features Ptiy«lca Wofld D«omb«r 1994 27 Einstein's equations of relativity do not rule out "closed time-like curves", bizarre trajectories in space-time that might allow us to travel backwards in time. What are the physical constraints on such "time machines" and what are the possible repercussions? The physics of time travel JONATHAN Z SIMON The Queen bawled out, "He's murdering the time! Off with his much less passage of time than the stationary twin, has head!" Lewis Carroll "gone into the future" relative to the first twin and all other occupants of Earth. TIME travel has traditionally been the domain of science This form of forward time travel happens quite naturally fiction, not physics. Fortunately, however, at least within all the time. High-energy particle showers formed in the Einstein's theories of relativity, discussions of time travel upper atmosphere by cosmic rays contain particles whose are open to physicists as well. Special relativity unifies the lifetimes are much shorter than the time it takes them to concepts of time and space. General relativity goes beyond reach ground level, but whose speeds are extremely close unification and allows time and space to warp together in to the speed of light. The very fact that we observe these the presence of matter. General relativity even permits particles at ground level means that their rate of time sufficient warping to allow "closed time-like curves". passage has dilated or, equivalently, that they have These seemingly perverse trajectories describe paths travelled into their future. through space-time that always move forward in local Although going into the future is straightforward, we time (i.e. an observer's watch always runs forward), but also need to go into the past - which is difficult - to make a eventually end up back where and when they started. A closed time-like curve. This is where general relativity - or, space-time that contains closed time-like curves, localized more precisely, the possibility of non-trivial space-times - in one region, can be said to have a "time machine". is required. Conversely, if closed time-like curves exist, Closed time-like curves appear in explicit analytical then a traveller can go back into the past. (Note that there solutions to the Einstein equation of general relativity. is a distinction between going back into the past, discussed Previously such solutions were deemed "unphysical", here, and going "backwards in time", that is having one's simply because they contained closed time-like curves. watch tick backwards, which is a separate issue with its Nevertheless, since these solutions obey the field equa- own physics). There is currently no evidence that closed tions, they should not be rejected out of hand. Indeed, time-like curves exist. For instance, we do not see future interest in closed time-like curves has increased in the past tourists coming back to visit the present. However, it is decade. The reasons for this are varied, ranging from not in the tradition of physics to turn the argument around the practical (if time machines can be built, they would and use this lack of evidence to argue that they cannot have a lot of potential uses), to the theoretical (perhaps exist. quantum gravity can say something about the existence of What about the paradoxes of time travel? Can they be closed time-like curves, or vice versa) to the philosophical used as evidence against the existence of closed time-like (do the laws of physics allow or prohibit closed time-like curves? The most serious of these is the "grandparent" curves?). paradox. In this scenario a time traveller goes back in time Travelling forward in time is easy and does not require and kills his or her grandparent before he or she has any much new physics. In Newtonian physics with its absolute children. This would be a true paradox. One resolution to background time we all travel forward in time at the rate of this paradox is to postulate that only self-consistent one second per second. Special relativity allows us to travel histories are allowed. With this postulate the rules of the forward in time at faster rates. The so-called twin paradox game are turned around. We can ask whether a particular is an excellent example. While one twin remains at home physical theory allows self-consistent evolution that is also in an inertial frame, the other zooms off into the Universe, consistent with its equation of motion. and then back home, at relativistic speeds with time dilated We should also be aware that many physical theories by a factor of 1/(1 -v2lcy each way, where v is the with different equivalent formulations (such as the travelling twin's speed and c is the speed of light. On Schrodinger and Feynman formulations of quantum returning home the travelling twin, having experienced mechanics) can give different results in the presence of 28 Physics World December 1994 time machine might break in antici- pation of the event. The problem here seems to be that human beings have too many degrees of freedom to keep track of. Fortunately the funda- mental physical issues are still pre- sent for a much simpler system - the billiard ball. As a simple example of a time machine, due to Joseph Polchinski while he was at the University of Texas, we use a specialized billiard table with two pockets. Any object that falls into the right-hand pocket at time t is shot out of the left-hand pocket at an earlier time, t-At, with the same speed but a new direction (figure la). This time machine could be implemented using a space-time "wormhole" - a shortcut in space- time (of non-trivial topology) which connects two distant points by a shorter path. The points connected may be separated in time as well as space (see figure 2). Wormholes have not been observed yet but are allowed by the Einstein equations. However, the trajectory in figure la is not self-consistent - if the ball enters the right-hand pocket at time t, it exits the left-hand pocket at an earlier time, t-At, and this earlier version of the ball collides with the current version, preventing it from reaching the right-hand pocket in the first place. This captures the bare essence of the grandparent paradox. 1 (a) A billiard table time machine. When a ball enters the right-hand pocket at time ( an earlier This system has been analysed in version of the same ball Is released from the left-hand pocket with the same speed at an earlier time great detail by Fernando Echeverria, t-At. The direction of the earlier version Is the reflection of the incident direction about a line Gunnar Klinkhammer and Kip connecting the point of entry into the pocket with the centre of the pocket. Although there is only one Thorne at Caltech who have found ball, two or more versions of the same ball can be on the table at certain times. The trajectory shown here Is not self-consistent because the ball collides with the earlier version of itself at such an angle answers to these questions, but have that it cannot enter the right-hand pocket, (b) A self-consistent trajectory for the Initial conditions raised new, even more interesting, shown in (a). The ball sets off in the same direction as before but collides with the earlier version at questions in the process. Their an angle which does not prevent it from entering the right-hand pocket, {c-f) Four (of an infinite analysis is completely classical and number of) self-consistent trajectories with the same Initial conditions. The number of passages non-relativistic (with the obvious through the time machine varies from solution to solution exception of the time machine link- ing the pockets). closed time-like curves. Therefore when determining the The resolution of the paradox in figure la is straightfor- effect of closed time-like curves on a particular theory, we ward - if the ball moves off at the same angle, but only have to define the theory very carefully. receives a glancing blow from the earlier version of itself, it will enter the right-hand pocket at a slightly different angle to that in figure la. The direction of the ball leaving the Classical behaviour left-hand pocket will also be slightly different - hence the In avoiding the grandparent paradox, the self-consistency collision will be glancing, rather than head on as before. postulate at first seems to violate our concept of free will. It This self-consistent trajectory is shown in figure \b. It is difficult to imagine oneself as a time traveller when one should be emphasized that, although the initial conditions is not allowed to make choices which might cause a in figures la and b are exactly the same, the evolution of paradox. But this is a bit of a red herring - this same lack of the latter is self-consistent whilst that of the former is not. free will already exists in ordinary Newtonian mechanics, It turns out that every initial condition for this model has or in any deterministic theory: once the initial values of the a self-consistent solution. In fact, for many initial fields and derivatives are specified, there is no room for conditions there is more than one self-consistent solu- free will. In this sense free will is not usually addressed in tion, and often an infinite number, which all obey physics. Still, what might happen if a time traveller did Newton's laws of classical mechanics.