ J/K is Boltzmann's constant. The second law says that in an isolated system entropy never decreases: it stays constant or, almost always, increases.
The cup on the table. A cup standing on a table is a highly ordered, low-entropy arrangement: there are few ways for the pieces to be arranged as a whole cup. It falls and shatters into a high-entropy state: there are vastly more ways to be scattered fragments. The fragments do not spontaneously reassemble. You can glue the cup back together, but the gluing uses energy, which ends up as waste heat; the total entropy of cup, glue, hands and room still goes up. The direction in which entropy rises is the direction we call the future.
The second law is statistical, not absolute. Nothing in the laws of motion forbids the fragments leaping back onto the table; it is just overwhelmingly improbable, because disordered arrangements outnumber ordered ones by factors like 10^(10²³). Boltzmann saw this in the 1870s. It means the arrow of time is not written into the laws themselves but into the starting conditions of the universe.
The film run backwards. Film two billiard balls colliding and play it in reverse: nobody can tell which version is real, because Newton's laws work equally well both ways. Now film the break shot that scatters fifteen racked balls, and reverse it: everyone sees at once that it is backwards. The individual collisions are just as reversible; what gives the game away is that the reversed film starts from a scattered state and ends in perfect order. Josef Loschmidt made this point against Boltzmann in 1876 (the reversibility objection): from time-symmetric laws you cannot derive a time-asymmetric law without a time-asymmetric assumption.
The three arrows Visintainer names
Following Hawking, the book identifies three arrows and argues they all point the same way.
- The thermodynamic arrow. The direction in which entropy increases.
- The psychological arrow. The direction in which we feel time pass: we remember the past, not the future.
- The cosmological arrow. The direction in which the universe expands.
Hawking's argument that the psychological arrow is the thermodynamic arrow uses a computer memory, which works like an abacus: each element can sit in one of two states. Before recording, the memory is in a disordered state; to record, it must be put into a definite ordered state, and doing so requires energy, which ends up as heat. The heat raises the entropy of the surroundings by more than the memory's order lowered it. So any recording device, brain or computer, can only lay down memories in the direction in which total entropy rises. We remember the past because the past is the direction of lower entropy.
Hawking's memory argument has a precise modern form: Landauer's principle (Rolf Landauer, IBM, 1961). Erasing one bit of information must dissipate at least k_B T ln 2 of heat, about 2.9 × 10⁻²¹ J at room temperature. Bérut and colleagues measured this bound directly in 2012 with a single colloidal bead in a laser trap; it has since been confirmed in nanomagnetic and quantum (trapped-ion and superconducting) bits. Information is physical, and keeping records costs entropy.
Why the universe began in order
The cosmological arrow is where Visintainer gets most interesting. Because the universe began in a big bang singularity, where the known laws fail, nobody can say with certainty how it began. But it almost certainly began in a state of very low entropy. If it had started disordered, entropy could not increase much and there would be no clear thermodynamic arrow at all. Philosopher David Albert calls this assumption the Past Hypothesis (2000).
- Present entropy of the observable universe: about 10¹⁰⁴ k_B, dominated by supermassive black holes.
- Maximum possible entropy (all matter in one black hole): about 10¹²³ k_B.
- Roger Penrose's estimate of the odds of a starting state as special as ours by chance: about 1 in 10^(10¹²³).
- Cosmic microwave background temperature variations: about 1 part in 100,000, showing how smooth (low gravitational entropy) the early universe was.
Two of the book's claims need care. It says the thermodynamic arrow "is transformed into" the cosmological arrow because entropy rises in the universe too; the two are different arrows (one is disorder, the other is expansion), and their alignment is a fact to be explained, not a definition. It also repeats the idea that if entropy were to decrease, the cosmological arrow would point opposite to the thermodynamic one. Hawking once argued (1985) that in a recollapsing universe the thermodynamic arrow would reverse during contraction; Don Page and his student Raymond Laflamme showed the argument was wrong, and Hawking later called it his "greatest mistake". Since 1998 we also know the expansion is accelerating, so a recollapse looks unlikely in any case.
More arrows than three
Physicists now list several further asymmetries, each with its own status.
| Arrow | What runs one way | Source of asymmetry |
|---|---|---|
| Thermodynamic | Entropy increases | Low-entropy initial state |
| Psychological | Memory of past only | Thermodynamic (records cost entropy) |
| Cosmological | Universe expands | Initial conditions of the big bang |
| Radiative | Waves spread out from sources, never converge on them | Thermodynamic and initial conditions |
| Quantum measurement | Outcomes become definite, records form | Decoherence, thermodynamic |
| Weak interaction | Kaon and B-meson decays differ under time reversal | Genuine law-level T-violation (CPLEAR 1998, BaBar 2012), far too small to explain everyday arrows |
| Black holes | Matter falls in, never out | Initial conditions (white holes not formed) |
For time travel this matters directly. A world with closed timelike curves must be globally consistent: entropy cannot rise all the way around a loop and arrive back at a lower value. Time machines therefore clash with the thermodynamic arrow, and Chapter 8's consistency paradoxes are, among other things, conflicts with the second law.
Spacetime
Minkowski's union of space and time
Visintainer credits Einstein with discovering the link between space and time, and notes that Hermann Minkowski, Einstein's former mathematics teacher in Zurich, gave it its geometric form in a lecture at Cologne on 21 September 1908. His most quoted sentence, in the standard translation: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality." He called the union the world; we call it spacetime.
Two slips in the book's account of spacetime. The quotation rendered as "From 'hour' on, space for itself and time for itself shall sink completely into shadows" is a word-for-word translation of the German "Von Stund an", an idiom meaning "henceforth", not a reference to an hour. And the claim that special relativity uses "a two-dimensional plane like a chessboard" is wrong: the spacetime of special relativity is four-dimensional and flat (Minkowski spacetime). It is often drawn with two dimensions suppressed, as in the light-cone diagram of Chapter 2, which is probably the source of the confusion. In general relativity the same four-dimensional spacetime is curved by mass and energy.
In Minkowski spacetime every event has four coordinates (three of space, one of time) and the separation between events is given by the interval s² = c²Δt² − Δx² − Δy² − Δz². The minus signs are what make time different from space: they create the light-cone structure and fix the distinction between timelike and spacelike directions.
Solutions of Einstein's equations are spacetimes
Every solution of the field equations (published November 1915) is an entire spacetime: a geometry with its matter content. Visintainer names the two most important:
- The Schwarzschild solution (1916). The spacetime around a spherical, non-rotating, uncharged mass: planets, stars and the simplest black holes (Chapter 4).
- The Friedmann universe. Alexander Friedmann (1922) and Georges Lemaître (1927) found expanding solutions describing a universe uniformly filled with matter; with later work by Robertson and Walker, this FLRW spacetime is the basis of big bang cosmology. It defines a natural cosmic time, the time kept by observers at rest relative to the average matter, which reads about 13.8 billion years since the big bang.
Other solutions are far stranger. Kurt Gödel's 1949 rotating universe and the wormhole spacetimes of Chapter 6 are also solutions, and in them the light cones tilt so far that world lines can loop back in time.
What time is: the philosophers' map
The book stops at physics, but the question "is time travel possible?" depends on which philosophical picture of time is right. John McTaggart's 1908 paper "The Unreality of Time" set the terms still used.
| A-theory (tensed) | B-theory (tenseless) | |
|---|---|---|
| Basic facts | Events are past, present or future, and these change | Events are earlier than, simultaneous with, later than each other; these never change |
| The present | Objectively special | As special as "here": a perspective |
| Typical metaphysics | Presentism (only the present exists) or growing block (past and present exist) | Eternalism: the block universe, all times equally real |
| Fit with relativity | Strained: whose "now"? | Natural |
| Time travel to the past | Hard to define: there is no past to go to | Coherent: the past exists; a CTC is just a shape |
The argument from relativity, made by Wim Rietdijk (1966) and Hilary Putnam (1967) and dramatised by Roger Penrose as the Andromeda paradox, runs:
- Events simultaneous with my present are real (now).
- Two observers passing each other on Earth, walking in opposite directions, have different planes of simultaneity; on Andromeda, 2.5 million light years away, their "nows" differ by days.
- If reality is "what is simultaneous with me", and both observers are equally real, then events that are future for one are present for the other.
- Therefore future events are already real: the block universe.
The middle view, the growing block (C. D. Broad, 1923), gives a curious result: the past and present exist but the future does not yet. On that picture travel to the past has a destination, while a traveller jumping forward would be heading for events that have not yet come into being, and could only "arrive" by waiting for them. Special relativity's forward travel fits it anyway, because the traveller never skips events; she only experiences fewer of her own seconds on the way.
For us believing physicists, the distinction between past, present and future is only a stubbornly persistent illusion.Albert Einstein, 1955
Where would you arrive? Imagine stepping into a machine set for "yesterday, noon". On presentism, yesterday no longer exists: the machine has no destination, and "travelling there" means at most rebuilding a copy of yesterday's world, not visiting it. On the block view, yesterday noon is still there, as real as any other spacetime point; a time machine is simply a world line shaped so that it reaches that region again. Physics' time machines are block-universe objects. Every one of them is a geometry in which a world line meets its own past, which only makes sense if the past is a place in the four-dimensional whole.
The cosmological arrow's long-term future is being revised. Results from the Dark Energy Spectroscopic Instrument (DESI) in 2024 and 2025, combined with supernova and microwave-background data, show a 2.8–4.2σ preference for dark energy that weakens over time rather than staying constant. If confirmed, the universe's ultimate fate (eternal expansion or eventual recollapse) is open again. The thermodynamic arrow is not affected: entropy keeps rising either way.
Time Machines
This is the heart of the book: concrete designs for machines that move people through time. Visintainer examines three, after Andreas Müller and Rüdiger Vaas: a turbo capsule flying near light speed, a parking capsule resting near a neutron star or black hole, and the wormhole. The first two are physically sound but carry travellers only forwards, at absurd cost. Only geometries containing closed timelike curves (wormholes, Gödel's rotating universe, spinning cylinders, cosmic strings and warp drives) can reach the past, and every one of them needs either negative energy, infinite structures or conditions that quantum effects seem to destroy.
Physicists have proposed many designs. Some were dropped because they contradicted Einstein's equations; others satisfy the equations but demand matter that may not exist. Michio Kaku sorts them by how far they are from being built. The useful dividing line is the one from Chapter 1: designs that slow the traveller's clock (forward only) versus designs that close a loop in spacetime (backward possible).
The special-relativistic turbo capsule
The simplest time machine is a fast spaceship. By the time dilation of Chapter 2, a capsule moving at v ≈ 0.995c has a Lorentz factor of about 10. Keep it at that speed for one hour of Earth time and only six minutes pass on board. The traveller steps out having aged 6 minutes while the world aged 60: a jump of 54 minutes into the future.
The one-g starship. Instead of an instant leap to 0.995c, accelerate at a comfortable 1 g (9.81 m/s²) all the way, flip halfway and decelerate at 1 g. Because the crew's proper acceleration is constant, their speed approaches c exponentially in rapidity. The results are startling: Alpha Centauri (4.4 light years) takes about 3.6 years of ship time; the galactic centre (about 26,000 light years) takes about 20 years of ship time; the Andromeda galaxy (2.5 million light years) about 29 years. Return from Andromeda and the crew have aged under 60 years, while five million years have passed on Earth. Special relativity offers a one-way ticket to the deep future to anyone who can pay for the fuel.
The bill: energy
The fuel is the problem. Visintainer notes that doubling a speed quadruples the kinetic energy; at relativistic speeds the cost is worse, because kinetic energy is (γ − 1)mc² and γ grows without limit.
- Kinetic energy of a 1-tonne capsule at γ = 10: (10 − 1) × 1,000 kg × c² ≈ 8 × 10²⁰ J.
- World primary energy use (2023): about 6 × 10²⁰ J per year, so the capsule needs more than a year of all human energy, delivered with perfect efficiency.
- Ideal photon rocket (exhaust at c): mass ratio e^(arccosh γ) ≈ 20 just to reach γ = 10, 400 to reach it and stop, 160,000 for a round trip with braking at both ends.
- A chemical rocket cannot do it at all: the required mass ratio exceeds the number of atoms in the universe.
Visintainer cites the Large Hadron Collider at CERN as evidence that relativistic speeds are reachable. Its 27 km ring uses superconducting dipole magnets of about 8.3 tesla to keep protons, accelerated to 6.8 TeV, on track at γ ≈ 7,250, or 99.999999% of c. The total energy stored in each full beam is about 360 megajoules, roughly the kinetic energy of a high-speed train.
Three loose claims in the turbo-capsule section. (1) The LHC is not "a very big step in the right direction" for time travel: it accelerates particles of about 10⁻²⁷ kg, while a crewed capsule is some 10³⁰ times heavier. (2) The fastest human-made objects are no longer "just above Earth's escape velocity, about 40,000 km/h" (that is the speed to leave Earth, 11.2 km/s). New Horizons left Earth at about 58,000 km/h in 2006, and NASA's Parker Solar Probe reached about 690,000 km/h (192 km/s) at perihelion on 24 December 2024, still only 0.064% of light speed. (3) A circular, accelerator-style track does not make a relativistic capsule safer: holding a mass on a circle at near light speed needs a centripetal force multiplied by γ, and the crew would feel it as crushing acceleration.
The bill: safety
Speed also turns the near-empty space between stars into a weapon.
- Interstellar gas. Space contains roughly one hydrogen atom per cubic centimetre. At γ = 10 each proton strikes with about 8.4 GeV, and the flux on the capsule's nose carries about 400 kilowatts per square metre of hard radiation, lethal within minutes without massive shielding.
- Dust. A one-microgram grain at γ = 10 carries about 8 × 10⁸ J, the energy of 190 kg of TNT.
- Doppler blueshift. Light from ahead arrives multiplied in frequency by √((1 + β)/(1 − β)), about 20 at γ = 10. The 2.7 K microwave background ahead becomes a 54 K glow; ordinary starlight is pushed into the extreme ultraviolet. At still higher speeds, radio and visible light become X-rays and gamma rays, which is the danger the book describes.
- Steering. At these speeds a navigation error cannot be corrected in time, and any manoeuvre requires enormous transverse forces.
Relativistic aberration: the starbow
Aberration follows from the velocity-addition rule: a ray arriving at angle θ in the rest frame appears at θ′ with cos θ′ = (cos θ + β)/(1 + β cos θ). At γ = 10, a star that was exactly abeam (90°) appears only 5.7° from the direction of travel. The whole forward hemisphere of the sky is compressed into a small, intensely bright, blueshifted disc: the beaming effect. The same effect makes the approaching side of the M87* ring in Chapter 4 brighter.
Visintainer's verdict on the turbo capsule is that technology, safety and control make it impossible today. The physics is sound and has been proven on particles, muons and atomic clocks. The barrier is engineering on an astronomical scale, not a law of nature. But even a perfect turbo capsule only goes forward.
The general-relativistic parking capsule
The second design uses gravity instead of speed. Fly to a very massive, compact object, stay close to it for a while, then leave. Because clocks run slower deep in a gravitational field (Chapter 3), the traveller returns to a world that has aged more than she has.
How much does each object slow time?
Müller's table, reproduced by Visintainer, gives the time that passes on the surface of various bodies while 60 seconds pass far away, using the factor √(1 − 2GM/(rc²)).
| Object | Mass (kg) | Radius | Clock factor | Seconds per far-away minute |
|---|---|---|---|---|
| Earth | 6 × 10²⁴ | 6,367 km | ≈ 1 − 7 × 10⁻¹⁰ | 59.99999996 |
| Jupiter | 2 × 10²⁷ | 70,000 km | 0.99999998 | 59.9999988 |
| Sun | 2 × 10³⁰ | 700,000 km | 0.999998 | 59.99987 |
| White dwarf | 2 × 10³⁰ | 5,000 km | 0.9997 | 59.9823 |
| Neutron star | 4 × 10³⁰ | 10 km | 0.64 | 38.39 |
| Black hole (horizon) | 10 × 10³⁰ | 15 km | 0 | 0 |
The Earth is useless as a time machine: per minute, a surface clock falls behind by 4 × 10⁻⁸ s. Only a neutron star or a black hole gives a noticeable effect.
The book's neutron-star arithmetic is off. It says a year on a neutron-star surface sends the traveller one year into the future because "twice as much time has passed" far away. With the table's own factor of 0.64, one year on the surface corresponds to 1 / 0.64 ≈ 1.56 years outside, a gain of about 7 months, not a year. Also, the "black hole" row lists a 15 km surface radius: a black hole has no surface, and 15 km is its Schwarzschild radius, where no one can park.
Getting there, landing and leaving
The parking capsule inherits every problem of the turbo capsule and adds new ones:
- Distance. The nearest known neutron stars, such as RX J1856.5−3754, are about 400 light years away; the nearest known black hole, Gaia BH1 (found in 2022), is about 1,560 light years away. Getting there at sub-relativistic speeds takes millennia; at relativistic speeds the turbo-capsule hazards return.
- Interstellar craft. The British Interplanetary Society's Project Daedalus (1973–1978) designed a two-stage fusion probe to reach Barnard's Star (5.9 light years) at 12% of light speed in about 50 years. It was an uncrewed flyby and was never built; the technical and financial hurdles were too great.
- Gravity. Surface gravity on a neutron star is about 10¹¹ g. A landing is a crash, and any body on the surface is flattened to a film atoms thick.
- Radiation and fields. Neutron stars bathe their surroundings in X-rays, and their magnetic fields reach 10⁸ tesla (magnetars 10¹¹ T), enough to distort atoms.
- Escape. Leaving requires the escape velocity, about 0.75c for the table's neutron star, which brings back the turbo-capsule problem.
For black holes the difficulties are even worse, because there is no solid surface to park on. A traveller must either hover with rockets or orbit.
Hovering near the horizon. Hold position at 1.0001 r_s above a black hole, and your clock runs 100 times slower than a distant one: ten days for you is three years outside. But hovering means resisting gravity with rockets, and the required thrust grows without limit at the horizon. Above Sgr A* (4.3 million solar masses) it is about 36 million g; above the far heavier M87* it is still about 24,000 g. Orbiting instead costs nothing, but for a non-rotating hole the innermost stable orbit, at 3 r_s, slows clocks by only a factor of 1.41. Large factors need an orbit close to a near-maximally spinning Kerr hole, as in Interstellar. Nature charges for time dilation in acceleration or in rarity.
Visintainer's conclusion: parking on a massive object works in thought experiments, but practical feasibility is very far off. Like the turbo capsule, it moves travellers only into the future.
Wormholes
From bridges to wormholes
A wormhole is a tunnel in spacetime connecting two distant regions, or two different times. The name is misleading in one way, as Visintainer notes: the tunnel is not empty, it is made of space itself.
- 1916 Ludwig Flamm notices that the Schwarzschild solution can be drawn as a tunnel joining two sheets.
- 1935 Einstein and Rosen describe the "bridge" (now the Einstein–Rosen bridge) as a model of elementary particles, not as a route for travel.
- 1955 John Wheeler draws the first wormhole sketch in his paper on "geons" and proposes that spacetime at the smallest scales is a foam of tiny wormholes.
- 1957 Charles Misner and Wheeler coin the word "wormhole", after a worm eating a shortcut through an apple.
- 1962 Robert Fuller and Wheeler prove the Einstein–Rosen bridge pinches off too fast for even light to cross.
- 1985 Carl Sagan asks Kip Thorne for a physically plausible shortcut for his novel Contact.
- 1988 Michael Morris and Thorne publish the rules for traversable wormholes; Morris, Thorne and Ulvi Yurtsever show how to turn one into a time machine.
White holes
Visintainer suggests a wormhole could have a black hole as its entrance and a white hole as its exit. A white hole is the time reverse of a black hole: nothing can enter it, and matter can only come out. Reversing the direction of time in the Schwarzschild solution yields one. Some physicists have speculated that the big bang was a kind of white hole. None has been observed, and calculations (Douglas Eardley, 1974) show they would be violently unstable, swallowing incoming radiation and collapsing into black holes.
Morris and Thorne's requirements
Thorne and Morris asked what a wormhole would need to be traversable by a human being. Visintainer lists their conditions.
Requirements for a traversable wormhole (Morris and Thorne, 1988)
- Static. It must not move or evolve on the timescale of the trip.
- Stable. Small disturbances, including the traveller, must not collapse it.
- No event horizon. Otherwise the traveller is trapped, as in a black hole.
- Bearable tidal forces. Gravitational stretching must stay within about 1 g across a human body.
- Reasonable crossing time. Days or less, measured by both traveller and outside observers.
- Finite, physically possible matter. The geometry must not require infinite amounts of energy or matter.
The simplest solution satisfying these conditions has an hourglass shape. Its spatial geometry can be written dl² + (b₀² + l²)(dθ² + sin²θ dφ²), where l is proper distance through the tunnel and b₀ is the radius of the narrowest point, the throat (Visintainer's "gullet"). Far from the throat, on either side, space is flat.
Exotic matter
The catch is in requirement 6. Light rays entering a wormhole converge towards the throat and must diverge again on the other side. Ordinary matter and energy always focus light (gravity attracts). Making light defocus requires matter whose energy density, as measured by some observers, is negative: exotic matter, which violates the null energy condition (ρ + p ≥ 0 for every light ray). Visintainer illustrates with surfaces: ordinary matter curves space like a sphere (positive curvature), while a wormhole throat needs the saddle shape of negative curvature.
Topological censorship (Friedman, Schleich and Witt, 1993) makes this a theorem: in general relativity, any wormhole that can be traversed from one outside region to another requires a violation of the averaged null energy condition. No exotic matter, no traversable wormhole. Everything therefore hinges on whether negative energy exists in the needed amount.
The Casimir effect: negative energy is real, but tiny
Negative energy is not pure fantasy. In 1948 the Dutch physicist Hendrik Casimir predicted that two uncharged, parallel metal plates in vacuum would attract each other. Quantum field theory fills empty space with fluctuating fields, pictured as virtual particle–antiparticle pairs that appear, separate and annihilate again.
Between the plates, only field modes whose wavelengths fit the gap are allowed; outside, all modes exist. The vacuum between the plates therefore has less energy than empty space outside, i.e. negative energy density, and the outside vacuum pushes the plates together.
- Casimir energy density: u = −π²ħc / (720 a⁴); at a gap of 1 µm, about −4 × 10⁻⁴ J/m³.
- Casimir pressure: P = −π²ħc / (240 a⁴), about 1.3 mPa at 1 µm, rising steeply as the gap shrinks.
- First precise measurement: Steve Lamoreaux, 1997, plate gaps of 0.6–6 µm, agreement within about 5%; Mohideen and Roy (1998) reached about 1%.
- Needed at a 1 m wormhole throat: a tension of about c⁴/(8πG b₀²) ≈ 5 × 10⁴² pascals, some 46 orders of magnitude beyond the Casimir effect at 1 µm.
- Total exotic mass for a 1 m throat (Matt Visser's estimate): about minus one Jupiter mass.
Quantum inequalities limit negative energy. Larry Ford and Thomas Roman showed (1995–1996) that quantum fields permit negative energy only in a trade-off: the larger the negative energy, the shorter it can last and the more it must be followed by compensating positive energy. Applied to wormholes, the exotic matter would have to be confined to a band only a few Planck lengths thick around a throat that could be macroscopic. The Casimir effect proves negative energy densities exist; it does not show they can be gathered in the amounts and shapes a wormhole needs. The book's statement that Lamoreaux's measurement shows "this must be negative energy" is correct, but its suggestion that this brings wormholes closer skips this problem.
Cosmic strings are mixed up with superstrings. The book says wormholes created after the big bang could have been stabilised by cosmic strings with negative mass, and then illustrates strings with a picture of an open and a closed string sweeping out world-sheets, which is the superstring of string theory, a fundamental object about 10⁻³⁵ m long. Cosmic strings are a different idea: macroscopic line-like defects in fields, possibly light years long, left over from phase transitions in the early universe (Tom Kibble, 1976). Ordinary cosmic strings have positive mass and tension; negative-mass versions are a speculative extension. None has been detected.
Hawking radiation as "microscopic time travel"
The book takes a detour through Hawking's reading of virtual pairs. In Feynman's picture (Chapter 7) an antiparticle moving forward in time is mathematically equivalent to a particle moving backward in time, so a virtual pair can be redrawn as one particle travelling in a closed loop in time. Near a black hole, one member of a pair can fall in while the other escapes; equivalently, a single particle comes out of the black hole by travelling backward in time inside it, then forward outside.
Hawking concluded that black-hole radiation suggests quantum theory allows microscopic movement into the past, and that it is in principle observable.
This is an interpretation, not an observation. The Feynman–Stückelberg picture of antiparticles as backward-moving particles is a mathematically equivalent bookkeeping device; it makes no prediction different from the ordinary forward-in-time description, and no message or information can be sent to the past with it. Hawking radiation has also never been detected from a real black hole (Chapter 4). The claim illustrates how naturally time loops appear in quantum theory, not that past-directed travel has been seen.
Finding or building a wormhole
Visintainer, following Vaas, describes two routes.
- The quantum route. If Wheeler's spacetime foam exists, Planck-scale wormholes (about 10⁻³⁵ m) appear and vanish constantly. One might catch one and inflate it, as cosmic inflation stretched the universe by a factor of at least 10²⁶ in a tiny fraction of a second after the big bang (Thomas Roman studied "inflating Lorentzian wormholes" in 1993). The difficulty is stopping the inflation at the right size, and doing all this needs a theory of quantum gravity that we do not have.
- The classical route. Deform spacetime like plasticine, cut it, and glue the edges to form a handle, like turning a ball of clay into a ring. Robert Geroch proved in 1967 that changing the topology of space in this way, inside a bounded region, is impossible without either closed timelike curves or singularities. So the "classical" route produces a time machine or a naked singularity (a singularity not hidden behind a horizon) as a by-product, which is the problem Visintainer flags.
The coffee mug and the doughnut. Topologists say a mug and a doughnut are "the same" because one can be smoothly deformed into the other; but a ball cannot become a doughnut without punching a hole. Geroch's theorem says spacetime behaves like the clay: to punch a wormhole into space without tearing, time itself must be twisted into loops along the way. Building a wormhole classically is already a form of time travel. That is why many physicists suspect that if wormholes exist at all, they were made by quantum processes or left over from the big bang.
How a wormhole would look: negative-mass lensing
A wormhole could be found by its effect on light. Normal mass converges light like a magnifying lens; the negative energy at a wormhole's throat makes it act like a diverging lens. The calculation, which Visintainer attributes to Igor Novikov, was published in detail by John Cramer, Robert Forward, Michael Morris, Matt Visser, Gregory Benford and Geoffrey Landis in 1995. A star passing behind an ordinary lens brightens once, smoothly. Passing behind a negative-mass lens produces caustics: the star brightens sharply, dims below normal, then brightens sharply again. That signature has been searched for in microlensing surveys and in galaxy catalogues; nothing has been found.
Seeing the past through a shortcut
Visintainer's first wormhole time-travel scenario uses two light beams. One travels through the wormhole; the other takes the long way around. The beam through the shortcut arrives first, and a traveller riding with it could look back and see the slower beam still on its way, perhaps even see herself preparing for the journey.
Seeing an old image is not travelling to the past. Looking at light that left a scene earlier is what every astronomer does when observing a star; it shows what happened, but you cannot intervene. A shortcut wormhole lets you outrun light travelling around it, which is faster-than-light travel in a global sense; Chapter 2 showed that this is what opens the door to backward time travel. It becomes a genuine time machine only when the wormhole's two mouths are given a time shift relative to each other, as in the next scenario.
The Morris–Thorne–Yurtsever time machine
The decisive idea, published in Physical Review Letters in 1988, applies the twin paradox to the wormhole's mouths.
- Start with a short wormhole whose two mouths, A and B, sit side by side on Earth. Through the tunnel, the clocks at both mouths agree.
- Load mouth B onto a spaceship and take it on a fast round trip, say at γ = 10 for 10 years of Earth time. When B returns, only one year has passed for it; through the tunnel, B is still connected to A as it was one year after departure.
- Now the mouths, side by side again, show a 9-year time shift. Step into B in the year 2040 and walk out of A in 2031. Walk back through A and you exit B nine years in the future.
The book's illustration shows exactly this: enter the mouth on Earth at 12:00, leave the mouth in the returned spaceship at 10:00.
Valery Frolov and Igor Novikov (1990) showed the same can be done with gravity: park one mouth near a neutron star for a while and its clock falls behind.
Two limits follow directly from the construction. You can never travel back to before the time shift existed: the earliest exit is the moment the machine was completed. And the machine only "switches on" when the time shift exceeds the time light needs to travel between the mouths through ordinary space. At that moment the first closed light ray forms, the boundary of the time-travel region, called the chronology horizon (a kind of Cauchy horizon). This is where, according to Hawking, physics fights back (Chapter 8).
Feedback at the chronology horizon. In 1991 Sung-Won Kim and Kip Thorne calculated what happens to vacuum fluctuations as a wormhole time machine forms. Light leaving mouth A, travelling to B through space and back through the tunnel, can arrive at A at the moment it left, and so circulate endlessly, each loop focused by the wormhole. Kim and Thorne found the energy of the quantum vacuum diverges at the chronology horizon, but argued that quantum gravity might cut it off in time. Hawking (1992) read the same calculation as evidence that the wormhole would be destroyed. The disagreement fuelled Hawking's chronology protection conjecture.
Is there evidence for wormholes?
None so far. Visintainer mentions one intriguing object: a double Einstein ring in the constellation Leo, found with the Hubble Space Telescope by a team including Adam Bolton, then at the University of Hawaii. An Einstein ring forms when a background galaxy lies almost exactly behind a foreground lens, so its light is spread into a ring. Here there are two concentric rings. Pedro González-Díaz of Madrid's Institute of Fundamental Physics proposed instead that the system might be a ringhole, a wormhole with a ring-shaped throat.
The ringhole interpretation is a fringe proposal. The system, SDSS J0946+1006 (nicknamed the "Jackpot", reported in 2008), is explained by two background galaxies at different distances lined up behind one massive foreground galaxy: each produces its own ring. Astronomers have since used it as a tool to measure cosmological parameters and dark-matter distribution, and a further background source has been identified. Nothing about it requires a wormhole.
Gödel's rotating universe
The first spacetime known to contain closed timelike curves everywhere came from logic, not cosmology. Kurt Gödel, the logician and Einstein's walking companion at Princeton, presented it in 1949 for Einstein's seventieth birthday volume.
- The model. A universe uniformly filled with dust that rotates as a whole, held in equilibrium by a negative cosmological constant. It does not expand.
- The loops. Because of the rotation, light cones tilt progressively with distance from any observer. Beyond a critical radius they tilt so far that a circle around the observer becomes timelike: a rocket following it returns before it left. Every event in Gödel's universe lies on such a loop.
- The cost. Gödel calculated that the trip requires speeds above about 71% of light (1/√2) and fuel far beyond any practical rocket. The loops are cosmologically large.
- The status. Our universe expands and shows no measurable rotation; microwave-background data limit any cosmic rotation to a tiny fraction of the expansion rate. So, as Visintainer says, the solution had to be rejected as a description of our world.
Gödel's significance is logical: Einstein's equations do not by themselves forbid travel into the past. Einstein admitted in his reply that the result had troubled him since he built the theory. Gödel himself drew a philosophical conclusion: if a universe consistent with relativity can contain loops in time, there is no objective lapse of time, and time as we experience it is an illusion.
Spinning cylinders: van Stockum and Tipler
The same frame-dragging trick works on a smaller scale.
- Willem Jacob van Stockum (1937) solved Einstein's equations for an infinitely long cylinder of rotating dust. Later analysis showed that outside a fast enough cylinder there are closed timelike curves.
- Frank Tipler (1974) proposed building one: a cylinder roughly 100 km long and 10 km in radius, with the density of a neutron star, spinning about twice per millisecond, so its surface moves at about half the speed of light. A spaceship spiralling around it in the right way would return to its own past.
- Hawking (1992) proved that a finite cylinder cannot create closed timelike curves in a bounded region without violating the weak energy condition: in other words, it would need exotic matter again. Tipler's infinite cylinder escapes this only by being infinite.
Gott's cosmic strings
J. Richard Gott of Princeton found a more elegant geometry in 1991. A straight, infinitely long cosmic string does not curve space around it in the usual way; it removes a thin wedge, a deficit angle of 8πGμ/c², where μ is the string's mass per unit length. Space around it is like a cone.
- The design. Let two infinite, parallel cosmic strings fly past each other at very nearly the speed of light. A spaceship looping around the pair, using the wedge "shortcuts" each string provides, can return to its starting point before it left.
- The objection, part 1. Stanley Deser, Roman Jackiw and Gerard 't Hooft (1992) showed that the two-string system as a whole has a tachyonic total momentum: it behaves like an object moving faster than light, so it cannot be assembled from ordinary slower-than-light matter.
- The objection, part 2. Sean Carroll, Edward Farhi and Alan Guth (1992) showed that in an open universe a Gott time machine cannot be built from realistic parts, and in a closed universe the energy needed would collapse it into a black hole before any loop formed.
- Gott's own estimate. Using a collapsing loop of string to travel back just one year would require a string loop with the mass-energy of more than half a galaxy.
The Alcubierre warp drive
In 1994 the Mexican physicist Miguel Alcubierre wrote down a spacetime inspired by Star Trek: a bubble of flat space that moves at any speed, even faster than light, because spacetime contracts in front of it and expands behind it. Inside the bubble the crew float in free fall, feel no acceleration and suffer no time dilation. Locally nothing beats light; the bubble surfs on a wave of geometry, much as the universe's expansion can carry distant galaxies apart faster than c.
- 1994 Alcubierre: the drive works on paper but needs negative energy in the bubble wall.
- 1996 Allen Everett: two superluminal warp trips can be combined into a closed timelike curve; a faster-than-light warp drive is a time machine.
- 1997 Michael Pfenning and Larry Ford: with quantum inequalities limiting the wall's thickness, a 100 m bubble needs negative energy around 10⁶² kg, about ten orders of magnitude more than the mass of the visible universe.
- 1997–1998 Allen Everett, Thomas Roman and Sergei Krasnikov: a superluminal bubble's crew cannot steer or stop it, because the front of the bubble is causally disconnected from its interior.
- 1999 Chris Van Den Broeck: reshaping the bubble (tiny outer surface, large interior) cuts the requirement to a few solar masses of negative energy.
- 2009 Finazzi, Liberati and Barceló: quantum effects make superluminal warp bubbles unstable, with energy piling up at the front.
- 2011–2013 Harold White (NASA Eagleworks) claims an oscillating, thick-walled bubble could need about 700 kg; the claim was never confirmed in the peer-reviewed literature, and the lab's interferometer test was inconclusive.
- 2021 Erik Lentz claims a positive-energy "soliton" warp drive; Shaun Fell and Lavinia Heisenberg reduce negative-energy needs; Alexey Bobrick and Gianni Martire show physical warp drives built from ordinary matter must be subluminal shells; Santiago, Schuster and Visser argue every superluminal warp drive violates the null energy condition for some observer.
- 2024 Jared Fuchs, Christopher Helmerich, Bobrick and colleagues construct a constant-velocity, subluminal warp-drive solution from a matter shell that satisfies all the standard energy conditions: a real but modest step, since it moves no faster than a spaceship would.
Ori's vacuum time machine and Mallett's light ring
- Amos Ori (Technion) published a family of time-machine solutions in 1993, and in 2005 (Physical Review Letters) a design in which closed timelike curves form inside a doughnut-shaped vacuum region surrounded by ordinary matter that satisfies the weak energy condition. This removes the need for exotic matter in the construction itself. Whether the configuration is stable, and whether quantum effects destroy it as the loops form, remain open.
- Ronald Mallett (University of Connecticut) proposed in 2000–2003 that a circulating beam of laser light drags spacetime enough to create closed timelike curves, and planned a tabletop experiment with neutrons. Ken Olum and Allen Everett (2005) showed that the solution needs an infinitely long light cylinder, contains a singularity, and would require enormous energies; the experiment was never carried out.
The designs side by side
| Design | Year, author | Direction | Mechanism | Exotic matter? | Fatal problem |
|---|---|---|---|---|---|
| Turbo capsule | 1905, Einstein | Future | Speed, γ | No | Energy, dust, radiation |
| Parking capsule | 1915, Einstein | Future | Gravity well | No | Distance, crushing gravity, escape |
| Gödel universe | 1949, Gödel | Past | Cosmic rotation | No (needs Λ < 0) | Universe does not rotate, does not match ours |
| Rotating cylinder | 1937 van Stockum; 1974 Tipler | Past | Frame dragging | Yes if finite | Must be infinitely long |
| Kerr interior | 1963 Kerr; 1968 Carter | Past | Ring singularity | No | Inner horizon unstable |
| Wormhole | 1988 Morris, Thorne, Yurtsever | Past | Time-shifted mouths | Yes | Quantum inequalities, vacuum feedback |
| Cosmic strings | 1991 Gott | Past | Deficit angles | No | Tachyonic total momentum; black hole forms |
| Warp drive | 1994 Alcubierre | Past (if superluminal) | Moving bubble | Yes | Energy, unsteerable, unstable |
| Vacuum torus | 2005 Ori | Past | Vacuum-core CTCs | No | Stability unknown |
| Light ring | 2000 Mallett | Past | Laser frame dragging | No | Infinite length, singularity |
Wormholes since 2016. A new line of work found wormholes made traversable by quantum effects rather than by bulk exotic matter. Ping Gao, Daniel Jafferis and Aron Wall (2016) showed that coupling the two ends of an anti-de Sitter wormhole briefly opens it; Juan Maldacena, Alexey Milekhin and Fedor Popov built a four-dimensional example using Standard Model-like fermions and magnetically charged black holes. In all such constructions the trip through the wormhole takes longer than the trip around it, so they are not shortcuts and cannot become time machines. Blázquez-Salcedo, Knoll and Radu (2021) found wormholes supported by classical Dirac and Maxwell fields, whose physical acceptability is disputed. In November 2022 a team led by Jafferis, Joseph Lykken and Maria Spiropulu reported in Nature a "holographic wormhole" teleportation protocol run on nine qubits of Google's Sycamore processor; in 2023 Bryce Kobrin, Thomas Schuster and Norman Yao argued the learned model was too small and too simple to show genuine gravitational behaviour. The consensus: a quantum simulation of a toy model's dynamics, not a wormhole in spacetime. Surveys of wormhole physics up to 2025 still list exotic matter and stability as the unsolved core problems.
The Sum Over All Histories
Richard Feynman's formulation of quantum mechanics says a particle does not take one path from A to B; it takes every possible path, and the probabilities come from adding them all up. Hawking extended the idea to entire spacetimes, including ones that curve back in time. In quantum theory, histories that loop in time are not excluded in advance; they are added into the sum, and the question becomes how much they contribute. Since 1991, David Deutsch and Seth Lloyd have turned this into precise models of quantum time travel that can be simulated in the laboratory.
Feynman's idea
Visintainer's chapter is short but its idea is deep. In 1948 Feynman published a third way of doing quantum mechanics, alongside Heisenberg's matrices and Schrödinger's waves. A particle travelling from A to B has many histories: it explores every possible route through spacetime at once, straight or wildly zigzagging.
The recipe for a sum over histories
- List every path from A to B, not only the physically "sensible" ones.
- Give each path a phase, a little rotating arrow whose angle is its action S divided by ħ: the contribution is e^(iS/ħ).
- Add the arrows of all paths, head to tail. The result is the amplitude for going from A to B.
- Square its length to get the probability: P(A → B) = |Σ e^(iS/ħ)|².
Why big objects take one path
The book notes that for macroscopic objects all paths cancel except one, so we feel that a thrown ball follows a single trajectory. The reason is the size of ħ ≈ 1.05 × 10⁻³⁴ J·s. For a 1 g ball thrown across a room, the action is of order 10⁻³ J·s, so its phase is about 10³¹ radians. Changing the path by a hair changes the phase by billions of full turns, and neighbouring paths cancel each other out. Only near the path where the action is stationary (unchanged to first order by small variations) do neighbouring arrows line up and reinforce. That path is precisely the one given by Newton's laws, the principle of least action. For an electron, whose action is comparable to ħ, many paths contribute and the result is interference.
The double slit, path by path. Fire electrons one at a time at a screen with two slits. Each electron arrives as a single dot, yet after thousands of electrons the dots form interference fringes, as if each electron passed through both slits. In Feynman's picture this is exactly what happens: the sum includes paths through the left slit and paths through the right slit, and at some points on the screen their arrows point opposite ways and cancel. Block one slit and the fringes disappear, because half of the histories are gone. The particle does not choose a path; the arrows of all paths compete.
Antiparticles as particles moving backward in time
Feynman's diagrams carry a second idea that bears directly on time travel. Ernst Stückelberg (1941) and Feynman (1949) showed that an antiparticle moving forward in time is mathematically identical to the corresponding particle moving backward in time. A positron is, on paper, an electron running into the past. The pair-creation loops of Chapter 6 can be redrawn as a single particle circling in a closed loop in time.
The one-electron universe. In 1940 John Wheeler telephoned his student Feynman with a wild idea: perhaps all electrons in the universe have the same mass and charge because they are one electron, zigzagging forward and backward through time, appearing as electrons on the forward legs and as positrons on the backward legs. Feynman pointed out the flaw: there should then be as many positrons as electrons, and there are not. He kept the useful part, the backward-in-time description of antiparticles, which became part of his Nobel-winning quantum electrodynamics.
Summing over spacetimes
Hawking's step was to apply the sum not to a particle's paths inside a fixed spacetime but to spacetimes themselves. In quantum gravity the geometry of spacetime is a quantum variable, so the full calculation would sum over all possible geometries, weighted by their action. That sum includes geometries that do not satisfy Einstein's classical equations, and among them are curved spacetimes with closed timelike curves, histories in which particles travel faster than light or move back in time.
Hawking concluded that, on the microscopic scale, time travel occurs everywhere: tiny loops in time are part of the quantum fluctuations of every region of space. The real question becomes how much loops on the macroscopic scale contribute. Hawking's estimate was that the chance of a macroscopic object such as Kip Thorne going back in time is about one in 10 with a trillion trillion trillion trillion trillion zeros after it, about 1 in 10^(10⁶⁰).
"Time travel occurs everywhere" is a manner of speaking. The loops in Feynman diagrams and the CTC geometries in a gravitational sum over histories are terms in a calculation, not events anyone could observe or use. No signal can be sent to the past with them. The sum over spacetime geometries is also not a finished theory: it is ill-defined without a full theory of quantum gravity, and Hawking's own version used "imaginary time" tricks whose validity is debated. The book's chapter presents the idea in one paragraph as if it settled the question; it shows only that quantum theory does not forbid time loops a priori.
Deutsch's quantum time travel (1991)
In 1991 David Deutsch of Oxford proposed the first concrete model of how quantum systems would behave if a closed timelike curve existed. He represented a CTC as a quantum circuit in which one system (a qubit, say) leaves the circuit's future and re-enters at its past, interacting with ordinary "chronology-respecting" systems along the way.
- The rule. The state of the system on the loop must be a fixed point: whatever state it enters the loop in, the interaction must return it in exactly the same state. In mathematical form, ρ_CTC = Tr_sys[U (ρ_in ⊗ ρ_CTC) U†].
- The guarantee. Deutsch proved that a fixed point always exists. So there is always a consistent history: no paradox can arise.
- The grandfather paradox, quantum style. Send a qubit around a loop through a gate that flips it (if it arrives as 0 it leaves as 1, and vice versa). Classically, no consistent value exists. Deutsch's fixed point is a 50/50 mixture of 0 and 1: in half the cases the grandfather lives, in half he dies, and each is consistent. Deutsch reads this through the many-worlds interpretation (Chapter 8): the traveller arrives in a different branch from the one she left.
Deutsch's model is nonlinear: the fixed-point condition makes outputs depend on inputs in a way ordinary quantum mechanics never allows. That has dramatic consequences. Todd Brun, Jim Harrington and Mark Wilde (2009) showed a Deutsch CTC could distinguish non-orthogonal quantum states, breaking quantum cryptography; Scott Aaronson and John Watrous (2009) showed it would make a computer as powerful as the complexity class PSPACE, solving problems believed far beyond any ordinary computer. Time travel, if it exists, would be the ultimate computational resource, which many take as a sign that it does not.
Lloyd's post-selected time travel (2011)
Seth Lloyd of MIT and colleagues proposed a rival model in 2011, based on quantum teleportation and post-selection, building on ideas of Charles Bennett and Benjamin Schumacher.
- Create a pair of entangled particles. One enters the "time loop" in the future; the other plays the part of the system emerging in the past.
- Let the emerging particle interact with the ordinary world.
- At the loop's end, perform a joint measurement on the particle and its entangled partner, and keep only the runs where the outcome matches the one that would make the teleportation perfect.
- In those runs, the particle's state has effectively been teleported back to the start of the loop.
The consequence for paradoxes is the opposite of Deutsch's: histories that would be inconsistent have zero amplitude and simply do not occur. A traveller who tries to shoot her grandfather finds the gun jams, the bullet misses, or she slips on a banana peel, and the probability of each outcome is set by the quantum rules. Lloyd's model agrees with the gravitational path integral and is equivalent in computing power to the class PP (post-selected quantum computing), weaker than Deutsch's PSPACE.
| Deutsch CTC (D-CTC), 1991 | Post-selected CTC (P-CTC), Lloyd et al. 2011 | |
|---|---|---|
| Consistency rule | State on loop is a fixed point | Inconsistent histories have zero amplitude |
| Grandfather paradox | Mixed outcome: 50% each, in different branches | Cannot happen: some unlikely event prevents it |
| Linearity | Nonlinear | Linear, but renormalised |
| Computing power | PSPACE | PP |
| Fits with | Many-worlds reading | Path integrals, Novikov consistency |
| Lab simulation | Photons, Ringbauer et al. 2014 | Photons, Lloyd, Steinberg et al. 2011 |
Simulations in the laboratory
Neither model needs an actual CTC to be tested; both can be simulated with ordinary quantum optics, because the loop's behaviour can be reproduced by entanglement plus measurement.
- 2011 Lloyd, with Aephraim Steinberg's group in Toronto, sends entangled photons through a P-CTC circuit modelling a "suicidal" time traveller. The paradoxical outcome never occurs; the surviving runs show the predicted probabilities.
- 2014 Martin Ringbauer, Tim Ralph and colleagues (University of Queensland, Nature Communications) simulate Deutsch's model with photons, including its ability to distinguish non-orthogonal states.
- 2023 David Arvidsson-Shukur, Aidan McConnell and Nicole Yunger Halpern (Physical Review Letters) show that a simulated P-CTC can improve quantum metrology: a sensor's optimal input state can be chosen after the measurement and "sent back" by entanglement, succeeding in a predictable fraction of runs.
- 2024–2025 Superconducting-qubit and further experiments implement this metrology protocol (PRL 2024, follow-ups in 2025), turning simulated time travel into a measurement technique.
None of these experiments sends anything into the past. They reproduce the statistics a CTC would produce, by entangling systems and discarding runs after the fact. Their value is in testing the mathematics of the two models and, unexpectedly, in metrology. Press headlines such as "physicists simulate time travel" are accurate only if the word simulate is read strictly.
The post-selection idea has a close cousin in everyday analytics: if you discard every run that does not match a chosen outcome, the survivors can look as if the future shaped the past. Survivorship bias in data and model evaluation works the same way. Knowing when a result is conditioned on a filter, and what fraction of runs survive it, is the practical lesson of P-CTCs: the cost of "retrocausal" control is the discarded majority.
Time Paradoxes
Time travel into the past is controversial above all because of its paradoxes: kill your grandfather and you are never born; carry a theorem back to its author and nobody ever invented it. Visintainer surveys the grandfather paradox, the twin paradox, meetings with oneself and violations of causality, then the proposed escapes: Hawking's chronology protection, Novikov's self-consistency, Kaku's argument from world lines and Everett's many worlds. Physics offers three exits: forbid closed timelike curves altogether, allow only self-consistent histories, or let the traveller arrive in a different branch of reality. Calculations since 1990 show the second exit is less paradoxical than it sounds.
Why paradoxes matter
Michio Kaku imagines a world where time machines are as common as cars: history would be rewritten constantly, with "time travel chaos" the result. The worry is not only practical. Physical laws are supposed to determine, from conditions now, what happens next. A closed timelike curve feeds the future back into the past, and if the feedback contradicts itself, the laws have no solution at all.
The two families of time-travel paradox
- Consistency paradoxes (grandfather type): a traveller's action in the past removes the cause of her own trip. There seems to be no consistent history.
- Bootstrap or information paradoxes: an object or piece of knowledge circulates in a loop with no origin. There is a consistent history, but something in it came from nowhere.
The grandfather paradox
Visintainer's version: fly through a wormhole and arrive before you set off. Now you could destroy your own spaceship before it launches. The classic form has the traveller kill her grandfather before her parent is born.
The paradox as an argument
- Time travel to the past is possible.
- If it is possible, a traveller can reach a time before her grandfather had children.
- A traveller there is physically able to kill him.
- If she kills him, she is never born, so she never travels, so she does not kill him.
- A situation in which she both kills and does not kill him is a contradiction.
- Therefore either premise 1 is false, or something prevents premise 3 from being acted on.
The philosopher David Lewis gave the classic analysis in "The Paradoxes of Time Travel" (1976). Lewis separated personal time (the traveller's own clock) from external time (the world's) and argued that the paradox trades on an ambiguity in "can". In one sense Tim can kill his grandfather: he has a rifle, good aim and the opportunity. In a broader sense, given all the facts, including that the grandfather lived, he cannot; so he will fail, for some ordinary reason. Nothing logically impossible happens; history just contains a failed attempt.
The bootstrap paradox
Who wrote Hamlet? A literature student takes a complete edition of Shakespeare back to 1590 and hands it to a young William Shakespeare, who copies the plays out and publishes them. Copies survive to the twenty-first century, and the student buys the edition she will take back. The history is perfectly consistent: nobody is killed, nothing contradicts itself. But who composed the plays? The information exists without ever having been created. Robert Heinlein built the stories "By His Bootstraps" (1941) and "All You Zombies" (1959) around such loops. Deutsch argues they violate a principle as basic as consistency: knowledge must be created by some process, such as evolution or thought, and cannot simply circulate.
The twin paradox
Visintainer calls the twin paradox the most famous paradox connected with time travel and works an example.
Twins and Vega. Twins A and B turn 20. B flies to the star Vega, taken as 26 light years away, at 99% of light speed (γ ≈ 7.09), turns around and returns. On Earth 2 × 26 / 0.99 ≈ 52.5 years pass; on board, 52.5 / 7.09 ≈ 7.4 years. A is 72.5 when B returns at 27.4. Dividing 52 light years by B's 7.4 years gives "7 times the speed of light", which, the book notes, relativity forbids, so B's journey must have looked different from Earth.
The book stops at the puzzle. The resolution is straightforward:
- Why it seems paradoxical. By the principle of relativity, B can say A was the one moving, so A's clock should run slow. Each twin sees the other's clock slowed; how can only B be younger?
- Why it is not. The situation is not symmetric. A stays in one inertial frame throughout. B changes frames at the turnaround, feeling acceleration. Proper time along a straight world line between two events (A's) is always longer than along a bent one (B's), just as in Euclidean geometry a straight line is the shortest path, except that in spacetime the minus sign makes it the longest.
- What B measures. In B's frame the distance to Vega is length-contracted to 26 / 7.09 ≈ 3.7 light years, covered at 0.99c in 3.7 years each way. B never exceeds c.
- The "7c". Distance in Earth's frame divided by time on B's clock is the proper velocity γv ≈ 7.0c: a legitimate quantity that no observer measures as a speed.
The twin paradox is neither a paradox nor time travel to the past. It is a counter-intuitive consequence of time dilation that has been confirmed with atomic clocks (Hafele–Keating 1971) and muons. It belongs to forward time travel, which raises no consistency problem: B cannot come back to an earlier event. Also, Vega is about 25 light years away, not 26; the conclusion is unaffected. Grouping it with the grandfather paradox, as the book does, mixes two different kinds of puzzle.
Meeting yourself, and broken causality
Travelling into the past creates further oddities. In Back to the Future Part II, Marty's girlfriend Jennifer meets her older self, with no catastrophic consequences beyond confusion. A traveller who goes back, returns, then goes back again would find two copies of herself in that past. Travelling into the future, on the other hand, she is unlikely to meet an older self, because her own future was cut short when she left.
The deeper issue is the principle of causality: every effect has a cause that comes before it. In a world with time loops, cause and effect can coincide, or the order can reverse. This is why many physicists suspect that the laws of nature forbid closed timelike curves, and so forbid travel into the past.
Where are the tourists?
The absence of visitors from the future, Hawking's argument from Chapter 1, has several possible explanations, and Visintainer lists them:
- Time travel to the past is impossible.
- It is possible but forbidden by a law of nature that prevents loops on macroscopic scales.
- It has happened, but travellers do not reveal themselves or are not noticed.
- (From Chapter 6) It is possible, but only back to the moment the first time machine was built, which has not yet happened.
A theory of quantum gravity, with string theory as the best-known candidate, should eventually say whether the closed timelike curves of general relativity can exist in reality.
Hawking's chronology protection conjecture
In 1992 Hawking proposed what he called the chronology protection conjecture: "The laws of physics conspire to prevent time travel on the macroscopic scale," or, in his lighter phrasing, they make the universe safe for historians.
Hawking's argument for chronology protection
- Any time machine built in a finite region must develop a chronology horizon, a boundary where the first closed light rays appear.
- Near this horizon, light and vacuum fluctuations can circulate around near-closed loops, returning focused on each pass.
- The energy density of the quantum vacuum therefore grows without bound as the horizon is approached (as Kim and Thorne had found for wormholes).
- Through Einstein's equations, this energy curves spacetime so strongly that the geometry is destroyed or a singularity forms before any loop can be used.
- Therefore macroscopic time machines cannot be completed.
- 1991 Kim and Thorne: vacuum energy diverges at a wormhole's chronology horizon, but maybe too weakly to matter before quantum gravity takes over.
- 1992 Hawking: the divergence is enough; proposes chronology protection.
- 1997 Bernard Kay, Marek Radzikowski and Robert Wald prove that the quantum stress–energy is mathematically ill-defined at some points of any compactly generated chronology horizon: semiclassical physics breaks down there, it does not decide.
- 1998 Li-Xin Li and Richard Gott find a special quantum state in a simple CTC spacetime (Misner space) whose energy does not diverge, a counterexample to the simplest version of the argument.
- 2003 Matt Visser reviews the field: semiclassical arguments make chronology protection plausible, but a proof needs quantum gravity.
The conjecture is still unproven. What has been established is that quantum fields behave badly wherever a time machine is about to switch on, which is exactly the regime where general relativity and quantum theory must be combined. Chronology protection is the default bet of most physicists, not a theorem.
Novikov's self-consistency principle
Igor Novikov, with John Friedman, Kip Thorne and colleagues in 1990, proposed the alternative: the only histories that can occur, even in a universe with time loops, are globally self-consistent ones. History cannot be changed because it already includes the traveller's visit. If you go back, you were always there, and whatever you do helped bring about the present you left. This explains, as Visintainer notes, why we are not visited by history-changing travellers: history is safe without any special conspiracy.
Polchinski's billiard ball
In 1989 Joseph Polchinski sent Thorne a sharp test case. A billiard ball rolls towards a wormhole time machine, aimed so that it enters one mouth, emerges from the other slightly earlier, and strikes its own younger self, knocking it off course so it never enters the wormhole. The ball is a grandfather with no free will to worry about: just Newton's laws.
Echeverria, Klinkhammer and Thorne, 1991. Fernando Echeverria, Gunnar Klinkhammer and Kip Thorne analysed Polchinski's setup in detail. For every initial trajectory they examined, including the "paradoxical" one, they found at least one fully self-consistent solution: the emerging ball strikes its younger self a glancing blow, which nudges it onto a slightly different path into mouth A, from which it emerges on exactly the path that delivers the glancing blow. No contradiction. The surprise was the opposite problem: many initial conditions allow infinitely many consistent solutions, so the classical laws no longer predict a unique future. Quantum mechanics, via the sum over histories, can assign each solution a probability, restoring predictability in a statistical sense.
The billiard-ball calculations turned the question around. Consistency paradoxes are not the real danger; under-determination is. In a world with time loops, the laws of physics may not fix a single history from present conditions. That is a far stranger consequence than a paradox, and it is also exactly what Lloyd's post-selected CTCs (Chapter 7) formalise: inconsistent histories get zero weight, the rest get probabilities.
World lines (Kaku's argument)
Visintainer follows Kaku in using world lines to dissolve paradoxes. Your world line is your path through spacetime. Stay in bed from 7 to 11 a.m. And your world line is vertical on a space–time diagram: you do not move in space, but time passes. Walk to work at 11 and it slopes to the right. According to Kaku, world lines have no beginning and no end and do not break: at birth you are made from the world lines of your parents' molecules; at death your molecules carry on drawing their lines. A time traveller's world line bends into a closed timelike curve.
The argument: if you prevented your parents from meeting, your own world line would have to disappear, as Marty begins to fade in Back to the Future. But a world line cannot simply vanish in relativity. So you cannot change the past. This is the self-consistency principle expressed geometrically.
Kaku's world-line argument is a picture, not a proof. World lines of particles can end, at singularities and in particle–antiparticle annihilation, and the "world line of a person" made from molecules is a loose bundle. The real content is the claim that spacetime is a single, fixed four-dimensional history (the block universe of Chapter 5), in which a change to the past has no meaning because each event happens once. That is a substantive assumption, consistent with Novikov but not something relativity forces on you.
The many-worlds escape
The alternative to a fixed history is branching. Hugh Everett III proposed in 1957 that the quantum wave function never collapses; every quantum measurement splits the universe into branches, one per outcome. Bryce DeWitt popularised it in the 1970s as the many-worlds interpretation. Applied to time travel: a traveller who goes back and changes things creates, or enters, a new branch. The history she left is untouched.
Back to the Future Part II illustrates the idea: Doc Brown draws the timeline on a blackboard, then a branch that splits off to an "alternate 1985" after a change in the past. On this view a traveller who kills "her grandfather" kills someone genetically identical to her grandfather in another branch, not her own ancestor.
Two cautions. First, in Everett's theory branching is caused by quantum decoherence, not by time travel, and the standard theory contains no way to travel between branches. The time-travel version needs an extra mechanism, such as Deutsch's CTC model (Chapter 7), which gives a precise sense in which a qubit on a loop arrives in a different branch. Second, the book reports that Hawking developed a theory in which wormholes allow travel between universes. Hawking's Euclidean wormholes and baby universes (late 1980s) are quantum-gravity fluctuations that connect regions of spacetime, but Hawking stated explicitly that nobody could travel through them; their role was to affect the values of constants of nature.
The responses side by side
| Response | Who, when | Grandfather paradox | Bootstrap paradox | Cost |
|---|---|---|---|---|
| Chronology protection | Hawking 1992 | Cannot arise: no CTCs | Cannot arise | Needs quantum gravity to prove |
| Self-consistency | Novikov, Thorne et al. 1990 | Attempt fails for ordinary reasons | Allowed | Loss of unique prediction |
| Post-selected CTCs | Lloyd et al. 2011 | Zero probability | Allowed | Probabilities renormalised |
| Deutsch CTCs | Deutsch 1991 | Mixed outcome across branches | Forbidden by Deutsch's principle | Nonlinear quantum mechanics |
| Many worlds | Everett 1957, DeWitt | Happens in another branch | Can arise | Needs inter-branch travel |
| Lewis's compossibility | Lewis 1976 | "Can" in one sense, will not | Allowed, but odd | Philosophical, no physics |
The debate has moved from words to models. Quantum-information work since 2009 has studied how each CTC model handles paradoxes, computation and cloning; photonic simulations (2011, 2014) and the 2023–2025 metrology experiments test their predictions. A 2020 analysis by Germain Tobar and Fabio Costa (University of Queensland, Classical and Quantum Gravity) argued that even systems with many interacting agents can follow self-consistent dynamics around time loops with freedom of action preserved, supporting Novikov's view against the "free will" objection; the claim is debated. No result has decided between chronology protection and consistent time travel, because that requires quantum gravity.
Conclusion: What Is Possible, What Is Not, What Is Unknown
Visintainer's verdict is cautious: a Back to the Future time machine cannot be ruled in or out, physicists are divided, and nothing macroscopic can travel through time with today's technology. Forward time travel is real and measured; backward time travel is permitted by classical general relativity in exotic geometries but probably blocked by quantum effects, and only a theory of quantum gravity can settle it. The scoreboard below sharpens her conclusion with the numbers from the preceding chapters.
The verdict, claim by claim
Visintainer pulls the threads together.
- Small trips into the future already happen. Astronauts who spend long periods in orbit move fast enough for special-relativistic time dilation to outweigh the gravitational effect, so they return slightly younger than if they had stayed home. On the International Space Station (7.66 km/s, 400 km up) the net rate is about −25 microseconds per day: Scott Kelly's 340-day mission left him about 8.4 ms younger relative to his twin Mark, and Gennady Padalka's 879 days about 22 ms.
- General relativity allows time travel in theory. It permits wormholes and closed timelike curves.
- The turbo capsule (near-light speed) and the parking capsule (deep in a gravity well) both send travellers into the future and are sound physics; both fail on energy, safety and distance.
- Wormholes are the most popular and most controversial design. They exist only on paper; they could link distant places, and with time-shifted mouths they could link different times.
- Paradoxes (grandfather, causality violations) argue against travel to the past. The book lists the twin paradox among them, though it is a feature of forward travel and has been verified.
- Two ways out: Hawking's chronology protection, under which nature keeps the order of events and forbids macroscopic time machines, and many worlds, under which a traveller changes a different branch of history.
- The decisive unknown is a quantum theory of gravity, combining general relativity and quantum theory.
- Fastest human-made object: 192 km/s (Parker Solar Probe, 2024), γ − 1 ≈ 2 × 10⁻⁷.
- Largest human time-dilation "jump": about 22 milliseconds (Padalka, orbit).
- Smallest measured gravitational time dilation: across 1 mm of height (JILA, 2022).
- Time shift needed for a useful wormhole machine: years; exotic mass for a 1 m throat: about −1 Jupiter mass.
- Hawking's odds of macroscopic time travel: about 1 in 10^(10⁶⁰).
The scoreboard
| Question | Status in 2026 | Basis |
|---|---|---|
| Do moving clocks run slow? | Established | Muons, Hafele–Keating, GPS, ion clocks at 10 m/s |
| Do clocks run slow in gravity? | Established | Pound–Rebka, Gravity Probe A, GPS, mm-scale optical clocks |
| Can a human travel years into the future? | Allowed, not achievable | Energy of order a year of world output per tonne at γ = 10 |
| Does general relativity allow CTCs? | Yes, in many exact solutions | Gödel, Kerr, Tipler, Morris–Thorne–Yurtsever, Gott, Ori |
| Does nature contain exotic matter in bulk? | No evidence | Casimir energy real but tiny; quantum inequalities |
| Do quantum effects destroy time machines? | Probably, not proven | Hawking 1992; Kay–Radzikowski–Wald 1997 |
| Are paradoxes logically fatal? | No | Novikov consistency; Echeverria–Klinkhammer–Thorne; Lewis |
| Can quantum CTCs be simulated? | Yes, as simulations | Photonic 2011, 2014; metrology 2023–2025 |
| Has anything travelled to the past? | No | No observation of any kind |
Why the door is not closed
Visintainer ends on historical modesty. In the nineteenth century few would have believed that a message could cross from Europe to America in less than a second; it happened within a few decades of the first transatlantic telegraph cable (1858, permanently from 1866). Impossibilities of one age become technologies of another, so time travel might be real in a few hundred years.
The telegraph analogy cuts both ways. Faster messaging broke no law of physics; it only needed engineering. Travel to the past runs into energy conditions, quantum instabilities and possibly logical constraints, which are not engineering problems. History also has many confident predictions that never came true: perpetual motion machines, for instance, were ruled out by the laws of thermodynamics rather than by insufficient effort. The honest form of the book's closing optimism is that the question is open in principle, not that progress will open it in practice.
A message from the future. Suppose tomorrow a quantum-gravity theory is completed and shows that wormhole time machines are stable. Nothing in our present would change: by every known design, the earliest anyone could travel back to is the moment the first machine is switched on. Now suppose instead the theory proves chronology protection. Again nothing visible changes. Our present is compatible with both answers, which is why the question cannot be settled by looking around for tourists and has to be settled by theory.
What would settle the question
- A theory of quantum gravity. String theory, loop quantum gravity and other approaches each aim to describe spacetime where quantum effects dominate. A complete theory would compute what happens at a chronology horizon.
- Holography. The AdS/CFT correspondence and the ER = EPR idea (Maldacena and Susskind, 2013), which links wormholes to entanglement, already let theorists study traversable wormholes as quantum systems; so far they find they cannot be used as shortcuts.
- Laboratory analogues. Fluid and Bose–Einstein-condensate "analogue horizons", quantum-processor simulations and post-selected CTC circuits test the mathematics, though not spacetime itself.
- Astronomy. Searches for negative-mass lensing, exotic compact objects, and deviations from Kerr black holes in gravitational waves (as with GW250114) and Event Horizon Telescope images.
The book's method, which this edition makes explicit, is a good template for judging any extraordinary claim. Separate the established from the permitted from the speculative. Ask what the claim needs (here, negative energy and stability), whether that requirement has ever been observed, and what independent results would change your mind. In technology strategy the same discipline separates a measured capability from a demo, and a demo from a roadmap.
The laws of physics conspire to prevent time travel on the macroscopic scale.Stephen Hawking, chronology protection conjecture, 1992
Master Takeaways
The whole book in its strongest claims, with the frameworks a reader needs to judge any future headline about time travel. The future is reachable with enough energy; the past is reachable only if nature permits closed timelike curves, and every route there found so far runs into negative energy or quantum instability.
- Forward time travel is a measured fact. Moving clocks run slow by γ = 1/√(1 − v²/c²) and lower clocks run slow by √(1 − 2GM/rc²); muons, airliner clocks, GPS and optical clocks a millimetre apart all confirm it.
- Backward time travel needs a closed timelike curve. Every serious "time machine" is a spacetime geometry in which a world line returns to its own past, not a vehicle.
- General relativity permits such curves, in Gödel's universe, Kerr interiors, rotating cylinders, wormholes with time-shifted mouths and Gott's strings; the equations alone do not forbid the past.
- The price is negative energy. Traversable wormholes and warp drives need exotic matter; the Casimir effect shows negative energy exists, but quantum inequalities cap it at tiny, brief amounts.
- Quantum effects probably destroy time machines as they form. Vacuum energy piles up at the chronology horizon; Hawking's chronology protection conjecture (1992) is the majority bet, but it is unproven.
- Paradoxes are not logically fatal. Novikov's self-consistency and the Polchinski billiard-ball analysis show consistent histories always exist; the deeper problem is too many solutions, not none.
- Faster than light equals backward in time. By the relativity of simultaneity, any superluminal signal or warp drive can be turned into a time machine.
- No time machine reaches back before it was built. This answers Hawking's "where are the tourists?" for every known design.
- GPS is a working relativity experiment. +45.85 µs/day from gravity, −7.21 µs/day from speed: +38.6 µs/day net, about 11 km/day of error if ignored.
- Relativistic travel costs absurd energy. One tonne at γ = 10 needs about 8 × 10²⁰ J, more than a year of world energy use; interstellar gas and dust become lethal radiation.
- Gravity wells are poor time machines. Earth gains you 4 × 10⁻⁸ s per minute; a neutron star gives a factor of 1.56, at the cost of 10¹¹ g of surface gravity.
- Black holes stop clocks at the horizon for distant observers and tip light cones inward; the singularity is a moment in the infaller's future, not a place.
- The arrow of time is an initial condition. Laws are almost perfectly time-symmetric; the arrow comes from the universe's low-entropy start (the Past Hypothesis).
- Memory follows entropy. Recording information costs k_B T ln 2 per bit (Landauer), so we remember only the low-entropy direction: the past.
- Time travel to the past presupposes a block universe. On presentism there is no past to arrive at; on eternalism a CTC is just a shape in spacetime.
- In quantum theory time loops enter the sum over histories. Hawking estimated the odds of macroscopic time travel at about 1 in 10^(10⁶⁰).
- Two quantum models of CTCs disagree. Deutsch gives mixed outcomes across branches and huge computing power; Lloyd's post-selection makes paradoxical histories have zero probability.
- Simulated CTCs are useful. Photonic simulations (2011, 2014) and 2023–2025 metrology experiments use "retroactive" state choice, without sending anything to the past.
- The twin paradox is not a paradox. The traveller changes frames; the stay-at-home twin's straight world line has the most proper time.
- Many worlds escapes the grandfather paradox only with extra machinery, such as Deutsch's model; standard Everett branching gives no way to travel between branches.
- Quantum-processor "wormholes" are simulations. The 2022 Sycamore experiment ran a toy model; critics showed it lacked genuine gravitational behaviour.
- Every known wormhole made traversable by quantum effects is longer than the way around, so it cannot serve as a shortcut or a time machine.
- The final word belongs to quantum gravity. Until general relativity and quantum theory are unified, the past is neither proven reachable nor proven closed.
Frameworks & mental models
| Name | What it says | Use it when |
|---|---|---|
| Forward vs backward travel | Slowing your clock (proven) differs in kind from closing a time loop (open) | Reading any time-travel claim |
| Lorentz factor γ | All special-relativistic effects scale with 1/√(1 − v²/c²) | Estimating dilation, contraction, energy |
| Light cone | Each event's future, past and elsewhere; order of spacelike events is relative | Asking whether one event can cause another |
| Equivalence principle | Gravity is locally indistinguishable from acceleration | Deriving light bending and gravitational redshift |
| GPS clock budget | +45.85 (GR) − 7.21 (SR) = +38.6 µs/day | Showing relativity is engineering |
| Closed timelike curve | A world line that returns to its own past | Defining a time machine |
| Energy conditions | Normal matter focuses light; defocusing needs negative energy | Testing wormhole and warp designs |
| Quantum inequalities | Negative energy: the bigger, the briefer | Judging Casimir-based hopes |
| Chronology horizon | Boundary where the first closed light ray appears | Locating where time machines fail |
| Chronology protection | Physics conspires against macroscopic CTCs | The default sceptical position |
| Self-consistency (Novikov) | Only globally consistent histories occur | Answering the grandfather paradox |
| D-CTC vs P-CTC | Fixed-point mixing vs post-selected zero amplitude | Quantum models of time loops |
| Arrows of time | Thermodynamic, psychological, cosmological, radiative, quantum | Explaining why time has a direction |
| A- vs B-theory | Tensed flow vs tenseless block | Asking whether the past exists to visit |
On the book as a whole. Visintainer's text is a short, student-level survey that relies on a handful of popular sources and repeats some of their slips: a gravitational time-dilation formula with reversed labels, the outdated relativistic mass, a mistranslated Minkowski quotation, "two-dimensional" special-relativistic spacetime, wrong neutron-star arithmetic, an outdated speed record, cosmic strings confused with superstrings, Hawking radiation as a detection method, and the twin paradox filed among the paradoxes of travel to the past. It does not mention the Morris–Thorne–Yurtsever time-machine construction by name, Gott's strings, the Alcubierre drive, chronology protection physics in detail, Novikov's billiard-ball analysis or the Deutsch/Lloyd quantum models, all of which this edition adds. Its framing and its conclusion, that the answer awaits quantum gravity, are sound.
Who's who
| Person | Contribution | Year |
|---|---|---|
| Albert Einstein | Special (1905) and general (1915) relativity | 1905–1915 |
| Hermann Minkowski | Spacetime as a four-dimensional union | 1908 |
| Karl Schwarzschild | First exact black-hole solution | 1916 |
| Kurt Gödel | Rotating universe with closed timelike curves | 1949 |
| John Wheeler | Spacetime foam; named wormholes and black holes | 1955–1967 |
| Richard Feynman | Sum over histories; antiparticles as backward particles | 1948–1949 |
| Roy Kerr | Rotating black hole with ring singularity | 1963 |
| Frank Tipler | Rotating-cylinder time machine | 1974 |
| Stephen Hawking | Hawking radiation; chronology protection | 1974; 1992 |
| Kip Thorne, Michael Morris, Ulvi Yurtsever | Traversable wormholes and the wormhole time machine | 1988 |
| Igor Novikov | Self-consistency principle | 1990 |
| J. Richard Gott | Cosmic-string time machine | 1991 |
| David Deutsch | Quantum model of CTCs | 1991 |
| Miguel Alcubierre | Warp-drive spacetime | 1994 |
| Seth Lloyd | Post-selected CTCs and photonic test | 2011 |
| Hugh Everett III | Many-worlds interpretation | 1957 |
| David Lewis | Philosophical analysis of time-travel paradoxes | 1976 |
Connect it
- Distributed systems and clocks. GPS-disciplined clocks and time synchronisation in trading systems and data centres (precision time protocols, timestamp ordering for regulatory reporting) depend on exactly the relativistic corrections in Chapter 3; at nanosecond precision, "simultaneous" across sites is a convention, as Einstein's train shows.
- Causal ordering in architecture. Light cones are the physical version of happens-before in event-driven systems: events outside each other's cones are concurrent, and only causally linked events have a fixed order. Lamport's logical clocks were inspired by special relativity.
- Post-selection and survivorship bias. Lloyd's CTCs show how conditioning on outcomes can make results look retrocausal; the same trap appears when AI model evaluations or credit back-tests keep only runs that "worked".
About this book
Time Travel Physics: Understanding the Possibilities of Time Travel by Lisa Visintainer. 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