Julian Barbour - The End of Time - The Next Revolution in Physics

Several profound ideas are unified and taken to the extreme in Figure 31: Einstein’s relativity of simultaneity, Minkowski’s fusion of time with space, Poincaré’s idea that the relativity principle should be realized through perfect Laplacian determinism, Poincaré’s idea that duration is defined so as to make the laws of nature take the simplest form possible, and the astronomers’ realization that it is measured by an average of everything that changes. Since best matching in general relativity holds throughout the universe in all conceivable directions, both time and space appear as the distillation of all differences everywhere in the universe. Machian relationships are manifestly part of the deep structure of general relativity. But are they the essential part?

If the world were purely classical, I think we would have to say no, and that the unity Minkowski proclaimed so confidently is the deepest truth of space-time. The 3-spaces out of which it can be built up in so many different ways are knitted together by extraordinarily taut interwoven bonds. This is where the deep dilemma lies. Four decades of research by some of the best minds in the world have failed to resolve it. On the one hand, dynamics presupposes – at the foundation of things – three-dimensional entities. Knowing nothing about general relativity, someone like Poincaré could easily have outlined a form of dynamics that was maximally predictive, flexible, refined and made no use of eternal space or time. Such dynamics, constrained only by the idea that there are distinct things, must have a certain general form. A whole family of theories can be created in the same Machian mould.

On the other hand, a truly inspired genius might just have hit on one further condition. Let dynamics do all those things with whatever three-dimensional entities it may care to start from. But let there be one supreme overarching principle, an even deeper unity. All the three-dimensional things are to be, simultaneously with all their dynamical properties, mere aspects of a higher four-dimensional unity and symmetry.

If certain simplicity conditions are imposed, only one theory out of the general family meets this condition. It is general relativity. It is this deeper unity that creates the criss-cross fabric of space-time and the great dilemma in the creation of quantum gravity. As we shall see, quantum mechanics needs to deal with three-dimensional things. The dynamical structure of general relativity suggests – and sufficiently strongly for Dirac to have made his ‘counter-revolutionary’ remark – that this may be possible. Yet general relativity sends ambivalent signals. Its dynamical structure says ‘Pull me apart’, but the four-dimensional symmetry revealed by Minkowski says ‘Leave me intact.’ Only a mighty supervening force can shatter space-time.

Note added for this printing . New work summarized on p. 358 could significantly change the situation discussed in this final section of the chapter. It suggests that the timeless Machian approach is capable of leading to a complete derivation of general relativity and that it is not necessary to presuppose ‘a higher tour-dimensional unity and symmetry.’ Since this new work has only just been published and has not yet been exposed to critical examination, I decided to leave the original text intact. However, as already indicated in the note at the end of the Preface, this new work does have the potential to strengthen considerably the arguments for the nonexistance of time.

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.

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.