Slightly More Realistic Models of Time Travel

Echeverria, Klinkhammer and Thorne (1991) considered the case of 3-dimensional single hard spherical ball that can go through a single time travel wormhole so as to collide with its younger self.

Figure 9
The threat of paradox in this case arises in the following form. There are initial trajectories (starting in the non-time travel region of space-time) for the ball such that if such a trajectory is continued (into the time travel region), assuming that the ball does not undergo a collision prior to entering mouth 1 of the wormhole, it will exit mouth 2 so as to collide with its earlier self prior to its entry into mouth 1 in such a way as to prevent its earlier self from entering mouth 1. Thus it seems that the ball will enter mouth 1 if and only if it does not enter mouth 1. Of course, the Wheeler-Feynman strategy is to look for a ‘glancing blow’ solution: a collision which will produce exactly the (small) deviation in trajectory of the earlier ball that produces exactly that collision. Are there always such solutions?[3]

Echeverria, Klinkhammer & Thorne found a large class of initial trajectories that have consistent ‘glancing blow’ continuations, and found none that do not (but their search was not completely general). They did not produce a rigorous proof that every initial trajectory has a consistent continuation, but suggested that it is very plausible that every initial trajectory has a consistent continuation. That is to say, they have made it very plausible that, in the billiard ball wormhole case, the time travel structure of such a wormhole space-time does not result in constraints on states on spacelike surfaces in the non-time travel region.

In fact, as one might expect from our discussion in the previous section, they found the opposite problem from that of inconsistency: they found underdetermination. For a large class of initial trajectories there are multiple different consistent ‘glancing blow’ continuations of that trajectory (many of which involve multiple wormhole traversals). For example, if one initially has a ball that is traveling on a trajectory aimed straight between the two mouths, then one obvious solution is that the ball passes between the two mouths and never time travels. But another solution is that the younger ball gets knocked into mouth 1 exactly so as to come out of mouth 2 and produce that collision. Echeverria et al.

do not note the possibility (which we pointed out in the previous section) of the existence of additional balls in the time travel region. We conjecture (but have no proof) that for every initial trajectory of A there are some, and generically many, multiple ball continuations.
Friedman et al. 1990 examined the case of source free non-self-interacting scalar fields traveling through such a time travel wormhole and found that no constraints on initial conditions in the non-time travel region are imposed by the existence of such time travel wormholes. In general there appear to be no known counter examples to the claim that in ‘somewhat realistic’ time-travel space-times with a partial Cauchy surface there are no constraints imposed on the state on such a partial Cauchy surface by the existence of CTC's. (See e.g. Friedman and Morris 1991, Thorne 1994, Earman 1995, Earman and Smeenk 1999.)

How about the issue of constraints in the time travel region T? Prima facie, constraints in such a region would not appear to be surprising. But one might still expect that there should be no constraints on states on a spacelike surface, provided one keeps the surface ‘small enough’. In the physics literature the following question has been asked: for any point p in T, and any space-like surface S that includes p is there a neighborhood E of p in S such that any solution on E can be extended to a solution on the whole space-time? With respect to this question, there are some simple models in which one has this kind of extendibility of local solutions to global ones, and some simple models in which one does not have such extendibility, with no clear general pattern. (See e.g. Yurtsever 1990, Friedman et. al. 1990, Novikov 1992, Earman 1995, Earman and Smeenk 1999). What are we to think of all of this?

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