Closed Timelike Curve: Time Loops in General Relativity
A closed timelike curve (CTC) is a worldline in spacetime that returns to its own starting point, allowing an object to travel into its own past. Several exact solutions of {{general relativity}} permit CTCs, and they raise consistency and determinism problems that David Deutsch's quantum model attempts to resolve.
A closed timelike curve (CTC) is a worldline of a material particle in a Lorentzian spacetime that loops back to its starting point in both space and time, effectively permitting travel into one's own past. CTCs were first identified by Willem Jacob van Stockum in 1937, and Kurt Gödel confirmed their mathematical possibility in 1949 with a rotating-universe solution of Einstein's field equations. Several exact solutions of general relativity contain CTCs, including the Gödel metric, the Kerr metric for rotating black holes, the Tipler cylinder, traversable wormholes, and Misner space. Their existence undermines straightforward determinism, because an event can in a sense be simultaneous with, or even cause, itself, the structure behind the bootstrap paradox and the causal loop. Two influential responses exist. The Novikov self-consistency principle holds that only globally self-consistent histories can occur near a CTC. In 1991, David Deutsch proposed a quantum treatment, often called the D-CTC model, in which the requirement is not that a returning system be bit-for-bit identical to the system sent back, but that the probability distribution of quantum states be a fixed point of the loop, a stable equilibrium under one circuit. This dissolves the grandfather paradox by enforcing a consistency condition on distributions rather than on individual outcomes. Separately, Stephen Hawking's chronology protection conjecture proposes that quantum-gravitational effects may prevent CTCs from forming in nature at all. See Novikov Self-Consistency Principle: Why Time Travel Need Not Create Paradoxes.