Closed timelike curve

A closed timelike curve is a closed curve in spacetime that is everywhere timelike, meaning a massive particle or observer can follow it without exceeding the speed of light. In general relativity, the existence of CTCs is not ruled out by the Einstein field equations; they arise in certain exact solutions. The first explicit example was the Gödel metric, discovered by Kurt Gödel in 1949, which describes a rotating universe containing CTCs¹. CTCs also appear in other solutions such as the Kerr black hole (inside the inner horizon), the Tipler cylinder, and traversable wormholes. The global structure of these spacetimes typically involves a violation of the strong energy condition or a nontrivial topology.

Gödel's rotating universe is a homogeneous solution where the entire spacetime rotates, and CTCs exist for all distant observers. The Tipler cylinder, an infinitely long rotating mass cylinder, can produce CTCs through frame dragging². Traversable wormholes, if they exist, can be manipulated to create CTCs by moving one mouth relative to the other in a gravitational field, exploiting time dilation. The Deutsch–Politzer spacetime demonstrates CTCs in a flat spacetime with a topological handle. These models highlight that CTCs are not artifacts of exotic matter alone but also of global topology and boundary conditions.

CTCs raise the possibility of causal loops and paradoxes, such as the grandfather paradox. To avoid logical contradictions, the Novikov self-consistency principle proposes that any actions on a CTC must be consistent with the history of the universe. Hawking formulated the chronology protection conjecture, arguing that quantum effects near the Cauchy horizon of a CTC would amplify vacuum fluctuations, preventing the formation of CTCs³. In quantum mechanics, CTCs allow for closed timelike loops that can be modeled with density matrices, leading to nonlinear effects. The Deutsch model of quantum computation on CTCs shows that classical and quantum information can be processed in ways that violate the no-cloning theorem⁴.

CTCs are not entirely forbidden by known physics; the Alcubierre warp drive, if superluminal, can also produce CTCs. The electromagnetic analog of CTCs appears in certain optical metamaterials: so-called “optical” closed timelike curves have been simulated in the lab using waveguides, allowing closed loops of light in time⁵. In the context of Mach's principle, Gödel's solution was partly motivated by showing that rotation is not absolute. The concept of a “time machine” is often confused with CTCs, but not all CTCs allow macroscopic traversal—some are confined to mathematical boundaries. The chronology protection conjecture remains unproven, and quantum gravity theories like string theory and loop quantum gravity have not yet settled the issue. A 2010 experiment using a quantum computer simulated a “simulated” CTC that misbehaved in ways consistent with self-consistency⁶.

worldline

The path of an object through spacetime.

Cauchy horizon

A boundary beyond which the spacetime becomes unpredictable, often associated with CTCs.

chronology protection conjecture

A hypothesis by Stephen Hawking that quantum effects prevent the formation of closed timelike curves.

Gödel metric

An exact solution to Einstein's equations describing a rotating universe that contains CTCs.

Novikov self-consistency principle

The idea that a time traveler cannot change the past; any action on a CTC must be consistent with the known history.