The End of Time: The Next Revolution in Physics

First, the classical (non-quantum) theory as it stands seems to make nonsense of my claim that time does not exist. The space-time of general relativity really is just like a curved surface except that it has four and not two dimensions. A two-dimensional surface you can literally see: it is a thing extended in two dimensions. In their mind’s eye, mathematicians can see four-dimensional space-time, one dimension of which is time, just as clearly. It is true that time-like directions differ in some respects from space-like directions, but that no more undermines the reality of the time dimension than the difference between the east-west and north-south directions on the rotating Earth makes latitude less real than longitude. However, the qualification ‘as it stands’ at the start of this paragraph is important. In the next chapter we shall see that there is an alternative, timeless interpretation of general relativity.

Next, there is the matter of the distinguished coordinate systems. In one sense, Einstein did abolish them. Picture yourself in some beautiful countryside with many varied topographic features. They are the things that guide your eye as you survey the scene. The real features in space-time are made of curvature, and hills and valleys are very good analogies of them. Imagined grid lines are quite alien to such a landscape. In general relativity, the coordinated lines truly are merely ‘painted’ onto an underlying reality, and the coordinates themselves are nothing but names by which to identify the points of space-time.

For all that, space-time does have a special, sinewy structure that needs to be taken into account. Distinguished coordinate systems still feature in the theory. This is because the theory of measurement and the connection between theory and experiment is very largely taken over from special relativity. In fact, much of the content of general relativity is contained in the meaning of the ‘distance’ that exists in space-time. This is where the analogy between space-time and a landscape is misleading. We can imagine wandering around in a landscape with a ruler in our pocket. Whenever we want to measure some distance, we just fish out the ruler and apply it to the chosen interval. But measurement in special relativity is a much more subtle and sophisticated business than that. In general, we need both a rod and a clock to measure an interval in space-time. Both must be moving inertially in one of the frames of reference distinguished by that theory, otherwise the measurements mean nothing. The theory of measurement in general relativity simply repeats in small regions of space-time what is done in the whole of Minkowski space-time in special relativity. No measurements can be contemplated in general relativity until the special structure of distinguished frames that is the basis of special relativity has been identified in the small region in which the measurements are to be made.

This is something that is often not appreciated, even by experts. It comes about largely because of the historical circumstances of the discovery of general relativity and the absence of an explicit theory of rods and clocks. There is also the stability of our environment on the Earth and the ready availability in our age of clocks. It is easy for us to stand at rest on the Earth, watch in hand, and perform a measurement of a purely timelike distance. But nature has given us the inertial frame of reference for nothing, and skilful engineers made the watch. Finally, because we can and very often do see a three-dimensional landscape spread out before our eyes, it is very easy to imagine four-dimensional space-time displayed in the same way. All textbooks and popular accounts of the subject positively encourage us to do so. They all contain ‘pictures’ of space-time. Now the picture is indeed there, and very wonderful it is too. But it arises in an immensely sophisticated manner hidden away within the mathematical structure of the Ricci tensor. The story of time as it is told by general relativity unfolds within the Ricci tensor. It performs the miracle – the construction of the cathedral of space-time by intricate laying and interweaving of the bricks of time. I shall try to explain this in qualitative terms in the next chapter. Let me conclude this one by highlighting again the importance of the historical development. It made possible the discovery of a theory without full appreciation of its content.