Imagine that two philosophers meet on a walk. Each believes in a present that sweeps through instants of time. But that implies a unique succession of instants, or Nows. Which Nows are they? If the two philosophers are to make such claims, they should be able to ‘produce’ the Nows through which time flows. Unfortunately, they face the problem of the relativity of simultaneity. Each can define simultaneity relative to themselves, but, since they are walking towards each other, their Nows are different, and that puts paid to any idea that there is a unique flow of time. There is no natural way in which time can flow in Minkowski’s space-time. At least within classical physics, space-time is a block – it simply is. This is known as the block universe view of time. Everything – past, present and future – is there at once. Some authors claim that nothing in relativity corresponds to the experienced Now: there are just point-like events in space-time and no extended Nows. At the psychological level, Einstein himself felt quite disturbed about this. Reporting a discussion, the philosopher Rudolf Carnap wrote:
Einstein said the the problem of the Now worried him seriously. He explained that the experience of the Now means something special for man, something essentially different from the past and the future, but that this important difference does not and cannot occur within physics. That this experience cannot be grasped by science seemed to him a matter of painful but inevitable resignation. So he concluded ‘that there is something essential about the Now which is just outside the realm of science’.
The block universe picture is in fact close to my own, but the idea that Nows have no role at all to play in physics, and must be replaced by point-like events, would destroy my programme. However, it is only absolute simultaneity that Einstein denied. Relative simultaneity was not overthrown.
We are all familiar with flat surfaces (two-dimensional planes) in three-dimensional space. Such planes have one dimension fewer than the space in which they are embedded, and are flat. Hyperplanes are to any four-dimensional space what planes are to space. In Newtonian physics, space at one instant of time is a three-dimensional hyperplane in four-dimensional Newtonian space-time. It is a simultaneity hyperplane: all points in it are at the same time. Such hyperplanes also exist in Minkowski space-time, but they no longer form a unique family. Each splitting of space-time into space and time gives a different sequence of them.
Now, what is Minkowski space-time made of? The standard answer is events, the points of four-dimensional space-time. But there is an alternative possibility in which three-dimensional configurations of extended matter are identified as the building blocks of space-time. The point is that the three-dimensional hyperplanes of relative simultaneity are vitally important structural features of Minkowski space-time. It is an important truth that special relativity is about the existence of distinguished frames of reference. And an essential fact about them is that they are ‘painted’ onto simultaneity hyperplanes. As a consequence, simultaneity hyperplanes, which are Nows as I define them, are the very basis of the theory. They are distinguished features. You cannot begin to talk about special relativity without first introducing them. At this point, the way both Einstein and Minkowski created special relativity becomes significant.
The question is this: how is a four-dimensional structure built up from three-dimensional elements? To make this easier to visualize, consider the analogous problem of building up a three-dimensional structure from cards with marks on them representing the distribution of matter. From one set of cards with given marks, many different structures can be built simply by sliding the cards horizontally relative to one another and changing their vertical spacings. Tait’s problem shows that in general the markings in a structure built without special care will not satisfy the laws of motion. What is more, to find the correct positionings we have to use the complete extended matter distributions. These are what I have identified as instants of time. You simply cannot make the space-time structure without using them.
The interesting thing is that neither Einstein nor Minkowski gave serious thought to this problem – they simply supposed it had been solved. They started their considerations at the point at which space-time had already been put together. A comment by Minkowski, more explicit than Einstein, makes this clear: ‘From the totality of natural phenomena it is possible, by successively enhanced approximations, to derive more and more exactly a system of reference x, y, z, t, space and time, by means of which these phenomena then present themselves in agreement with definite laws.’ He then points out that one such reference system is by no means uniquely determined, and that there are transformations that lead from it to a whole family of others, in all of which the laws of nature take the same form. However, he never says what he means by ‘the totality of natural phenomena’ nor what steps must be taken to perform the envisaged successive approximations. But how is it done? This is a perfectly reasonable question to ask. We are told how to get from one reference system to another but not how to find the first one. Had either Einstein or Minkowski asked this question explicitly, and gone through the steps that must be taken, then the importance of extended matter configurations, and with them instants of time as I define them, would have become apparent. This is a key part of my argument. The accidents of the historical development have obscured the vital role of extended Nows and given the erroneous impression that events are primary.
I am not claiming that the description of space-time given by Einstein or Minkowski is wrong. Far from it – they got it right, but they described the finished product, and the complete story must also include the construction of the product. This is best done directly for the space-time of general relativity, the topic of the next chapters. As preparation for them, I conclude this chapter with a summary of the most important points.
Minkowski space-time is not some amorphous bulk in which there is no simultaneity structure at all. We can ‘paint coordinate lines’ – and an associated simultaneity structure – on space-time in many different ways. But the whole content of the theory would be lost if we could not do it one way or the other. There is no doubt about it – simultaneity hyperplanes exist out there in space-time as distinguished features.
Moreover, to give any content to relativity, we must, almost paradoxically, assume a universality of three-dimensional things. The clocks we can find in one Lorentz frame must be identical to the clocks we can find in any other. This is a prerequisite of the relativity principle, for it says that the laws of physics are identical in any such frame. That would be impossible if a particular kind of clock could exist in one frame but not in another. We can go further. On any hyperplane in any Lorentz frame, the actual things in the world (electric and magnetic fields, charged particles, etc.) can have any one of a huge number of different arrangements. Each of them is just like the possible distributions of particles from which we constructed Platonia for Newtonian physics.
Exactly the same thing can be done in relativity. There is a Minkowskian Platonia, whose points are all possible distributions of fields and matter that one can find on any simultaneity hyperplane in Minkowski space. Whatever Lorentz frame we choose, the Minkowskian Platonia always comes out the same. If it were not, the relativity principle, with its insistence that the laws of nature are identical in all Lorentz frames, would be meaningless. To be identical, the laws must operate on identical things, which are precisely the distributions that define the points of Platonia. For all its four-dimensional integrity, space-time is built of three-dimensional bricks. The beautiful four-dimensional symmetries hide the vital role of the bricks.