If only relative quantities count, then Newton assumed too much structure. In a universe of just three particles, only the three distances between them (the triangle they form) should count. The triangle universe cannot have an overall position and orientation in some invisible containing space. Similarly, since the idea of an external ‘grandstand clock’ is absurd, we cannot say ‘how fast’ the universe travels along the curve in Figure 9. It simply occupies all the points along it.
If Mach is right, so that time is nothing but change and all that really counts in the world is relative distances, there should be a perfect analogue of Laplace’s scenario with a divine intelligence that contemplates Platonia. Machian dynamics in Platonia must be about the determination of paths in that timeless landscape. It should be possible to specify an initial point in Platonia and a direction at that point, and that should be sufficient to determine the entire path. Nothing less can satisfy a rational mind. The history in Figure 10 starts at the centre of Shape Space, so that there the particles form an equilateral triangle, and set off in a certain direction. In Machian dynamics, the initial position and initial direction (strictly in Triangle Land, not Shape Space) should determine the complete curve uniquely. Now we can test this idea in the real world. The heavens provide plenty of triple-star systems, and astronomers have been observing their behaviour for a long time. They certainly meet the Laplace-type condition when described in Newtonian terms. But are their motions comprehensible from a Machian point of view? This is the question Poincaré posed.
Figure 9. The (computer-generated) path traced in Shape Space by the triangles formed by three mutually gravitating particles. It is important to realize that the winding path shown is not traced by a single particle moving over the page, but that each point on the path represents the shape of a complete triangle. The curve shows the succession of triangle shapes. As explained in the text, it is impossible to say that time increases as you go in a particular direction along this curve. However, suppose we imagine it starts in the top left corner. This corresponds to particles A and C nearly colliding, while particle B is far away. Then they move to a configuration that is quite close to an equilateral triangle, after which A and B get very close together near the bottom of the diagram. Then the triangle shape evolves along the curve up to the top right. Where the curve nearly touches the top line, all three particles are almost on a line, with C between A and B. Finally the curve returns to the bottom of the figure, where the wiggles indicate that particles A and B are orbiting around each other, while C is far away. You see how history is all coded in one curve, but you just cannot tell in which direction it unfolds!
The answer is very curious. The motions are nearly but not quite comprehensible. This can be highlighted by showing how different possible Newtonian motions look when represented as curves in Triangle Land, our model Platonia – or rather Shape Space, since this is much easier to represent. To create a vivid picture, let us imagine that we are holding two cardboard triangles that are slightly different. These can represent the relative configurations of three mutually gravitating bodies at two slightly different instants of Newton’s absolute time.
Figure 10. Another possible path traced by the same three particles as in Figure 9 (I refer to them now as bodies). This history starts (or ends if time is assumed to run the other way) at the configuration in which the three bodies form an equilateral triangle. The shape of the triangle changes in a definite way at all points along the curve. If it left the initial equilateral triangle along one of the dotted lines, that would mean two sides of the triangle remaining equal in length while the third changes – the equilateral triangle would become an isosceles triangle. In fact, for the example shown, the ratios of all three sides change. For readers used to thinking of motions in ordinary space, this example corresponds to particles that constantly orbit each other in a fixed plane. The positions at which the curve touches the dotted line that bound Shape Space correspond to eclipses, when one particle is between the other two and on the line joining them. Such a configuration is called a syzygy (that’s a nice word to show off with). In ordinary space, the one particle passes through the line joining the other two and comes out the other side. But the points on the curve in Figures 9 and 10 stand for the complete triangle, not one of the three particles. This is why the curve approaches the syzygy frontier and then returns into the interior of Shape Space. There are no triangles outside the syzygies!
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.