Julian Barbour - The End of Time - The Next Revolution in Physics

In Figure 25, the original grid is ‘painted’ onto space-time with the dashes, while the dots show an alternative. As we saw, the law of nature that describes the behaviour of light pulses allows them to travel along the diagonals of either grid. A transformation of this law from one coordinate grid to another is called a Lorentz transformation , and the grids themselves are called Lorentz frames . I have already mentioned that you should not think in terms of there being one rectangular coordinate grid, and all the others oblique. Alice thinks Alice* has an oblique system compared to her, but Alice* thinks the same about Alice’s system. This is a consequence of the relativity principle, and a special property of space-time that we shall come to shortly. Minkowski pointed out that the transformation shown in Figure 25 is a kind of rotation in four dimensions. The possibility of making rotations in ordinary space is a deep reflection of its unitary, block-like nature. Minkowski saw the possibility of making a kind of rotation in space-time, which is impossible in Newtonian space-time, as the clearest evidence for the intimate fusion of space and time, even though the need for ‘articulation’ showed that time was still somewhat different in nature from space.

Einstein, Minkowski and others were able to show that all the laws of nature known in their time (except initially for gravitation) either already had a form that was exactly the same in all Lorentz frames or could be relatively easily modified so that they did. Even though the modifications were relatively easy once the idea was clear, their implications, including Einstein’s famous equation E = mc 2 (a prediction at that time), were mostly very startling. Minkowski, like Einstein and Poincaré, made a strong prediction that all laws of nature found in future would accord with the relativity principle, and emphasized that the guiding principle for finding such laws was to treat time exactly as if it were space.

Except for the intermingling of space and time and the distinguished role played by light, Minkowski’s space-time strongly resembles Newtonian space-time. Matter neither creates nor changes its rigid and absolute structure. It is like a football field, complete with markings, on which the players must abide by rules they cannot change.

It is often said that relativity destroyed the concept of Now. In Newtonian physics the axes can never be tilted as they are in Figure 25. The simultaneity levels stay level, and there is a unique sequence of instants of time, each of which applies to the complete universe. This is overthrown in relativity, where each event belongs to a multitude of Nows. This has important implications for the way we think about past, present and future.

Even in Newtonian theory we can picture world history laid out before us. In this ‘God’s-eye’ view, the instants of time are all ‘there’ simultaneously. The alternative idea of a ‘moving present’ passing through the instants from the past to the future is theoretically possible but impossible to verify. It adds nothing to the scientific notion of time. Special relativity makes a ‘moving present’ pretty well untenable, even as a logical possibility.

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.