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.
The ‘location in time’ is equally difficult to pin down. We started our experiment at noon according to Newton’s clock. In fact, the starting time has no influence whatsoever: all the ‘carrots’ come out just the same. So, as far as the position of the first triangle in both absolute space and time is concerned, it has no influence whatsoever. We begin to wonder whether they play any role at all. This doubt is strengthened when we consider where to place the second triangle. It turns out that we can position its centre of mass anywhere in absolute space relative to the first triangle. This too has no effect at all on the sequence of triangles that then follow. This absence of effect is due to so-called Galilean relativity, which is one of the most fundamental principles of physics (Box 5).
BOX 5 The Galilean Relativity Principle
Galileo noted that all physical effects in the closed cabin of a ship sailing at uniform speed on a calm sea unfold in exactly the same way as in a ship at rest. Unless you look out of the porthole, you cannot tell whether the ship is moving. Quite generally, in Newtonian mechanics the uniform motion of an isolated system has no effect on the processes that take place within it. The left-hand diagram in Figure 12 shows (in perspective) the triangles formed by the three gravitating bodies in the history of Figure 10 at equal intervals of absolute time. The individual bodies move along the ‘spaghetti’ tubes. The centre of mass moves uniformly up the z axis. (Despite appearances, the triangles are always horizontal, i.e. parallel to the xy plane.) The right-hand diagram has two physically equivalent interpretations. First, it is how observers moving uniformly to the left past the system on the left would see that system receding behind them. Second, it is also how observers at rest relative to the system on the left would see a system identical to that system except for a uniform motion of the centre of mass to the right. This is how the happenings in the cabin of Galileo’s galley would be ‘sheared’ to the right for observers standing on the shore. Depending on the speed of the system, the centre of mass will be shifted in unit time by different amounts, but the actual sequence of triangles remains the same. This corresponds to the freedom mentioned in the text.
Because of the relativity principle, the laws of motion satisfied by bodies take exactly the same form in any frame of reference moving uniformly through absolute space as they do in absolute space itself. Although Newton did not like to admit it, this fact makes it impossible to say whether any such frame, which is called an inertial frame of reference, is at rest in absolute space or moves through it with some uniform velocity. Bodies with no forces acting on them move in a straight line with uniform speed in any inertial frame of reference (hence the name). It is impossible to say that you are at rest in absolute space, only that you are at rest in some inertial frame of reference. For historical reasons I use ‘absolute space’ in the text, but strictly I should be using ‘any inertial frame of reference’.
Figure 12. Unlike Figures 9 and 10 (and the later Figure 14), the lines followed by the spaghetti strands in this figure (and also Figure 13) show the tracks of the three individual particles in space. This is why there are three strands and not a single curve. It will help you a lot if you can get used to thinking about these two different ways of representing one and the same state of affairs. Here we see individual particles moving in absolute space. In Figures 9, 10 and 14 we ‘see’ (in our mind’s eye) the ‘world’ or ‘universe’ formed by the three particles moving in Platonia.
There are only four freedoms that remain. Having placed the centre of mass of the second triangle at some position, we can change its orientation (three freedoms). We can also change the amount of Newton’s absolute time that elapses between the instants at which the three bodies occupy the two positions (one freedom, the fourth). If the time difference is shortened, this means that the bodies travel farther in less of Newton’s time – that is, they are moving faster initially. In fact, since the motion of the centre of mass does not matter, we can keep it fixed and change only the orientation. Now, at last, we come to something that does matter. Both these changes – in the time difference and in the relative orientation – have dramatic consequences, which are illustrated in Figures 13 and 14.
Figures 13 and 14 express the entire mystery of absolute space and time. Both of Newton’s absolutes are invisible, yet their effects show up in the evolutions of the triangles, which are more or less directly visible. The astronomers do see stars and the spaces between them (admittedly in projection) when they look through telescopes. If time were merely change and only distances had dynamical effect, a decent Machian mechanics – one that would satisfy Laplace’s divine intelligence – should lead to exactly the same evolutions in all nine cases. This is manifestly not true for the real triple-star systems that astronomers observe. All the different kinds of evolution shown in Figures 13 and 14, and many more, are found. All the facts that enabled Newton to win his argument against Leibniz are contained in these diagrams, but it took about two centuries before Poincaré found the best way to demonstrate them. He concluded regretfully that a mechanics that uses only relative quantities, as Mach advocated, cannot get off the ground. It lacks perfect Laplacian determinism. Nevertheless, the failure is curious. Absolute space and time could have had an effect through all the freedoms allowed in the placing of the two triangles. There are fourteen degrees of freedom in total, of which ten have no effect whatsoever. This is just what the invisibility of space and time would lead us to expect. Yet four degrees of freedom do have a profound influence. Three are associated with twists in space, the fourth with the overall speed put into the system. These strange mismatches between expectation and reality have kept the philosophers arguing and the physicists puzzling for centuries.
The fact is that Newton’s absolute space and time play a decidedly odd role. The first problem is their invisibility. The more serious problem is what little part they play in the whole story, and how irrationally they enter the stage when they do participate in the action. Once we have chosen the relative orientation and time separation of the two triangles, we can take them anywhere in absolute space and time. They will always give rise to the same evolution. Absolute space and time seem to matter very little; only the relative orientation and time separation count.