CHAPTER 11
General Relativity: The Timeless Picture
Strange as it may seem, general relativity was little studied for about forty years. This was not for want of admiration, for it was soon recognized as a supreme achievement. Confirmation of the predicted bending of starlight near the Sun by Arthur Eddington’s eclipse expedition in 1919, communicated by telegram to The Times, made Einstein into a world celebrity overnight. The problem was that there seemed to be little one could do except wonder at the miracle of the theory he had created.
The main difficulty was the extreme weakness of all readily accessible gravitational fields. Apart from three small differences from Newtonian theory, which were all reasonably well confirmed, no further experimental tests seemed possible. A further problem was the mathematical complexity of the theory. Its solutions contained fascinating structures, above all black holes, but it was decades before these were discovered and fully understood. Finally, interest in general relativity was overshadowed by the discovery in 1925/6 of quantum mechanics. In fact, truly active research in general relativity commenced only in 1955, ironically the year Einstein died, with a conference held in Bern (where Einstein had worked as a patent clerk in 1905) to mark the fiftieth anniversary of special relativity.
Since then, research has concentrated in three main fields. First, there have been tremendous experimental advances, made possible above all by technological developments, including space exploration. The foundations and some detailed predictions of the theory have been tested to a very high accuracy. Particularly important was the discovery a quarter of a century ago of the first binary pulsar, observations of which have provided strong evidence for the existence of the gravitational waves predicted by the theory. General relativity has also played a crucial role in observational astronomy and cosmology.
There have been two broad avenues of theoretical research. First, general relativity has been studied as a classical four-dimensional geometrical theory of space-time, a systematical and beautiful development of Minkowski’s pioneering work. Roger Penrose has probably done more than anyone else in this field, though many others, including Stephen Hawking, have made very important contributions. Second, the desire to understand the connection between general relativity and quantum mechanics (Box 2) has stimulated much work. Here it is necessary to distinguish two programmes. The less ambitious one accepts space-time as a classical background and seeks to establish how quantum fields behave in it. This work culminated in the amazing discovery by Hawking that black holes have a temperature and emit radiation. In Black Holes and Time Warps, Kip Thorne has given a gripping account of this story. Although the full significance of Hawking’s discovery is still far from understood, nobody doubts its importance for the more ambitious programme, which is to transform general relativity itself into a quantum theory (Box 2). This transformation, which has not yet been achieved, is called the quantization of general relativity.
In fact, many researchers believe that it is a mistake to try to quantize general relativity directly before gravity has been unified with the other forces of nature. This they hope to achieve through superstring theory. However, a substantial minority believe that general relativity contains fundamental features likely to survive in any future theory, and that a direct attempt at its quantization is therefore warranted. This is my standpoint. In particular, I regard general relativity as a classical theory of time. It must surely be worth trying to establish its quantum form. Even if we have to await a future theory for the final details, the quantization of general relativity should give us important hints about the quantum theory of time.
It was the desire to quantize general relativity that led to the work described in this chapter. One important approach, called canonical quantization, is based on analysis of the dynamical structure of the classical theory. This is how general relativity came to be studied in detail as a dynamical theory nearly half a century after its creation as a geometrical space-time theory. The ‘hidden dynamical core’, or deep structure, of the theory was revealed. The decisive analysis was made in the late 1950s by Paul Dirac and the American physicists Richard Arnowitt, Stanley Deser and Charles Misner. They created a particularly elegant theory, now known universally as the ADM formalism. (Because it is regarded as controversial by some, the initials are occasionally reshuffled as MAD or DAM.)
The dynamical form of general relativity is often called geometrodynamics. The term, like ‘black hole’ and several others, was coined by John Wheeler, who, together with his many students at Princeton, did much to popularize this form of the theory. The interpretation of it proposed in this chapter is very close to one put forward by Wheeler in the early 1960s. However, I believe it brings out the essentially timeless nature of general relativity rather more strongly than Wheeler’s well-known writings of that period. What is at stake here is the plan of general relativity. What are its ultimate elements when it is considered as a dynamical theory, and how are they put together?
This is what Dirac and ADM set out to establish. The answer was manifestly a surprise for Dirac at least, since it led him to make the remarkable statement quoted in the Preface. They found that if general relativity is to be cast into a dynamical form, then the ‘thing that changes’ is not, as people had instinctively assumed, the four-dimensional distances within space-time, but the distances within three-dimensional spaces nested in space-time. The dynamics of general relativity is about three-dimensional things: Riemannian spaces.
To connect this with the topics of Part 2, let me tell you about the work that Bruno Bertotti and I did after the work described there. We began to wonder whether we could be more ambitious and construct not merely a non-relativistic, Machian mechanics, but perhaps an alternative to general relativity. At the time, we believed that Einstein’s theory did not accord with genuine Machian principles. Experimental support for it was beginning to seem rather convincing, but tiny effects have often led to the replacement of a seemingly perfect theory by another with a very different structure. We were aware of quite a lot of the work of Wheeler and ADM, and various arguments persuaded us that the geometry of three-dimensional space might well be Riemannian, possess curvature and evolve in accordance with Machian principles. We wanted to find a Machian geometrodynamics, which we did not think would be general relativity. The first task was to select the basic elements of such a theory. What structures should represent instants of time and be the points of the theory’s Platonia?
This question was easily answered. Any class of objects that differ intrinsically but are all constructed according to the same rule can form a Platonia. So far, we have considered relative configurations of particles in Euclidean space. There is nothing to stop us considering three-dimensional Riemannian spaces, especially if they are finite because they close up on themselves. This is difficult for a non-mathematician to grasp, but the corresponding things in two dimensions are simply closed, curved surfaces like the surface of the Earth or an egg. The points of Platonia for this case are worth describing. The surface of any perfect sphere is one point; each sphere with a different radius is a different point. Now imagine deforming a sphere by creating puckers on its surface. This can be done in infinitely many ways. There can be all sorts of ‘hills’ and ‘valleys’ on the surface of a sphere, just as there are on the Earth and the Moon. And there is no reason why the surface should remain more or less sphericaclass="underline" it can be distorted into innumerable different shapes to resemble an egg, a sausage or a dumbbell. On all of these there can be hills and valleys. Each different shape is just one point in Platonia, and could be a model instant of time. In this case you can form a very concrete image of what each point in Platonia looks like. These are things you could pick up and handle. Note that only the geometrical relationships within the surface count. Surfaces that can be bent into each other without stretching, like the sheet of paper rolled into a tube, count as the same. However, this is a mere technicality. The important thing is that the points of any Platonia are real structured things, all different from one another.