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Closed timelike curve
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In a Lorentzian manifold, a closed timelike curve (CTC) is a worldline of a material particle in spacetime that is "closed," returning to its starting point. 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. CTCs are related to frame dragging and the Tipler time machine, one of the many interesting side-effects in general relativity.
discussing the evolution of a system in general relativity, or more specifically Minkowski space, physicists often refer to a "light cone".
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In a Lorentzian manifold, a closed timelike curve (CTC) is a worldline of a material particle in spacetime that is "closed," returning to its starting point. 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. CTCs are related to frame dragging and the Tipler time machine, one of the many interesting side-effects in general relativity.
Light cones
When discussing the evolution of a system in general relativity, or more specifically Minkowski space, physicists often refer to a "light cone". A light cone represents any possible future evolution of an object given its current state, or every possible location given its current location. An object's possible future locations are limited by the speed that the object can move, which is at best the speed of light. For instance, an object located at position at time can only move to locations within in time .
This is commonly represented on a graph with physical locations along the horizontal axis and time running vertically, with units of for time and for space. Light cones in this representation appear as lines at 45 degrees centered on the object, as light travels at per . On such a diagram, every possible future location of the object lies within the cone. Additionally, every space location has a future time, implying that an object may stay at any location in space indefinitely.
Any single point on such a diagram is known as an event. Separate events are considered to be timelike if they are separated across the time axis, or spacelike if they differ along the space axis. If the object were in free fall it would travel up the axis, if it accelerates it moves across the x axis as well. The actual path an object takes through spacetime, as opposed to the ones it could take, is known as the worldline. Another definition is that the light cone represents all possible worldlines.
In "simple" examples of spacetime metrics the light cone is directed forward in time. This corresponds to the common case that an object cannot be in two places at once, or alternately that it cannot move instantly to another location. In these spacetimes, the worldlines of physical objects are, by definition, timelike. However this orientation is only true of "locally flat" spacetimes. In curved spacetimes the light cone will be "tilted" along the spacetime's geodesic. For instance, while moving in the vicinity of a star, the star's gravity will "pull" on the object, affecting its worldline, so its possible future positions lie closer to the star. This appears as a slightly tilted lightcone on the corresponding spacetime diagram. An object in free fall in this circumstance continues to move along its local axis, but to an external observer it appears it is accelerating in space as well – a common situation if the object is in orbit, for instance.
In extreme examples, in spacetimes with suitably high-curvature metrics, the light cone can be tilted beyond 45 degrees. That means there are potential "future" positions, from the object's frame of reference, that are spacelike separated to observers in an external rest frame. From this outside viewpoint, the object can move instantaneously through space. In these situations the object would have to move, since its present spacial location would not be in its own future light cone. Additionally, with enough of a tilt, there are event locations that lie in the "past" as seen from the outside. With a suitable movement of what appears to it its own space axis, the object appears to travel though time as seen externally.
A closed timelike curve 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. An object in such an orbit would repeatedly return to the same point in spacetime if it stays in free fall. 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.
General relativity
CTCs 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:
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.
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.
No CTC can be continuously deformed as a CTC to a point (that is, a CTC and a point are not timelike homotopic), as the manifold would not be causally well behaved at that point. The topological feature which prevents the CTC from being deformed to a point is known as a timelike topological feature.
Existence of CTCs places restrictions on physically allowable states of matter-energy fields in the universe. Propagating a field configuration along the family of closed timelike worldlines must eventually result in the state that is identical to the original one. This has been explored by some scientists as a possible approach towards disproving the existence of CTCs.
See also
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