The End of Time: The Next Revolution in Physics

One of the big unsolved problems of physics is the origin of distance and whether it is absolute. Since we need a measuring rod to measure any distance, this suggests rather strongly that distance is relative (to the chosen rod). If we tried to double every distance in the universe, the length of the rod would be doubled too, and nothing actually observable would be changed. For reasons like this, physicists have a hunch that absolute scale should have no objective meaning. However, this is not confirmed by existing theories and experimental facts, which do suggest that distance is in a well-defined sense absolute. The hope remains that a physics completely without scale will be found. If so, Shape Space gives an idea of what the corresponding Platonia will be like. There is still a uniquely distinguished point in it – the central point in Figure 8. It takes the place of Alpha in Triangle Land. If you ‘touched’ the central point, the equilateral triangle would ‘light up’. This is the most symmetrical configuration the three-body universe can have. Symmetry is beautiful in one way but bland in another. The boundaries of Shape Space are somewhat unappealing too, because they represent improper triangles that flatten into a line (mathematicians call such configurations, collinear in this case, degenerate). The vertices of Shape Space correspond to collinear configurations in which one particle is infinitely far from the other two. The interesting structures in this case lie between the bland centre and the degeneracy of the frontiers.

In modern theoretical cosmology, distance is absolute and the universe expands. For reasons that are not yet understood, it simultaneously becomes more richly structured. In a cosmology without both time and scale, this would correspond, in a realistic scale-free Platonia, to going from the bland centre to the more interestingly structured ‘instants of time’ situated between it and the frontiers. That is where the mist I introduced in Chapter 3 must collect most thickly – have the highest intensity – at time capsules structured so that they seem to record evolution from the symmetric centre. This would be a cosmology of pure structure, an appealing thought. The scalene and obtuse triangles that inhabit the ‘favoured belt’ in Shape Space remind me of the line in Gerard Manley Hopkins’s poem ‘Pied Beauty’, in which he praises ‘All things counter, original, spare, strange’. In such a scheme, the bland centre and degenerate frontiers still have a vital role to play in the scheme of things. This is because a kind of resonance between all the instants of time determines where the mist settles. Any acoustician will recognize the importance of the walls and centre of a building in determining its harmonies. Platonia, shown here as Shape Space, is a ‘heavenly vault’ in which the music of the spheres is played. However, I should emphasize that the more realistic Shape Spaces corresponding to universes with more than three particles are most definitely ‘open ended’ and should not be thought of as enclosed spaces, as might appear from the simple example of Triangle Shape Space. Platonia is not a claustrophobic vault but an ‘echoing canyon’ open to the sky. What is more, there is a sense in which its echo, heard at any point within it, is what we call the past.

In completing this box, I note it has a nice unplanned symbolism. Box 3 considers Shape Space, which is represented by the perfect (equilateral) triangle, which itself, through each of its points, represents all triangles, all of which are unities in the sense mentioned on p.18. Shape Space illustrates Giordano Bruno’s monas monadum, the unity of the unities.

If Laplace’s divinity contemplates a three-particle universe, its history will be a curve in Triangle Land, which, omitting the triangle size, we can show as a curve in Shape Space. The example of real Newtonian three-body gravitational interaction shown in Figure 9 brings out very clearly the main fact that we associate with time, that its instants come in a unique succession. This translates beautifully into the winding path. (You can see why I am so indebted to Dierck Liebscher, who crafted this diagram.) However, the other two important attributes of time, duration and direction, are not yet reflected in Figure 9. There are no marks along the curve to indicate how much time elapses between any two points on it. It is also impossible to say in which direction along the curve time increases.

What information must we give to determine a history of the kind encoded in the path in Figure 9? According to Laplace, simply the positions and velocities of the bodies at one instant (together with their masses). Poincaré remarked, however, that the positions and motions of the bodies are defined in absolute space, and the speeds of the bodies are defined using absolute time. This is a subtle and important qualification.