Julian Barbour - The End of Time - The Next Revolution in Physics

Now we can talk about distance. In ordinary space it is always positive. The distance relationships are reflected in Pythagoras’ theorem: the square of the hypotenuse in any right-angled triangle is equal to the sum of the squares of the other two sides: H 2= A 2+ B 2.

Minkowski was led to introduce a ‘distance’ in space-time by noting a curious fact. For observers who use the xy frame in Figures 27 and 28, event A is separated from O by the space-like interval EA and by the time-like interval DA . For observers who use the starred frame, however, O and A are at the same space point and are merely separated by the time-like interval OA . The xy observers measure EA with a rod and DA with a clock, obtaining results X and T , respectively. With their clock, observers in the starred frame can measure only the time-like interval OA . Now, their clock runs at a different rate to the xy clock, so they will find that OA is not T but T starred. Using Einstein’s results, Minkowski found that ( T starred) 2= T 2− X 2. This is just like Pythagoras’ theorem, except for the minus sign.

There are several important things about this result. Einstein had shown that observers moving relative to each other would not agree about distances and times between pairs of events. However, Minkowski found something on which they will always agree. Measurements of the space-like separation (by a rod) and the time-like separation (by a clock) of the same two events O and A can be made by observers moving at any speed. They will all disagree about the results of the separate measurements, but they will all find the same value for the square of the time-like separation minus the square of the space-like separation. It will always be equal to the square of the time-like separation, called the proper time , of the unique observer for whom O and A are at the same space position. This result created a sensation. Space and time, like rods and clocks, seem to have completely different natures, but Einstein and Minkowski showed that they are inseparably linked.

What is more, Minkowski showed that it is very natural to regard space and time together as a kind of four-dimensional country in which any two points (events in space-time) are separated by a ‘distance’. This ‘distance’, found by measurements with both rods and clocks, is to be regarded as perfectly real because everyone will agree on its value. In fact, Minkowski argued that it is more real than ordinary distances or times, since different observers disagree on them. Only the ‘distance’ in space-time is always found to be the same. But it is a novel distance – positive for the time-like OA in Figure 27, zero for the light-like OF and negative for the space-like OC . (It is a convention, often reversed, to make time-like separations positive and space-like ones negative. What counts is that they have opposite signs. Also, if the units of space and time are not chosen to make the speed of light c equal to 1, the square of the space-time ‘distance’ becomes (cT) 2– X2 .)

Almost everything mysterious and exciting about special relativity arises from the enigmatic minus sign in the space-time ‘distance’. It causes the ‘skewing’ of both axes of the starred frame of the starred twins in Figure 25, and leads to the single most startling prediction – that it is possible, in a real sense, to travel into the future, or at least into the future of someone else, since the future as such is not uniquely defined in special relativity. What we call space and time simply result from the way observers choose to ‘paint coordinate systems’ on space-time, which is the true reality. Minkowski’s diagrams made all these mysteries transparent – and intoxicatingly exciting for physicists. However, this is not the place to discuss time travel and the other surprises of relativity, which are dealt with extensively in innumerable other books.

For physicists, ‘relativity’ has two different meanings. The more common is the one employed by Einstein when he created relativity. He related it to the empirical fact, first clearly noted by Galileo in 1632, that all observations made within an enclosed cabin on a ship sailing with uniform speed are identical to observations made when the ship is at rest. Einstein illustrated this fact with experiments on trains. The lesson he drew from it was that uniform motion as such could not be detected by any experiment. The laws of nature could therefore not be expressed in a unique frame of reference known to be at rest. They could be expressed only in any one of a family of distinguished frames in uniform motion relative to one another. The relativity principle states that the laws of nature have the identical form in all such frames. For reasons shortly to be explained, this later became known as the restricted or special relativity principle.

This meaning of relativity is tied to a special feature of the world – the existence of the distinguished frames and their equivalence for expressing the laws of nature. The other meaning of relativity is more primitive and less specific. It simply recognizes that space and time are invisible: all we ever see are objects and their relative motions. We can speak meaningfully of the position and motion of an object only if we say how far it is from other objects. Position and motion are relative to other objects. This is often called kinematic relativity , to distinguish it from Galilean relativity.

Both relativity principles have played important – often decisive – roles in physics. Copernicus and Kepler used kinematic relativity to great effect in the revolution they brought about. Galileo used the other relativity principle to explain how we can live on the Earth without feeling its motion. That was almost as wonderful a piece of work as Einstein’s, nearly three hundred years later. A natural question is this: what is the connection between the two relativity principles? Any satisfactory answer must grapple with and resolve the issue of the distinguished frames of reference. How are they determined? What is their origin? As we have seen, neither Einstein nor Minkowski addressed these questions when they created special relativity, and they have been curiously neglected ever since. This is a pity, since they touch upon the nature of time. We cannot say what time is – and whether it even exists – until we know what motion is.

Poincaré sought to unite the two relativity principles in a single condition on the structure of dynamics, as formulated in the two-snapshots idea. Had he succeeded, he would have derived the empirical fact of Galilean relativity solely on the basis of a natural criterion derived from kinematic relativity. He died without taking this idea any further, but in any case it is doubtful whether the two relativity principles can be fully fused into one. Poincaré formulated his idea in 1902, before the relativistic intermingling of space and time became apparent, and it is hard to see how that can ever be derived from the bare fact of kinematic relativity. It is, however, of great interest to see how far Poincaré’s idea can be taken. We shall come to this when we have seen how Einstein thought about and developed his own relativity principle and thereby created general relativity.

It is important not to be overawed by the genius of Einstein. He did have blind spots. One was his lack of concern about the determination in practice of the distinguished frames that play such a vital role in special relativity – he simply took them for granted. It is true that they are realized approximately on the reassuringly solid Earth in skilfully engineered railway carriages. But how does one find them in the vast reaches of space? This is not a trivial question. Matching this lack of practical interest, we find an absence of theoretical concern. Einstein asked only what the laws of nature look like in given frames of reference. He never asked himself whether there are laws that determine the frames themselves. At best, he sought an indirect answer and got into a muddle – but a most creative muddle.