C. D. Broad, Perception, Physics, and Reality, 1914

APPENDIX

NOTE ON THE MEASUREMENT OF THE VELOCITY OF LIGHT AND ON THE THEORY OF RELATIVITY

    A work on philosophy to be in the present fashion must talk about M. Bergson, and a work on space and time must mention the Theory of Relativity. We will confine ourselves to a few words about the more important of these two modish subjects.

    It is clear that the equations of electrodynamics, on any theory, raise the same questions about absolute or relative space and time as do the Newtonian laws of motion. For we have the old difficulty that the laws are supposed to be the results of empirical observation, and to be stated in terms of absolute space and time. They contain moreover a certain constant c which has the dimensions of a velocity, and is identified with the velocity of light, or any other electromagnetic disturbance, in vacuo. So the question arises whether this is an absolute or a relative velocity.

    If electrodynamics merely raises the same problems as ordinary mechanics, and provides no material for a further discussion, it will not be worth while to trouble about it. There will be nothing to discuss except the interesting question whether the laws of electrodynamics hold for the same axes (those defined by the fixed stars) as the laws of motion, and this can be left to the physicist. But the velocity of light or any electromagnetic disturbance is a somewhat different thing from that of a piece of matter; and electrodynamic considerations have given rise to the Theory of Relativity which is believed by some of its supporters (notably by the late distinguished mathematician Minkowski1, who did so much for its theoretical development) to necessitate entirely new views about space and time. It will therefore be worth our while to discuss shortly the assumptions involved in measuring the velocity of light, and the Theory of Relativity.

    Suppose you know the time at which a disturbance leaves a body A, and the time at which it reaches another body B, and the distance between the two, you have necessary -- though not perhaps sufficient -- factors for determining the velocity of propagation of the disturbance. But, in practice, the determination even of these factors involves certain assumptions which are not commonly made explicit. It is one of the greatest and least equivocal of the services of the Theory of Relativity to have called attention to this fact. Let us leave that theory out of the question for the present and consider for ourselves the problems involved in the determination of these three factors.

    If you want to measure the velocity of a body it is a truism that you must be able to identify it. It must be the same body that leaves one of your fixed points at one moment and reaches the other at the second moment. Similarly, in measuring the velocity of light, we want to deal with the same disturbance throughout. This identification is made in a simple way in Fizeau's method of using a toothed wheel to cover and uncover a source of light successively. The disturbance that we have to deal with is that emitted by the source during the time that the opening is faced by a definite one of the gaps in the rotating toothed wheel. This disturbance passes to a distant mirror, and is reflected back to the toothed wheel. If the wheel be rotating below a certain speed the disturbance will come back to find a part of the gap from which it came still open, and an image of the source will be seen. But, if the speed be increased, a velocity will be reached such that, when the light that has passed through one gap in the wheel returns from reflexion, it is faced by the tooth which forms one side of this gap; it cannot pass through and the image disappears. If the speed be further increased it returns to find the next gap opposite the source; it passes through; and the image is restored. Thus, if you first increase the speed till the image vanishes, and then increase it till it reappears, you know that the image that you see is due to the light that passed through the gap next before the one through which it is now returning. There is not absolute identification, but you know that you are dealing with some part of the light emitted during the very short time, that the opening is faced by any given gap; and that is enough. It is because there seem to be no bodies that are opaque to gravitation, as so many are to light, that we cannot measure the velocity of gravitation or prove that it has one at all; for we cannot identify the 'gravitational disturbance' emitted by any body at a given moment with that received elsewhere at another moment.

    So much for identification. I will continue to consider the Fizeau method, because, although it is not in practice so good as that of Fizeau and Foucault, which uses a rotating mirror, it is more straightforward and therefore better adapted as an example for a discussion of general principles. Let us turn then to the two positions involved in the measurement of a velocity. In this method the two positions at which we actually observe are indeed identical points on the same instrument. But here, as everywhere else, you must measure a distance, and so deal with two different points in order to measure a velocity. The two points are of course the gap and the centre of the mirror. In terrestrial experiments the distance of these would of course be measured by means of rigid bodies, whether in the gross form of meter scales or in the refinements of micrometers. But many long distances have to be measured by means of light signals, and, again, our most accurate measures of very small differences of length are by means of the shifting of interference bands. Both these methods involve a knowledge of the velocity of light, and would therefore be inadmissible in experiments to prove that light has a velocity or to determine its particular magnitude. But the presuppositions of ordinary straightforward measurement with scales have not always been made explicit, and it is a merit of the Theory of Relativity to call attention to them. They involve, in fact, time as well as space.

    It has always been recognised that measurement by superposition involves at least one assumption which cannot well be verified, and that is the constancy of length of the measure. The difficulty is of this nature. I use a scale to compare two lengths because I do not trust my immediate judgments of comparison under certain circumstances. The circumstances are the following. (i) Even if I can see two lengths at the same time, yet, unless I can put them side by side and notice whether there is any overlap, I cannot trust my immediate judgments of comparison. And (ii) the case is worse when I cannot perceive the two objects at the same time in any position. For I then have to compare what I see with what I remember, and memory in such matters is very untrustworthy. By the use of a portable scale I obviate the first difficulty altogether; but I only obviate the second on the assumption that my scale remains of the same length when I carry it about. Now my judgment on this point rests on memory, to correct the untrustworthiness of which we used a scale at all. This is to put the difficulty in its most striking way; but actually there are modifications to be made. (i) We happen to have the power of actually observing motion, if it be within certain limits of speed, as a special object. Now we can keep our scale under view while we are carrying it about, and, since we do not observe this phenomenon, we can be sure that, if it be changing in length, it must be either very quickly or very slowly. We can further be sure that it is not changing very quickly, for, if it were, even our imperfect comparisons based on memory would soon indicate a change in length to us. Hence we can at least conclude that, if our scale does change in length, it does so either very slowly or, at any rate, by a very small total amount. (ii) The untrustworthiness of judgments of comparison based on memory increases very quickly with the time that elapses between the perception of what is now remembered and the present memory of it. If we compare two distant lengths it is not in our power to reduce this period, which depends on their distance apart, beyond a certain amount; but it is possible to keep the scale under practically continuous observation. If I want to compare a length at Cambridge with a length in London by memory not less than 1½ hours can elapse between the two observations; but, if I carry a metre scale with me, I can (at the risk of alarming my fellow-passengers) contemplate it during the whole journey from Cambridge to King's Cross. I am well aware of course that this is not conclusive. Ln+1 may look as long as Ln, and Ln as Ln-1, and so on; and yet actually there may be such a shortening that, if I could directly compare Ln-1 and Ln+1, I should at once see that their length differed. Still I am at any rate less liable to error than if I trusted immediately to memory. For, be it noted, the length of the lapse, which decreases the trustworthiness of direct comparisons by memory of things in widely different places, may very well be a positive help here. For, if the slow change goes on continually in one direction, I shall surely be able to detect it even by memory sooner or later. If A and B, the two distant bodies, were nearly equal then memory is a bad guide; but, if A be the length of my metre scale when I left Cambridge, and B its length at King's Cross, and B is actually considerably shorter than A, though at any two observations made near together on the journey the lengths appeared the same, then the defect of memory due to lapse of time is being compensated by the actual growth of difference between what I observed at Cambridge and what I observe at King's Cross; for comparison is always more trustworthy for great differences than for small ones. (iii) Finally, there is a more general argument in favour of assuming the constancy of length of a scale when there is no special reason to doubt it. Suppose that we have done as we should do in all accurate physical measurement, and eliminated the effects of those causes which we know to influence the length of material bodies. We are left with unknown possible physical causes, and with the possibility that the mere fact of motion influences length. Now the unknown physical causes might depend on the path pursued in carrying the scale from A to B. These could be neglected if we found that our comparison of the lengths of A and B by the scale were the same no matter what path we pursued2. The main exceptions to this would be (1) if the change of length of the scale depended on bodies so distant that any difference of path that we could make would be negligible in regard to them, or (2) if the change were due to bodies within the earth, since we could not follow paths both in and out of the earth to see whether there were any difference. (1) may fairly be neglected, on the ground that in all science we are obliged to assume that the influence of bodies decreases with their distance, and that we know (as far as we can be said to know anything in the empirical world) that physical forces and disturbances, like light and sound and gravitation, diminish rapidly in intensity with distance. I do not know how (2) could be eliminated. Another possibility is that the change of length is not due to the path pursued, but depends on circumstances which are only operative quite near the bodies to be measured. Here we can only say that, since, by hypothesis, the change is sudden, we might fairly expect it to be observable.

    Lastly, the question of the possibility of length being dependent on velocity needs a moment's consideration. It will meet us again in the Theory of Relativity. We must begin by noticing that it is a suggestion that has a much more definite meaning on the theory of absolute than on that of relative motion. It is perfectly possible that, if there be absolute space and time, there may be laws connecting the lengths of bodies with their absolute velocities. In Lorentz's3 and Fitzgerald's theories of electromagnetic phenomena in moving bodies it is found necessary to suppose that bodies alter their linear dimensions with their velocities along the direction of motion but not at right angles to it. But the velocities contemplated are absolute ones, or, at any rate, relative to the ether only. If it were suggested that there is only motion relative to other bodies, and yet that length depends on velocity, we should have a strange position; for every body in the universe has relative motions of the most various magnitudes and in the most various directions at the same time. If I walk across the room my metre scale has a velocity relative to me, and, at the same time, it has velocities relative to all the white ants who are moving about in South Africa. It were surely very wild to suppose that these relative velocities -- which are just as real as any others -- genuinely condition the length of bodies, unless we have the strongest grounds for doing so. But it will be best to defer the further discussion of this theory until we have seen why it is supposed to be needful to assume it.

    The upshot of this discussion is that here, as in the choice of axes for mechanics, and the determination of equality of times, we start with a belief in the general trustworthiness of our judgments of comparison, then proceed to criticise and modify on the basis of what seems probable from our general knowledge of the 'make' of the world, and finally take as equal those qualities which at once seem equal to our senses as corrected by the best means at our disposal, and, by being taken as equal, enable us to state verifiable general laws about the real world in the simplest possible terms. Here, as everywhere, there is always the possibility of our being wrong -- and I would add the possibility of theoretically incorrigible mistakes; but, until we are proved wrong, we shall act reasonably in believing ourselves to be right. What I would particularly deprecate is that perverted kind of idealism of which the philosophic works of the late M. Poincaré are full, which holds that, when we can imagine a physical law which would make our measurements in error and yet would not allow us ever to detect the mistake, we must say that there is nothing absolute to measure and (what certainly follows) that there can be no right or wrong in the matter. We may surely recognise, as a matter of theory, that we may be liable to genuine errors which we shall never discover; but, having made this acknowledgement, we need never trouble further about them.

    The difficulty which I have been discussing about measurement is, as I said, an old one, but it is worthy of consideration. There remains, however, another connexion between the measurement of lengths and time, the implications of which have only lately been recognised. This connexion is very relevant to the velocity of light and the Theory of Relativity. If there be absolute space the distances between points in it are not functions of time but are eternal relations. But, whether there be absolute space or not, it is certain that we cannot measure directly the distances between its points, but must confine ourselves to the distances4 between bodies in space. Now the distances between bodies are not eternal but are liable to change. What we want to know is the distance between two bodies at a definite moment, for this alone is certainly definite. Hence time enters essentially into the measurement of spatial magnitudes, though not, so far as I can see, into the notion of space, as Minkowski would have us believe.

    It is evident that no process of measurement can directly give us the distance between two bodies at any definite moment, though, under certain circumstances, the results of our measurement may as a matter of fact coincide numerically with the value of the distance at some definite moment. Unless my measuring-rod exactly fits the distance to be measured, it is clear that the moment at which, in the course of measuring, one end of the rod reaches the body B cannot be the same as that at which the other end is at A, but will be later. But what we want to know is the distance between A at one moment and B at the same moment. We might feel tempted to say that, if A and B be relatively at rest, then the number of times that I lay the measuring-rod down in passing from A to B will give their distance at any moment during the period for which they remain relatively at rest. But this forgets an essential factor. The measuring-rod itself is a material object, and has to be laid on something -- call it a third body X. If there be no relative motion between A, B, and X, and none between X and the rod at the moments when the positions of the latter's end-points are being marked on X, what we measure will give the distance at any moment between A and B. If, on the other hand, whilst the same conditions are fulfilled, as between A and B on the one hand, and X and the rod on the other, but A and B are moving relatively to X, what we measure will not be the distance between A and B at any moment. It will, in both cases, be a measure of the distance between the point on X with which A coincided when we began to measure and the point on X with which B coincided when we ceased to measure, but this will no longer coincide with the distance between A and B either at the moment when we began or at that when we ceased measuring. If we forget this consideration, and if A and B both move in the direction AB-- > with the velocity v relative to X, whilst the rod when in position for reading is at rest relative to X, it is clear that we shall reach different results according as we measure from A to B or from B to A. If we measure from A to B the distance from where A was when we started to where B is when we end is greater than that between A and B at any moment, and the difference will depend on how fast we measure. If we measure from B to A our measured distance will be too short. Finally, if v be greater than the rate at which we measure, we shall never overtake B, and shall therefore judge that the distance between A and B is infinite as compared with the length of our rod. If, on the other hand, we measure from B to A, it may be that A will have reached the end of the rod that was at B before we move the rod, and then we shall conclude that there is no distance between A and B.

    All this sounds silly enough when we put it in terms of rods. We say that we should see A moving up the rod as we measured from A to B, or B moving away from it as we measured from B to A; and that, anyhow, we could tell that something was wrong, because, if the measured distance from A to B differed from that from B to A, we should know at once that we could not be dealing with the distances at the same moment, since distance at the same moment is symmetrical. This is doubtless true, but it would not help us if the velocities with which we had to deal were very small; and it is only because we do not use rods for measurement in the cases where errors of this kind are at once important and difficult to detect that the problem does not arise with measuring-rods. But it arises acutely when we deal with the velocity of light, and this in two ways. In the first place, it enters into the actual determination of the velocity of light, and, further, it is involved in all measurements of distance made by means of that velocity when it is supposed to be known.

    Let us return once more to our Fizeau experiment to illustrate these statements. We take as the distance that the light has travelled twice the measured length between the wheel and the mirror. We have good grounds for holding that these are relatively at rest to each other and also that both are relatively at rest to the earth on which we lay our scale. Hence the distance measured is actually that between the two at any moment. But did the light travel this distance when it left the gap and reached the mirror, and does it travel the same distance when it leaves the mirror and returns to the gap? We are likely to say that the actual distance travelled by the light is not from where the gap is when it leaves it to where the mirror is at that moment; nor from where the gap was when the light reached the mirror to where the mirror is then; but that it is the distance between where the gap was when the light left it and where the mirror is when it reaches it. But here the consistent relativist is on very slippery ground, and, if he thinks that he is merely using the language of common-sense, he must remember that commonsense is not relativistic. By the 'place where A was' the relativist must of course mean a place defined by distances from some material axes. Hence the 'place where A was' will depend on your axes. If you take axes fixed in the earth the place where the gap was when the light left it is the same as the place where it is when the light returns to it, and the place where the mirror was when the light left the gap is the same as the place where it is when the light reaches it. For there is no relative motion between the instrument and the earth. If, on the other hand, we take the fixed stars to define our axes the place where the gap was when the light left it will no longer be the same as the place where the gap is when the light returns to it; and naturally the same remarks apply to the mirror. This is of course because of the rotation of the earth relative to the fixed stars. What we have to be careful to remember is that the relativist has no right to the phrase 'the place where A is' because he must mean its place with respect to some definite set of material axes, and, according as he chooses a set X or a set Y, the place where A was at t1 may be the same as or different from the place where it is at t2. What the relativist must talk about is always 'the place of A relative to X' and 'the place of A relative to Y'; there is no inconsistency in one of these changing and the other being unaltered in an interval t. Common-sense on the other hand can talk of 'the place of A' at a given moment, because it means its place in absolute space. It follows that the relativist has no right to talk of the distance travelled by the light. The light left the gap, reached the mirror, and returned to the gap. Relative to the earth the distance travelled is twice that measured between the gap and the mirror; relative to the fixed stars it is different from this. If you object that, after all, there must be some definite distance that the light has travelled independent of your choice of axes, I can only answer that, if you think so, you must drop your theory of relative space. No doubt the distance between two bodies at any moment is independent of choice of axes; but, when the distance required is that from where A was at one moment to where B is at another moment, it is no longer the distance between A and B at any moment that you are measuring but the distance between two hypothetical pieces of matter A' and B', rigidly attached to your particular system of axes, and such that A coincided with A' at the first moment and B with B' at the second moment.

    We might expect then that, if we divide twice the measured distance by the time elapsed between leaving the first gap and re-entering the next, we shall get the velocity of light relative to the earth, but not relative to the fixed stars. (This of course assumes that the velocity of light is constant and that no time is taken up by reflexion at the mirror. These assumptions I do not propose to discuss.) We can now pass to the celebrated Michelson-Morley experiment. Stated in its simplest terms the experiment comes to this. Light from a source fixed in the earth is sent out in a direction tangential to the earth's orbit. It is then partially reflected and partially transmitted by a mirror fixed at a point P in this line. One part continues in its old direction, and the other goes at right angles to it and therefore to the tangent to the earth's orbit. Both parts are reflected back along their respective paths by mirrors placed normally to them at the same measured distances from P. Now, owing to the motion of the earth in its orbit, the light that has travelled straight on in the tangent and back again has described a slightly longer path relative to the fixed stars than that which was deflected at right angles. Since the velocity relative to the fixed stars is constant in all directions the two disturbances that meet after the reflexions will not have started from the common source at exactly the same time. They will therefore interfere. If we turn the whole apparatus through a right angle in its own plane the interference bands ought to have shifted, and the amount of shifting should enable us to compare the velocity of the earth in its orbit relative to the fixed stars with that of light. Michelsen and Morley's apparatus was quite delicate enough to detect changes of this order, but none were detected.

    Another and an equivalent way of stating the result is as follows. If the velocity of light relative to the fixed stars be the same in all directions, whilst the earth has a velocity relative to them in a certain direction and none at right angles to it, we should expect the velocity of light relative to the earth to be different for paths along and at right angles to its direction of motion. Hence, if the paths on the earth are equal, light ought to take different times to traverse them, and so there ought to be this interference phenomenon which actually is not observed.

    It is the non-observability of such effects which is the motive of the Theory of Relativity. We can already see that one of the alleged paradoxes that flow from this Theory, viz. that if A has velocity u relative to B and B has velocity v relative to C in the same direction then the velocity of A relative to C is not u+v, is realised in the experiment; for, if we call c the velocity of light relative to the fixed stars and u that of the earth relative to them, it would follow from this law that the velocity of light relative to the earth would be c-u in the direction of the earth's motion and c at right angles to it, whereas both velocities are found to be equal. But before we actually pass to the Theory itself there are two points worth mentioning. (i) In the Michelsen-Morley experiment we are sending out light from a source which is in motion relative to the fixed stars. If we supposed light to consist of particles obeying the ordinary laws of mechanics and shot off from the emitting body by impulsive forces which are the same in all directions, then those particles which travelled parallel to the earth's direction of motion would start with the common velocity due to the impulse + the velocity that they already had as parts of a body moving relative to the fixed stars. Those which were shot off at right angles to the earth's motion would only have the velocity due to the impulse in that direction, though they would of course drift with the velocity of the earth in the direction of its motion. The result would be that the velocity of light from a source moving relatively to the fixed stars would not be the same relative to the fixed stars in all directions, and this might explain the negative result of the experiment without resort to anything more complicated. We should further, I take it, have to suppose that on passing through the glass plate at P the direct ray would be retarded and that this retardation would be permanent. On the wave-theory it is of course retarded, but only for as long as it is travelling in the glass. It is worth while to mention this alternative theory, but hardly to work it out, since all the arguments by which physicists have persuaded themselves that the undulatory theory is to be preferred to the emission theory are against it. (ii) The purely physical theory to account for the result of the Michelsen-Morley experiment is that of Lorentz and Fitzgerald that bodies in motion relative to the fixed stars undergo a contraction in the direction of their motion, which depends on their velocity, but none at right angles to it. This contraction cannot be detected, because it affects the measuring scale too; so that lengths measured on the earth in various directions, which appear equal, are not really so. It can be shown that, if a length which is l when there is no motion relative to the fixed stars becomes l x (1- (v2/c2))½ when it moves with velocity v along its own direction relative to the stars, but remains unchanged if moved at right angles to itself, the observed results can be predicted. (The letter c stands here for the velocity of light relative to the fixed stars.) Of course Lorentz does not mention the fixed stars but talks about motion relative to the ether, which be supposes to be fixed in absolute space. Since you can no more determine velocities or positions relative to the ether than to absolute space, and since the former is supposed to be fixed in the latter, I see no advantage in retaining both conceptions. If either is to be retained absolute space has much the better claim. But, in any case, the empirical laws must have been discovered by reference to actual bodies, and these are here, as in mechanics, ultimately the fixed stars, or a system of coordinates so imagined as to make a form of statement of the laws which is approximately true in terms of motions relative to the fixed stars completely true in terms of motions with respect to it. The objection to the Lorentz-Fitzgerald theory as it stands is its apparent arbitrariness. I do not think that it is a legitimate objection to say with Poincare that the contraction is postulated ad hoc. Of course it is postulated ad hoc; so is everything in physics that cannot be directly observed. Nor is it relevant to say that there is a very great difference between postulating something to account for what you do observe and postulating something to account for your observing nothing. To observe nothing where, on theories which have so far accounted wonderfully well for all the phenomena, you ought to observe something, is a very positive fact; it shows that your theories are either wrong, or, at any rate, incomplete; and, in as far as they have been so successful over so large a field, it is reasonable, not to reject them entirely and start afresh, but to seek for some supplementary hypothesis that shall account for their present failure. My objection to the theory is that, when you leave out of account ether and absolute space, it seems strange and improbable. Let us grant at once that this is no conclusive reason against it, and that no causal law that you can possibly suggest is a priori impossible, and let us then consider what is involved in this suggested law.

    If we are to be consistent relativists we must say that length, on this theory, depends on motion relative to the fixed stars. We shall then probably be asked why motion relative to the fixed stars should be connected with length rather than motions relative to (say) the most hard-roed codfish in the Atlantic, since one set has no prerogative of reality over the other. But our discussions about dynamics will enable us to dispose of this question with something more than the general answer that you cannot prescribe a priori what in nature is going to be relevant to what. The answer in fact is that length will be as much connected with motions relative to the codfish, or any other body of reference, as with motions relative to the fixed stars. All that we have to add is that, if you refer the motions of all bodies to the fixed stars, the law connecting their lengths with their velocities will take the simple form given by Lorentz and Fitzgerald, whilst, if you refer their motions to axes fixed in the codfish, the law (though equivalent) will be of excessive complication. There is nothing arbitrary in an objectionable sense in the Lorentz-Fitzgerald law when stated in terms of the fixed stars; it gives the fixed stars no prerogative in nature but only a prerogative in human calculations. Leibniz's God, with his passion for complicated mathematics, would probably prefer the codfish as a body of reference, and would commit no error in doing so. You may, if you choose, make an objection somewhat like Mr Russell's question whether the fixed stars on the relativist view are subject to Newtonian forces. You can say: At the same moment the fixed stars cannot have two different sizes; when you refer motions to them they will not be shortened, since v in the Lorentz- Fitzgerald formula, will vanish. But, when you refer them and other bodies to the codfish, they will have a definite velocity, and they will be shortened. All this, however, is sheer confusion. If the Lorentz-Fitzgerald formula be strictly true for the bodies in the world when referred to the fixed stars then the equivalent law when you refer all bodies, including the fixed stars, to the codfish must be such that the absolute size of each body at the same moment is the same.

    So far then the Lorentz-Fitzgerald law, though stated by its authors in terms of absolute motion or (what is equivalent) of motion relative to a fixed ether, furnishes, even if true, no more ground for a belief in absolute space or a fixed ether than do the laws of mechanics when similarly stated. But where people feel a difficulty is here. Is it possible, they will ask, that an intrinsic quality of a body, like its length, can be causally connected with a mere change in its external relations like its relative motions? We do not, I must repeat, in ordinary life believe for an instant that motion is merely a change of distance either between two bodies or between bodies and points of absolute space; it is only by a great effort that we can bring ourselves to keep this view firmly before the mind in all its bearings, and the fallacies with which discussions on absolute and relative motion bristle (some of which I can hardly expect to have been fortunate enough to avoid) show that, with the slightest relaxation of attention, we fall back into the notions of common- sense. We regard the motion of each body as a quality of it as much as its size, and, whilst we grant that this quality is connected with changes of spatial relation between bodies, and, if we believe in absolute space, between them and it, I doubt whether even the acceptance of absolute space, when coupled with the assertion that the motion of a body is nothing but the change of its relation to points of absolute space, gives what common-sense thinks it means by motion. Now common-sense, holding this belief that the motion of a body is something more than the change of its spatial relations, finds no particular difficulty in thinking that this quality may be connected with the other quality of the body called length. It is surprising, like the properties of Radium, because unfamiliar; but that is the worst that can be said of it. But, if you ask common-sense to drop the idea that you can talk of the motion of a single body as a quality of it, and to confine itself to the change of distance between it and other bodies -- a perfectly reciprocal change -- and then tell it that this is causally connected with the size of the body, it will tell you that it is inconceivable. Nor would it be much comforted if you told it that the size was connected with the rate at which its distance was changing with respect to points of absolute space. It will say to the first suggestion: I am ready to believe that if I walk across the room my height alters because the motion and the height are both qualities of me; but I cannot believe that if you walk across the room the mere change of distance will alter my height, for my height is a quality of me and the change of distance a mere alteration in a reciprocal relation between us. I am even ready to believe that your motion, if you will let me take it as a quality of you, affects my height, for I know of plenty of cases of transeunt causation; but further than this I cannot go. Put into abstract form, the position of common-sense is that, if a reciprocal relation between two terms involves no qualities in those terms except that of standing in this relation, then a change in this relation cannot be causally connected with changes in those qualities which can be seen not to be dependent for their existence on the existence of other terms. Size is such a quality; common-sense is firmly convinced that, if there were only one body in the universe, it would have some definite size and shape at each moment; and, if you answer that size depends on relations, you simply commit the fallacy of confusing magnitude with the possibility of measuring magnitude. The same difficulty is not felt with mechanics, because here we have causal relations between changes of distance, not between them and qualities of terms.

    Frankly I am not prepared to pronounce on this dogma. When I reflect on it I do not see that it is self-evident; but one must attach a certain weight to general principles which many people find evident, unless one can either show that they are false, or point to causes which have actually produced the belief but include among them beliefs incompatible with it. The dogma remains the only serious objection to the Lorentz-Fitzgerald theory, assuming that the latter accounts for the facts. But, if we could show that there was no need for any special physical assumption, that the negative result of the Michelsen-Morley and numerous other experiments depends on assumptions involved in all measurement, which are commonly overlooked and only here begin to be relevant, we should gain a double advantage. We should no longer go against what may be an a priori law, and what, at any rate, renders such a theory as that of Lorentz intrinsically improbable; and we should not have to alter or supplement our old, and, so far, successful theory of electrodynamics, but merely to develop the consequences of something in it that had been overlooked. The Theory of Relativity claims these advantages, and we will now consider it.

    Let us consider two systems which are in relative motion to each other, and suppose that the people on them are trying to find the magnitude of some physical quantity like the velocity of light. It is clear that, if we call S1 and S2 the two systems, the question: Is the velocity of light relative to S1 the same as or different from its velocity relative to S2? has no very definite meaning till we are sure that the people on both systems are using the same units of length and of time, or, at any rate, know the relations between their respective units. Of course, in nature, light has a certain velocity relative to each system, and these velocities are either the same or different, whatever may be people's measures of them. But all that we can know is whether the numerical values found by the two sets of observers are the same; and it is clear that their sameness or difference will only allow us to infer to the sameness or difference of physical velocity if they are using the same units. If, again, they think they are using the same units, but their judgments on this point are in fault, sameness of measure will not mean sameness of magnitude measured, and difference of measure need not mean difference of magnitude measured. These points seem obvious; but, as most difficulties here spring from neglecting to make the obvious explicit, I mention them. Let us first consider their judgments about time. It is clear that they must not merely agree about their units of time, but must also agree as to when a physical process is uniform. The only thing they can do is to agree to take some definite physical process, which they can both observe, as uniform, and to standardise their measuring instruments by means of this. As we have already pointed out, there is an element of convention + an element of immediate judgment involved in taking a certain process as uniform. We begin with immediate judgments of comparison, and end by defining as uniform some process which (a) appears uniform to our judgments of immediate comparison, and (b) by being assumed uniform, simplifies as far as possible the statement of laws that are found to be true. We can suppose that each set of people has gone through this process for themselves independently of each other and that each has decided that the velocity of light relative to his system is uniform. This does not imply that each has found the same numerical value for it, for that would involve that they have agreed about units of length and time5. This being assumed, we can suppose that they both use the same methods of regulating their clocks so that they shall go uniformly. The method adopted by both will be as follows: When a clock on S1 marks t1 a light signal is sent out to some other point in the direction of relative motion and reflected back. It return{ when the clock marks t2. Another signal is sent out to the same point when the clock marks t3 and returns when it marks t4. If you always find that t2 - t1 = t4 - t3 the clock will be said to be going uniformly. The people on S2 will regulate their clocks in the same way.

    Now we can suppose clocks dotted about S1 and S2 wherever we please, and the next point that you must agree about is in your test for whether all the clocks in a system are going at the same rate, it being granted that each goes at a uniform rate. Here two methods present themselves. If you can carry the clocks about you have only to bring them up to the clock that you are going to take as standard on S1, and see whether, while this clock's hands pass from t1 to t2, those of all the others pass through the same angle. If so you can further completely synchronise the clocks by seeing that all mark t1 together. Assuming that the rates of your clocks will not be altered by carrying them about they can now be put back in their places on S1 and all the clocks on that system will be running at the same uniform rate and having the same zero. The people on S2 can synchronise their clocks with their standard one in the same way. But, if you cannot synchronise your clocks in this way (and it is very unlikely that you could do so for all the clocks of a system), another plan must be adopted by the people on each system. Let the standard clock on S1 send out a light signal to a point P on S1 when it marks t1. Let this reach P when P's clock marks t2. Let the standard clock send out a second signal when it marks t3, and let this reach P when P's clock marks t4. Then if t2 - t1 = t4 - t3 we say that the clock at P is going at the same rate as the standard clock. It only remains to set the zeros of the other clocks on S1 in accord with the standard one. If the standard one sends out a signal when it marks t1, and this is received at P when P's clock marks t2 and at once reflected back to the standard clock, reaching it when it marks t3, we say that the zeros of the two clocks agree, provided that t2 = ½ (t1 + t3). The people on S2 are supposed to set their clocks in the same way.

    So far we have assumed that both people have concluded that the velocity of light is uniform independently of each other. It is clear that, if this way of setting clocks be reasonable at all, it is as reasonable for the people on S1 as for those on S2. And the supposed uniformity6 of the velocity of light makes it reasonable for both. Let us now consider their measures of length. Granted that each holds that light travels uniformly relative to him, and remembering that they cannot possibly compare the velocities of light which each finds relative to his own system with each other until they have agreed on their units of length and time, we see that there must be an element of convention in the answer to the question whether the velocity of light relative to both is the same or different. It is clear that they can so choose their respective units that the numerical values of the two velocities will be the same, and, if they further have grounds for thinking that their units are the same under these circumstances, they will naturally conclude that the actual velocities of which their measures are equal are equal in nature. Now the people on S1 are at liberty to define their units in such a way that if a light signal leaves a point P when P's clock marks t1, and reaches a point Q when Q's clock marks t2 the measure of the distance PQ shall be c (t2 - t1). c is a mere number which they can choose as they like, and whatever they choose will be the numerical value of the velocity of light relative to them in their units. Now, if the people on S2 have synchronised their clocks in the same way as those on S1, and have decided to set out a coordinate system in the same way by light signals, their choice of units of space and time are still open to them. They can make their standard clock go at any rate they like compared with that of S1 and they can choose any unit they like for distance. Since, until they have chosen their units, they cannot say whether the velocity of light relative to them is the same as or different from that of light relative to S1, they are at liberty to choose their units so that the two velocities shall have the same measure. This will not of course supply them with any reason for supposing that the two velocities are equal in fact, unless they have grounds for believing that their units of time and space are the same multiples of S1's units of time and space respectively. Thus the fact that the measured velocity is the same relative to the two systems depends on a convention about units. Still the question whether the velocities themselves, as distinct from the measures of them, are equal is not a matter of convention; it depends on whether equal measures of length and time, as determined on these conventions, really correspond to equal lengths and times in nature.

    Let us suppose that the people on S, reckon that the standard clock on S2 is going r times as fast as theirs. Their method of judgment is as follows. They find that when they observe this clock to mark t'1 the clock on their system which is opposite to it marks t1, that when they observe it to mark t'2 the clock on their system opposite to it marks t2, and that (t'2 - t'1)/(t2 - t1) = r. Let us further suppose that the people on S1 reckon that S2 is moving with a velocity u relative to them along their x axis. Then it is easy to show, and it is shown by Prof. Huntington, that, if the clocks be synchronised and the coordinates set out by light signals in accordance with the rules laid down, the following relations will hold between the coordinates and clock-readings of the two systems. Let the standard clocks be at the respective origins of the two systems, and, when these origins are opposite each other, let the clocks mark 0. Further let the common direction of relative velocity be taken as the x- axis in both. Then, if x, y, z be the coordinates of a point on S1 and the clock there reads t1, whilst x', y', z' are the coordinates of the point on S2, opposite to this and t'2 its clock reading, the following equations hold:

x' = lk(x - ut), y' = ly, z' = lz

t' = lk (t -[(u/c2)x])

k =1/[(1- (u2/c2)½], l =r/[(1- (u2/c2)½]

    These are the usual equations of the Theory of Relativity. It can be shown that, with our conventions, the numerical value of the velocity of S1 relative to S2, as measured by people on S2, is - u, and that the rate of the standard clock on S1, as observed from S2, is 1/(k2r). It will then be seen that the equations are -- as they ought to be, seeing we are dealing simply with relative motion between the two systems -- completely reciprocal; i.e. we have

x = kl'(x'- u't') etc.

    A further very important result is that the numerical value of the velocity of light from a source in uniform motion relative to the two systems is found to be the same, viz. c by observers on each relative to his own system. The fact that this follows from simple and obvious considerations about measurement on Prof Huntington's method seems to me to be its great advantage over that of Einstein. Einstein7 takes this by no means obvious result as an independent postulate, and it is from this that he actually obtains the transformation equations by mathematical reasoning. I say that he takes it as independent; I should say that be ought to do so. Actually he seems to think (wrongly, as is well brought out by Mr Norman Campbell8) that it is deducible from the axiom that the velocity of light relative to a system from a source fixed in it and measured by people on it will be independent of the relative motion of the system to others9.

    The Lorentz-Fitzgerald formula follows at once from these equations, and therefore -- when measurements are made by light signals, at any rate from plausible conventions about measurement, and from these alone. Let P and Q be two points on the x-axis of S1, and let their coordinates be xp and xq. Then the people on S1 will call the measure of PQ xq -xp. The measure which people on S2 will give of their distance will be the measure that they give of the distance between two points P' and Q' on S2, such that, at the same moment by the clocks on S2 (say t'), P' is opposite to P and Q' to Q. Then we have:

x'p = kl (xp - utp),

and

x'q = kl (xq - utq)

when tp and tq are the readings of the clocks on S1 at P when P is opposite to P' and at Q when Q is opposite to Q' respectively. Thus

t' = kl (tp - (u/c2)xp) = kl (tq- (u/c2)xq).

Whence

tq - tp = (u/c2)(xq - xp),

Therefore x'2 - x'p = kl (xq - xp)(1 - (u2/c2))

=l/k (xq - xp).

    Suppose that R is a point on S, such that RP is at right angles to PQ and yr - yp = xq - xp. Then it is easy to show that the distance RP as reckoned by people on S, will be l (yr -yp). Hence (y'r - y'p)/(x'q - x'p) = l (yr- yp)/((l/k)(xq - xp)) = k (yr -yp)/(xq - xp) = k = 1/(1 - (u2/c2))½

    This is in accord with the Lorentz-Fitzgerald formula.

    Again, the people on S2 will not reckon two events which happen at different places on S1 as contemporary when the people on S1 count them as contemporary. Suppose an event happens at P on S1 when the clock there marks t and another at Q on S1 when the clock there marks the same. The times at which they will be judged to have happened by people on S2 will be those marked by their clocks which are opposite P and Q respectively when the clocks at P and Q mark t. Let these times be t'p and t'q. Then t'p = kl (t-(v/c2)xp) and t'q = kl (t-(v/c2)xq),

t'q - t'p = - kl (v/c2)(xq - xp)

= - (v/c2)(x'q -x'p).

    So the event at the place further along the axis of x is judged by the people on S2 to happen before that at the point nearer the origin, though the people on S1 judge them to be contemporary. Of course the complete reciprocity of the transformation equations involves that all these remarks apply, mutatis mutandis, to the judgment of people on S2 about times and distances on S1.

    But, now that we have seen some of the implications of setting clocks and measuring lengths according to the reasonable conventions set out above, the question arises: What would the people on S1 and S2 really believe about the physical world? We must assume that they can communicate with each other (e.g. by wireless telegraphy); indeed we have already assumed this when we said that they both took the same number c as the measure of the velocity of light relative to their own systems from sources fixed in them. But we will not assume at present that clocks or measuring rods can be carried about either system, or, a fortiori, from one system to the other.

    It is clear that the people on S1 will tell those on S2 that the latter's clocks are not really synchronous, but that they get slower the further they are along the x-axis from the origin. And the people on S2 will make exactly the same criticism on the clocks of S1. Yet both have synchronised their clocks by the same method, and therefore the opinion of neither is to be preferred to that of the other. What are they to conclude? The difference arises out of the employment of two different methods of judging synchronism. The people on S1 judge of the synchronism of their own clocks by the method of light signals, so do the people on S2. But the people on S1 apply a different test to S2's clocks. Their test is that, if a clock of theirs at P marks t and the clock opposite to it on S2 then marks t', then, if a clock of theirs at Q marks t, the clock opposite to this on S2 ought also to mark t'. The argument is that the readings at P and P' are certainly contemporary and the readings at Q and Q' are certainly contemporary, but identity of reading at P and Q indicates identity of time if the clocks on S2 are properly set, and therefore, if the clocks on S2 were properly set, there ought to be identity of readings at P' and Q'. It is difficult to find any flaw in this test, and therefore we must conclude that there is something wrong with the test which is used by the separate systems, in fact that identity of clock-readings on a given system set out by the conventions that we have adopted is not a mark of identity of time. Since both parties cannot be right, and since neither has the slightest claim to be right as against the other, the only conclusion is that both may be wrong, and that these conventions, plausible as they may seem, do not give us a system of clocks such that identity of reading on two clocks of the same system means identity of time.

    But the source of the perplexity would be quite clear to a believer in a fixed ether or absolute space. He would say: Your tests for synchronism in clocks and your measures of distance are not in general valid ones, they only are real tests for the particular case of systems that are at rest relative to the medium in which light travels. Now, when you have (as in your example) two systems in relative motion to each other, it is certain that one and possible that each of them is in motion relative to the ether. Hence your tests are bound to lead you to wrong beliefs in at least one of the systems, and the paradoxes that you find result as a matter of course. Unfortunately no experiments will detect or measure velocity relative to the ether, and therefore we cannot tell in what system, if any, the tests proposed do really ensure the synchronism of clocks.

    It is clear bow it happens that the test is not in general accurate. It will be remembered that, in regulating our clocks, we said that their zeros agreed if a light signal which left the standard clock when it marks t1 reaches a clock at P on the x-axis when it marks t2, is reflected, and returns to the standard clock when it marks t3 such that t2 = 1/2 (t1 + t3). We can see that this test is only valid if the velocity of light relative to our system be the same in both directions. We have assumed in fact in using the test that the velocity of light relative to each system is not merely uniform in any given direction, but is the same in all directions and in both senses. Since we have set out our clocks and coordinates on this assumption it is not surprising that our measures of the velocity of light accord with it; but, unless the assumption be true, agreement of numerical values of measures is no proof of identity of what is measured. On the other hand until we have made some assumption that will enable us to set out coordinates and regulate clocks, we can get no numerical measures at all. The result of the discussion seems to me to be this:

  1. Left to themselves the people on S1 and S2 would have no reason to doubt the correctness of the assumptions on which they regulated their clocks and set out their coordinates.
  2. Even when they can communicate and discover disagreement neither has any claim to be right as against the other, since both have made the same assumptions. Hence one must be wrong and both may be, and neither can tell which is.
  3. The assumption which must be wrong in at least one case is that the velocity of light relative to the system in nature (as distinct from the numerical value found for it) is the same in all directions and both senses.
  4. Since you cannot regulate your clocks and set out your coordinates until you have made some assumption on this point, and cannot get numerical values for the relative velocities of light in various directions until you have regulated your clocks and set out your coordinates, no numerical value gained by experiment can ever without circularity be claimed as a proof of your assumption.
  5. But disagreement between people on different systems who have made similar assumptions on points as to which there must be agreement if the assumptions be true disproves their truth in one case at least.
  6. What remains to be said is that these disagreements are of no direct importance to philosophy or to science.
They are of no direct importance to philosophy, because the reality of time and space and the existence of equal lengths and contemporary events are in no way touched by the fact that it is impossible to give an absolutely satisfactory test for equality of lengths or times. Indirectly they are of great interest because they force us to see that assumptions are made in all measurement which are often overlooked or taken as obviously true. And they are of no direct importance to science, because we are not there concerned to discover laws connecting absolute lengths, durations, and other magnitudes, but only laws connecting the values which we can read off our instruments. No doubt we call these values measures of the lengths, times, forces, etc. in nature, and so speak of the laws connecting the measured values as laws of what they are supposed to measure. But what we have to remember is that, whilst these laws do, no doubt, imply the existence of laws in nature, they are expressions of the latter subject to the assumptions on which our measured values are supposed to be measures of the natural magnitudes in question. If those assumptions be false, but internally consistent and consistently employed, we shall arrive at laws which are true and valuable, but are not direct transcriptions into numbers of the functional relations of those natural magnitudes in terms of which they are verbally stated. To put it more clearly, let us suppose that I discover a law connecting my measures of three magnitudes P, Q, and R. I shall no doubt talk of this as a law connecting the magnitudes P, Q, and R themselves. Now, all my measurements are made on certain assumptions about what is a test for equality and inequality among such magnitudes. If these assumptions be true, i.e. if my assumptions lead me to believe that two quantities of the kind P are equal when they actually have the same magnitude, and to suppose that one is greater than the other when it actually has a greater magnitude, and so on, then my law connecting the measures of P, Q, and R can be called a direct transcription into numbers of the law of nature connecting the magnitudes themselves. If, on the other hand, my assumptions be false, but consistent and consistently used, i.e. if they lead me to take quantities with different magnitudes as equal, and so on, yet so that my mistakes obey a fixed law, then the law connecting my measures will not be direct transcription into numbers of that which in nature connects the magnitudes of the kind involved, but it will be an indirect transcription of it which will be just as useful, and -- since it is connected with the direct transcription in a fixed, though to us unknown, way -- in a perfectly definite sense, true. The Principle of Relativity, as a physical theory based on empirical evidence, now simply comes to this. If the people on two systems S1 and S2 in uniform relative motion to each other have both laid out their coordinate systems and regulated their clocks on the assumptions described above, what each takes to be the measure of any given physical magnitude will be the same function of what he takes to be the measures of other physical magnitudes. We know that one man's laws, at any rate, cannot be a direct transcription of the laws of these magnitudes in nature, and that perhaps the laws of neither are direct transcriptions; but this need worry no one, since no one can ever prove that his laws are direct transcriptions of those of nature, and all purposes of science are served if they be only indirect ones.

Notes

1 Raum und Zeit. For a full exposition of the Theory of Relativity see Das Relativitätsprinzip by M. Laue, and, for a more elementary account, Das Relativitätsprinzip der Elektrodynamik by G. O. Berg. The best discussion of the subject in English is the very able article by Prof. Huntington, Phil. Mag. April, 1912. Two interesting pamphlets, called Optical Geometry of Motion and A Theory of Time and Space, by Mr A. A. Robb, may also be mentioned.

2 Of course this is only true if you are using a scale to measure and compare two distant bodies; if you are measuring the distance between two bodies this method of elimination is no longer possible because you must push the scale along the line joining them.

3 Loretz, The Theory of Electrons.

4 I do not attempt here to discuss what precisely the distance between two bodies can mean. But I mention the matter as being one full of difficulties that have hardly been touched as yet.

5 In what follows about the regulation of clocks and the comparison of rates and zeros I am very much indebted to Prof. Huntington's article already quoted.

6 Prof. Huntington does not mention this point, which seems to me important. Unless they both assumed this it would be an unreasonable way of setting clocks for the ones who did not assume it.

7 Jahrbuch d. Radioaktivität u. Electronik, Jan. 1908.

8 'Common Sense of Relativity,' Phil. Mag. April, 1911.

9 This axiom may be compared with our remarks on the Lorentz-Fitzgerald Theory, p. 374 et seq.

Contents