THE theory of relativity has resulted from a combination of the three elements which were called for in a reconstruction of physics; first, delicate experiment; secondly, logical analysis; and thirdly, epistemological considerations. These last played a greater part in the early stages of the theory than in its finished form, and perhaps this is fortunate, since their scope and validity may be open to question, or at least would be but for the successes to which they have led. One may say, broadly, that relativity, like earlier physics, has assumed that when different observers are doing what is called "observing the same phenomenon," those respects in which their observations differ do not belong to the phenomenon, but only those respects in which their observations agree. This is a principle which common sense teaches at an early age. A young child, seeing a ship sailing away, thinks that the ship is continually growing smaller; but before long he comes to recognize that the diminution in size is only "apparent," and that the ship "really" remains of the same size throughout its voyage. In so far as relativity has been inspired by epistemological considerations, they have been of this common-sense kind, and the apparent paradoxes have resulted from the discovery of unexpected differences between our observations and those of other hypothetical observers. Relativity physics, like all physics, assumes the realistic hypothesis, that there are occurrences which different people can observe. For the present, we may ignore epistemology, and proceed to consider relativity simply as theoretical physics. We may also ignore the experimental evidence, and regard the whole theory as a deductive system, since that is the point of view with which we are concerned in Part I.

The most remarkable feature of the theory of relativity, from a philosopher's standpoint, was already present in the special theory: I mean the merging of space and time into space-time. The special theory has now become only an approximation, which is not exactly true in the neighbourhood of matter. But it remains worth understanding, as a stage towards the general theory. Moreover, it does not demand the abandonment of nearly such a large proportion of our common-sense notions as is discarded by the general theory.

Technically, the whole of the special theory is contained in the Lorentz transformation. This transformation has the advantage that it makes the velocity of light the same with respect to any two bodies which are moving uniformly relatively to each other, and, more generally, that it makes the laws of electromagnetic phenomena (Maxwell's equations) the same with respect to any two such bodies. It was for the sake of this advantage that it was originally introduced; but it was afterwards found to have wider bearings and a more general justification. In fact, it may be said that, given sufficient logical acumen, it could have been discovered at any time after it was known that light is not propagated instantaneously. It has grown by this time very familiar—so familiar that I have even seen it quoted (quite correctly) in an advertisement of Fortnum and Mason's. Nevertheless, it is, I suppose, desirable to set it forth. In its simplest form it is as follows:

Suppose two bodies, one of which () is moving relatively to the other () with velocity v parallel to the -axis. Suppose that an observer on observes an event which he judges to have taken place at time , by his clocks, and in the place whose co-ordinates, for him, are , , . (Each observer takes himself as origin.) Suppose that an observer on judges that the event occurs at time and that its co-ordinates are , , . We suppose that at the time when the two observers are at the same place, and also . It would formerly have seemed axiomatic that we should have . Both observers are supposed to employ faultless chronometers, and, of course, to allow for the velocity of light in estimating the time when the event occurs. It would be thought, therefore, that they would arrive at the same estimate as to the time of the occurrence. It would also have been thought that we should have: Neither of these, however, is correct. To obtain the correct transformation, put: where is, as always, the velocity of light. Then: For the other co-ordinates , , we still have, as before: It is the formulæ for and that are peculiar. These formulæ contain, implicitly, the whole of the special theory of relativity.

The formula for embodies the FitzGerald contraction. Lengths on either body, as estimated by an observer on the other, will be shorter than as estimated by an observer on the body on which the lengths are: the longer length will have to the shorter the ratio . More interesting, however, is the effect as regards time. Suppose that an observer on the body judges two events at and to be simultaneous, and both at time . Then an observer on will judge that they occur at times , where: and therefore: This is not zero unless ; thus in general events which are simultaneous for one observer are not simultaneous for the other. We cannot therefore regard space and time as independent, as has always been done in the past. Even the order of events in time is not definite: in one system of co-ordinates an event A may precede an event B, while in another B may precede A. This, however, is only possible if the events are so separated that, no matter how we choose our co-ordinates, light starting from either could not reach the place of the other until after the other had occurred.

The Lorentz transformation yields the result that: Since and , we have: or, putting , for the distances of the event from the two observers: This result is general—i.e. given any two reference-bodies in uniform relative motion, if is the distance between two events according to one system, the distance according to the other, and if , are the corresponding time-intervals between the events, equation (2) will always hold. Thus represents a physical quantity, independent of the choice of co-ordinates; it is called the square of the "interval" between the two events. There are two cases, according as it is positive or negative. When it is positive, the interval between the events is called "time-like"; when negative, "space-like." In the intermediate case in which it is zero, the events are such that one light-ray can be present at each. In this case, one event might be the seeing of the other. The time-order of two events will be different in different reference-systems when their interval is space-like, but when it is time-like the time-order is the same in all systems, though the magnitude of the time-interval varies.

When the interval between two events is time-like, it is possible for a body to move in such a way as to be present at both events. In that case, the interval is what clocks on that body will show as the time. When the interval between two events is space-like, it is possible for a body to move in such a way that, by its clocks, the two events will be simultaneous; in that case, the interval is what, in relation to that body, appears as their distance. (In these remarks, we are taking the velocity of light as the unit of velocity, which is convenient in relativity theory.) Both these are consequences of the Lorentz transformation. From the first of them it follows that, if two events both happen to me, the time between them as measured by my watch (assuming it to be a good watch) is the "interval" between them, and has still a physical significance. Thus the time that is concerned in psychology is unaffected by relativity, assuming that everything that psychology is concerned with happens, from a physical point of view, in the body of the person whose mental events are being considered. This is an assumption for which grounds will be given at a later stage.

It follows from the ambiguity of simultaneity between distant events that we cannot speak unambiguously of "the distance between two bodies at a given time." If the two bodies are in relative motion, a "given time" will be different for the two bodies and different again for other reference-bodies. It follows that such a conception cannot enter into the correct statement of a physical law. On this ground alone, we can conclude that the Newtonian form of the law of gravitation cannot be quite right. Fortunately, Einstein has supplied the necessary correction.

It will be observed that, as a consequence of the Lorentz transformation, the mass of a body will not be the same when it is in motion relatively to the reference-body as when it is at rest relatively to it. The mass of a body is inversely proportional to the acceleration produced in it by a given force, and two reference-bodies in uniform relative motion will give different results for the acceleration of a third body. This is obvious as a consequence of the FitzGerald contraction. The increase of mass with rapid motion was known experimentally before the special theory of relativity had explained it; it is very marked for velocities such as those attained by -particles (electrons) emitted by radio-active bodies, since these velocities may be as great as 99 per cent, of the velocity of light. This change of mass, like the FitzGerald contraction, seemed strange and anomalous until the special theory of relativity explained it.

One more point is important as showing how easily what seems axiomatic may be false: it concerns the composition of velocities. Suppose three bodies moving uniformly in the same direction: the velocity of the second relatively to the first is , that of the third relatively to the second is . What is the velocity of the third relatively to the first? One would have thought it must be , but in fact it is: It will be seen that this ; if or , it is , otherwise it is less than . This is an illustration of the way in which the velocity of light plays the part of infinity in relation to material motions.

The special theory set itself the task of making the laws of physics the same relatively to any two co-ordinate systems in uniform rectilinear relative motion. There were two sets of equations to be considered: those of Newtonian dynamics, and Maxwell's equations. The latter are unaltered by a Lorentz transformation, but the former require certain adaptations. These, however, are such as experimental results had already suggested. Thus the solution of the problem in hand was complete, but of course it was obvious from the first that the real problem was more general. There could be no reason for confining ourselves to two co-ordinate systems in uniform rectilinear motion; the problem ought to be solved for any two co-ordinate systems, no matter what the nature of their relative motion. This is the problem which has been solved by the general theory of relativity.