The End of Time: The Next Revolution in Physics

What is more, however we choose the ‘direction of attempted evolution’, Einstein’s equations always have a very characteristic structure. There are ten equations in all. One of them does not contain any derivative with respect to the variable in which we are going to attempt the evolution. Three of them contain only first derivatives with respect to that variable. The remaining six equations contain second derivatives with respect to it and have the form of equations that are suitable for evolution in the chosen direction. But we must first solve the other four equations, which are so-called constraints. Unless the initial data satisfy these four equations, evolution is impossible.

There are two ways to look at a space-time that satisfies Einstein’s equations: either as a structure obtained from initial data that have been (somehow) obtained in a form that satisfies the constraints and then built by the more or less conventional evolution equations, or as a structure that satisfies everywhere the constraints however we choose to draw the coordinate lines. In the second way of looking at space-time, conventional evolution does not come into the picture at all. Much suggests that this is the more fundamental way of looking at Einstein’s equations (see, in particular, Kuchaf’s beautiful 1992 paper).The connection with my timeless way of thinking about general relativity is expressed by the fact that the three constraint equations containing only first derivatives of the evolution variable are precisely the expression of the fact that a best-matching condition holds along the corresponding ‘initial’ hypersurface, while the fourth constraint equation, containing no derivatives of the evolution variable, expresses the fact that proper time is determined in geometrodynamics as a local analogue of the astronomers’ ephemeris time. It is this complete freedom to draw coordinate lines as we wish and, at least formally, to attempt evolution in any direction, that makes me feel that the second alternative envisaged in the Platonia for Relativity note is appropriate. I think it is also very significant that Einstein’s equations have the same form whatever the signature of space-time. The signature is not part of the equations, it is a condition normally imposed on the solutions. The demonstration that Einstein’s general relativity is the unique theory that satisfies the criterion (mentioned at the end of this section) of a higher four-dimensional symmetry was given by Hojman et al. (1976).

1 mentioned on p. 346 at the end of the notes on Chapter 4 my recent discovery of a way to create dynamical theories of the universe in which absolute distance is no longer relevant. My Irish colleague Niall Ó Murchadha, of University College Cork, and I are currently working on the application of the new idea to theories like general relativity, in which geometry is dynamical. There is a possibility that this work will not only give new insight into the structure of general relativity, in which a kind of residual absolute distance does play a role, but also lead to a rival alternative theory in which no distance of any kind occurs.

The key step is to extend the principle of best matching from superspace to so-called conformal superspace. In the context of geometrodynamics, this is analogous to the passage from Triangle Land to Shape Space as described in Box 3. However, whereas in Box 3 it is only the overall scale that is removed, and it is still meaningful to talk about the ratios of lengths of sides, the transition to conformal superspace is much more drastic and removes from physics all trace of distance comparison at spatially separated points.

In more technical terms, for people in the know, each point of conformal superspace has a given conformal geometry and is represented by the equivalence class of metrics related by position-dependent scale transformations.

The potentially most interesting implication of this work is that it could resolve the severe problem of the criss-cross fabric of space-time illustrated by Figure 31. At the level of conformal superspace, the universe passes through a unique sequence of states. For latest developments, please consult my website (www.julianbarbour.com) and the final entries in these notes and the notes on p.358.

CHAPTER 12: THE DISCOVERY OF QUANTUM MECHANICS

(p. 191) On the connection between particles and fields, let me mention here that I assume the appropriate ‘Platonic’ representation at the level of quantum field theory to be in terms of the states of fields, not particles.

CHAPTER 13: THE LESSER MYSTERIES

(p. 202) Wheeler and Zurek (1983) have published an excellent collection of original papers on the interpretational problems of quantum mechanics.

CHAPTER 14: THE GREATER MYSTERIES