A paradox arises when, from the same premises P, two (or more, of course) apparently contradictory conclusions, X and Y, seem inescapably to follow. It can be resolved only if one of the following four things can be shown: (1) the conclusions are not, in fact, contradictory; (2) the conclusion X does not follow; (3) the conclusion Y does not follow; (4) the premises P contain an internal contradiction so that X and Y follow from incompatible parts of them. In the famous (or infamous, as Professor Bondi calls it,¹ and I would not quarrel with the description) clock paradox, the premises P are the special (or sometimes general) theory of relativity; since the many who have discussed it have differed on the question whether the general theory needs to be brought into the matter, this ambiguity must be admitted: the conclusion X is that, if two similar docks separate and re-unite and their readings agree at the moment of separation, they will agree at the moment of re-union since the theory allows the motion to be ascribed with equal right to either, and no influence on their readings other than their relative motion can be dealt with by the theory: and the conclusion Y is that one will read an earlier time than the other on re-union, because the special theory of relativity, by virtue of the Lorentz transformation, requires that their rates of working differ. X and Y are obviously contradictory, so solution (1) is impossible, and we have to choose between the others.

From this it follows that Einstein chose Y as the correct solution, and therefore must have rejected X. But he did not disprove X, which seems to follow from the postulate of relativity which is an integral part of the theory P; hence he did not resolve the paradox. I think it is true to say that no one has ever disproved X - at least, I have never seen a disproof of it, and I have read innumerable treatments of the problem; nevertheless, nearly everyone accepts Y, on grounds that are almost unbelievably diverse. They involve Doppler effects, observations by external observers in countless varieties of circumstances, the influence of the rest of the universe, electromagnetic considerations, and more ingenious situations than one would have thought possible. They have one thing only in common, apart from their conclusion - they are all unaccompanied by a disproof of X. Had that been given, one would have sufficed; without it their total contribution to the resolution of the paradox is precisely nothing.

The commonest attempt to justify the ignoring of X invokes the fact that, in order that 'the clock moved from A to B' shall return to its starting point, it must undergo an acceleration, which removes the problem from the scope of the special theory. But there are three comments that may be made on this. First, Y, like X, is drawn from the special theory, so that if X is nullified by this consideration, so is Y. Second, Einstein at this time evidently considered that the acceleration did not affect the matter, for he wrote:² 'It is at once apparent that this result [concerning uniform motion] still holds good if the clock moves from A to B in any polygonal line' (but see the comment made later - p. 200 - on his general failure to look beyond the immediate object of attention), and it cannot do this without being accelerated. And third, the conclusion X is drawn from the postulate of relativity alone, without the postulate of constant light velocity, and in his general theory Einstein generalised the former postulate to cover both accelerated and uniform motion, so that, if we accept the generalisation, the acceleration cannot invalidate X.

Although the 'paradox' is obvious to anyone who reads Einstein's paper, it did not at first attract much attention, for the reason I have already given, namely, that Einstein's theory was regarded as merely a recondite form of Lorentz's, and on Lorentz's theory it does not arise. For, despite the name 'relativity theory' given to it, Lorentz's theory was not, strictly speaking, a relativity theory at all; that is to say, it did not regard the relative motion of two bodies as, with equal validity, divisible between them in any of the various conceivable ways; each body had its own absolute motion — i.e. its motion with respect to the ether — and, although we had not discovered how to find what that was, it was nevertheless real. Consequently, on Lorentz's theory, 'asymmetrical ageing', as Y has been called, would actually occur, and the clock showing the earlier time on re-union would be the one whose velocity through the ether had been the greater. Hence on Lorentz's theory there was no difficulty in disposing of X, because it followed from the postulate of relativity which that theory rejected.

This is, in fact, one of many examples of the way in which the two theories are confused. When the clocks A and B are only in uniform motion, and so receding steadily from one another, it is usual to emphasise that there is no 'real' difference between their rates, but each merely appears to go slow from the point of view of the other. ('Here is a paradox,' wrote Eddington,⁸ 'beyond even the imagination of Dean Swift. Gulliver regarded the Lilliputians as a race of dwarfs; and the Lilliputians regarded Gulliver as a giant. That is natural. If the Lilliputians had appeared dwarfs to Gulliver, and Gulliver had appeared a dwarf to the Lilliputians - but no! that is too absurd for fiction, and is an idea only to be found in the sober pages of science.') But when the motion ceases to be uniform, the reciprocity of Einstein's theory is abandoned, and the asymmetry of Lorentz is invoked. A traveller to Arcturus at uniform speed, pictured by Eddington, merely appears to age slowly to an observer on the Earth, and the observer on the Earth likewise appears to age slowly to the traveller, but 'if in some way his [the traveller's] motion were reversed so that he returned to the Earth again, he would find that centuries had elapsed here, while he himself did not feel a day older.' Why a retardation of ageing before reversal is only apparent, so that 'really' the traveller ages at the normal rate, and then, having decided to reverse, he regains his lost youth, is explained, according to Eddington, by the claim that the motion 'must be reversed by supernatural means or by an intense gravitational force'. What happens if the traveller reverses by the natural means of suitably operating the engine of his space vehicle is not explained. Needless to say, this was written after Einstein's general theory had re-named 'the relativity theory of Lorentz' as 'the special relativity theory', so that conclusions could be drawn indiscriminately from either, according to what happened to be necessary to preserve it from refutation.

I do not propose here to survey the enormous mass of literature on this subject. As I have said, the 'paradox' did not become a matter for serious consideration until after the general relativity theory had made its appearance, although, as also I have said, opinions were divided on the question whether the general theory needed to be brought into the matter at all. In practically all the treatments of the problem, Einstein's original 'polygonal line' has been simplified by reduction to a single to-and-fro journey in a straight line: the traveller is supposed to set out from the Earth at uniform speed, and after a while to reverse his motion and return along the same path at the same uniform speed. He would, of course, have to accelerate in order to start his motion and to reverse it (and to stop it on return if he did so, but of course this would be unnecessary for the problem), but the duration of these accelerations is always regarded as negligibly brief compared with that of the motion at uniform velocity. One of the chief objectors to the view that asymmetrical ageing is compatible with the relativity postulate was the philosopher Bergson, who wrote a book on the subject, Durйe et Simultaneity, in 1922; this has recently been translated into English by Professor L. Jacobson, and published, under the title, Duration and Simultaneity,* with a long Introduction by me on the modern phases of the controversy, most of which it will be unnecessary to repeat here. Also, in my discussion with the late Viscount Samuel, A Threefold Cord,5 I described the earlier stages of the modern revival, and that also I can leave out of account. I shall therefore here restrict my remarks to a brief statement of my own relation to the problem, to a discussion of Einstein's treatment of it, which appears to be unknown to the great majority of those who affect to solve it, and to a mere mention of a recent treatment by Professor H. Bondi,⁶ with whom, many years ago, I debated the 'paradox' on the BBC radio. He has recently given a new approach to the problem (leading, however, to the same conclusion — that Y is right - but still with no attempt to dispose of X), and I have since realised what I did not see at that rime - that solution (4) gives the key to the problem. At the time of our debate I believed the special theory of relativity to be valid, but held that it made Y impossible, so the debate, if held now, would take a very different form. It would, however, be of little help to present our divergence, which still exists, in its new form.

Bondi's argument depended on the fact that the mathematics of the theory (the Lorentz transformation), which required the clocks to work at different rates, necessarily compelled a difference of reading on re-union. In other words, I argued for solution X and Bondi for solution Y. But, as I have pointed out, it is not sufficient to 'prove' X or Y; one must also disprove Y or X in order that such 'proof shall resolve the paradox. I cannot remember the details of our debate, but my disproof of Y was essentially this. The Lorentz transformation certainly required that the traveller's clock, when it reached the end of its outward journey, should be behind the 'time' (instant) prescribed by the theory for its arrival, according to the Earth clock. But that 'time' was freely defined, and the fact that the actual reading of the traveller's clock differed from it told you nothing about the rates of the clocks: a clock at the distant point had been artificially set to agree with the freely adopted definition, and the fact that the traveller's clock disagreed with it did not mean that it would disagree with the Earth clock, which had not been artificially set to agree with any definition. What the 'slowing down' required by the Lorentz transformation meant, therefore, was that the common reading of the two clocks on re-union was behind that calculated on pre-relativity principles, so that the journey had had a shorter duration than would have been expected from the distance and velocity of travel according to Newtonian kinematics.

Neither Bondi nor anyone else offered a disproof of X, so, although I was not altogether satisfied with my disproof of Y, it did seem to me free from fatal objection, and I worked out the mathematics in detail in a paper published in the Proceedings of the Physical Society,¹⁰ showing that the mathematics of the theory was not inconsistent with the agreement of the clocks on re-union. But I soon realised that there was a fatal objection to this disproof of Y. If the traveller moved at a speed greater than c/v2, but less than c, where c is the velocity of light, the calculation showed that the common reading of the clocks on re-union would be earlier than the reading of the Earth clock when a beam of light, starting at the same instant as the traveller and covering the same distance, would return: in other words, the traveller, moving always more slowly than the light, would nevertheless get back first. This was clearly impossible; hence my disproof of Y had failed. I could see no alternative disproof, so I was faced with the situation that neither X nor Y could be disproved. All that was left was solution (4) - that the special relativity theory was self-contradictory.

It was then comparatively easy to prove this in other ways, of which the one I have chosen for this book seems to me the simplest and most direct. It seems to me quite unanswerable, but what is absolutely certain is that it has not been answered. It leaves the question of the possibility of asymmetrical ageing at present quite open, although one may incline with a variety of degrees of probability to one side or the other. The original question, to which either X or Y was an answer, was: is asymmetrical ageing compatible with relativity theory? and that was a purely abstract question, the answer to which was quite independent of the truth of relativity theory or the reality of asymmetrical ageing. But if, as I now hold, special relativity is false, then the reality or otherwise of asymmetrical ageing depends on which of its postulates is wrong. If the postulate of relativity is wrong, then there is a Lorentzian ether and asymmetrical ageing is possible. If, on the other hand, the relativity postulate is right and the postulate of constant light velocity wrong, then asymmetrical ageing is impossible. These possibilities will be discussed at greater length in the next chapter.

I turn now to Einstein's paper on the clock paradox, which, though in one sense it is not an attempt to solve the problem at all but aims merely at showing that the relativity postulate can survive either solution, does dispose of a large number of arguments for solution Y - in particular, all those which attempt to dismiss X by claiming that the effect of the acceleration on reversal invalidates it. In a paper in Naturwissenschaften in 1918,¹¹(shortly after he had published his general theory, Einstein discussed this problem — as he was forced to do because, having committed himself to the postulate of relativity with respect to accelerated as well as uniform motion, he had to show that the extended postulate was not violated by the asymmetrical ageing which he had originally deduced from 'the special theory. He puts his argument into the form of a dialogue between a relativist (who, of course, is Einstein himself) and a critic who argues that the general postulate cannot be true because asymmetrical ageing violates it. This paper, as I say, is surprisingly little known, and as it has not, to my knowledge, been published in English translation, I shall quote extensively from a rendering made for me by a competent translator.

Now it is clear from this, first of all, that Eddington's remark which I quoted earlier, that the travelling clock is reversed by 'an intense gravitational force' (which presumably is taken from this account of Einstein's, since it is Einstein's theory that he is propounding) is based on a misreading. The travelling clock is reversed in a normal way by the traveller who, if we suppose him to remain at rest all the time, must have the motion which his action would otherwise give him neutralised by a postulated gravitational field which has no other source than our imagination. For, as Einstein says, 'exactly the same process is described' in the two cases. If we choose the co-ordinate system K, in which the traveller moves, there is no gravitational field, for the traveller's engine causes his motion, and the Earth remains at rest because there is nothing to move it; but if we choose the system K', then something must keep the 'traveller' at rest despite the working of this engine, and something must make the Earth move. The gravitational field that serves this double purpose must therefore be purely ad hoc.

At first this seems a wholly arbitrary procedure: if one is at liberty freely to invent agencies to perform whatever functions are necessary to save a theory, then science becomes a farce; we can prove anything at all. But this, in fact, is not so, because Einstein is looking at the problem from the opposite side, so to speak, from most of those who have discussed it. Whereas the usual procedure is to try to show that asymmetrical ageing is possible, notwithstanding the relativity postulate, Einstein's aim is to show that the (general) relativity postulate is tenable, notwithstanding asymmetrical ageing, which he takes for granted, as though it were an established fact. 'I have noticed with regret,' says the relativist near the beginning, 'that some authors try to escape from this unavoidable result.' In these circumstances he is no more open to criticism for introducing ad hoc fields than Newton is open to criticism for introducing his gravitational force to explain the acceleration of a falling body. Newton takes his first law of motion - that a free body moves uniformly - for granted, and when it is observed that a naturally falling apple does not move uniformly, he invents gravitational force to accelerate it. That force is no more observable than Einstein's fields, and has no other justification than that it is necessary to preserve an already accepted axiom (the first law of motion and asymmetrical ageing in the two cases) from violation.

Furthermore, Einstein's treatment has the unique merit that it does, in anticipation, give a direct answer to the question posed by my syllogism (p. 190). Whereas all other treatments either evade that question or give it a palpably spurious answer, Einstein's answer is straightforward - it is item (2) that is wrong; asymmetrical ageing does not enable one to say which body has moved, for it is compatible with both suppositions. No one else (except, of course, the few who reproduce Einstein's argument, with more or less amplification - Tolman, Meiller, and Born and Biem are the only ones I can think of, and none of these relates it to my syllogism) has ventured even tacitly to imply that item (2) is wrong.

First of all, consider the arbitrariness of the postulated gravitational fields. As I have said, their introduction in itself is no more invalid than Newton's similar procedure, but there are attendant circumstances that make it altogether different and quite inadmissible. In the first place, Newton's gravitational force was not at all arbitrary; it was defined in terms of quantities - mass and distance - defined and measured quite independendy of gravitation, so that the fact that a particular combination of these things did indeed give an acceleration agreeing with that observed was a discovery of the highest importance. We are not concerned here with general relativity, but it would be a culpable waste of an opportunity not to point out in this connection that, in the usual presentation of Newton's theory by those (including Einstein himself) who purports to show its inferiority to Einstein's, this discovery of Newton's is totally misrepresented as a defect. It is said that Newton's theory includes two kinds of mass - inertial and gravitational, which are mysteriously identical - a fact, which the theory is at fault in leaving unexplained. But Newton's theory does not contain two kinds of mass. He defines only one, the so-called inertial mass - 'it is this quantity that I mean hereafter everywhere under the name of body or mass': he says, and his magnificent discovery that this quantity plays a major part in determining the actual accelerations of bodies observed in nature is totally misrepresented by the assertion that gravitation requires a second mass which happens, in a magical way, to be identical with the mass that measures inertia.

However, the point at the moment is that, not only is Newton's gravitational force not arbitrary but something calculable in terms of independently measurable quantities while Einstein's 'gravitational fields' are wholly ad hoc — but also it is, in the nature of the case, impossible that Einstein's fields ever can be any other. For the one essential characteristic of such a field is that it keeps the clock U₂ permanently at rest in spite of the 'external force’, which we may regard as, applied by a traveller carrying the clock. But that traveller can apply the reversing force as he likes - steadily, jerkily, rapidly, slowly... - and the gravitational field must therefore also be one of infinite variety. Obviously it is impossible for such a field to be expressible in terms of any independently known quantities at all, as Newton's force was expressed in terms of inertial mass and distance.

Even less is it permissible to say what, if any, effect the field necessary in any particular case will have on the rates of the clocks. Einstein says: 'Calculation shows that the consequent advancement amounts to exactly twice as much as the retardation' given by the Lorentz transformation during the uniform motion; but he does not make the calculation, here or anywhere else, and it is obvious that it cannot possibly be done. For it is misleading to call these ad hoc fields 'gravitational fields', since they are essentially different from the fields represented by Einstein's law of gravitation, which are applied to calculate the motions of the planets and such things. Those fields are not ad hoc, and they are related to the distribution of matter in the universe, otherwise they could not be applied to observation, whereas the fields postulated in the 'clock paradox' case are necessarily 'homogeneous', i.e. of me same strength throughout the universe, so that they apply equally to U₁ and U₂ however far apart they may be. In his 'Autobiographical Notes' Einstein claims that his approach to mechanics is justified 'if one regards as possible, gravitational fields of arbitrary extension which are not initially restricted by spatial limitations'.¹² It is certainly possible to imagine such things, but not to suppose that they 'exist' in the sense in which the field in which the apple falls 'exists', or to call them by the same name.

Although, according to Einstein's law of gravitation properly so called, the rate of a clock is dependent in a calculable way on the potential of the natural gravitational field at the place where it is situated, it by no means follows that the same, or any, effect on the rate will occur in these infinitely variable artificial fields. M?ller, nevertheless, gives a calculation, on the assumption that this must be so." He chooses, moreover, a very special case in which simplifying assumptions are made, and, like all such simplifications, it breaks down when these are removed. It will be sufficient to give but one. M?ller supposes that the external forces by which U₂ is accelerated at the beginning, middle and end of the journey (and therefore the gravitational fields by which they are neutralised when U₁ is supposed to move) are all constant and equal to one another. In this 'way certain terms are made to cancel out, and on re-union U₂ is indeed found to be behind U₁ by the same amount whichever is supposed to move. But all that the traveller with U₂ has to do to upset this is to use different forces on starting and reversing. The agreement is then destroyed, and the clock-readings will then reveal which clock has 'really' moved.

And, worst of all, even if we allow that the hypothetical fields affect the clock-rates in any way at all, that effect would at once enable the motion be ascribed uniquely to one of the clocks. For, if U₂ moves there is no field at the reversal of motion, while if U₁ moves there is one. Now suppose a third clock U₃ at B, stationary with respect to U₁ and synchronised with it. Then, if U₂ moves, U₁ and U₃ remain synchronised throughout, but if U₁ (and U₃ of course) moves, the field that comes into play on reversal puts U₁ and U₃ out of synchronisation. An observer of U₃ from U₁ will therefore in due course see it go wrong, so he will know that it is U₁ that has moved and that the relativity principle is false.

This is specially emphasised by a particular case raised by Lenard as an objection to the general relativity principle, which Einstein claims to have answered in this paper. Lenard imagined a moving railway train brought to a sudden halt by collision with a station buffer. The relativity principle would require that the sudden change of motion could be ascribed either to the train or to the rest of the world, but it is the train that is damaged, not the Church steeple outside the station. Einstein replies, as with the clocks, that we may suppose that the rest of the world has its motion suddenly stopped by a gravitational field, so the relativity principle is preserved. But he does not proceed to the necessary accompaniment of this stoppage (according to his explanation), that then all the previously synchronised clocks of the world go out of kilter, and that fact, if it is observed to occur, fixes the motionuniquely on the rest of the world.

This relates to the fact that Einstein considers only the durations of his five stages, and not the instants (clock-readings) at which they begin and end. Had he given those, as they are required by his explanation, he would have seen that although, when the clocks U₁ and U₂ alone are considered, the only such instants that are actually observed are those of the beginning and end of the whole process, and he had succeeded in getting these to agree, the matter is quite different if U₃ is included. For in that case there are two coincidences of U₂ and U₃ (at the end of stage 2 and the beginning of stage 4), and not only is it impossible to make these (which are observable) agree in the two co-ordinate systems, but also, in the K' system, quite impossible changes of reading would be needed - corresponding, if the clocks are actually human twins, to a change of one of them from old age back to babyhood. I think this needs no further comment.

All these considerations show, as I say, both the impossibility of this description of the process and the extraordinary manner in which the clay mingled with the gold in Einstein's remarkable intellectual make-up. It is impossible to imagine Newton overlooking such points as these. In perceiving possibilities of solution of the problem immediately before him, Einstein was without a rival in his generation, but he seems not to have thought of looking beyond the immediate solution to its necessary implications, or even of maintaining I consistency between his various achievements. Having, as he considered, solved a problem, he no longer gave it further thought. That is why, for instance, he does not here consider asymmetrical ageing as open to question: it has been established once for all by special relativity, and what he has to do now is to defend the generalised relativity postulate against disproof by this established fact. Born tells us that when he first met Einstein in 1909, Einstein 'had already proceeded beyond special relativity which he left to minor prophets'.⁷

A very striking example of this is shown by his complete, and apparently unconscious, change of attitude to the whole meaning of relativity between 1905, when special relativity was first (and for him finally) formulated, and 1918, when he gave the above justification of the general relativity postulate. At the opening of his 1905 paper he states quite plainly that the blemish in the existing statement of the electromagnetic theory, which prompted his proposed reform of it, was the fact that its description of phenomena dependent only on relative motion differed when the standard of rest was changed, whereas the phenomena themselves remained the same. He cites as an example 'the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion.' He then states that 'examples of this sort' prompt the formulation of the (special) relativity postulate.

This point is emphasised by Born in his reminiscences of Einstein; he writes: 'The second peculiar feature of the first relativity paper by Einstein is his point of departure, the empirical facts on which he built his theory. It is of surprising simplicity. He says that the usual formulation of the law of induction contains an asymmetry, which is artificial and does not correspond to facts. According to observation the current induced depends only on the relative motion... while the... theory explains the effect in quite different terms according to whether the wire is at rest and the magnet moving or vice versa.'⁷