Many more things could be said about general relativity and its discovery. However, what I want to do now is identify the aspects of the theory and the manner of its discovery that have the most bearing on time.
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.
At the end of November 1915, Einstein wrote an ecstatic letter to his lifelong friend Michele Besso, telling him that his wildest dreams had come true: ‘General covariance. Mercury’s perihelion with wonderful accuracy.’ These two verbless sentences say it all. Einstein was convinced that general covariance had deep physical consequences and had led him to one of the greatest triumphs of all time. Yet, barely two and a half years later, he admitted, in response to a quite penetrating criticism from a mathematician called Erich Kretschmann, that general covariance had no physical significance at all.
In a way, this is obvious. Space-time is a beautiful sculpture. What makes it beautiful is the way in which its parts are put together. The fact that one can paint coordinate lines on the finished product and measure distance on the sculpture between points on it labelled by the arbitrary coordinates clearly leaves the sculpture exactly the same. All this changing of coordinates is purely formal. It tells you nothing about the true rules that make the sculpture.
Belatedly, Einstein came to see that his whole drive to achieve general covariance as a deep physical principle had no foundation in fact. It was just a formal mathematical necessity. Ever determined to find new and even more beautiful laws of nature, he never felt the need to go back and see exactly how his sculpture was actually created. In a book I wrote some years ago on the discovery of dynamics, I commented on the fact that Kepler (so very like Einstein in his dogged holding on to an idea that eventually transformed physics) never realized quite what a wonderful discovery he had made. I likened him to
a boy who finds for the first time a ripe horse-chestnut with the outer shell intact. Cherishing the golden and curiously shaped object, he might take it home, quite unaware of the shiny brown and perfectly smooth conker ready to spring from the shell on application of a little directed pressure. That was Kepler’s fate: he died without an inkling of what his nut really contained.
The same thing happened to Einstein. Realizing while still at Prague the sort of thing he needed, he hurried to a shop called ‘Mathematics’ owned by his friend Grossmann in Zurich. Straight off the shelf, at a bargain price, he bought a wonderful device called the Ricci tensor. Three years later, after agonizing struggles, he learned how to turn the handles properly, and out popped the advance of Mercury’s perihelion and the exact light deflection at eclipses.
But it never entered his head to ask how the device actually worked. He died only half aware of the miracle he had created.
Einstein’s Way to General Relativity (p. 151) Einstein’s papers and correspondence are currently being published (with translations into English) by Princeton University Press. The letter to his wife mentioned in this section can be found in the first volume of correspondence (Stachel et al. 1987).