The End of Time: The Next Revolution in Physics

Playing the role of Laplace’s divinity, we place the first triangle, at the instant when Newton’s grandstand clock says it is noon, at some position in absolute space. A second later, we place the second triangle somewhere near it in a slightly different position. The first triangle defines the initial positions of the three bodies. Given the position of the second triangle one second later, we can calculate the initial motions, since we know where the particles have gone and how long it took them. (Strictly, to calculate the instantaneous velocities we must take an infinitesimal time interval, not one second, but that is a minor detail). Imagine now that a strobe light illuminates the bodies with a flash once every second, corresponding to the seconds ticking on Newton’s clock, so that we can watch how the triangle formed by the bodies moves through absolute space. We have seen this already, in Figure 1. We can also plot the points corresponding to the triangles in either Triangle Land or Shape Space, obtaining a curve like those in Figures 9 and 10. This abstracts away the extra Newtonian information – the positions in absolute space and the time separations – that we possessed originally.

Now, wherever we place the two triangles, the resulting curves in either Triangle Land or Shape Space will all start at the same point, since we always begin with the same triangle, and that corresponds to just one fixed point in Platonia. The curve must also have the same initial direction, since that is determined by the position of the second triangle in Platonia, which is also fixed. This is explained in the caption to Figure 10. The question is, how does the curve run after that? What effect do the positioning of the first two triangles in absolute space and the time separation have on the subsequent evolution?

To answer this question, we need the notion of centre of mass (Box 4). For a given triangle, there are two different things to bear in mind when it comes to placing it in absolute space. First, we can place its centre of mass anywhere. Since space has three dimensions, this means that we can shift the centre of mass along three different directions. Physicists say that in such cases there are three degrees of freedom. Second, holding the centre of mass fixed, we can change the orientation of the triangle in space. This introduces three more degrees of freedom. To see this, picture an arrow passing through the centre of mass perpendicular to the triangle. It will point to somewhere on the two-dimensional sky, giving two degrees of freedom. The third arises because one can, keeping the arrow fixed, rotate the triangle around it as an axis.

BOX 4 Centre of Mass

The centre of mass of a system of bodies is the position of a fictitious mass equal to the sum of the masses of the system. For two unequal masses m and M, it lies on the line joining them at the position that divides the line in the ratio M/m – that is, closer to the heavier mass in that proportion. For any isolated system of bodies, the centre of mass either remains at rest or moves uniformly in a straight line through absolute space. The centre of mass for three bodies is shown in Figure 11.

Figure 11 Masses of 1, 2 and 3 units (indicated by their sizes) are shown at the vertices of the two slightly different triangles (which correspond to the two triangles discussed in the main text). The two centres of mass are shown by the big blob (mass 6 units). The position of the centre of mass is found by finding the centre of mass of any pair and then the centre of mass of it and the remaining third mass.

Now, wherever we place the centre of mass of the first triangle, and however we orient it, the sequence of triangles that then arises is always the same. The path traced out in Platonia is the same. The starting position in absolute space does not matter an iota. This is rather remarkable. It is as if you could grow identical carrots in your garden, at the bottom of the sea, and in outer space. Different locations in absolute space have a decidedly shadowy reality. Unlike real locations on the Earth, they do not have any observable effects.