The End of Time: The Next Revolution in Physics

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.

PLATONIA FOR RELATIVITY