What is so remarkable about these results – and it seems so impossible that many quite intelligent people still refuse to accept it – is their mutual nature. How can it be that each family finds that the clocks of the other family run slower than their own? Box 10 explains.
BOX 9 Relativity in One Diagram and 211 Words
Figure 25 Alice (A) and Bob (B) believe they are at rest in the aether, and therefore draw a grid (dashes) with time vertical and space (only one dimension shown) horizontal. To synchronize their clocks, Alice sends Bob a light signal (solid line), which reaches him at X, where it is reflected back to Alice at T. Alice concludes that the signal reached Bob when she was at H. Their identical twins Alice* (A*) and Bob* (B*) are moving uniformly relative to them, but also believe they are at rest. Alice* sends a light signal, just as her twin does, at the moment they meet. It reaches Bob* at X*, and the reflection of it returns to Alice* at T*. She therefore concludes that her signal reached Bob* at H*, so she and Bob* have a grid (dots) inclined relative to their twins’ grid. The pairs do not agree on which events are simultaneous. Alice and Bob think H and X are, their twins think H* and X* are. However, they confirm the relativity principle, since both find that light travels along rays parallel to the diagonals (AX, XT and A*X*, X*T*) of their respective coordinate grids. Despite appearances, the situation is symmetric – in Alice* and Bob*’s grid their twins’ grid appears skew.
THE FORGOTTEN ASPECTS OF TIME
Fascinating as the results of Einstein’s two relativity theories are, many of them are not directly relevant to my main theme. Popular accounts that cover topics I omit are recommended in the section on Further Reading. My aim in Part 3 is to show how Einstein’s approach to relativity led him to an explicit theory of simultaneity but an implicit theory of duration. It is the latter that is important for this book, but it never got properly treated in relativity. The point is that remarkable facts about duration, as revealed through the clocks of different observers, follow inescapably from the definition of simultaneity and the relativity principle. Einstein did not need to create a theory of clocks and duration from first principles in order to learn some facts about them: they already followed from his two primary postulates.
BOX 10 The Impossible Becomes Possible
Figure 26 The horizontal and inclined strips, in which time increases vertically and the horizontal represents one space dimension, show the histories of two physically identical rods moving uniformly relative to each other. For Bob and Alice, points on the continuous line PPQQ are simultaneous and show the positions and lengths of the rods at the corresponding times. Their rod, PP, appears longer than Bob* and Alice*’s rod, QQ. But the starred twins think that points on the line P’P’Q’Q’ are simultaneous, and conclude that their rod Q’Q’ is longer than P’P’. A similar illustration could be given for clocks. Such diagrams do not explain this behaviour of rods and clocks, but do show that there is no outright logical contradiction. Einstein’s conclusions are as secure as his premises. His confidence in them has so far been totally vindicated.
The method Einstein used to create his relativity theories is an important factor. During the nineteenth century, mainly through the development of thermodynamics, physicists began to distinguish between, on the one hand, theories of the world in terms of truly basic laws and constituents (e.g. atoms and fields) and, on the other hand, so-called principle theories. In the latter, no attempt would be made to give an ultimate theory of things. Instead, the idea was to seek principles that seemed to hold with great generality and include them in the foundations of the description of phenomena. The repeated failures of all attempts to build perpetual-motion machines, of which two distinct types could be envisaged, became the basis of the first and second laws of thermodynamics. In the form in which it was developed on this basis, thermodynamics was a theory of the second kind – a principle theory.
In contrast, Lorentz’s combined theory of the electromagnetic field, electric charges and the aether was basically a theory of the first kind – it aspired to a fundamental description of the world in terms of its ultimate constituents. Einstein deliberately decided not to follow such a path in his own work on electrodynamics, from which the special theory of relativity emerged. He based it as far as possible on general principles. The fact is that Max Planck’s quantum discoveries (Box 1) and Einstein’s own development of them a few months before the relativity paper had persuaded Einstein that something very strange was afoot. Despite his admiration for Maxwell’s equations, he felt sure that they could not be the true laws of electromagnetism because they completely failed to explain the quantum effects. He had no confidence in his ability to find correct alternatives. Then, and to the end of his days, Einstein found the quantum baffling. He felt deeply that it was a huge mystery. By comparison, relativity (the special theory at least) was almost child’s play.
It was this attitude that largely shaped Einstein’s strategy in approaching the problem of the electromagnetic aether. He resolved to make no attempt at a detailed description of microscopic phenomena. Instead, he would rely on the relativity principle, for which there seemed to be strong experimental support, and make as few additional assumptions as possible. In the event, he was able to limit these to his assumption about the nature of light propagation. This was the one part of Maxwell’s scheme that he felt reasonably confident would survive the quantum revolution.
This had important consequences for the theory of time. Poincaré’s 1898 paper showed that it must answer two main questions: how simultaneity is to be defined, and what duration is. Associated with the second question is another, almost as important: what is a clock? Because of his approach, Einstein answered only the first question at a fundamental level. He gave at best only partial answers to the other two, and gave no explicit theories of either rods or clocks. Instead, he tacitly assumed the minimal properties they should possess. Otherwise, he relied to a very great extent on the relativity principle. It took him far. Few things in physics are more beautiful than the way he postulated the universal relativity principle and the one particular law of light propagation, and then deduced, from their combination, extraordinary properties of rods, clocks and time. If the premises were true, rods and clocks had to behave that way.
During his protracted creation of general relativity, Einstein used this trick several times. The strategy was always to avoid specific assumptions, and instead to seek principles. In this way he avoided ever having to address the physical working of rods and clocks: they were always treated separately as independent entities in both relativity theories. Their properties were not deduced from the inner structure of the theory, but were simply required to accord with the relativity principle. Einstein was well aware that this was ultimately unsatisfactory, and said so in a lecture delivered in 1923. He made similar comments again in 1948 in his Autobiographical Notes.
However, the tone of his comments does not suggest that he expected any great insight to spring from the rectification of this ‘sin’ (Einstein’s own expression). Only a ‘tidying up’ operation was needed. This gap in the theory of duration and clocks has still not been filled. I know of no study that addresses the question of what a clock is (and how crucially it depends on the determination of an inertial frame of reference) at the level of insight achieved in non-relativistic physics by James Thomson, Tait and Poincaré. Throughout relativity, both in its original, classical form and in the attempts to create a quantum form of it (which we come to in Part 5), clocks play a vital role, yet nobody really asks what they are. A distinguished relativist told me once that a clock is ‘a device that the National Bureau of Standards confirms keeps time to a good accuracy’. I felt that, as the theorist, he should be telling them, not the other way round.