The Man in the Lift. About 1915 Einstein made a further development of his theory of relativity extending it to non-uniform motion. The easiest way to approach this subject is by considering the Man in the Lift.

Suppose that this room is a lift. The support breaks and down we go with ever-increasing velocity, falling freely.

Let us pass the time by performing physical experiments. The lift is our laboratory and we shall start at the beginning and try to discover all the laws of Nature—that is to say, Nature as interpreted by the Man in the Lift. To a considerable extent this will be a repetition of the history of scientific discovery already made in the laboratories on terra firma. But there is one notable difference.

I perform the experiment of dropping an apple held in the hand. The apple cannot fall any more than it was doing already. You remember that our lift and all things contained in it are falling freely. Consequently the apple remains poised by my hand. There is one incident in the history of science which will not repeat itself to the men in the lift, viz. Newton and the apple tree. The magnificent conception that the agent which guides the stars in their courses is the same as that which in our common experience causes apples to drop, breaks down because it is our common experience in the lift that apples do not drop.

I think we have now sufficient evidence to prove that in all other respects the scientific laws determined in the lift will agree with those determined under more orthodox conditions. But for this one omission the men in the lift will derive all the laws of Nature with which we are acquainted, and derive them in the same form that we have derived them. Only the force which causes apples to fall is not present in their scheme.

I am crediting our observers in the lift with the usual egocentric attitude, viz. the aspect of the world to me is its natural one. It does not strike them as odd to spend their lives falling in a lift; they think it much more odd to be perched on the earth’s surface. Therefore although they perhaps have calculated that to beings supported in this strange way apples would seem to have a perplexing habit of falling, they do not take our experience of the ways of apples any more seriously than we have hitherto taken theirs.

Are we to take their experience seriously? Or to put it another way—What is the comparative importance to be attached to a scheme of natural laws worked out by observers in the falling lift and one worked out by observers on terra firma? Is one truer than the other? Is one superior to the other? Clearly the difference if any arises from the fact that the schemes are referred to different frames of space and time. Our frame is a frame in which the solid ground is at rest; similarly their frame is a frame in which their lift is at rest. We have had examples before of observers using different frames, but those frames differed by a uniform velocity. The velocity of the lift is ever-increasing—accelerated. Can we extend to accelerated frames our principle that Nature is indifferent to frames of space and time, so that no one frame is superior to any other? I think we can. The only doubt that arises is whether we should not regard the frame of the man in the lift as superior to, instead of being merely coequal with, our usual frame.

When we stand on the ground the molecules of the ground support us by hammering on the soles of our boots with force equivalent to some ten stone weight. But for this we should sink through the interstices of the floor. We are being continuously and vigorously buffeted. Now this can scarcely be regarded as the ideal condition for a judicial contemplation of our natural surroundings, and it would not be surprising if our senses suffering from this treatment gave a jaundiced view of the world. Our bodies are to be regarded as scientific instruments used to survey the world. We should not willingly allow anyone to hammer on a galvanometer when it was being used for observation; and similarly it is preferable to avoid a hammering on one’s body when it is being used as a channel of scientific knowledge. We get rid of this hammering when we cease to be supported.

Let us then take a leap over a precipice so that we may contemplate Nature undisturbed. Or if that seems to you an odd way of convincing yourself that bodies do not fall,[13] let us enter the runaway lift again. Here nothing need be supported; our bodies, our galvanometers, and all measuring apparatus are relieved of hammering and their indications can be received without misgiving. The space- and time-frame of the falling lift is the frame natural to observers who are unsupported; and the laws of Nature determined in these favourable circumstances should at least have not inferior status to those established by reference to other frames.

I perform another experiment. This time I take two apples and drop them at opposite ends of the lift. What will happen? Nothing much at first; the apples remain poised where they were let go. But let us step outside the lift for a moment to watch the experiment. The two apples are pulled by gravity towards the centre of the earth. As they approach the centre their paths converge and they will meet at the centre. Now step back into the lift again. To a first approximation the apples remain poised above the floor of the lift; but presently we notice that they are drifting towards one another, and they will meet at the moment when (according to an outside observer) the lift is passing through the centre of the earth. Even though apples (in the lift) do not tend to fall to the floor there is still a mystery about their behaviour; and the Newton of the lift may yet find that the agent which guides the stars in their courses is to be identified with the agent which plays these tricks with apples nearer home.

It comes to this. There are both relative and absolute features about gravitation. The feature that impresses us most is relative—relative to a frame that has no special importance apart from the fact that it is the one commonly used by us. This feature disappears altogether in the frame of the man in the lift and we ought to disregard it in any attempt to form an absolute picture of gravitation. But there always remains something absolute, of which we must try to devise an appropriate picture. For reasons which I shall presently explain we find that it can be pictured as a curvature of space and time.

A New Picture of Gravitation. The Newtonian picture of gravitation is a tug applied to the body whose path is disturbed. I want to explain why this picture must be superseded. I must refer again to the famous incident in which Newton and the apple-tree were concerned. The classical conception of gravitation is based on Newton’s account of what happened; but it is time to hear what the apple had to say. The apple with the usual egotism of an observer deemed itself to be at rest; looking down it saw the various terrestrial objects including Newton rushing upwards with accelerated velocity to meet it. Does it invent a mysterious agency or tug to account for their conduct? No; it points out that the cause of their acceleration is quite evident. Newton is being hammered by the molecules of the ground underneath him. This hammering is absolute—no question of frames of reference. With a powerful enough magnifying appliance anyone can see the molecules at work and count their blows. According to Newton’s own law of motion this must give him an acceleration, which is precisely what the apple has observed. Newton had to postulate a mysterious invisible force pulling the apple down; the apple can point to an evident cause propelling Newton up.

The case for the apple’s view is so overwhelming that I must modify the situation a little in order to give Newton a fair chance; because I believe the apple is making too much of a merely accidental advantage. I will place Newton at the centre of the earth where gravity vanishes, so that he can remain at rest without support—without hammering. He looks up and sees apples falling at the surface of the earth, and as before ascribes this to a mysterious tug which he calls gravitation. The apple looks down and sees Newton approaching it; but this time it cannot attribute Newton’s acceleration to any evident hammering. It also has to invent a mysterious tug acting on Newton.

We have two frames of reference. In one of them Newton is at rest and the apple is accelerated; in the other the apple is at rest and Newton accelerated. In neither case is there a visible cause for the acceleration; in neither is the object disturbed by extraneous hammering. The reciprocity is perfect and there is no ground for preferring one frame rather than the other. We must devise a picture of the disturbing agent which will not favour one frame rather than the other. In this impartial humour a tug will not suit us, because if we attach it to the apple we are favouring Newton’s frame and if we attach it to Newton we are favouring the apple’s frame.[14] The essence or absolute part of gravitation cannot be a force on a body, because we are entirely vague as to the body to which it is applied. We must picture it differently.

The ancients believed that the earth was flat. The small part which they had explored could be represented without serious distortion on a flat map. When new countries were discovered it would be natural to think that they could be added on to the flat map. A familiar example of such a flat map is Mercator’s projection, and you will remember that in it the size of Greenland appears absurdly exaggerated. (In other projections directions are badly distorted.) Now those who adhered to the flat-earth theory must suppose that the map gives the true size of Greenland—that the distances shown in the map are the true distances. How then would they explain that travellers in that country reported that the distances seemed to be much shorter than they “really” were? They would, I suppose, invent a theory that there was a demon living in Greenland who helped travellers on their way. Of course no scientist would use so crude a word; he would invent a Graeco-Latin polysyllable to denote the mysterious agent which made the journeys seem so short; but that is only camouflage. But now suppose the inhabitants of Greenland have developed their own geography. They find that the most important part of the earth’s surface (Greenland) can be represented without serious distortion on a flat map. But when they put in distant countries such as Greece the size must be exaggerated; or, as they would put it, there is a demon active in Greece who makes the journeys there seem different from what the flat map clearly shows them to be. The demon is never where you are; it is always the other fellow who is haunted by him. We now understand that the true explanation is that the earth is curved, and the apparent activities of the demon arise from forcing the curved surface into a flat map and so distorting the simplicity of things.

What has happened to the theory of the earth has happened also to the theory of the world of space-time. An observer at rest at the earth’s centre represents what is happening in a frame of space and time constructed on the usual conventional principles which give what is called a flat space-time. He can locate the events in his neighbourhood without distorting their natural simplicity. Objects at rest remain at rest; objects in uniform motion remain in uniform motion unless there is some evident cause of disturbance such as hammering; light travels in straight lines. He extends this flat frame to the surface of the earth where he encounters the phenomenon of falling apples. This new phenomenon has to be accounted for by an intangible agency or demon called gravitation which persuades the apples to deviate from their proper uniform motion. But we can also start with the frame of the falling apple or of the man in the lift. In the lift-frame bodies at rest remain at rest; bodies in uniform motion remain in uniform motion. But, as we have seen, even at the corners of the lift this simplicity begins to fail; and looking further afield, say to the centre of the earth, it is necessary to postulate the activity of a demon urging unsupported bodies upwards (relatively to the lift-frame). As we change from one observer to another—from one flat space-time frame to another—the scene of activity of the demon shifts. It is never where our observer is, but always away yonder. Is not the solution now apparent? The demon is simply the complication which arises when we try to fit a curved world into a flat frame. In referring the world to a flat frame of space-time we distort it so that the phenomena do not appear in their original simplicity. Admit a curvature of the world and the mysterious agency disappears. Einstein has exorcised the demon.

Do not imagine that this preliminary change of conception carries us very far towards an explanation of gravitation. We are not seeking an explanation; we are seeking a picture. And this picture of world-curvature (hard though it may seem) is more graspable than an elusive tug which flits from one object to another according to the point of view chosen.

A New Law of Gravitation. Having found a new picture of gravitation, we require a new law of gravitation; for the Newtonian law told us the amount of the tug and there is now no tug to be considered. Since the phenomenon is now pictured as curvature the new law must say something about curvature. Evidently it must be a law governing and limiting the possible curvature of space-time.

There are not many things which can be said about curvature—not many of a general character. So that when Einstein felt this urgency to say something about curvature, he almost automatically said the right thing. I mean that there was only one limitation or law that suggested itself as reasonable, and that law has proved to be right when tested by observation.

Some of you may feel that you could never bring your minds to conceive a curvature of space, let alone of space-time; others may feel that, being familiar with the bending of a two-dimensional surface, there is no insuperable difficulty in imagining something similar for three or even four dimensions. I rather think that the former have the best of it, for at least they escape being misled by their preconceptions. I have spoken of a “picture”, but it is a picture that has to be described analytically rather than conceived vividly. Our ordinary conception of curvature is derived from surfaces, i.e. two-dimensional manifolds embedded in a three-dimensional space. The absolute curvature at any point is measured by a single quantity called the radius of spherical curvature. But space-time is a four-dimensional manifold embedded in—well, as many dimensions as it can find new ways to twist about in. Actually a four-dimensional manifold is amazingly ingenious in discovering new kinds of contortion, and its invention is not exhausted until it has been provided with six extra dimensions, making ten dimensions in all. Moreover, twenty distinct measures are required at each point to specify the particular sort and amount of twistiness there. These measures are called coefficients of curvature. Ten of the coefficients stand out more prominently than the other ten.

Einstein’s law of gravitation asserts that the ten principal coefficients of curvature are zero in empty space.

If there were no curvature, i.e. if all the coefficients were zero, there would be no gravitation. Bodies would move uniformly in straight lines. If curvature were unrestricted, i.e. if all the coefficients had unpredictable values, gravitation would operate arbitrarily and without law. Bodies would move just anyhow. Einstein takes a condition midway between; ten of the coefficients are zero and the other ten are arbitrary. That gives a world containing gravitation limited by a law. The coefficients are naturally separated into two groups of ten, so that there is no difficulty in choosing those which are to vanish.

To the uninitiated it may seem surprising that an exact law of Nature should leave some of the coefficients arbitrary. But we need to leave something over to be settled when we have specified the particulars of the problem to which it is proposed to apply the law. A general law covers an infinite number of special cases. The vanishing of the ten principal coefficients occurs everywhere in empty space whether there is one gravitating body or many. The other ten coefficients vary according to the special case under discussion. This may remind us that after reaching Einstein’s law of gravitation and formulating it mathematically, it is still a very long step to reach its application to even the simplest practical problem. However, by this time many hundreds of readers must have gone carefully through the mathematics; so we may rest assured that there has been no mistake. After this work has been done it becomes possible to verify that the law agrees with observation. It is found that it agrees with Newton’s law to a very close approximation so that the main evidence for Einstein’s law is the same as the evidence for Newton’s law; but there are three crucial astronomical phenomena in which the difference is large enough to be observed. In these phenomena the observations support Einstein’s law against Newton’s.[15]

It is essential to our faith in a theory that its predictions should accord with observation, unless a reasonable explanation of the discrepancy is forthcoming; so that it is highly important that Einstein’s law should have survived these delicate astronomical tests in which Newton’s law just failed. But our main reason for rejecting Newton’s law is not its imperfect accuracy as shown by these tests; it is because it does not contain the kind of information about Nature that we want to know now that we have an ideal before us which was not in Newton’s mind at all. We can put it this way. Astronomical observations show that within certain limits of accuracy both Einstein’s and Newton’s laws are true. In confirming (approximately) Newton’s law, we are confirming a statement as to what the appearances would be when referred to one particular space-time frame. No reason is given for attaching any fundamental importance to this frame. In confirming (approximately) Einstein’s law, we are confirming a statement about the absolute properties of the world, true for all space-time frames. For those who are trying to get beneath the appearances Einstein’s statement necessarily supersedes Newton’s; it extracts from the observations a result with physical meaning as opposed to a mathematical curiosity. That Einstein’s law has proved itself the better approximation encourages us in our opinion that the quest of the absolute is the best way to understand the relative appearances; but had the success been less immediate, we could scarcely have turned our back on the quest.

I cannot but think that Newton himself would rejoice that after 200 years the “ocean of undiscovered truth” has rolled back another stage. I do not think of him as censorious because we will not blindly apply his formula regardless of the knowledge that has since accumulated and in circumstances that he never had the opportunity of considering.

I am not going to describe the three tests here, since they are now well known and will be found in any of the numerous guides to relativity; but I would refer to the action of gravitation on light concerned in one of them. Light-waves in passing a massive body such as the sun are deflected through a small angle. This is additional evidence that the Newtonian picture of gravitation as a tug is inadequate. You cannot deflect waves by tugging at them, and clearly another representation of the agency which deflects them must be found.

The Law of Motion. I must now ask you to let your mind revert to the time of your first introduction to mechanics before your natural glimmerings of the truth were sedulously uprooted by your teacher. You were taught the First Law of Motion—

“Every body continues in its state of rest or uniform motion in a straight line, except in so far as it may be compelled to change that state by impressed forces.”

Probably you had previously supposed that motion was something which would exhaust itself; a bicycle stops of its own accord if you do not impress force to keep it going. The teacher rightly pointed out the resisting forces which tend to stop the bicycle; and he probably quoted the example of a stone skimming over ice to show that when these interfering forces are reduced the motion lasts much longer. But even ice offers some frictional resistance. Why did not the teacher do the thing thoroughly and abolish resisting forces altogether, as he might easily have done by projecting the stone into empty space? Unfortunately in that case its motion is not uniform and rectilinear; the stone describes a parabola. If you raised that objection you would be told that the projectile was compelled to change its state of uniform motion by an invisible force called gravitation. How do we know that this invisible force exists? Why! because if the force did not exist the projectile would move uniformly in a straight line.

The teacher is not playing fair. He is determined to have his uniform motion in a straight line, and if we point out to him bodies which do not follow his rule he blandly invents a new force to account for the deviation. We can improve on his enunciation of the First Law of Motion. What he really meant was—

“Every body continues in its state of rest or uniform motion in a straight line, except in so far as it doesn’t.”

Material frictions and reactions are visible and absolute interferences which can change the motion of a body. I have nothing to say against them. The molecular battering can be recognised by anyone who looks deeply into the phenomenon no matter what his frame of reference. But when there is no such indication of disturbance the whole procedure becomes arbitrary. On no particular grounds the motion is divided into two parts, one of which is attributed to a passive tendency of the body called inertia and the other to an interfering field of force. The suggestion that the body really wanted to go straight but some mysterious agent made it go crooked is picturesque but unscientific. It makes two properties out of one; and then we wonder why they are always proportional to one another—why the gravitational force on different bodies is proportional to their inertia or mass. The dissection becomes untenable when we admit that all frames of reference are on the same footing. The projectile which describes a parabola relative to an observer on the earth’s surface describes a straight line relative to the man in the lift. Our teacher will not easily persuade the man in the lift who sees the apple remaining where he released it, that the apple really would of its own initiative rush upwards were it not that an invisible tug exactly counteracts this tendency.[16]

Einstein’s Law of Motion does not recognise this dissection. There are certain curves which can be defined on a curved surface without reference to any frame or system of partitions, viz. the geodesics or shortest routes from one point to another. The geodesics of our curved space-time supply the natural tracks which particles pursue if they are undisturbed.

We observe a planet wandering round the sun in an elliptic orbit. A little consideration will show that if we add a fourth dimension (time), the continual moving on in the time-dimension draws out the ellipse into a helix. Why does the planet take this spiral track instead of going straight? It is because it is following the shortest track; and in the distorted geometry of the curved region round the sun the spiral track is shorter than any other between the same points. You see the great change in our view. The Newtonian scheme says that the planet tends to move in a straight line, but the sun’s gravity pulls it away. Einstein says that the planet tends to take the shortest route and does take it.

That is the general idea, but for the sake of accuracy I must make one rather trivial correction. The planet takes the longest route.

You may remember that points along the track of any material body (necessarily moving with a speed less than the velocity of light) are in the absolute past or future of one another; they are not absolutely “elsewhere”. Hence the length of the track in four dimensions is made up of time-like relations and must be measured in time-units. It is in fact the number of seconds recorded by a clock carried on a body which describes the track.[17] This may be different from the time recorded by a clock which has taken some other route between the same terminal points. On p. 39 we considered two individuals whose tracks had the same terminal points; one of them remained at home on the earth and the other travelled at high speed to a distant part of the universe and back. The first recorded a lapse of 70 years, the second of one year. Notice that it is the man who follows the undisturbed track of the earth who records or lives the longest time. The man whose track was violently dislocated when he reached the limit of his journey and started to come back again lived only one year. There is no limit to this reduction; as the speed of the traveller approaches the speed of light the time recorded diminishes to zero. There is no unique shortest track; but the longest track is unique. If instead of pursuing its actual orbit the earth made a wide sweep which required it to travel with the velocity of light, the earth could get from 1 January 1927 to 1 January 1928 in no time, i.e. no time as recorded by an observer or clock travelling with it, though it would be reckoned as a year according to “Astronomer Royal’s time”. The earth does not do this, because it is a rule of the Trade Union of matter that the longest possible time must be taken over every job.

Thus in calculating astronomical orbits and in similar problems two laws are involved. We must first calculate the curved form of space-time by using Einstein’s law of gravitation, viz. that the ten principal curvatures are zero. We next calculate how the planet moves through the curved region by using Einstein’s law of motion, viz. the law of the longest track. Thus far the procedure is analogous to calculations made with Newton’s law of gravitation and Newton’s law of motion. But there is a remarkable addendum which applies only to Einstein’s laws. Einstein’s law of motion can be deduced from his law of gravitation. The prediction of the track of a planet although divided into two stages for convenience rests on a single law.

I should like to show you in a general way how it is possible for a law controlling the curvature of empty space to determine the tracks of particles without being supplemented by any other conditions.

Two “particles” in the four-dimensional world are shown in Fig. 5, namely yourself and myself. We are not empty space so there is no limit to the kind of curvature entering into our composition; in fact our unusual sort of curvature is what distinguishes us from empty space. We are, so to speak, ridges in the four-dimensional world where it is gathered into a pucker. The pure mathematician in his unflattering language would describe us as “singularities”. These two non-empty ridges are joined by empty space, which must be free from those kinds of curvature described by the ten principal coefficients. Now it is common experience that if we introduce local puckers into the material of a garment, the remainder has a certain obstinacy and will not lie as smoothly as we might wish. You will realise the possibility that, given two ridges as in Fig. 5, it may be impossible to join them by an intervening valley without the illegal kind of curvature. That turns out to be the case. Two perfectly straight ridges alone in the world cannot be properly joined by empty space and therefore they cannot occur alone. But if they bend a little towards one another the connecting region can lie smoothly and satisfy the law of curvature. If they bend too much the illegal puckering reappears. The law of gravitation is a fastidious tailor who will not tolerate wrinkles (except of a limited approved type) in the main area of the garment; so that the seams are required to take courses which will not cause wrinkles. You and I have to submit to this and so our tracks curve towards each other. An onlooker will make the comment that here is an illustration of the law that two massive bodies attract each other.

We thus arrive at another but equivalent conception of how the earth’s spiral track through the four-dimensional world is arrived at. It is due to the necessity of arranging two ridges (the solar track and the earth’s track) so as not to involve a wrong kind of curvature in the empty part of the world. The sun as the more pronounced ridge takes a nearly straight track; but the earth as a minor ridge on the declivities of the solar ridge has to twist about considerably.

Suppose the earth were to defy the tailor and take a straight track. That would make a horrid wrinkle in the garment; and since the wrinkle is inconsistent with the laws of empty space, something must be there—where the wrinkle runs. This “something” need not be matter in the restricted sense. The things which can occupy space so that it is not empty in the sense intended in Einstein’s law, are mass (or its equivalent energy) momentum and stress (pressure or tension). In this case the wrinkle might correspond to stress. That is reasonable enough. If left alone the earth must pursue its proper curved orbit; but if some kind of stress or pressure were inserted between the sun and earth, it might well take another course. In fact if we were to observe one of the planets rushing off in a straight track, Newtonians and Einsteinians alike would infer that there existed a stress causing this behaviour. It is true that causation has apparently been turned topsy-turvy; according to our theory the stress seems to be caused by the planet taking the wrong track, whereas we usually suppose that the planet takes the wrong track because it is acted on by the stress. But that is a harmless accident common enough in primary physics. The discrimination between cause and effect depends on time’s arrow and can only be settled by reference to entropy. We need not pay much attention to suggestions of causation arising in discussions of primary laws which, as likely as not, are contemplating the world upside down.

Although we are here only at the beginning of Einstein’s general theory I must not proceed further into this very technical subject. The rest of this chapter will be devoted to elucidation of more elementary points.

Relativity of Acceleration. The argument in this chapter rests on the relativity of acceleration. The apple had an acceleration of 32 feet per second per second relative to the ordinary observer, but zero acceleration relative to the man in the lift. We ascribe to it one acceleration or the other according to the frame we happen to be using, but neither is to be singled out and labelled “true” or absolute acceleration. That led us to reject the Newtonian conception which singled out 32 feet per second per second as the true acceleration and invented a disturbing agent of this particular degree of strength.

It will be instructive to consider an objection brought, I think, originally by Lenard. A train is passing through a station at 60 miles an hour. Since velocity is relative, it does not matter whether we say that the train is moving at 60 miles an hour past the station or the station is moving at 60 miles an hour past the train. Now suppose, as sometimes happens in railway accidents, that this motion is brought to a standstill in a few seconds. There has been a change of velocity or acceleration—a term which includes deceleration. If acceleration is relative this may be described indifferently as an acceleration of the train (relative to the station) or an acceleration of the station (relative to the train). Why then does it injure the persons in the train and not those in the station?

Much the same point was put to me by one of my audience. “You must find the journey between Cambridge and Edinburgh very tiring. I can understand the fatigue, if you travel to Edinburgh; but why should you get tired if Edinburgh comes to you?” The answer is that the fatigue arises from being shut up in a box and jolted about for nine hours; and it makes no difference whether in the meantime I move to Edinburgh or Edinburgh moves to me. Motion does not tire anybody. With the earth as our vehicle we are travelling at 20 miles a second round the sun; the sun carries us at 12 miles a second through the galactic system; the galactic system bears us at 250 miles a second amid the spiral nebulae; the spiral nebulae.... If motion could tire, we ought to be dead tired.

Similarly change of motion or acceleration does not injure anyone, even when it is (according to the Newtonian view) an absolute acceleration. We do not even feel the change of motion as our earth takes the curve round the sun. We feel something when a railway train takes a curve, but what we feel is not the change of motion nor anything which invariably accompanies change of motion; it is something incidental to the curved track of the train but not to the curved track of the earth. The cause of injury in the railway accident is easily traced. Something hit the train; that is to say, the train was bombarded by a swarm of molecules and the bombardment spread all the way along it. The cause is evident—gross, material, absolute—recognised by everyone, no matter what his frame of reference, as occurring in the train not the station. Besides injuring the passengers this cause also produced the relative acceleration of the train and station—an effect which might equally well have been produced by molecular bombardment of the station, though in this case it was not.

The critical reader will probably pursue his objection. “Are you not being paradoxical when you say that a molecular bombardment of the train can cause an acceleration of the station—and in fact of the earth and the rest of the universe? To put it mildly, relative acceleration is a relation with two ends to it, and we may at first seem to have an option which end we shall grasp it by; but in this case the causation (molecular bombardment) clearly indicates the right end to take hold of, and you are merely spinning paradoxes when you insist on your liberty to take hold of the other.”

If there is an absurdity in taking hold of the wrong end of the relation it has passed into our current speech and thought. Your suggestion is in fact more revolutionary than anything Einstein has ventured to advocate. Let us take the problem of a falling stone. There is a relative acceleration of 32 feet per second per second—of the stone relative to ourselves or of ourselves relative to the stone. Which end of the relation must we choose? The one indicated by molecular bombardment? Well, the stone is not bombarded; it is falling freely in vacuo. But we are bombarded by the molecules of the ground on which we stand. Therefore it is we who have the acceleration; the stone has zero acceleration, as the man in the lift supposed. Your suggestion makes out the frame of the man in the lift to be the only legitimate one; I only went so far as to admit it to an equality with our own customary frame.

Your suggestion would accept the testimony of the drunken man who explained that “the paving-stone got up and hit him” and dismiss the policeman’s account of the incident as “merely spinning paradoxes”. What really happened was that the paving-stone had been pursuing the man through space with ever-increasing velocity, shoving the man in front of it so that they kept the same relative position. Then, through an unfortunate wobble of the axis of the man’s body, he failed to increase his speed sufficiently, with the result that the paving-stone overtook him and came in contact with his head. That, please understand, is your suggestion; or rather the suggestion which I have taken the liberty of fathering on you because it is the outcome of a very common feeling of objection to the relativity theory. Einstein’s position is that whilst this is a perfectly legitimate way of looking at the incident the more usual account given by the policeman is also legitimate; and he endeavours like a good magistrate to reconcile them both.

Time Geometry. Einstein’s law of gravitation controls a geometrical quantity curvature in contrast to Newton’s law which controls a mechanical quantity force. To understand the origin of this geometrisation of the world in the relativity theory we must go back a little.

The science which deals with the properties of space is called geometry. Hitherto geometry has not included time in its scope. But now space and time are so interlocked that there must be one science—a somewhat extended geometry—embracing them both. Three-dimensional space is only a section cut through four-dimensional space-time, and moreover sections cut in different directions form the spaces of different observers. We can scarcely maintain that the study of a section cut in one special direction is the proper subject-matter of geometry and that the study of slightly different sections belongs to an altogether different science. Hence the geometry of the world is now considered to include time as well as space. Let us follow up the geometry of time.

You will remember that although space and time are mixed up there is an absolute distinction between a spatial and a temporal relation of two events. Three events will form a space-triangle if the three sides correspond to spatial relations—if the three events are absolutely elsewhere with respect to one another.[18] Three events will form a time-triangle if the three sides correspond to temporal relations—if the three events are absolutely before or after one another. (It is possible also to have mixed triangles with two sides time-like and one space-like, or vice versa.) A well-known law of the space-triangle is that any two sides are together greater than the third side. There is an analogous, but significantly different, law for the time-triangle, viz. two of the sides (not any two sides) are together less than the third side. It is difficult to picture such a triangle but that is the actual fact.

Let us be quite sure that we grasp the precise meaning of these geometrical propositions. Take first the space-triangle. The proposition refers to the lengths of the sides, and it is well to recall my imaginary discussion with two students as to how lengths are to be measured (p. 23). Happily there is no ambiguity now, because the triangle of three events determines a plane section of the world, and it is only for that mode of section that the triangle is purely spatial. The proposition then expresses that

“If you measure with a scale from to and from to the sum of your readings will be greater than the reading obtained by measuring with a scale from to .”

For a time-triangle the measurements must be made with an instrument which can measure time, and the proposition then expresses that

“If you measure with a clock from to and from to the sum of your readings will be less than the reading obtained by measuring with a clock from to .”

In order to measure from an event to an event with a clock you must make an adjustment of the clock analogous to orienting a scale along the line . What is this analogous adjustment? The purpose in either case is to bring both and into the immediate neighbourhood of the scale or clock. For the clock that means that after experiencing the event it must travel with the appropriate velocity needed to reach the locality of just at the moment that happens. Thus the velocity of the clock is prescribed. One further point should be noticed. After measuring with a scale from to you can turn your scale round and measure from to , obtaining the same result. But you cannot turn a clock round, i.e. make it go backwards in time. That is important because it decides which two sides are less than the third side. If you choose the wrong pair the enunciation of the time proposition refers to an impossible kind of measurement and becomes meaningless.

You remember the traveller (p. 39) who went off to a distant star and returned absurdly young. He was a clock measuring two sides of a time-triangle. He recorded less time than the stay-at-home observer who was a clock measuring the third side. Need I defend my calling him a clock? We are all of us clocks whose faces tell the passing years. This comparison was simply an example of the geometrical proposition about time-triangles (which in turn is a particular case of Einstein’s law of longest track). The result is quite explicable in the ordinary mechanical way. All the particles in the traveller’s body increase in mass on account of his high velocity according to the law already discussed and verified by experiment. This renders them more sluggish, and the traveller lives more slowly according to terrestrial time-reckoning. However, the fact that the result is reasonable and explicable does not render it the less true as a proposition of time geometry.

Our extension of geometry to include time as well as space will not be a simple addition of an extra dimension to Euclidean geometry, because the time propositions, though analogous, are not identical with those which Euclid has given us for space alone. Actually the difference between time geometry and space geometry is not very profound, and the mathematician easily glides over it by a discrete use of the symbol . We still call (rather loosely) the extended geometry Euclidean; or, if it is necessary to emphasise the distinction, we call it hyperbolic geometry. The term non-Euclidean geometry refers to a more profound change, viz. that involved in the curvature of space and time by which we now represent the phenomenon of gravitation. We start with Euclidean geometry of space, and modify it in a comparatively simple manner when the time-dimension is added; but that still leaves gravitation to be reckoned with, and wherever gravitational effects are observable it is an indication that the extended Euclidean geometry is not quite exact, and the true geometry is a non-Euclidean one—appropriate to a curved region as Euclidean geometry is to a flat region.

Geometry and Mechanics. The point that deserves special attention is that the proposition about time-triangles is a statement as to the behaviour of clocks moving with different velocities. We have usually regarded the behaviour of clocks as coming under the science of mechanics. We found that it was impossible to confine geometry to space alone, and we had to let it expand a little. It has expanded with a vengeance and taken a big slice out of mechanics. There is no stopping it, and bit by bit geometry has now swallowed up the whole of mechanics. It has also made some tentative nibbles at electromagnetism. An ideal shines in front of us, far ahead perhaps but irresistible, that the whole of our knowledge of the physical world may be unified into a single science which will perhaps be expressed in terms of geometrical or quasi-geometrical conceptions. Why not? All the knowledge is derived from measurements made with various instruments. The instruments used in the different fields of inquiry are not fundamentally unlike. There is no reason to regard the partitions of the sciences made in the early stages of human thought as irremovable.

But mechanics in becoming geometry remains none the less mechanics. The partition between mechanics and geometry has broken down and the nature of each of them has diffused through the whole. The apparent supremacy of geometry is really due to the fact that it possesses the richer and more adaptable vocabulary; and since after the amalgamation we do not need the double vocabulary the terms employed are generally taken from geometry. But besides the geometrisation of mechanics there has been a mechanisation of geometry. The proposition about the space-triangle quoted above was seen to have grossly material implications about the behaviour of scales which would not be realised by anyone who thinks of it as if it were a proposition of pure mathematics.

We must rid our minds of the idea that the word space in science has anything to do with void. As previously explained it has the other meaning of distance, volume, etc., quantities expressing physical measurement just as much as force is a quantity expressing physical measurement. Thus the (rather crude) statement that Einstein’s theory reduces gravitational force to a property of space ought not to arouse misgiving. In any case the physicist does not conceive of space as void. Where it is empty of all else there is still the aether. Those who for some reason dislike the word aether, scatter mathematical symbols freely through the vacuum, and I presume that they must conceive some kind of characteristic background for these symbols. I do not think any one proposes to build even so relative and elusive a thing as force out of entire nothingness.