## Notes to Time Travel and Modern Physics

[1.] Multiple collisions are handled in the obvious way by continuity considerations: just continue straight lines through the collision point and identify which particle is which by their ordering in space.

[2.] The dynamics here is radically non-time-reversible. Indeed, the dynamics is deterministic in the future direction but not in the past direction.

[3.]
One might
hope that fixed point theorems can be used to prove the existence of
solutions in this type of cases too. Consider, for instance, a fixed
initial state of motion I of the ball. Then consider all the possible
velocities and locations and times
<*v*,*x*,*t*> at which such a ball could
enter mouth 1 of the wormhole. Each such triple
<*v*,*x*,*t*> will determine the trajectory
of that ball out of mouth 2. One can then look at the continuation of
the trajectory from state I and that from state s, and see whether
these trajectories collide. Then one can see for each possible triple
<*v*,*x*,*t*> whether the ball that starts
in state I will be collided into mouth 1, and if it is, with which
speed at what location and at which time this will occur. Thus given
state I, each triple <*v*,*x*,*t*> maps onto
another triple
<*v*′,*x*′,*t*′>. One
might then suggest appealing to a fixed point theorem to argue that
there must be a solution for each initial state I. However, in the
first place the set of possible speeds and times are open sets. And in
the second place there can be multiple wormhole traversals. Thus the
relevant total state-space of wormhole mouth crossings consists of
discretely many completely disconnected state-spaces (with increasing
numbers of dimensions). So standard fixed point theorems do not apply
directly. It should be noted that the results that have been achieved
regarding this case do make use of fixed points theorems quite
extensively. But their application is limited to certain sub-problems,
and do not yield a fully general proof of the lack of constraints for
arbitrary I.

[4.] This argument, especially the second illustration of it, is similar to the one in Horwich 1987, p 124-128. However, we do not share Horwich's view that it only tells against time travel of humans into their local past.