The End of Time: The Next Revolution in Physics

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.

It is just that space-time is not constructed from a unique set. The analogy with a pack of cards is again quite apt. Newtonian space-time is an ordinary pack; Minkowski space-time is a magical pack. Look at it one way, and cards run through the block with one inclination. Look at it another way, and different cards run with a different inclination. But whichever way you look, cards are there.

CHAPTER 10

The Discovery of General Relativity

FUNNY GEOMETRY