Definition
A closed timelike curve (CTC) is a path through spacetime that is everywhere timelike — moving slower than light — and that loops back on itself, returning to its own starting event.
An observer following a CTC experiences continuous personal time but eventually meets their own younger self. CTCs are the precise geometric meaning of "time travel into the past."
Why it matters
How it works
In general relativity, spacetime is a four-dimensional manifold and each event has a light cone defining its possible future and past. A timelike curve stays inside the future light cone at every point. In flat spacetime, no such curve can return to its starting event — the future light cone always opens forward. But curved spacetime allows the light cones to tilt and twist enough that a timelike curve can loop around.
Concrete examples include the inside of a Kerr (rotating) black hole below the inner horizon, where CTCs exist for any observer; Kurt Gödel's 1949 rotating universe, where CTCs pass through every event; Tipler's infinite rotating cylinder; and traversable wormholes whose mouths have been moved relative to each other (Morris-Thorne-Yurtsever). Each of these requires either an idealized geometry impossible to assemble, or exotic matter with negative energy density.
The deeper question is whether CTCs can form dynamically from initial conditions that themselves have no time travel. The answer involves the Cauchy horizon — the surface beyond which the initial data no longer determines the future uniquely. Hawking's chronology protection conjecture asserts that quantum effects make this horizon unstable, preventing the loops from ever closing. Until quantum gravity is understood, CTCs remain a tantalizing loophole: not ruled out by classical physics, but quite possibly excluded by deeper laws.