The End of Time: The Next Revolution in Physics

This is an arbitrary quantity, since the relative positioning of the two triangles is arbitrary. It is, however, possible to consider all relative positionings and find the one for which d is minimized. This is a very natural quantity to find, and it is not arbitrary. Two different people setting out to find it for the same two triangles would always get the same result. It measures the intrinsic difference between the two matter distributions represented by the triangles. It is completely determined by them, and does not rely on any external structure like absolute space.

The intrinsic difference between two arbitrary matter distributions can be found similarly. One distribution is supposed fixed, and the other moved relative to it. In any trial position, the analogue of the above expression is calculated, and the position in which it is minimized is sought. Because this special position reduces the apparent difference between the two matter configurations to a minimum, it may be called the best-matching position.

Figure 21 A trial relative placing of the two triangles.

Using the intrinsic difference defined in Box 8, we can determine ‘shortest paths’ or ‘histories’ in Platonia as explained above. However, the intrinsic difference by itself does not lead to very interesting histories, and it is more illuminating to consider a related quantity. The potential energy of any matter distribution (Figure 17) is determined by its relative configuration, and is therefore already ‘Machian’. Each matter distribution has its own Newtonian gravitational potential energy. Two nearly identical matter distributions have almost the same potential. Now, the intrinsic difference is determined by two nearly identical configurations. To obtain more interesting histories we can simply multiply the intrinsic differences by the potential (strictly speaking, by the square root of minus the potential). This will change the definition of ‘distance’, but it will still enable us to determine ‘shortest distances’. You do not need to worry about these details, but I do want to give you a flavour of what is involved.

I think you will agree that finding shortest paths in an imagined timeless landscape bears little direct resemblance to our powerful sense of the passage of time. Yet the outcome turned out to be remarkably like what happens in Newtonian theory. Let me explain, taking again the example of a three-body universe, for which Platonia is Triangle Land.

Any continuous path in it corresponds to a sequence of triangles: they are the ‘points’ through which the path passes. But this is very similar to what comes out of Newtonian theory (Figure 1) – which, however, yields not only the triangles but also their positions in absolute space and separations in time. Remember that the triangles in Figure 1 are ‘lit up’ by flashes at each unit of absolute time, and that we see them, in perspective, in absolute space at those times. However, these are invisible aids. The astronomers see neither when they look through telescopes, all they see are stars. Thus, as far as observable things are concerned, both theories yield the same kind of thing – sequences of triangles. The question is, what kind of sequences do the two theories yield? In what respects do the theories differ when it comes down to what is actually observable?

The major difference is that the Machian theory makes more definite predictions. As a theory of geodesics, it determines the shortest path between any two fixed points in Platonia. It covers Platonia with such paths. These geodesics have the following important property: at any one point in Platonia, many of them pass through it. In fact, for every direction that you can go from a point, there is just one geodesic. This is the crucial difference. As Figures 13 and 14 highlighted, when the Newtonian histories are represented as paths in Platonia, it turns out that many can pass through the same point, and have at that point the same direction. However, these paths then ‘splay out’ and go to quite different places in Platonia. In Newtonian terms, they differ in energy and angular momentum. The difference is not apparent in the initial point and direction, but comes to light dramatically in the later evolution of the paths. This defect is absent in the Machian theory. For any given point and direction at that point, there is just one geodesic. Bruno and I constructed the theory precisely to achieve that aim.

What did quite surprise us was to discover that the unique Machian history with a given direction through a point is identical to one of the many Newtonian histories through the point with the same direction. It is, in fact, the Newtonian history for which the energy and angular momentum are both exactly zero. The small fraction of Newtonian solutions with this property are all the solutions of a simpler timeless and frameless theory.