[Physics FAQ] - [Copyright]
Original by Jon J. Thaler.
We define time travel to mean departure from a certain place and time followed (from the traveller's point of view) by arrival at the same place at an earlier (from the sedentary observer's point of view) time. Time travel paradoxes arise from the fact that departure occurs after arrival according to one observer and before arrival according to another. In the terminology of special relativity time travel implies that the timelike ordering of events is not invariant. This violates our intuitive notions of causality. However, intuition is not an infallible guide, so we must be careful. Is time travel really impossible, or is it merely another phenomenon where "impossible" means "nature is weirder than we think?" The answer is more interesting than you might think.
The B-movie image of the intrepid chrononaut climbing into his time machine and watching the clock outside spin backwards while those outside the time machine watch the him revert to callow youth is, according to current theory, impossible. In current theory, the arrow of time flows in only one direction at any particular place. If this were not true, then one could not impose a 4-dimensional co-ordinate system on space-time, and many nasty consequences would result. Nevertheless, there is a scenario which is not ruled out by present knowledge. This usually requires an unusual spacetime topology (due to wormholes or strings in general relativity) which has not yet seen, but which may be possible. In this scenario the universe is well behaved in every local region; only by exploring the global properties does one discover time travel.
It is sometimes argued that time travel violates conservation laws. For example, sending mass back in time increases the amount of energy that exists at that time. Doesn't this violate conservation of energy? This argument uses the concept of a global conservation law, whereas relativistically invariant formulations of the equations of physics only imply local conservation. A local conservation law tells us that the amount of stuff inside a small volume changes only when stuff flows in or out through the surface. A global conservation law is derived from this by integrating over all space and assuming that there is no flow in or out at infinity. If this integral cannot be performed, then global conservation does not follow. So, sending mass back in time might be all right, but it implies that something strange is happening. (Why shouldn't we be able to do the integral?)
The possibility of time travel in GR has been known at least since 1949 (by Kurt Godel, discussed in , page 168). The GR spacetime found by Godel has what are now called "closed timelike curves" (CTCs). A CTC is a worldline that a particle or a person can follow which ends at the same spacetime point (the same position and time) as it started. A solution to GR which contains CTCs cannot have a spacelike embedding - space must have "holes" (as in donut holes, not holes punched in a sheet of paper). A would-be time traveller must go around or through the holes in a clever way.
The Godel solution is a curiosity, not useful for constructing a time machine. Two recent proposals, one by Morris, et al.  and one by Gott , have the possibility of actually leading to practical devices (if you believe this, I have a bridge to sell you). As with Godel, in these schemes nothing is locally strange; time travel results from the unusual topology of spacetime. The first uses a wormhole (the inner part of a black hole, see figure 1 of ) which is held open and manipulated by electromagnetic forces. The second uses the conical geometry generated by an infinitely long string of mass. If two strings pass by each other, a clever person can go into the past by travelling a figure-eight path around the strings. In this scenario, if the string has non-zero diameter and finite mass density, there is a CTC without any unusual topology.
With the demonstration that general relativity contains CTCs, people began studying the problem of self-consistency. Basically, the problem is that of the "grandfather paradox": What happens if our time traveller kills her grandmother before her mother was born? In more readily analyzable terms, one can ask what are the implications of the quantum mechanical interference of the particle with its future self. Boulware  shows that there is a problem - unitarity is violated. This is related to the question of when one can do the global conservation integral discussed above. It is an example of the "Cauchy problem" [1, chapter 7].
How does one avoid the paradox that a simple solution to GR has CTCs which QM does not like? This is not a matter of applying a theory in a domain where it is expected to fail. One relevant issue is the construction of the time machine. After all, infinite strings aren't easily obtained. In fact, it has been shown  that Gott's scenario implies that the total 4-momentum of spacetime must be spacelike. This seems to imply that one cannot build a time machine from any collection of non-tachyonic objects, whose 4-momentum must be timelike. There are implementation problems with the wormhole method as well.
Finally, a diversion on a possibly related topic.
If tachyons exist as physical objects, causality is no longer invariant. Different observers will see different causal sequences. This effect requires only special relativity (not GR), and follows from the fact that for any spacelike trajectory, reference frames can be found in which the particle moves backward or forward in time. This is illustrated by the pair of spacetime diagrams below. One must be careful about what is actually observed; a particle moving backward in time is observed to be a forward moving anti-particle, so no observer interprets this as time
t One reference | Events A and C are at the same frame: | place. C occurs first. | | Event B lies outside the causal | B domain of events A and C. -----------A----------- x (The intervals are spacelike). | C In this frame, tachyon signals | travel from A-->B and from C-->B. | That is, A and C are possible causes of event B.
Another t reference | Events A and C are not at the same frame: | place. C occurs first. | | Event B lies outside the causal -----------A----------- x domain of events A and C. (The | intervals are spacelike) | | C In this frame, signals travel from | B-->A and from B-->C. B is the cause | B of both of the other two events.
The unusual situation here arises because conventional causality assumes no superluminal motion. This tachyon example is presented to demonstrate that our intuitive notion of causality may be flawed, so one must be careful when appealing to common sense. See the FAQ article on tachyons, for more about these weird hypothetical particles.
The possible existence of time machines remains an open question. None of the papers criticizing the two proposals are willing to categorically rule out the possibility. Nevertheless, the notion of time machines seems to carry with it a serious set of problems.