Space Time and Gravitation/Chapter 4

by Arthur Eddington

 

CHAPTER IV

FIELDS OF FORCE

For whenever bodies fall through water and thin air, they must quicken their descents in proportion to their weights, because the body of water and subtle nature of air cannot retard everything in equal degree, but more readily give way overpowered by the heavier; on the other hand empty void cannot offer resistance to anything in any direction at any time, but must, as its nature craves, continually give way; and for this reason all things must be moved and borne along with equal velocities though of unequal weights through the unresisting void.

Lucretius, De Natura Rerum.

The primary conception of force is associated with the muscular sensation felt when we make an effort to cause or prevent the motion of matter. Similar effects on the motion of matter can be caused by non-living agency, and these also are regarded as due to forces. As is well known, the scientific measure of a force is the momentum that it communicates to a body in given time. There is nothing very abstract about a force transmitted by material contact; modern physics shows that the momentum is communicated by a process of molecular bombardment. We can visualise the mechanism, and see the molecules carrying the motion in small parcels across the boundary into the body that is being acted on. Force is no mysterious agency; it is merely a convenient summary of this flow of motion, which we can trace continuously if we take the trouble. It is true that the difficulties are only set back a stage, and the exact mode by which the momentum is redistributed during a molecular collision is not yet understood; but, so far as it goes, this analysis gives a clear idea of the transmission of motion by ordinary forces.

But even in elementary mechanics an important natural force appears, which does not seem to operate in this manner. Gravitation is not resolvable into a succession of molecular blows. A massive body, such as the earth, seems to be surrounded by a field of latent force, ready, if another body enters the field, to become active, and transmit motion. One usually thinks of this influence as existing in the space round the earth even when there is no test-body to be affected, and in a rather vague way it is suspected to be some state of strain or other condition of an unperceived medium.

Although gravitation has been recognised for thousands of years, and its laws were formulated with sufficient accuracy for almost all purposes more than 200 years ago, it cannot be said that much progress has been made in explaining the nature or mechanism of this influence. It is said that more than 200 theories of gravitation have been put forward; but the most plausible of these have all had the defect that they lead nowhere and admit of no experimental test. Many of them would nowadays be dismissed as too materialistic for our taste—filling space with the hum of machinery—a procedure curiously popular in the nineteenth century. Few would survive the recent discovery that gravitation acts not only on the molecules of matter, but on the undulations of light.

The nature of gravitation has seemed very mysterious, yet it is a remarkable fact that in a limited region it is possible to create an artificial field of force which imitates a natural gravitational field so exactly that, so far as experiments have yet gone, no one can tell the difference. Those who seek for an explanation of gravitation naturally aim to find a model which will reproduce its effects; but no one before Einstein seems to have thought of finding the clue in these artificial fields, familiar as they are.

When a lift starts to move upwards the occupants feel a characteristic sensation, which is actually identical with a sensation of increased weight. The feeling disappears as soon as the motion becomes uniform; it is associated only with the change of motion of the lift, that is to say, the acceleration. Increased weight is not only a matter of sensation; it is shown by any physical experiments that can be performed. The usual laboratory determination of the value of gravity by Atwood's machine would, if carried out inside the accelerated lift, give a higher value. A spring-balance would record higher weights. Projectiles would follow the usual laws of motion but with a higher value of gravity. In fact, the upward acceleration of the lift is in its mechanical effects exactly similar to an additional gravitational field superimposed on that normally present.

Perhaps the equivalence is most easily seen when we produce in this manner an artificial field which just neutralises the earth's field of gravitation. Jules Verne's book Round the Moon tells the story of three men in a projectile shot from a cannon into space. The author enlarges on their amusing experiences when their weight vanished altogether at the neutral point, where the attraction of the earth and moon balance one another. As a matter of fact they would not have had any feeling of weight at any time during their journey after they left the earth's atmosphere. The projectile was responding freely to the pull of gravity, and so were its occupants. When an occupant let go of a plate, the plate could not "fall" any more than it was doing already, and so it must remain poised.

It will be seen that the sensation of weight is not felt when we are free to respond to the force of gravitation; it is only felt when something interferes to prevent our falling. It is primarily the floor or the chair which causes the sensation of weight by checking the fall. It seems literally true to say that we never feel the force of the earth's gravitation; what we do feel is the bombardment of the soles of our boots by the molecules of the ground, and the consequent impulses spreading upwards through the body. This point is of some importance, since the idea of the force of gravitation as something which can be felt, predisposes us to a materialistic view of its nature.

Another example of an artificial field of force is the centrifugal force of the earth s rotation. In most books of Physical Constants will be found a table of the values of "g," the acceleration due to gravity, at different latitudes. But the numbers given do not relate to gravity alone; they are the resultant of gravity and the centrifugal force of the earth's rotation. These are so much alike in their effects that for practical purposes physicists have not thought it worth while to distinguish them.

Similar artificial fields are produced when an aeroplane changes its course or speed; and one of the difficulties of navigation is the impossibility of discriminating between these and the true gravitation of the earth with which they combine. One usually finds that the practical aviator requires little persuasion of the relativity of force.

To find a unifying idea as to the origin of these artificial fields of force, we must return to the four-dimensional world of space-time. The observer is progressing along a certain track in this world. Now his course need not necessarily be straight. It must be remembered that straight in the four-dimensional world means something more than straight in space; it implies also uniform velocity, since the velocity determines the inclination of the track to the time-axis.

The observer in the accelerated lift travels upwards in a straight line, say 1 foot in the first second, 4 feet in two seconds, 9 feet in three seconds, and so on. If we plot these points as $x$ and $t$ on a diagram we obtain a curved track. Presently the speed of the lift becomes uniform and the track in the diagram becomes straight. So long as the track is curved (accelerated motion) a field of force is perceived; it disappears when the track becomes straight (uniform motion).

Again the observer on the earth is carried round in a circle once a day by the earth's rotation; allowing for steady progress through time, the track in four dimensions is a spiral. For an observer at the north pole the track is straight, and there the centrifugal force is zero.

Clearly the artificial field of force is associated with curvature of track, and we can lay down the following rule:—

Whenever the observer's track through the four-dimensional world is curved he perceives an artificial field of force.

The field of force is not only perceived by the observer in his sensations, but reveals itself in his physical measures. It should be understood, however, that the curvature of track must not have been otherwise allowed for. Naturally if the observer in the lift recognises that his measures are affected by his own acceleration and applies the appropriate corrections, the artificial force will be removed by the process. It only exists if he is unaware of, or does not choose to consider, his acceleration.

The centrifugal force is often called "unreal." From the point of view of an observer who does not rotate with the earth, there is no centrifugal force; it only arises for the terrestrial observer who is too lazy to make other allowance for the effects of the earth's rotation. It is commonly thought that this "unreality" quite differentiates it from a "real" force like gravity; but if we try to find the grounds of this distinction they evade us. The centrifugal force is made to disappear if we choose a suitable standard observer not rotating with the earth; the gravitational force was made to disappear when we chose as standard observer an occupant of Jules Verne's falling projectile. If the possibility of annulling a field of force by choosing a suitable standard observer is a test of unreality, then gravitation is equally unreal with centrifugal force.

It may be urged that we have not stated the case quite fairly. When we choose the non-rotating observer the centrifugal force disappears completely and everywhere. When we choose the occupant of the falling projectile, gravitation disappears in his immediate neighbourhood; but he would notice that, although unsupported objects round him experienced no acceleration relative to him, objects on the other side of the earth would fall towards him. So far from getting rid of the field of force, he has merely removed it from his own surroundings, and piled it up elsewhere. Thus gravitation is removable locally, but centrifugal force can be removed everywhere. The fallacy of this argument is that it speaks as though gravitation and centrifugal force were distinguishable experimentally. It presupposes the distinction that we are challenging. Looking simply at the resultant of gravitation and centrifugal force, which is all that can be observed, neither observer can get rid of the resultant force at all parts of space. Each has to be content with leaving a residuum. The non-rotating observer claims that he has got rid of all the unreal part, leaving a remainder (the usual gravitational field) which he regards as really existing. We see no justification for this claim, which might equally well be made by Jules Verne's observer.

It is not denied that the separation of centrifugal and gravitational force generally adopted has many advantages for mathematical calculation. If it were not so, it could not have endured so long. But it is a mathematical separation only, without physical basis; and it often happens that the separation of a mathematical expression into two terms of distinct nature, though useful for elementary work, becomes vitiated for more accurate work by the occurrence of minute cross-terms which have to be taken into account.

Newtonian mechanics proceeds on the supposition that there is some super-observer. If he feels a field of force, then that force really exists. Lesser beings, such as the occupants of the falling projectile, have other ideas, but they are the victims of illusion. It is to this super-observer that the mathematician appeals when he starts a dynamical investigation with the words "Take unaccelerated rectangular axes, $O_{x}$, $O_{y}$, $O_{z}$…." Unaccelerated rectangular axes are the measuring-appliances of the super-observer.

It is quite possible that there might be a super-observer, whose views have a natural right to be regarded as the truest, or at least the simplest. A society of learned fishes would probably agree that phenomena were best described from the point of view of a fish at rest in the ocean. But relativity mechanics finds that there is no evidence that the circumstances of any observer can be such as to make his views preeminent. All are on an equality. Consider an observer $A$ in a projectile falling freely to the earth, and an observer $B$ in space out of range of any gravitational attraction. Neither $A$ nor $B$ feel any field of force in their neighbourhood. Yet in Newtonian mechanics an artificial distinction is drawn between their circumstances; $B$ is in no field of force at all, but $A$ is really in a field of force, only its effects are neutralised by his acceleration. But what is this acceleration of $A$? Primarily it is an acceleration relative to the earth ; but then that can equally well be described as an acceleration of the earth relative to $A$, and it is not fair to regard it as something located with $A$. Its importance in Newtonian philosophy is that it is an acceleration relative to what we have called the super-observer. This potentate has drawn planes and lines partitioning space, as space appears to him. I fear that the time has come for his abdication.

Suppose the whole system of the stars were falling freely under the uniform gravitation of some vast external mass, like a drop of rain falling to the ground. Would this make any difference to phenomena? None at all. There would be a gravitational field; but the consequent acceleration of the observer and his landmarks would produce a field of force annulling it. Who then shall say what is absolute acceleration?

We shall accordingly give up the attempt to separate artificial fields of force and natural gravitational fields; and call the whole measured field of force the gravitational field, generalising the expression. This field is not absolute, but always requires that some observer should be specified.

It may avoid some mystification if we state at once that there are certain intricacies in the gravitational influence radiating from heavy matter which are distinctive. A theory which did not admit this would run counter to common sense. What our argument has shown is that the characteristic symptom in a region in the neighbourhood of matter is not the field of force; it must be something more intricate. In due course we shall have to explain the nature of this more complex effect of matter on the condition of the world.

Our previous rule, that the observer perceives an artificial field of force when he deviates from a straight track, must now be superseded. We need rather a rule determining when he perceives a field of force of any kind. Indeed the original rule has become meaningless, because a straight track is no longer an absolute conception. Uniform motion in a straight line is not the same for an observer rotating with the earth as for a non-rotating observer who takes into account the sinuosity of the rotation. We have decided that these two observers are on the same footing and their judgments merit the same respect. A straight-line in space-time is accordingly not an absolute conception, but is only defined relative to some observer.

Now we have seen that so long as the observer and his measuring-appliances are unconstrained (falling freely) the field of force immediately round him vanishes. It is only when he is deflected from his proper track that he finds himself in the midst of a field of force. Leaving on one side the question of the motion of electrically charged bodies, which must be reserved for more profound treatment, the observer can only leave his proper track if he is being disturbed by material impacts, e.g. the molecules of the ground bombarding the soles of his boots. We may say then that a body does not leave its natural track without visible cause; and any field of force round an observer is the result of his leaving his natural track by such cause. There is nothing mysterious about this field of force; it is merely the reflection in the phenomena of the observer's disturbance; just as the flight of the houses and hedgerows past our railway-carriage is the reflection of our motion with the train. Our attention is thus directed to the natural tracks of unconstrained bodies, which appear to be marked out in some absolute way in the four-dimensional world. There is no question of an observer here; the body takes the same course in the world whoever is watching it. Different observers will describe the track as straight, parabolical, or sinuous, but it is the same absolute locus.

Now we cannot pretend to predict without reference to experiment the laws determining the nature of these tracks; but we can examine whether our knowledge of the four-dimensional world is already sufficient to specify definite tracks of this kind, or whether it will be necessary to introduce new hypothetical factors. It will be found that it is already sufficient. So far we have had to deal with only one quantity which is independent of the observer and has therefore an absolute significance in the world, namely the interval between two events in space and time. Let us choose two fairly distant events $P_{1}$ and $P_{2}$. These can be joined by a variety of tracks, and the interval-length from $P_{1}$ to $P_{2}$ along any track can be measured. In order to make sure that the interval-length is actually being measured along the selected track, the method is to take a large number of intermediate points on the track, measure the interval corresponding to each subdivision, and take the sum. It is virtually the same process as measuring the length of a twisty road on a map with a piece of cotton. The interval-length along a particular track is thus something which can be measured absolutely, since all observers agree as to the measurement of the interval for each subdivision. It follows that all observers will agree as to which track (if any) is the shortest track between the two points, judged in terms of interval-length.

This gives a means of defining certain tracks in space-time as having an absolute significance, and we proceed tentatively to identify them with the natural tracks taken by freely moving particles.

In one respect we have been caught napping. Dr A. A. Robb has pointed out the curious fact that it is not the shortest track, but the longest track, which is unique^[1]. There are any number of tracks from $P_{1}$ to $P_{2}$ of zero interval-length; there is just one which has maximum length. This is because of the peculiar geometry which the minus sign of $(t_{2}-t_{1})^{2}$ introduces. For instance, it will be seen from equation (1), p. 53, that when $(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}=(t_{2}-t_{1})^{2}$, that is to say when the resultant distance travelled in space is equal to the distance travelled in time, then $s$ is zero. This happens when the velocity is unity—the velocity of light. To get from $P_{1}$ to $P_{2}$ by a path of no interval-length, we must simply keep on travelling with the velocity of light, cruising round if necessary, until the moment comes to turn up at $P_{2}$. On the other hand there is evidently an upper limit to the interval-length of the track, because each portion of $s$ is always less than the corresponding portion of $(t_{2}-t_{1})$, and $s$ can never exceed $t_{2}-t_{1}$.

There is a physical interpretation of interval-length along the path of a particle which helps to give a more tangible idea of its meaning. It is the time as perceived by an observer, or measured by a clock, carried on the particle. This is called the proper-time; and, of course, it will not in general agree with the time-reckoning of the independent onlooker who is supposed to be watching the whole proceedings. To prove this, we notice from equation (1) that if $x_{2}=x_{1}$, $y_{2}=y_{1}$, and $z_{2}=z_{1}$, then $s=t_{2}-t_{1}$. The condition $x_{2}=x_{1}$, etc. means that the particle must remain stationary relative to the observer who is measuring $x$, $y$, $z$, $t$. To secure this we mount our observer on the particle and then the interval-length $s$ will be $t_{2}-t_{1}$, which is the time elapsed according to his clock.

We can use proper-time as generally equivalent to interval-length; but it must be admitted that the term is not very logical unless the track in question is a natural track. For any other track, the drawback to defining the interval-length as the time measured by a clock which follows the track, is that no clock could follow the track without violating the laws of nature. We may force it into the track by continually hitting it; but that treatment may not be good for its time-keeping qualities. The original definition by equation (1) is the more general definition.

We are now able to state formally our proposed law of motion—Every particle moves so as to take the track of greatest interval-length between two events, except in so far as it is disturbed by impacts of other particles or electrical forces.

This cannot be construed into a truism like Newton's first law of motion. The reservation is not an undefined agency like force, whose meaning can be extended to cover any breakdown of the law. We reserve only direct material impacts and electromagnetic causes, the latter being outside our present field of discussion.

Consider, for example, two events in space-time, viz. the position of the earth at the present moment, and its position a hundred years ago. Call these events $P_{2}$ and $P_{1}$. In the interim the earth (being undisturbed by impacts) has moved so as to take the longest track from $P_{1}$ to $P_{2}$—or, if we prefer, so as to take the longest possible proper-time over the journey. In the weird geometry of the part of space-time through which it passes (a geometry which is no doubt associated in some way with our perception of the existence of a massive body, the sun) this longest track is a spiral—a circle in space, drawn out into a spiral by continuous displacement in time. Any other course would have had shorter interval-length.

In this way the study of fields of force is reduced to a study of geometry. To a certain extent this is a retrograde step; we adopt Kepler's description of the sun's gravitational field instead of Newton's. The field of force is completely described if the tracks through space and time of particles projected in every possible way are prescribed. But we go back in order to go forward in a new direction. To express this unmanageable mass of detail in a unified way, a world-geometry is found in which the tracks of greatest length are the actual tracks of the particles. It only remains to express the laws of this geometry in a concise form. The change from a mechanical to a geometrical theory of fields of force is not so fundamental a change as might be supposed. If we are now reducing mechanics to a branch of natural geometry, we have to remember that natural geometry is equally a branch of mechanics, since it is concerned with the behaviour of material measuring-appliances.

Reference has been made to weird geometry. There is no help for it, if the longest track can be a spiral like that known to be described by the earth. Non-Euclidean geometry is necessary. In Euclidean geometry the shortest track is always a straight line; and the slight modification of Euclidean geometry described in Chapter iii is found to give a straight line as the longest track. The status of non-Euclidean geometry has already been thrashed out in the Prologue; and there seems to be no reason whatever for preferring Euclid's geometry unless observations decide in its favour. Equation (1), p. 53, is the expression of the Euclidean (or semi-Euclidean) geometry we have hitherto adopted; we shall have to modify it, if we adopt non-Euclidean geometry.

But the point arises that the geometry arrived at in Chapter iii was not arbitrary. It was the synthesis of measures made with clocks and scales, by observers with all kinds of uniform motion relative to one another; we cannot modify it arbitrarily to fit the behaviour of moving particles like the earth. Now, if the worst came to the worst, and we could not reconcile a geometry based on measures with clocks and scales and a geometry based on the natural tracks of moving particles—if we had to select one or the other and keep to it—I think we ought to prefer to use the geometry based on the tracks of moving particles. The free motion of a particle is an example of the simplest possible kind of phenomenon; it is unanalysable; whereas, what the readings of any kind of clock record, what the extension of a material rod denotes, may evidently be complicated phenomena involving the secrets of molecular constitution. Each geometry would be right in its own sphere; but the geometry of moving particles would be the more fundamental study. But it turns out that there is probably no need to make the choice; clocks, scales, moving particles, light-pulses, give the same geometry. This might perhaps be expected since a clock must comprise moving particles of some kind.

A formula, such as equation (1), based on experiment can only be verified to a certain degree of approximation. Within certain limits it will be possible to introduce modifications. Now it turns out that the free motion of a particle is a much more sensitive way of exploring space-time, than any practicable measures with scales and clocks. If then we employ our accurate knowledge of the motion of particles to correct the formula, we shall find that the changes introduced are so small that they are inappreciable in any practical measures with scales and clocks. There is only one case where a possible detection of the modification is indicated; this refers to the behaviour of a clock on the surface of the sun, but the experiment is one of great difficulty and no conclusive answer has been given. We conclude then that the geometry of space and time based on the motions of particles is accordant with the geometry based on the cruder observations with clocks and scales; but if subsequent experiment should reveal a discrepancy, we shall adhere to the moving particle on account of its greater simplicity.

The proposed modification can be regarded from another point of view. Equation (1) is the synthesis of the experiences of all observers in uniform motion. But uniform motion means that their four-dimensional tracks are straight lines. We must suppose that the observers were moving in their natural tracks; for, if not, they experienced fields of force, and presumably allowed for these in their calculations, so that reduction was made to the natural tracks. If then equation (1) shows that the natural tracks are straight lines, we are merely getting out of the equation that which we originally put into it.

The formula needs generalising in another way. Suppose there is a region of space-time where, for some observer, the natural tracks are all straight lines and equation (1) holds rigorously. For another (accelerated) observer the tracks will be curved, and the equation will not hold. At the best it is of a form which can only hold good for specially selected observers.

Although it has become necessary to throw our formula into the melting-pot, that does not create any difficulty in measuring the interval. Without going into technical details, it may be pointed out that the innovations arise solely from the introduction of gravitational fields of force into our scheme. When there is no force, the tracks of all particles are straight lines as our previous geometry requires. In any small region we can choose an observer (falling freely) for whom the force vanishes, and accordingly the original formula holds good. Thus it is only necessary to modify our rule for determining the interval by two provisos (1) that the interval measured must be small, (2) that the scales and clocks used for measuring it must be falling freely. The second proviso is natural, because, if we do not leave our apparatus to fall freely, we must allow for the strain that it undergoes. The first is not a serious disadvantage, because a larger interval can be split up into a number of small intervals and the parts measured separately. In mathematical problems the same device is met with under the name of integration. To emphasise that the formula is strictly true only for infinitesimal intervals, it is written with a new notation $ds^{2}=-dx^{2}-dy^{2}-dz^{2}+dt^{2}\qquad \ldots \ldots \ldots \ldots$(2) where $dx$ stands for the small difference $x_{2}-x_{1}$, etc.

The condition that the measuring appliances must not be subjected to a field of force is illustrated by Ehrenfest's paradox. Consider a wheel revolving rapidly. Each portion of the circumference is moving in the direction of its length, and might be expected to undergo the FitzGerald contraction due to its velocity; each portion of a radius is moving transversely and would therefore have no longitudinal contraction. It looks as though the rim of the wheel should contract and the spokes remain the same length, when the wheel is set revolving. The conclusion is absurd, for a revolving wheel has no tendency to buckle—which would be the only way of reconciling these conditions. The point which the argument has overlooked is that the results here appealed to apply to unconstrained bodies, which have no acceleration relative to the natural tracks in space. Each portion of the rim of the wheel has a radial acceleration, and this affects its extensional properties. When accelerations as well as velocities occur a more far-reaching theory is needed to determine the changes of length.

To sum up—the interval between two (near) events is something quantitative which has an absolute significance in nature. The track between two (distant) events which has the longest interval-length must therefore have an absolute significance. Such tracks are called geodesics. Geodesics can be traced practically, because they are the tracks of particles undisturbed by material impacts. By the practical tracing of these geodesics we have the best means of studying the character of the natural geometry of the world. An auxiliary method is by scales and clocks, which, it is believed, when unconstrained, measure a small interval according to formula (2).

The identity of the two methods of exploring the geometry of the world is connected with a principle which must now be enunciated definitely. We have said that no experiments have been able to detect a difference between a gravitational field and an artificial field of force such as the centrifugal force. This is not quite the same thing as saying that it has been proved that there is no difference. It is well to be explicit when a positive generalisation is made from negative experimental evidence. The generalisation which it is proposed to adopt is known as the Principle of Equivalence.

A gravitational field of force is precisely equivalent to an artificial field of force, so that in any small region it is impossible by any conceivable experiment to distinguish between them.

In other words, force is purely relative.

1. It is here assumed that $P_{2}$ is in the future of $P_{1}$ so that it is possible for a particle to travel from $P_{1}$ to $P_{2}$. If $P_{1}$ and $P_{2}$ are situated like $O$ and $P$ in Fig. 3, the interval-length is imaginary, and the shortest track is unique.