By Simon Horsley

I had a bit of a weird few days about a month ago; all of a sudden nearly everyone I talked to seemed to be bothered by what time was. Of course neither was this representative of the population as a whole, nor was it particularly long lived. As someone interested in both physics and philosophy though, I was quite excited. I also remembered that ‘time travel’ has always bothered me. The physicist in me is convinced that ‘whatever goes for time, goes for space also; if I can walk backwards, surely then I can go through time backwards’, while the philosopher hates the idea. In particular, if I can prevent my own existence—and thereby prevent my own going back in time—through going back and neutering my own ancestor, then something must be very wrong.

Happily enough, immediately after I remembered that I was a ‘time travel’ schizophrenic, a third year physics–philosopher introduced me to a way in which all of the paradoxes of ‘time travel’ might be avoided, and this is really what I feel like telling you about. The central idea is so simple and representative of how most of us probably think the physical world should be, that it’s hard to reject. At the same time, some of the consequences are hard to swallow. Apparently, if we are to allow ‘time travel’, and avoid all of the possible paradoxes that we might run into, then not only must our laws of nature turn out to be horribly staged, but there is also definitely no possibility for the existence of free will.

As usual, some claim that quantum mechanics might help us out of this mess, but this is probably only because quantum mechanics lets us do pretty much anything. You’ve probably noticed that I’ve, annoyingly, so far put scare quotes around any reference to motion through time. This is for quite a good reason, and it’s something I’d like to make clear before we deal with the paradoxes. When we think of motion, this is always made up of a series of different locations in space ordered throughout a period of time. ‘Time travel’ can’t be quite like this; if we change location in time then we have no means by which to order this into something we might call motion, or travel. In fact it’s quite hard to find a suitable vocabulary from the ordinary English language which might describe what we do mean. We must realise that a ‘motion’ through time really is something not entirely like anything we are used to thinking about. Because of this, I’d rather not get confused and instead sometimes I’ll talk in terms of the more obfuscated concept of a ‘closed time–like curve’. This is a concept that clearly needs explanation; being borrowed from physics, it’s designed with complete inaccessibility in mind.

In modern physics we work, not with time and space separated, but with them both united within a single, four dimensional space–time. Compared with ordinary space, this is a bit like the difference between a square and a cube. To make a cube from a square, we need to be able to create lines at right angles to the surface of the square. Similarly, to make space–time from space, we must draw lines at right angles to space itself (I’ve tried to draw this as best I can in figure 1). The direction which is at a right angle to space is taken to be that of time. The actual situation is somewhat more complex than this, but as it is, it is quite a nice idea. Space–time enables us to view the totality of existence as a single four dimensional entity. In accordance with this idea, an object’s complete existence is represented by a line—or a curve—in this space– Time Space Object time.



1 Time Space Object

This line is said to be ‘time–like’, if the object always observes effects as following from causes during its lifetime. Normally, your life is causally ordered along only one direction in time, so most of us are represented by time–like lines which have both ends dangling and don’t have any knots in them (see figure 1). However, if you are allowed to go back to an earlier time (and keep causality happy), then you allow for the possibility of closed time–like curves in space– time; that is, your ‘life–line’ may have both ends fastened to one another, or
be tied in knots. In the context of possible ‘motion’ through time, the concept of a closed time–like curve thus represents an object’s going back to a previous time and interacting with it’s earlier self.

With this in mind, suppose for the moment that we allow for the existence of a time–machine. There isn’t any immediate reason why we can’t use this machine; if it happens to lead to a contradiction, then we’d expect that it shouldn’t be physically possible anyway. I’ll split the uses of the time–machine up into three classes.

1. Going to a very distant future point in time (one which you can’t just sit and wait for).

2. Going to a past, but isolated point in time (not interacting with your past self, either directly, or indirectly).

3. Going to a past, but non–isolated point in time (interacting in some way with your past self ).

Some of the uses of the time–machine seem perfectly fine. For example, if you jump forward to a time far in the future (use 1) and live there until your death, then there doesn’t appear to be any obvious inconsistency. Similarly, let’s speculate that you climb into the machine and appear at some earlier point in time in some place where you will never interact with your earlier self, either directly, or indirectly (use 2). If you then either go to a point far in the future, or live carefully isolated from your earlier self until death, then, besides the fact that you now allow for yourself to exist in two places at once, there appears to be no contradiction.

However, if an individual is allowed to interact with his or her earlier self (use 3) and so form a closed time–like curve , then we get into all sorts of bother. The clearest example of this involves an impatient man’s suicide. Consider someone so desperately unhappy that he wishes to die without delay. There are no deadly implements immediately to hand, so instead he works tirelessly in order to construct a time machine. After a great deal of effort it is finished, and it seems to work. Why did he bother? Surely in constructing the machine, he could have found a blunt enough instrument to end it all. Well, it was all quite clever. Upon entering the machine, he sets the controls such that he re–appears moments before his earlier self’s suicidal thought, to then enact his own fatal wishes. In doing this his earlier self didn’t even have to wait the insignificant amount of time that it would have taken to find a suitable death– device! But hang on, if he died at that earlier moment, then he didn’t build the time machine, and so he didn’t die, but if he didn’t die then he built the time machine, and so he managed to kill himself, and so on. Clearly there is a horrible paradox here.

So it is the third use that’s the annoyance to anyone wishing to use a time–machine. How do we come to terms with this? Well, as we said earlier, we should suppose that this particular use just isn’t physically reasonable from the start. Oddly enough, there is no immediate reason for supposing this, in fact if we look at Einstein’s theory of General Relativity—the theory which tells us just what we are and are not allowed to do with space–time—then we find that, while use 1 appears barred (so we can’t peak into the future without waiting), use 3 is most certainly allowed. Moreover, several eminent physicists have put forward designs of machines that could feasibly carry out uses 2 and 3!

This is all very worrying; logical possibility usually precedes physical possibility. Here though, we appear to have a physically viable situation which leads directly to logical inconsistency; the problem I mentioned at the start. The solution is beautifully simple, all we have to do is apply a single a priori restriction to all of our theories of physical processes. Quite plainly this is the condition that all physical processes must be continuous. By this I mean that, if we have some process, for instance a human life, and if we then look at snapshots of that life over smaller and smaller periods of time, then we can eventually pick a small enough period of time such that nothing happens. A discontinuous process would involve something like magic. For example, if I could instantaneously make tea appear in my cup, then—apart from feeling very pleased with myself—however short a snapshot I took of the moment when the tea appeared, the tea would still just miraculously jolt into existence. What this continuity rule really amounts to is a ‘no magic please’ law.

How does this restriction affect our suicidal man? Well, at the moment, if we take him to have both built the time machine and have been successful in killing his earlier self, then there must clearly be a moment in his existence where he goes from being dead to being suddenly alive. The assumption of continuity has at least ruled out the physical possibility of this messy kind of time travel. The question naturally arises though, as to just what we are allowed to do with our machine. Time travel is already looking a bit dull; we’ve said that 1 appears to be impossible, in our universe at least. Use 2 still seems fine for the moment, but we’ve cut a good deal out of 3.

It turns out that the ‘no magic please’ law is an awfully restrictive requirement. To illustrate this, let’s consider a typical example. Suppose we have another time machine. A black and white camera will be pointed at the exit of this device. Take for granted that the film of this camera is taken out and is put through the machine such that it exits at an earlier point in time where the camera (with the film’s earlier self inside) takes a picture of it. This series of events consequently forms an indirectly closed time–like curve. In taking the picture, the earlier film so becomes the negative of its later self.

Just like in the case of our suicidal man, this must introduce a discontinuity in nearly all cases. At some point in time the film must have ‘jumped’ from being one image to being that image’s negative. However, we should notice that this need not happen in every situation. If the film was its own negative then everything would be perfectly continuous. Is this possible? Yes, but in only a single case, where the film is uniformly grey. As a consequence our physical laws must be such that something stops this particular film from going back to the particular moment where is is photographed unless that film is in every way uniform and of the exact right shade of grey. We must have some amazingly ugly physical laws at work!

To rule out possible paradoxes, not only must the universe work in this rather unpleasant way, but along with this an individual’s freedom of choice must also be removed. To observe this we return again to our suicidal man. Clearly it is his choice that he must die, and as we have already established, this is not a possible choice that he can make. Such a feature alone is not unfamiliar; I can choose to outrun a tiger, but that doesn’t mean that when I attempt to, the physical laws will comply. The requirements of continuity are apparently stronger than those of leg physics. It is not just that he can’t kill himself, it’s that he must follow a very specific set path of events throughout the closed curve. The example often given is one where the suicidal man aims to shoot his earlier self. In such a case he must miss. But why? If he missed to hit his earlier self in the eye, then this may impede his later self ’s vision. This may thus cause him to miss. It’s all perfectly consistent, but there is certainly very little room for this individual to have made any real choices.

It might be taken that the consequences of fixing the paradoxes of ‘time travel’ in this way are so unappealing that the idea should just be outright rejected. The problem is, besides gut feeling and aesthetics I can’t really see any reasonable argument for this.