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

To explain the problem, let me first recall what best matching does and achieves in the Newtonian case of a large (but fixed) number of particles. Each instant of time, each Now, is defined by a relative configuration of them in Euclidean space. We modelled each Now as a ‘megamolecule’, and compared two such Nows, without reference to any external space or time, by moving one relative to the other until they were brought as close as possible to coincidence as measured by a suitable average. This is where the real physics resides, since the residual difference between the Nows in the best-matching position defines the ‘distance’ between them in Platonia. Once we possess all such ‘distances’ between neighbouring Nows, we can determine the geodesics in Platonia that correspond to classical Machian histories. Besides defining these ‘distances’, the best matching automatically brings the two Nows into the position they have in Newton’s absolute space, if we want to represent things in that way.

However, to complete that Newtonian-type picture, we have still to determine ‘how far apart in time’ the two Nows are. This is the problem of finding the distinguished simplifier , the time separation that unfolds the dynamical history in the simplest or most uniform way. As we saw in the final section of Chapter 6, in the discussion of ephemeris time, the choice of distinguished simplifier is unique if we want to construct clocks that will enable their users to keep appointments. Our ability to keep appointments is a wonderful property of the actual world in which we find ourselves, and we must have a proper theoretical understanding of its basis. This is achieved if we insist that a clock is any mechanism that measures, or ‘marches in step with’, the distinguished simplifier. This is the theory of duration and clocks that Einstein never addressed explicitly. However, the most important thing is that history itself is constructed in a timeless fashion. The distinguished simplifier is introduced after the event to make the final product look more harmonious. Duration is in the eye of the beholder.

In Newtonian best matching, the compared Nows are moved rigidly relative to one another. We could conceive of a more general procedure, but since the Nows are defined by particles in Euclidean space its flatness and uniformity make that an additional complication. We should always try to keep things simple.

However, if we adopt curved three-dimensional spaces, or 3-spaces as they are often called, as Nows, any best-matching procedure for them will have to use a more general pairing of points between Nows. For example, two 3-spaces (which may or may not contain matter) may have different sizes. It will then obviously be impossible to pair up all points as if they were sitting together in the same space. More generally, the mere fact that both spaces are curved – and curved in different ways – forces us to a much more general and flexible method for achieving best matching.

In a talk, I once illustrated what has to be done by means of two magnificent fungi of the type that grow on trees and become quite solid and firm. For reasons that will become apparent, I called them Tristan and Isolde. Tristan was a bit larger than Isolde, and both were a handsome rich brown, the darkness of which varied over their curved and convoluted surfaces. I wanted to explain how one could determine a ‘difference’ between the two by analogy with the best-matching for mass configurations in flat space. In some way, this would involve pairing each point on Tristan to a matching point on Isolde. A little reflection shows that the only way to do this is to consider absolutely all possible ways of making the matching.

I took lots of pins, numbered 1, 2, 3, ..., and stuck them in various positions into Tristan. I then took a second set, also numbered 1, 2, 3, ..., and stuck them into Isolde. Since they had similar shapes, I placed the pins in corresponding positions, as best as I could judge. I could then say that, provisionally, pin 1 on Tristan was ‘at the same position’ as pin 1 on Isolde. All the other points on them were imagined to be paired similarly in a trial pairing .

This made it possible to determine a provisional difference . For example, I could compare the two fungi using the darkness of their brown surfaces. Alternatively, and much closer to what happens in general relativity, I could compare the curvatures at matching points. The essential point is that some intrinsic property is compared at each pair of matched points, and an average of all the resulting differences is then determined. This average, one number, is the provisional difference. I leave out the mathematical details, which are intricate even though the underlying idea is simple.

This provisional difference is clearly arbitrary since the pairing on which it is based could have been made differently. To find an intrinsic difference that can have real physical meaning, we must now embark – in imagination at least – on an immensely laborious task. Keeping the pins on Tristan fixed, we need to rearrange the pins (reasonably continuously so that the mathematics works) on Isolde in every conceivable way. For each trial pairing of all points on Isolde to all points on Tristan, we must find the provisional difference. We shall know that we have found the best-matching pairing and corresponding intrinsic difference when the provisional difference remains unchanged if we go from the given pairing to any other pairing that differs from it ever so slightly. (In mathematics, the fulfilment of this condition indicates that one has found a maximum, a minimum or a so-called stationary point of the quantity being considered. It turns out that a stationary point is what is found in this case, but that is a mere technicality.) Since there is an immense – indeed infinite – number of ways of changing the pairings, the best-matching requirement imposes a very strong condition. It is impossible to conceive of a more refined and delicate comparison of two things that are different but of the same kind. However, as Bruno and I realized, it is made necessary by the nature of the compared things.

It leads immediately to the ne plus ultra of best matching – and rationality.

It was around 1979 that Bruno and I developed the new best-matching idea. We did quite a lot of technical work, and were beginning to get quite hopeful. We knew that we could construct various forms of Machian geometrodynamics, and we began to think that one of them might be a serious rival to general relativity. But it is not easy to beat Einstein, as we were soon to find. This came about through the intervention of another friend, Karel Kuchař, whom I had got to know in 1972, when we had several discussions. Karel is Czech and studied physics at the Charles University in Prague, specializing in relativity. In 1968 he won an award to study at Princeton with John Wheeler, where he quickly established himself as a leading expert in the canonical quantization of gravity (the most straightforward quantization procedure (Box 2) that can be used in the attempt to quantize gravity), in which Dirac and ADM had been the pioneers. Some years later he became a professor of physics at the University of Utah in Salt Lake City, where he still works. Over the years I have profited greatly from discussions with Karel, and certainly would not have been in the position to write this book without assistance from him at some crucial points. However, I hasten to add that Karel is sceptical about my idea that time does not exist at all. As we shall see, general relativity presents a great dilemma. Karel gives more weight to one Born of this dilemma, I to the other.

The issue came into clear focus for me in 1980. In April of that year, Karel gave a memorable review talk at an international conference in Oxford, during which I had an opportunity to discuss with him the ideas that Bruno and I were developing. He invited me to come to Salt Lake City, which I did in the late fall, just in time to see the pale gold of the aspens in the Wasatch mountains. Getting to know Utah and the magnificent deserts of the western United States has been a great bonus from the study of physics for me and my family. But as this is a book about physics, not travel, I had better not digress.