Closed Timelike Curves

In 1949, in a journal issue marking Albert Einstein's seventieth birthday, his friend Kurt Gödel published an unusual present. Gödel, better known for proving that mathematics contains true statements it cannot prove, had found a new solution to Einstein's equations of gravity. It described a universe filled with rotating matter. In that universe, a traveler who stayed in a rocket, never exceeded the speed of light, and followed a long enough loop could arrive back at the moment of departure. ^[1]

Nothing in the calculation was exotic in the usual sense. Gödel had not added faster-than-light particles or new forces. He had simply asked what General Relativity allows, and the answer included a universe with paths into its own past.

The result was taken seriously from the start, and more than seventy years later it still troubles physicists. The paths are permitted by mathematics, burdened by severe physical obstacles, and absent from every observation so far.

A Path Back to Its Start

In everyday language, you are always moving into the future. A clock you carry ticks forward, you remember yesterday and not tomorrow, and nothing you do seems to reach backward.

Relativity describes this with a picture. Every event, meaning a place at a time, sits inside a pair of cones drawn by light rays. The future light cone contains every event you could possibly reach or influence without traveling faster than light. The past light cone contains every event that could have influenced you. Your path through spacetime, called your worldline, always stays inside your future light cone. Physicists call such a path timelike.

In flat spacetime, the kind described by Special Relativity, the cones everywhere point the same way. Following them always takes you further into the future, and a timelike path can never meet itself.

General Relativity changes one thing: spacetime can curve, and the light cones can tilt from place to place. A traveler still obeys the local rule, never outrunning light, but the direction that counts as "forward" is set by the local geometry. If the cones tilt enough along a loop, a traveler who always moves locally forward can nevertheless come back to an earlier event.

Picture a staircase in an Escher drawing. Each step climbs upward, and yet the staircase closes on itself. No individual step is wrong. The strangeness lives in the global shape.

A timelike path that returns to its own starting event is called a Closed Timelike Curve, or CTC.

The Formal Description

Spacetime geometry is encoded in the metric, $g_{\mu\nu}$ , which tells you the interval $ds$ between two nearby events:

ds^2 = g_{\mu\nu}\,dx^\mu dx^\nu .

Here $x^\mu$ are four coordinates (one of time, three of space), and repeated indices are summed. In the flat spacetime of Special Relativity, with the sign convention used on this page,

ds^2 = -c^2\,dt^2 + dx^2 + dy^2 + dz^2 ,

where $c$ is the speed of light. A path is timelike when $ds^2<0$ along it: the time part outweighs the space part, so the traveler moves slower than light.

A Closed Timelike Curve is then a curve $\gamma(\tau)$ , parametrized by the traveler's own clock time $\tau$ (proper time), that is timelike everywhere and returns to its starting point:

g_{\mu\nu}\,\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} < 0 \quad \text{along } \gamma, \qquad \gamma(\tau_0) = \gamma(\tau_1), \quad \tau_1 > \tau_0 .

Closed Timelike Curve

The first condition says the traveler never moves faster than light. The second says the traveler's clock has advanced from $\tau_0$ to $\tau_1$ , yet the traveler is back at the very same event. The traveler experiences a normal passage of time. The universe, globally, is arranged so that this passage ends where it began.

What decides whether such curves exist is the metric, and the metric is determined by the Einstein field equations:

R_{\mu\nu} - \frac{1}{2}R\,g_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu}

Einstein Field Equations

The left side describes the curvature of spacetime ( $R_{\mu\nu}$ and $R$ are built from the metric; $\Lambda$ is the cosmological constant). The right side describes matter and energy through the stress–energy tensor $T_{\mu\nu}$ , with $G$ the gravitational constant.

The key point is that these equations are local. They relate curvature at each point to matter at that point. They say nothing directly about the global shape of spacetime, and nothing in them forbids light cones from tilting all the way around a loop. Whether a solution contains CTCs is a question about the whole solution, not about any single point.

Where the Mathematics Allows Them

Several families of exact solutions to Einstein's equations contain CTCs. They differ in how physically reasonable their ingredients are, and that difference matters more than the mere existence of the curves.

Gödel's rotating universe (1949)

Gödel's solution describes a universe of uniformly rotating dust, with a cosmological constant whose value is tied to the dust density. ^[1] In a common choice of cylindrical coordinates $(t, r, \phi, z)$ , the coefficient of $d\phi^2$ in the metric is proportional to

g_{\phi\phi} \;\propto\; \sinh^2 r \,\bigl(1 - \sinh^2 r\bigr).

Here $r$ is a dimensionless radial distance and $\phi$ an angle around the rotation axis. When $\sinh r > 1$ , which means $r > \ln(1+\sqrt{2})$ , this coefficient becomes negative. A circle at constant $t$ , $r$ and $z$ is then a timelike curve that closes on itself: the light cones have tipped far enough that going "around" is a way of going into the past.

Two details reduce the practical force of Gödel's result. First, the CTCs are not free-fall paths. Chandrasekhar and Wright showed that no geodesic in Gödel's universe is a closed timelike curve, so a traveler would need a rocket, and Malament later derived how much total acceleration any such trip would require. ^[15,16] Second, our universe is not a Gödel universe. Measurements of the cosmic microwave background show no detectable global rotation, and a 2016 analysis placed tight limits on it. ^[20]

Status: mathematical possibility; excluded as a description of our universe.

Rotating cylinders: van Stockum and Tipler

In 1937–38 Willem van Stockum solved Einstein's equations for an infinitely long cylinder of rotating dust. ^[2] Nobody at the time noticed that the solution contained CTCs. Frank Tipler pointed this out in 1974: if the cylinder rotates fast enough, paths circling it can become closed timelike curves. ^[3]

The difficulty is the word "infinite." Tipler's construction relies on a cylinder of unbounded length. Hawking later proved a general theorem showing that creating CTCs in a finite region, starting from ordinary conditions, requires matter that violates the weak energy condition, which roughly says that every observer measures a non-negative energy density. ^[11] A finite spinning cylinder made of ordinary matter therefore cannot do the job.

Status: mathematical possibility; relies on idealized infinite matter.

Inside a rotating black hole: Kerr

Roy Kerr's 1963 solution describes the spacetime of a spinning mass, and it is believed to describe the exterior of real rotating black holes. ^[4] Its interior is much stranger. Brandon Carter analyzed the full structure in 1968 and found CTCs deep inside, near the ring-shaped singularity, beyond the inner horizon. ^[5]

In Boyer–Lindquist coordinates (with $G=c=1$ ), the angular part of the Kerr metric is

g_{\phi\phi} = \left(r^2 + a^2 + \frac{2Mr\,a^2\sin^2\theta}{\Sigma}\right)\sin^2\theta, \qquad \Sigma = r^2 + a^2\cos^2\theta .

Here $M$ is the mass, $a$ is the spin per unit mass, and $r$ , $\theta$ , $\phi$ are radial, polar and azimuthal coordinates. Outside the black hole this quantity is positive. But the full Kerr solution continues through the ring to a region where $r<0$ . Near the equatorial plane ( $\theta \approx \pi/2$ ) with small negative $r$ , the term $2Ma^2/r$ becomes large and negative, $g_{\phi\phi}$ turns negative, and circles around the axis become closed timelike curves.

This is widely regarded as a feature of an idealized solution rather than of real black holes. The inner horizon of a rotating black hole is expected to be unstable: infalling radiation is blueshifted without limit there, a process studied by Poisson and Israel as mass inflation. ^[17] If so, the smooth region containing CTCs would never form in a black hole produced by stellar collapse.

Status: mathematical possibility inside an idealized solution; probably not realized in astrophysical black holes.

Wormholes: Morris, Thorne and Yurtsever (1988)

The most discussed construction came from Kip Thorne and his students. Michael Morris and Thorne first asked what a traversable wormhole would require: a tunnel connecting two distant regions of space that a person could pass through and survive. ^[6] Their answer was that the throat must be held open by matter with negative energy density, at least as seen by some observers. Such matter is called exotic.

Morris, Thorne and Ulvi Yurtsever then showed that a traversable wormhole could be turned into a time machine. ^[7] The idea uses ordinary time dilation (see Time Dilation).

Start with a wormhole whose two mouths sit side by side, their clocks synchronized through the tunnel. Carry one mouth away at speed $v$ and bring it back, in a round trip lasting $T$ as measured outside. Like the traveling twin in the twin paradox, the moving mouth ages less. Its clock falls behind by

\Delta = T\left(1 - \sqrt{1 - v^2/c^2}\right).

Through the tunnel, the two mouths remain synchronized with each other. Through ordinary space, they now disagree by $\Delta$ . Suppose the mouths end a distance $D$ apart. If $\Delta > D/c$ , a traveler can go from one mouth to the other through ordinary space at less than light speed, enter the tunnel, and emerge before setting out. A closed timelike curve exists.

A striking consequence is that such a machine cannot take anyone back to a time before it was built. The earliest reachable moment is when the time shift first became large enough. Hawking once offered, with some humor, the absence of tourists from the future as evidence against time machines. ^[11] The wormhole construction shows why that evidence is weaker than it sounds: if no machine has been built yet, nobody could have arrived.

Status: mathematical possibility; requires exotic matter whose existence in the needed form and amount is unknown.

Cosmic strings: Gott (1991)

Richard Gott showed that two infinitely long cosmic strings, hypothetical thin tubes of trapped energy from the early universe, passing each other at very high speed produce CTCs. ^[8] Follow-up work found serious obstacles. Carroll, Farhi and Guth showed that a universe with the right total momentum cannot contain enough energy to build such a system from ordinary starting conditions, and Deser, Jackiw and 't Hooft showed that the configuration requires an unphysical total momentum. ^[9,10]

Status: mathematical possibility; strongly constrained.

The Evidence Question

It helps to be exact here, because the words "General Relativity predicts time travel" appear often and mislead.

Established fact: General Relativity has passed every experimental test so far, from planetary orbits to gravitational waves and black-hole images.
Established fact: Exact solutions of Einstein's equations contain closed timelike curves. This is a mathematical statement, and it is not in dispute. ^[1–8]
Established fact: No closed timelike curve has ever been observed, and no known astrophysical system produces one.
Accepted theory: Every known CTC solution depends on at least one ingredient that is idealized (infinite matter), excluded by observation (a rotating universe), unstable (the Kerr interior), or of unknown physical status (exotic matter).
Open question: Whether any physically realistic process can create a CTC.

The most relevant laboratory fact concerns negative energy. It is real in a limited sense: the Casimir effect, in which two close metal plates are pushed together because the quantum vacuum between them has lower energy than outside, has been measured. ^[19] But quantum field theory also appears to limit how much negative energy can gather, and for how long. Ford and Roman used these quantum inequalities to argue that a wormhole held open by quantum negative energy would either be absurdly small or require the negative energy to be confined to an extremely thin band at the throat. ^[18] Casimir experiments show that exotic energy is possible. They do not show that enough of it can be collected to build a wormhole.

In the plain language suggested by these pages: General Relativity permits closed timelike curves mathematically. No observation has shown that nature realizes one.

Chronology Protection

If CTCs are so troublesome, perhaps physics prevents them. In 1992, Stephen Hawking proposed the chronology protection conjecture: the laws of physics prevent the appearance of closed timelike curves. ^[11]

His argument had two parts. The first was the theorem mentioned above: making CTCs in a finite region requires violating the weak energy condition. The second concerned quantum effects. Imagine building a wormhole time machine. There is a boundary, the chronology horizon, separating the region where CTCs exist from the region where they do not. As a time machine is about to form, light and quantum fluctuations can circulate through the wormhole and return to nearly the same place and time, again and again. Hawking argued that these fluctuations pile up, so the energy density of the quantum vacuum grows without limit near the chronology horizon. The resulting energy would distort spacetime and could destroy the time machine before it formed.

The argument is not airtight. Kim and Thorne had argued, a year earlier, that the growth might be cut off at the Planck scale, where the classical picture of spacetime is expected to fail, before it becomes large enough to matter. ^[12] Kay, Radzikowski and Wald then proved a sharper result: on a chronology horizon of this type, the standard way of computing quantum field energies breaks down at some points and cannot give a well-defined answer. ^[13] That supports Hawking's intuition that something goes wrong. It also shows that the question cannot be settled without a theory of quantum gravity, which does not yet exist.

Matt Visser's review summarizes the situation well: there is a strong suggestion that quantum physics resists the formation of time machines, but no proof. ^[14] Chronology protection remains a conjecture.

Status: plausible conjecture; unproven.

What CTCs Would Allow

If CTCs existed, what could happen along them? Several consequences follow directly from the mathematics.

Paradoxes of consistency

A traveler could try to change the past, most famously by preventing their own journey. The standard physics response, the self-consistency principle, holds that only histories that are consistent around the whole loop can occur. Friedman and colleagues stated this formally in 1990 and studied its consequences for fields moving through wormhole spacetimes. ^[21] Philosophers of physics have examined how strange such global constraints are compared with ordinary physical laws. ^[24] This idea is explored on its own page: Novikov Self-Consistency Principle.

Loops without origin

CTCs also permit events that cause themselves: an object or piece of information that exists only because it was carried back in time. These loops are logically consistent, which makes them more puzzling in a way than the grandfather paradox. See Bootstrap Paradox.

Loss of prediction

In an ordinary spacetime, knowing the state of the universe at one moment lets you, in principle, compute everything that follows. This is the basis of physical determinism and the reason the future is predictable at all. A spacetime with CTCs usually lacks a Cauchy surface, a moment of time from which the whole history can be computed. Information can enter from the region containing CTCs in ways that no initial data determines. ^[21,25] The Earman, Smeenk and Wüthrich analysis argues that this makes the question "do the laws forbid time machines?" much more subtle than it first looks. ^[25]

Unusual computation

Physicists have asked what a computer could do if part of it ran on a CTC. Using David Deutsch's quantum model of CTCs, Aaronson and Watrous proved that such a computer could solve every problem in the complexity class PSPACE, which includes problems believed to be vastly harder than anything ordinary computers or quantum computers can do efficiently. ^[22,23] The result is a mathematical theorem about a hypothetical model. It is also one of the reasons many researchers suspect that CTCs do not exist: their consequences look too powerful.

A link to the block universe

CTCs fit naturally with the view that spacetime is a single four-dimensional object rather than a present moment moving forward. A loop in spacetime is simply part of that object's shape. See Block Universe.

Limitations and Open Problems

It is worth listing what the theory does not establish.

Existence in nature. Nothing observed requires a CTC. The absence is not a proof of impossibility, but there is no positive evidence.
Constructibility. Even granting exotic matter, no one knows how to build a traversable wormhole or move one of its mouths. Every wormhole-based time machine assumes a wormhole to begin with.
Energy conditions. Whether quantum fields can produce enough negative energy, in the required arrangement, is unresolved. Quantum inequalities suggest they cannot, at least not easily. ^[18]
Quantum gravity. The decisive calculations occur at the chronology horizon, where current theories break down. ^[13,14]
Other routes. CTCs are not the only way physics could allow signals into the past. Hypothetical faster-than-light particles would, combined with relativity, also permit backward signaling; see Tachyons.

The honest summary is that General Relativity, taken alone, is permissive about time travel, and our other physics appears to be less permissive, for reasons not yet fully understood.

Why Physicists Keep Studying Them

Closed timelike curves are unlikely to become a technology. Their value lies elsewhere.

They show that a successful physical theory can allow structures that seem to violate common sense, and that the right response is neither to embrace them uncritically nor to dismiss them, but to ask what else in physics would have to be true for them to exist. Asking that question about wormholes led to the study of energy conditions and negative energy. Asking it about the chronology horizon exposed limits of quantum field theory on curved spacetime. Asking it about computation connected gravity to complexity theory.

Einstein's equations unquestionably have solutions containing paths into the past. Whether any physical spacetime realizes those solutions is another matter: none has been observed, and the most usable constructions demand exotic matter or idealized conditions. The remaining question is whether quantum gravity forbids them or whether nature ever uses the permission General Relativity appears to grant.

As Gödel's birthday present showed, the mathematics allows it. Building it is another matter.

References

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[16] Malament, D. B. (1985). "Minimal acceleration requirements for 'time travel' in Gödel space-time." Journal of Mathematical Physics, 26, 774–777. https://doi.org/10.1063/1.526566

[17] Poisson, E., & Israel, W. (1990). "Internal structure of black holes." Physical Review D, 41, 1796–1809. https://doi.org/10.1103/PhysRevD.41.1796

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[19] Lamoreaux, S. K. (1997). "Demonstration of the Casimir Force in the 0.6 to 6 μm Range." Physical Review Letters, 78, 5–8. https://doi.org/10.1103/PhysRevLett.78.5

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[21] Friedman, J., Morris, M. S., Novikov, I. D., Echeverria, F., Klinkhammer, G., Thorne, K. S., & Yurtsever, U. (1990). "Cauchy problem in spacetimes with closed timelike curves." Physical Review D, 42, 1915–1930. https://doi.org/10.1103/PhysRevD.42.1915

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[23] Aaronson, S., & Watrous, J. (2009). "Closed timelike curves make quantum and classical computing equivalent." Proceedings of the Royal Society A, 465, 631–647. https://doi.org/10.1098/rspa.2008.0350

[24] Smeenk, C., Arntzenius, F., & Maudlin, T. "Time Travel and Modern Physics." Stanford Encyclopedia of Philosophy (first published 2000; revised 2023). https://plato.stanford.edu/entries/time-travel-phys/

[25] Earman, J., Smeenk, C., & Wüthrich, C. (2009). "Do the laws of physics forbid the operation of time machines?" Synthese, 169(1), 91–124. https://doi.org/10.1007/s11229-008-9338-2