The End of Time: The Next Revolution in Physics

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).

CHAPTER 11: GENERAL RELATIVITY: THE TIMELESS PICTURE

Platonia for Relativity (p. 167) This is a technical note about the definition of superspace. The equations of general relativity lead to a great variety of different kinds of solution, including ones in which there are so-called closed time-like loops. These are solutions in which a kind of time travel seems to be possible. The question then arises of whether a given solution of general relativity—that is, a space-time that satisfies Einstein’s equations—can be represented as a path in superspace, in technical terms, as a unique succession of Riemannian three-geometries. If this is always so, then superspace does indeed seem a natural and appropriate concept. Unfortunately, it is definitely not so. There are two ways in which we can attempt to get round this difficulty. We could say that classical general relativity is not the fundamental theory of the universe, since it is not a quantum theory. This allows us to argue that superspace is the appropriate quantum concept and that it will allow only certain ‘well-behaved’ solutions of general relativity to emerge as approximate classical histories. For these, superspace will be an appropriate concept. Alternatively, we could extend the definition of super-space to include not only proper Riemannian 3-geometries (in which the geometry in small regions is always Euclidean), but also pscudo-Riemannian 3-geometries (in which the local geometry has a Minkowski type signature), and also geometries in which the signature changes within the space. For the reasons given in the long note starting on p. 348 below, I prefer the second option.

The above note was written before my new insights mentioned at the end of the Preface. I now believe that there is a potentially much more attractive resolution of the difficulty: the true arena of the world is not superspace but conformal superspace, which I describe on p. 350.

Catching Up with Einstein (1) (p. 175) Figure 30 is modelled directly on well-known diagrams in Wheeler (1964) and Misner et al. (1973).

(2) Technical note: Einstein’s field equations relate a four-dimensional tensor formed from geometrical quantities to the four-dimensional energy-momentum tensor, which is formed from the variables that describe the matter. Machian geometrodynamics shows how these four-dimensional tensors are built up from three-dimensional quantities. The two principles by which this is done are best matching, and Minkowski’s rule that the space and time directions must be treated in exactly the same way (see the following note). As far as I know, the mathematics of how this is done when matter is present was first spelled out in a recent paper by Domenico Giulini (1999), to whom I am indebted for numerous discussions on this and many other topics covered in this book.

A Summary and the Dilemma (1) (p. 177) This is another technical note. My image of space-time as a tapestry of interwoven lovers rests on the following property of Einstein’s field equations. If, in any given space-time that is a solution of the field equations, we lay out an arbitrary four-dimensional grid in any small region of the space-time, we can then, in principle, attempt to take the data on one three-dimensional hypersurface and use Einstein’s equations to evolve these data and recover the space-time in the complete region. Normally, we attempt to do this in a time-like direction. However, the form of the equation is exactly the same whichever direction in which we choose to attempt the evolution from initial data. This is an immediate consequence of an aspect of the relativity principle that Minkowski gave a special emphasis: as regards the structure of the equations, whatever holds for space holds for time and vice versa.