nLab closed time-like curve

Clossed time-like curves

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Clossed time-like curves


Physical observers in spacetime seem to travel only slower than the speed of light. Mathematically, when we represent spacetime by a Lorentzian manifold, the paths of physical observers must be timelike. Our intuition is that travel along a timelike curve will always move either forwards or backwards in time (and it is the former that matches our subjective experience). Naïvely, we may think that travelling along timelike curves prevents observers from engaging in time travel.

However, in principle it is quite possible for a time-like curve to be closed! Locally, we are always travelling into the future when following such a curve (in the future-oriented direction), but after some time we find ourselves back where we started, in the past. This is not possible in Minkowski space, or indeed in any spacetime with a global time coordinate, but it is allowed under certain circumstances in general relativity.

In science-fiction descriptions of time travel through wormholes, the time travelling follow a portion of a closed time-like curves (CTC) through the wormhole. (However, this really only works for the you-can’t-change-the-past sort of time travel, and science fiction rarely limits itself to that.)

See Lorentzian manifold for a precise mathematical definition.


As with any time travel, there are numerous paradoxes that arise when considering such curves and many physicists doubt their existence.

Here are some references on the Deutsch–Bacon consistency condition for CTCs:

  • Dave Bacon, Quantum Computational Complexity in the Presence of Closed Timelike Curves (pdf)

  • Charles H. Bennett, Debbie Leung, Graeme Smith, John A. Smolin, Can closed timelike curves or nonlinear quantum mechanics improve quantum state discrimination or help solve hard problems? (pdf)

  • Ian T. Durham, Quantum communication on closed time-like curves (pdf)

  • T.C. Ralph, Unitary Solution to a Quantum Gravity Information Paradox (pdf)

Last revised on April 10, 2010 at 23:59:33. See the history of this page for a list of all contributions to it.