The Supersized Universe (A New Time Travel Paradox)
First, consider what would happen if you were to get into a time machine today at time t2 and travel back to some past time t1, say a year ago. Well, for one thing, at t1 there would be two of you, your current self, who is x years old, and your younger self, who is x-1 years old. Suppose, furthermore, that instead of getting back into the time machine and returning to the present, you decided to stay and relive the past year. The universe would then contain this one additional entity, you, between the times t1 and t2. That is, prior to t1, there was just your younger self. At t1, your current self arrived from the future and stayed. Between t1 and t2, the two of you existed side by side. And finally at t2, the younger of the two went back to t1, leaving the older to move on beyond today (one year older than you are now, but having experienced the past year twice). So in this scenario, the universe would contain additional matter between t1 and t2, the matter that is you at the moment you take the trip back in time.
Now, if it is the case that you can travel to the past, then it seems that, in principle, so can anything else. So what would happen if everything in the universe traveled back in time together? Well, for one thing, there would be nothing around – except perhaps spacetime – for everything would have left the present. But things get considerably weirder than that.
Suppose that, as in the above scenario, each thing travels back in time and then hangs around until the time of the original departure. That is, everything in the universe goes back in time from t2 to t1, and then relives the time from t1 to t2. Once again, the universe would have additional entities between t1 and t2. But by how much would it have increased? Although the answer may initially appear obvious – the universe is now twice its former size – the truth is actually much more complex than that.
To see this, let t0 be a time preceding t1, and suppose, first, that the universe has a finite positive mass at t0, t1, and t2; second, that this mass remains constant unless there are time travel departures or arrivals; and third, that the only time travel that occurs in the scenario is that from t2 to t1. It can be shown both that the mass of the universe at t1 equals the mass of the universe at t2, and that the mass of the universe at t1 is greater than the mass of the universe at t2.
Here is the argument that the mass at t1 is greater than the mass at t2:
1. Let the mass at t0 = m
2. Let the mass at t2 = n
3. Then the mass at t1 = m + n
4. But m + n > n
5. Therefore, the mass at t1 > the mass at t2
In other words, at t1, the universe is larger than at t2, for it includes everything at t2 as well as everything that was already around before t1.
Here's the argument that the mass at t1 equals the mass at t2:
1. From t1 to t2, inclusive, the mass of the universe remains constant
2. Therefore, the mass at t1 = the mass at t2
That is, since there are no time travel departures or arrivals between t1 and t2, the mass of the universe does not change from t1 to t2.
But of course, the mass at t1 cannot both be equal to and greater than the mass at t2. So what is the solution to this paradox?
Let's look at it again, but from your point of view as one of the entities that goes back in time. You exist between t0 and t1, then at t1 an older copy of you arrives from the future, and so two of you are around from t1 to t2. It follows that at t2both of you travel back to t1. Unlike in the first scenario we considered, the older "you" does not stay past t2, for everything at t2 goes back in time. It therefore is not the case that merely one copy of you travels back in time. Nor is it the case that merely two travel back in time, for if the two of you travel back to t1, to join the younger you that was there in the first place, there are in fact at least three of you between t1 and t2. So at least three of you travel back in time, only that is not all of you either, for in that case there will be at least four who travel back in time, and so on ad infinitum. The number of copies of you – and of each other entity that travels back in time at t2 – must therefore be infinite.
Now, if that is the case, then one can no longer prove in an similar manner to the above that the mass at t1 is greater than the mass at t2. For infinity plus any positive number is still infinity. The argument now becomes:
1. Let the mass at t0 = m
2. Let the mass at t2 = ∞
3. Then the mass at t1 = m + ∞
4. But m + ∞ = ∞
5. Therefore, the mass at t1 = the mass at t2
This does not really solve the problem, however. The fact remains that everything in the universe that existed prior to t1 is joined by the copies that travel back from t2, so there are copies of each entity (the "original" ones) between t1 and t2 that have not yet travelled back in time. And yet the total mass between t1 and t2 cannot change, which in turn means that all of it does travel back in time. There is therefore still a contradiction.
Another possible way out might be to argue that, if there are an infinite number of copies of each entity between t1 and t2 (each older than its immediately younger copy by t2-t1 years), the oldest ones would be infinitely old. And that in turn means that it would take an infinite amount of time for them to reach that state – or, in other words, that they never would reach that state. And that is why everything in the universe cannot travel back in time at t2.
And yet, if you can travel from t2 to t1 and stay until t2, it seems that in principle it should be possible for everything else to do the same.
Although the paradox has been presented using the example of everything travelling back in time, it does not depend on that. The underlying problem is actually simpler, and occurs whenever what remains at t2 (that is, what doesn't travel back in time) has different mass from what was originally around at t0. (If everything at t2 travels back, then obviously this is the case.) Consider the following scenario:
1. Let mass at t0 = m
2. Let mass at t2 = m + n
3. Let mass that travels back at t2 = p, where p ≠ n
4. Then mass at t1 = m + p
5. But m + n ≠ m + p
6. Therefore, mass at t1 ≠ mass at t2
And once again, since the mass at t1 must equal that at t2, there is a contradiction. Thus, if just a little more or a little less than n travels back in time, or what amounts to the same thing, a little more or a little less than m remains behind, there is a paradox.
If the mass at t2 is m + n, that means that n arrived at t1 (and stayed until t2). So only n can depart from t2. Put this way, the cause of the problem can be seen to be the supposition that a different amount of mass departed at t2 than arrived at t1. And that is, of course, impossible. The time travel proponent might therefore solve the paradox by pointing out that we are not free to decide at t2 what travels back in time independently of what already arrived at t1.
However, a problem remains, which is yet another version of the inconsistency between time travel and the autonomy principle. It seems that at t2, a time machine operator ought to be able to decide that what travels back in time is mass p. Why couldn't he? But if there are to be no paradoxes, he cannot do so.