Its row in the atlas →

Gott's cosmic-string time machine (1991)

Two infinitely long cosmic strings passing each other fast enough let an observer circle them and arrive in their own past, with no exotic matter anywhere.

Kind: physics · Loophole: L3 · Standing: K3 · Bill: B-caus, B-new, B-mass · Last reviewed: 2026-09-12

The claim

Gott, Phys. Rev. Lett. 66, 1126 (1991), presents "exact solutions of Einstein's field equations ... for the general case of two moving straight cosmic strings that do not intersect" [HIGH] S1. A single straight cosmic string is a conical defect: spacetime is flat everywhere except on the string, from which a wedge of angle $8\pi G\mu$ has been cut out and the edges glued. A single string cannot make a time machine. But when two parallel strings move past each other in opposite directions, each fast enough that $\gamma_s > (\sin 4\pi\mu)^{-1}$ in the laboratory frame (units $G = c = 1$), the solution "show[s] closed timelike curves (CTC's) that circle the two strings as they pass, allowing observers to visit their own past" [HIGH] S1. The same happens for non-parallel strings and for point masses in (2+1)-dimensional gravity [HIGH] S1.

In its strongest form the claim is that closed timelike curves need neither a wormhole nor negative energy. Cosmic strings are ordinary matter satisfying the weak energy condition; the spacetime is topologically Minkowski, free of singularities and horizons [HIGH] S2 (Carroll, Farhi, Guth and Olum §I). The device is nothing but two lumps of positive-energy matter moving quickly, and general relativity, taken at its word, permits it. Gott himself flagged the limit: for finite string loops, black-hole formation may prevent the closed curves from forming [HIGH] S1. The catalogue files the construction under L3 because it is the wormhole-free member of the time-machine family: the shortcut it offers is not through space but round the strings and back in time.

Origin and lineage

Cosmic strings as conical defects go back to Vilenkin (1981); their (2+1)-dimensional reduction to point particles in flat space with deficit angles is Deser, Jackiw and 't Hooft (1984), where the authors already noted that a spinning particle makes closed timelike curves and asserted that spinless moving particles do not [HIGH] S2 (Carroll et al. §I). Gott's paper is a counterexample to that assertion as stated. The replies came within a year: Deser, Jackiw and 't Hooft, Phys. Rev. Lett. 68, 267 (1992), and Carroll, Farhi and Guth, Phys. Rev. Lett. 68, 263 (1992), published back to back [HIGH] S1. Cutler (1992) mapped the global structure; Ori (1991) and Grant (1993) computed the semiclassical back reaction; 't Hooft (1992) treated the closed-universe case; Carroll, Farhi, Guth and Olum (1994) gave the general open-universe theorem; Headrick and Gott (1994) replied on the (2+1)-dimensional side. The sibling in this catalogue is the Morris, Thorne and Yurtsever time machine, which needs a wormhole and exotic matter; Gott's device is the argument that the exotic matter was never the essential ingredient, and the replies are the argument that something else always is.

The mechanism

One string. The metric outside an infinite straight string of mass per unit length $\mu$ along the $z$ axis is flat with a conical identification:

$$ds^2 = -dt^2 + dz^2 + dr^2 + r^2 d\phi^2, \qquad 0 \le \phi < 2\pi - 8\pi G\mu ,$$

so the deficit angle is $\alpha = 8\pi G\mu$ [HIGH] S2. For a grand-unified string $G\mu \sim 10^{-6}$ and the deficit is a few arcseconds. Because the spacetime is flat off the string, a path that goes round the string on the side of the missing wedge is shorter than the straight line: the string is a lens, and the two images of a source behind it are separated by the deficit angle. A single string gives a shortcut but not a time machine, because the shortcut is symmetric under time reversal and the geometry is static.

Two moving strings. Boost the string. The identification across the wedge, which in the string's rest frame is at equal times, now relates events at unequal times in the laboratory frame. Take two parallel strings moving in opposite directions along $x$ at speed $v$, with impact parameter $2d$ along $y$, and orient each wedge away from the string's direction of motion. Gott's condition for closed timelike curves is

$$\gamma_s \sin(4\pi G\mu) > 1, \qquad \gamma_s = (1 - v^2)^{-1/2},$$

so that each string's boost times the half-deficit-angle sine exceeds unity [HIGH] S1 (abstract). A traveller who crosses the wedge of the first string picks up a time shift that a fast enough string turns into a jump backward in laboratory time; crossing the second string's wedge on the return does it again; and for $\gamma_s$ above the threshold the loop round both strings closes as a timelike curve [HIGH] S2 (Carroll et al. §I). The closed curves circle the strings "in the sense opposite to their motion" and exist only during the passage [HIGH] S2. For $G\mu \sim 10^{-6}$ the threshold is $\gamma_s \gtrsim 8 \times 10^4$, that is $1 - v \lesssim 10^{-10}$ [MED] S2 (arithmetic on the published condition).

The (2+1) reduction and holonomy. Any spacetime of parallel infinite strings is invariant along $z$, so it is a (2+1)-dimensional spacetime of point particles, and in 2+1 dimensions vacuum is flat, so the whole content of the geometry is in the identifications. The energy-momentum of a system of particles is measured by the holonomy, the Lorentz transformation a vector suffers on being parallel-transported round a loop enclosing them, an element of $SO(2,1)$. For a single static particle it is a rotation by $8\pi G M$. Composition is the group product, which is nonlinear, so the total momentum of a system is not the sum of its parts [HIGH] S2 (Carroll et al. §II). Deser, Jackiw and 't Hooft showed that the holonomy of the Gott pair is boost-like, equivalent to a pure boost: the energy-momentum vector of the two-string system is spacelike, "despite the fact that each particle is moving slower than c" [HIGH] S2 (Carroll et al. §I). This is the mathematical content of the objection: the Gott pair has the total momentum of a tachyon.

The objections. Deser, Jackiw and 't Hooft: "such acausal behavior cannot be realized by physical, timelike, sources" [HIGH] S1 (abstract). Their argument is that a closed universe of point particles must have total deficit angle $4\pi$ and closes up before any closed curve can form, and that in the open case the tachyonic total momentum is unphysical. Carroll, Farhi and Guth: "there is never enough mass in an open universe to build the time machine from the products of decays of stationary particles. More generally, the Gott time machine cannot exist in any open (2+1)-dimensional universe for which the total momentum is timelike" [HIGH] S1 (abstract). Carroll, Farhi, Guth and Olum sharpened this to arbitrary initial data on an edgeless spacelike surface with timelike total momentum: no subsystem can ever acquire the spacelike momentum a Gott pair needs, because there is not enough energy; they also showed that in a closed universe a Gott pair, though not closed curves, can arise from decays, and that 't Hooft's result is that the universe then shrinks to zero volume before the curves close [HIGH] S1. Their reading of Deser, Jackiw and 't Hooft is that the tachyonic momentum is not itself unphysical, since it can be produced from static particles in a closed universe; the barrier in an open universe is energetic [HIGH] S1.

Semiclassical back reaction. Grant (1993) showed that away from the strings the Gott spacetime is a generalised Misner space, and computed the vacuum expectation value of the stress-energy of a conformally coupled scalar there: it diverges weakly on the chronology horizon and more strongly on the polarised hypersurfaces, strongly enough that the back reaction is of order one before quantum gravity is reached, supporting chronology protection for this non-compactly generated horizon [HIGH] S1 (abstract). Ori (1991) had argued along similar lines [MED] S2.

The headline number is the threshold $\gamma_s \sin(4\pi G\mu) = 1$, and the headline fact is that a system meeting it has spacelike total momentum.

What it costs

B-caus. Closed timelike curves are the purpose of the construction, not a side effect; observers circle the strings and reach their own past.

B-new. Cosmic strings are not in the Standard Model. They arise in grand-unified symmetry breaking and in string theory, and no observation has found one; the strings here are moreover infinite and straight. Finite loops, Gott notes, may collapse to black holes first.

B-mass. The threshold speed makes the total momentum of the pair spacelike. In an open universe with timelike total momentum that state cannot be reached from any initial data, however much energy is spent (Carroll, Farhi, Guth and Olum): the bill is not merely large but unpayable in a universe like ours. The alternative, a closed (2+1)-dimensional universe with total deficit angle $4\pi$, closes up before the curves form.

The bill carries no B-neg, and that is the historically important point: this is the construction that showed positive-energy closed timelike curves are consistent with the Einstein equations, and the replies showed they are still inaccessible.

Constraint scoring

ConstraintVerdictNote
CAU-1VIOLATESThe construction yields observers who visit their own past; a signal round the strings arrives before it was sent, in every frame, which is CAU-1's forbidden outcome by design.
CAU-2DODGESHawking's theorem assumes a compactly generated chronology horizon; Gott's horizon extends infinitely into the past because the strings are infinite, so the theorem does not apply, though Grant's divergence shows the conjecture still bites.
CAU-3N/ACAU-3 converts a wormhole or warp drive into a time machine; Gott's device is a time machine directly, with no wormhole to convert.
CAU-4N/AThe preferred-frame escape is not taken; the solution is fully Lorentz-invariant general relativity and accepts the closed curves.
CAU-5N/ANo entanglement enters.
ENE-1SATISFIESCosmic strings satisfy the weak energy condition, as Carroll et al. note; no negative energy appears anywhere in the solution.
ENE-2DODGESThe Olum and Gao–Wald theorems assume the generic condition, that every causal geodesic meets some curvature; the Gott spacetime is flat off the strings and fails it, which is exactly how positive-energy closed curves evade a theorem that says shortcuts need negative energy.
ENE-3N/ANo negative energy is invoked, so the quantum inequalities bound nothing here.
ENE-4N/AThe pocket geometry is a warp-bubble device.
ENE-5N/ANot a Natário-class warp metric.
ENE-6N/ANo negative energy is required.
ENE-7DODGESTwo moving strings in a spacetime that is topologically Minkowski and flat off the strings: no handle, so no topology for the theorem to censor, and the total momentum of the pair is spacelike, so the spacetime is not asymptotically flat in the sense the theorem needs. The closed curves evade Friedman, Schleich and Witt the way they evade ENE-2, by falling outside the hypotheses rather than by violating the achronal ANEC.
CON-1N/ANo bubble wall or horizon; the spacetime has no horizons at all.
CON-2N/AThe bootstrap problem concerns laying a metric along a route in advance; here the obstacle is different, that the strings can never be accelerated to threshold (Carroll et al.), and is recorded under B-mass and Status.
CON-3N/ANo Krasnikov-type pre-laid metric.
STA-1N/ANo superluminal bubble wall.
STA-2N/ANo throat to hold open.
STA-3VIOLATESGrant computes the renormalised stress-energy on the chronology horizon and polarised hypersurfaces of the Gott spacetime and finds it diverges strongly enough for order-one back reaction before the Planck scale; the machine requires that divergence not to matter.
HAZ-1N/ANo superluminal bubble sweeps up particles.
HAZ-2N/ANo horizons, so no Hawking bath.
HAZ-3N/ANo throat; the traveller circles strings through flat space, and tidal forces vanish off the strings.
LOR-1SATISFIESEach string moves at $v < c$ and no body is pushed to light speed; the observer's loop is timelike throughout.
LOR-2DODGESNo tachyon is present, yet the two-string system has the holonomy, and so the total energy-momentum, of a tachyon (Deser, Jackiw and 't Hooft); the construction obtains a spacelike total momentum from subluminal parts, which Carroll et al. show cannot be reached from timelike initial data in an open universe.
LOR-3N/ANo wave-propagation claim.
LOR-4N/ANo Scharnhorst-type effect.
WRP-1N/ANot a warp drive.
WRP-2N/AThe shell classification does not apply.
WRP-3N/AThe positive-energy warp argument does not bear on a conical spacetime.
MAN-1N/AThe (2+1)-dimensional reduction is a symmetry of four-dimensional space, not an extra dimension or a hyperspace.

Status of the argument

1991: Gott publishes the exact solution [HIGH] S1. Ori (PRD 44, R2214) argues that rapidly moving strings are subject to chronology protection [MED] S2.

1992: Deser, Jackiw and 't Hooft reply that physical timelike sources cannot produce the curves; Carroll, Farhi and Guth show there is never enough mass in an open universe to build the machine from decays of stationary particles and that the machine cannot exist in an open (2+1)-dimensional universe with timelike total momentum [HIGH] S1. Cutler maps the global structure and shows the spacetime has regions free of closed curves and a spacelike surface with none in its past [HIGH] S2. 't Hooft treats the closed universe and shows it collapses before the curves form [HIGH] S2.

1993: Grant shows the semiclassical stress-energy diverges on the chronology horizon and polarised hypersurfaces, supporting chronology protection for non-compactly generated horizons [HIGH] S1.

1994: Carroll, Farhi, Guth and Olum prove the general open-universe theorem for arbitrary initial data with timelike total momentum, and clarify that the tachyonic momentum is not itself unphysical, only unreachable [HIGH] S1. Headrick and Gott reply with a broader study of (2+1)-dimensional spacetimes containing closed timelike curves [MED] S2 (title record only).

Since then: no peer-reviewed construction has evaded the energy-momentum obstruction as of 2026-09-12. The solution is accepted as exact; the consensus is that it cannot be assembled in an open universe from any physical initial state, and that even the eternal version is semiclassically unstable. The standing K3 records an exact solution whose central claim, that the machine can exist, is contested in print and, on the energetic argument, lost.

Sources