Closed timelike curve

In a Lorentzian manifold, a closed timelike curve (CTC) is a worldline of a material particle in spacetime that is closed. This possibility was raised by Willem Jacob van Stockum in 1937 and by Kurt Gödel in 1949. If CTCs exist, their existence would seem to imply at least the theoretical possibility of making a time machine, as well as raising the spectre of the grandfather paradox.

Light cones

File:Tilted light cone.png
The lower light cone is characteristic of light cones in flat space - all spacetime coordinates included in the light cone have later times. The upper light cone not only includes other spacial locations at the same time, it doesn't include x=0 at future times, and includes earlier times.

In most spacetimes, every light cone is directed forward in time, which corresponds to the inability of known objects to move anywhere instantaneously. That is, since an object cannot escape its own light cone, it cannot enter a region of spacetime which has a different spacial location but the same time, as judged by any rest frame. It is however possible to construct a metric for a spacetime where this is not the case, and some light cones are not pointed forward in time but rather in some spacial direction. If a light cone points in the x direction, then that has become the timelike coordinate, and an object placed at that location would not only be able to travel in the x direction "instantaneously", but would have to move, since its present spacial location would not be in its own future light cone. A CTC can be created if a series of such light cones are set up so as to loop back on themselves, so it would be possible for an object to move around this loop and return to the same place and time that it started. Returning to the original spacetime location would be only one possibility, the object's future light cone would include spacetime points both forwards and backwards in time, and so it should be possible for the object to engage in time travel under these conditions. This is the mechanism the Tipler Cylinder uses to be a time machine.

General relativity

CTC's have an unnerving habit of appearing in locally unobjectionable exact solutions to the Einstein field equation of general relativity, including some of the most important solutions. These include:

• the Kerr vacuum (which models a rotating uncharged black hole)
• the van Stockum dust (which models a cylindrically symmetric configuration of dust),
• the Gödel lambdadust (which models a dust with a carefully chosen cosmological constant term).

Some of these examples are, like the Tipler cylinder, rather artificial, but the exterior part of the Kerr solution is thought to be in some sense generic, so it is rather unnerving to learn that its interior contains CTCs. Most physicists feel that CTCs in such solutions are artifacts.

J. Richard Gott has proposed a mechanism for creating CTCs using cosmic strings.

Consequences

One feature of a CTC is that it opens the possibility of a worldline which is not connected to earlier times, and so the existence of events that cannot be traced to an earlier cause. Ordinarily, causality demands that each event in spacetime is preceded by its cause in every rest frame. This principle is critical in determinism, which in the language of general relativity states complete knowledge of the universe on a spacelike Cauchy surface can be used to calculate the complete state of the rest of spacetime. However, in a CTC, causality breaks down, because an event can be "simultaneous" with its cause - in some sense an event may be able to cause itself. It is impossible to determine based only on knowledge of the past whether or not something exists in the CTC that can interfere with other objects in spacetime. A CTC therefore results in a Cauchy horizon, and a region of spacetime that cannot be predicted from perfect knowledge of some past time.

References

S. Carroll (2004). Spacetime and Geometry, Addison Wesley. ISBN 0805387323.