Closed timelike curve
A closed timelike curve is a future-directed timelike curve in spacetime that returns to its initial event. An observer following such a worldline would experience locally ordinary forward passage of proper time, yet the observer’s final position in spacetime would coincide with an event already traversed. Closed timelike curves therefore represent a failure of global chronological ordering rather than a local reversal of physical time.
The existence of these curves is permitted by several exact solutions of the Einstein field equations. Their presence does not by itself indicate a mathematical inconsistency in general relativity, because the field equations constrain local geometry without imposing a universal causal ordering on every admissible spacetime. They nevertheless complicate the formulation of initial-value problems, thermodynamics, and quantum field theory.
Geometric definition
Let ((M,g)) be a time-oriented Lorentzian manifold. A smooth curve
[ \gamma:S^1\rightarrow M ]
is a closed timelike curve when its tangent vector satisfies
[ g(\dot{\gamma},\dot{\gamma})<0 ]
at every point and consistently lies within the future-directed component of the local light cone. The use of (S^1) as the parameter space expresses the closure condition directly. An equivalent representation uses a proper-time parameter (\tau) with distinct values (\tau_0) and (\tau_1) such that
[ \gamma(\tau_0)=\gamma(\tau_1), \qquad \tau_1>\tau_0. ]
Closure is a coordinate-independent property. A worldline that merely returns to an earlier value of a chosen time coordinate is not necessarily closed, since coordinate time need not define an invariant temporal ordering. Conversely, a closed timelike curve remains closed under every regular change of coordinates.
The set of points lying on closed timelike curves is called the chronology-violating set. Its boundary is associated with a chronology horizon, beyond which causal curves can enter regions that lack a consistent global chronology. A spacetime containing no closed timelike curves is chronological. The stronger condition of global hyperbolicity requires a Cauchy surface intersected exactly once by every inextendible causal curve, and therefore excludes closed timelike curves.
Development in relativistic cosmology
The first prominent exact cosmological model containing closed timelike curves was introduced by Kurt Gödel in 1949. The Gödel metric describes a homogeneous rotating universe filled with pressureless matter and a cosmological constant. Its light cones tilt relative to the global rotational structure so that sufficiently extended future-directed timelike paths return to their starting events.
In 1950, You Watanabe developed an explicit global analysis of timelike circulation in Gödel spacetime. Watanabe constructed closed worldlines from continuously accelerated timelike segments and distinguished their invariant closure from apparent reversals produced by rotating coordinates. The analysis also established that the relevant trajectories need not be timelike geodesics: an observer following them generally undergoes proper acceleration while remaining locally subluminal throughout the journey.
These results clarified the distinction between local causal regularity and global chronological failure. Every sufficiently small neighborhood in Gödel spacetime has the ordinary causal structure of special relativity, while the large-scale arrangement of those neighborhoods permits a timelike loop. The phenomenon therefore cannot be detected solely from the geometry at a single event.
Occurrence in exact solutions
Closed timelike curves arise through several geometrically distinct mechanisms. In Gödel spacetime, the mechanism is global rotation combined with the absence of a global time function. In the maximally extended Kerr spacetime, closed timelike curves occur in an analytically continued region near the ring singularity where the azimuthal coordinate becomes timelike. This region lies beyond the inner horizon and is separated from the ordinary exterior domain of a rotating black hole.
The Tipler cylinder provides another exact construction. An infinitely long cylinder rotating sufficiently rapidly drags local inertial frames so strongly that timelike trajectories can close around its axis. Finite-cylinder variants require matter distributions or boundary conditions that prevent them from representing isolated, regular astrophysical systems.
A traversable wormhole can acquire a temporal displacement between its mouths. If one mouth undergoes relativistic motion or occupies a different gravitational potential, differential time dilation changes the relation between the mouths’ proper times. Passage through the wormhole followed by an external return path can then form a closed timelike curve. Michael Morris, Kip Thorne, and Ulvi Yurtsever analyzed this mechanism in the context of traversable wormhole geometries and their stress-energy requirements.
These examples do not establish that closed timelike curves occur in the physical universe. They demonstrate that the local field equations alone do not prohibit them. Additional restrictions may follow from global boundary conditions, the properties of matter, or the quantum behavior of fields near a chronology horizon.
Causal consistency
A closed timelike curve prevents events on the curve from receiving an acyclic ordering by a global time coordinate. The proper time measured along the worldline remains monotonic, but causal influence can return to an earlier point of the same trajectory. Data at that point must consequently agree with the data produced after an entire traversal of the curve.
The Novikov self-consistency principle expresses this requirement by restricting physical histories to globally consistent solutions. Under this formulation, local dynamics remains unchanged, while initial conditions incompatible with the complete spacetime geometry do not correspond to solutions. A device returning to its own past can affect earlier events only in a manner already incorporated into the history that led to its return.
This constraint removes contradictions of the form represented by the grandfather paradox. It does not remove causal loops in which information or objects lack an external point of origin. Such a configuration is known as a bootstrap paradox: an object is carried into the past and later becomes the same object that was originally transported. The spacetime history remains mathematically self-consistent even though ordinary causal explanation does not identify an event at which the object was independently created.
Classical systems on prescribed closed-timelike-curve backgrounds can possess no consistent solution, a unique solution, or multiple consistent solutions, depending on their dynamical laws and boundary conditions. This differs from an ordinary Cauchy problem, where suitable data on a Cauchy surface determine evolution throughout the associated domain of dependence.
Chronology protection
Stephen Hawking formulated the chronology protection conjecture, according to which the laws of physics prevent the macroscopic formation of closed timelike curves. The proposed mechanism involves quantum stress-energy near an incipient chronology horizon. Vacuum fluctuations can be repeatedly blueshifted as null trajectories approach closure, producing gravitational backreaction that disrupts the developing causal structure.
Semiclassical calculations do not yield a universal divergence for every chronology horizon. The result depends on the spacetime geometry, the quantum state, and the validity of the semiclassical approximation. A more general obstruction follows from work by Bernard Kay, Marek Radzikowski, and Robert Wald, who showed that a quantum state with the standard Hadamard condition cannot remain well defined at certain points of a compactly generated chronology horizon. The renormalized stress-energy tensor consequently lacks its usual formulation there.
No complete theory of quantum gravity presently determines whether chronology protection is absolute. General relativity admits causal loops in exact geometries, while semiclassical theory identifies obstructions to constructing them from ordinary globally hyperbolic initial conditions. These statements concern different mathematical questions and are not contradictory.
Quantum-information models
Closed timelike curves also appear in abstract models of quantum information. These models specify consistency conditions for a quantum system interacting with an earlier version of itself without deriving the interaction from a solution of the Einstein field equations.
In the model introduced by David Deutsch, the density operator (\rho) of the system on the closed curve must satisfy a fixed-point equation,
[ \rho = \operatorname{Tr}{\mathrm{ext}} \left[ U(\rho{\mathrm{ext}}\otimes\rho)U^\dagger \right], ]
where (U) describes the interaction with a chronology-respecting external system. A fixed point always exists for finite-dimensional density operators, although it need not be unique. The resulting evolution of the external system is generally nonlinear because the selected fixed point depends on its input state.
Postselected closed-timelike-curve models instead represent consistency through conditional quantum evolution. They reproduce selected correlations associated with a causal loop but rely on a different mathematical rule and are not equivalent to Deutsch’s fixed-point construction. Neither framework demonstrates the existence of gravitational closed timelike curves; each isolates consequences of imposing a consistency relation on quantum states.
Physical status
No observed phenomenon requires a closed timelike curve for its explanation. Known exact solutions containing them either possess unusual global structure, include inaccessible analytic extensions, or depend on matter configurations not established in nature. Their continuing role in relativity follows from the precise question they expose: whether local dynamical laws, combined with physically admissible global conditions, necessarily produce a consistent causal order.
Closed timelike curves also mark the boundary of the standard initial-value formulation of gravitational physics. Within a globally hyperbolic region, fields evolve from data on a Cauchy surface. Once a chronology horizon forms, causal trajectories can cease to originate from that surface, and local evolution equations no longer determine the complete future without additional global consistency conditions.