Time Machines and the Breakdown of Unitarity

We present a generic way of thinking about time machines from the view of a far away observer. In this model the universe consists of three (or more) regions: One containing the entrance of the time machine, another the exit and the remaining one(s) the rest of the universe. In the latter we know ordinary quantum mechanics to be valid and thus are able to write down a Hamiltonian describing this generic time machine. We prove the time-evolution operator to be non-symmetric. Various interpretations of this irreversibility are given. Introduction The question of whether time machines are possible or not has been studied by several authors in the last couple of years. This interest was spawned by the realization that topologically non-trivial space-times may exhibit closed time-like curves, or “time machines”. The most important example is an otherwise flat spacetime with a sufficiently short wormhole connecting two distant regions, see figure 1. This can be made to function as a time-machine, either by putting the two mouths of the wormhole in regions of different gravitational potential or by accelerating one with respect to the other, and then bringing it to rest again – both of these methods generate a time shift, which an object travelling through the wormhole experiences (Morris et al. (1988), Kim and Thorne (1991), Friedman et al. (1990), Novikov (1992)). The presence of closed time-like curves (time-machines) would make the past and the future fuse in the sense that ‘someone’ travelling on a closed time-like curve could influence his own past (the past and future light cones overlap). So time-machines makes distinguishing past and future impossible, right? Wrong! The Hamiltonian describing the action of the time-machine becomes non-symmetric making the evolution operator non-unitary, and thus time machines will be time-asymmetric in a quantum mechanical context. We can model such time machines very easily. First assume a 3+1 splitting of spacetime, i.e. the existence of a cosmic time (if the time machine is constructed from a wormhole, then this splitting will only be possible sufficiently far away from the mouths). Space will be divided into a number of regions. The time machine has its entrance (deep) inside region 1, its exit (deep) within region 2, see figure 2, and it operates in the following way: any