CHAPTER XXIVTHE POSTULATE OF EQUIVALENCE
EINSTEIN’SPostulate of Equivalenceconsists essentially in an identification of forces of gravitation and forces of inertia. This identification he considered permissible because of certain well-known empirical facts which we shall discuss presently.
Let us first recall that a field of force is exemplified by a region of space, at each and every point of which a definite force would be found to be acting on a test-body with which the field might be explored. Several different types of fields were known to classical science. Electric fields acted on electrified bodies, magnetic fields acted on magnets, and both inertial and gravitational fields acted on material bodies in general, whether electrified or not. For the present we shall be concerned solely with the inertial and gravitational fields.
These two species of fields of force were regarded by classical science as of a totally different nature. There appeared to be very good reasons for this distinction. Suppose, for instance, that a train is slowing down. If we are standing in the train, we shall feel a force pulling us towards the engine and it may require a certain effort on our part to resist its pull. As such, the force is obviously real, in that it is experienced. But suppose now that an observer on the embankment views these same happenings. He will argue as follows: No force is pulling the passenger; but as the train is slowing down and as the passenger’s body tends to maintain a constant velocity along a straight line, in conformity with the laws of motion (law of inertia), the net result is that he will overtake the engine unless he holds on to the seat. Thus we see that according to whether we judge these same happenings from the standpoint of the train or of the embankment, the force exists or becomes a fiction.
Similar conclusions would apply were we to consider those other species of inertial forces called centrifugal and Coriolis forces. From the standpoint of an observer who selects a rotating platform as frame of reference (just as prior to Copernicus men were wont to take the earth as frame of reference), centrifugal and Coriolis forces are real. But the moment we refer events to a Galilean frame, these forces become fictitious, and can be interpreted in terms of the law of inertia, just as the force in the train became fictitious when events were viewed from the embankment.
Contrast this situation with that of a stone falling towards the ground. Here, we may refer events to any frame we please, but in any case the law of inertia can never be made to account for the stone’s motion towards the earth. The situation appears to be entirely different from that of the passenger in the train. And so it followsthat we are compelled to conceive of a real force,a force of gravitationpulling the stone towards the centre of the earth. Thus it would appear that gravitational force, in contradistinction to inertial force, could not be ascribed to mere conditions of observation. It was assumed, therefore, by classical science that there existed around a body like the earth a real absolute field of force distributed radially through space, whereas in the accelerated enclosure the forces were merely fictitious or relative.
There is yet another way of bringing out the difference between forces of inertia and forces of gravitation. Forces of inertia are generated by motion, whereas forces of gravitation are generated by matter. Thus, if we arrest the rotation of a disk, the observer, fixed to the disk, will note that the forces of inertia have vanished. But to remove the force of gravitation, it would be necessary to annihilate the entire mass of the earth, and this would constitute an operation of an entirely different nature.
Now, in every type of field of force, the magnitude of the pull which is exerted on a body susceptible to the influence of the field is governed by a certain definite characteristic of the body. Thus, in a given electric field, the magnitude of the pull exerted on a body at a given point of the field is proportional to the electric charge of the body. Likewise, in an inertial field, the magnitude of the pull is proportional to what is known as theinertial massof the body, and in a gravitational field the pull is proportional to thegravitational massof the body. Every material body was assumed, therefore, to possess two types of masses, the inertial and the gravitational, corresponding to the two types of fields to which the body would react.
A concrete illustration of the difference between inertial and gravitational mass is given by the following example: Consider a billiard ball at rest on a table. We should discover that the more massive the ball, the greater would be the effort necessary to set it in motion, as also to arrest its motion once started. The type of mass with which we should here be concerned would be inertial mass; and we may say that it is inertial mass which opposes a departure from rectilinear uniform motion or rest, hence which opposes acceleration. If, now, we were to lift the ball from the table and hold it at arm’s length, the effort we should have to exert to prevent our arm and ball from falling would again vary with the mass of the ball. This time, however, the mass whose effects we were seeking to resist would be the gravitational mass,i.e., the mass which is responsible for weight. It is this same gravitational mass which enters into the expression of Newton’s law of attraction.
There is noa priorireason why any connection should exist between these two different types of masses. Thus, if a billiard ball is found to weigh twice as much as another, there is no logical necessity to anticipate that its inertial mass will also be twice that of the other. Yet it so happens that the most refined physical experiments have invariably proved the strictest proportionality between these two types of masses; so that by choosing our unitssuitably it was always possible to represent both the inertial and the gravitational mass of a given body by the same number. Classical science had accepted the equality of the two types of masses as an empirical fact, but had found itself incapable of suggesting any theoretical justification for it. It appeared to be due to some miraculous coincidence.
This equality in the numerical values of the two types of masses will entail important consequences. For suppose, indeed, that the equality did not hold. This would imply that two billiard balls might present exactly the same gravitational mass, yet differ in the value of their respective inertial masses. If, then, the two balls were to be released simultaneously and allowed to fall from the same height towards the earth, their gravitational masses being equal, both balls would be subjected to exactly the same pull by the earth. But their inertial masses being assumed unequal, the ball whose inertial mass was greater would oppose the acceleration of the fall more strenuously than would be the case with the ball of lesser inertial mass. As a result, the balls would not reach the earth’s surface simultaneously. On the other hand, if the two types of masses are identical, we may be assured that any two objects released from the same point will reach the earth’s surface at the same instant (provided we operatein vacuoso that the resistance of the air does not interfere with their fall). It was precisely because experiment had demonstrated the existence of the same rate of fall for all bodies released from the same height, that Galileo was led to recognise the equality of the two masses. It may be mentioned, however, that ultra-precise experiments (notably with Eötvös’ torsion balance) have since verified these results with extreme accuracy.[80]
We see, therefore, that the equality of the two types of masses allows us to anticipate that an observer situated in a falling elevator would have no weight. He might be standing on a weighing machine, but the needle would always register zero. The fact is that both observer and weighing balance would be falling with the same motion towards the earth; hence the observer’s feet would never press against the balance.
These anticipations may appear to be self-evident; but it is important to note that they are in no wise self-evident until we have recognised the perfect equality of the two types of masses. If such an equality did not exist, the observer in the elevator might still press against the scales of the weighing machine or might hit the ceiling of the elevator, and it would no longer be correct to anticipate that elevator, scales and observer would all fall with exactly the same velocity and acceleration.
We may illustrate the example of the falling elevator in a more general way by considering an observer situated in the interior of a hollow projectile describing its parabola under the influence of the earth’s attraction. The observer would detect no manifestation of weight in his enclosure. He might tip a glass full of water upside down: the water would not spill. If he threw a handful of pebbles into the air, each pebble would follow a straight course with rectilinear motion through the projectile, just as would be the case in a Galilean frame. In short, no mechanical experiment could ever reveal the presence of the earth’s field of gravitation in the interior of any frame of reference moving freely under the action of this field of gravitation. Experiments conducted inside such falling bodies are very difficult to perform, but since the results we have mentioned are the immediate consequences of the equality of the two types of masses, which highly refined experiment has established, we are able to anticipate indirectly the results of such experiments.
So far we have considered the cancellation of a field of gravitation, but we may also consider the creation of a field of force in empty space far from matter. Thus, if a hollow chest be subjected to a variable acceleration through empty space, a variable field of force (inertial force) will be experienced by an observer in the chest’s interior. If now we consider a second chest at rest in a varying gravitational field distributed throughout the chest in exactly the same way as were the forces of inertia and with the same intensity, the equality of the two masses proves that mechanical experiments conducted in either of the two enclosures would yield exactly the same results, and hence that bodies would fall towards the floor of the enclosure in exactly the same way. And so it would be impossible,on the basis of mechanical experimentsconducted in the interior of the enclosure, to decide whether we were in a field of inertial force generated by acceleration or in a field of gravitational force generated by matter. These are the conclusions which the equality of the two masses urges upon us. Inasmuch as the equality of the two types of masses constitutes a fact established by experiment, we may also regard as established empirically the inability of mechanical experiments to differentiate between a field of inertia and one of gravitation.[81]
Now, up to this point, all the facts we have mentioned were perfectly well known to classical science. It is solely in the interpretation of these facts that Einstein suggests highly original views. Let us first examine what classical science had to say. Its arguments ran somewhat as follows: Our inability to differentiate between a field of inertial forces generated by acceleration and one of gravitation generated by matter is due solely to a miraculous coincidence, that of the equality of the two masses. Again, when, in the elevator falling in the earth’s gravitational field, we experience no feeling of weight, it is not that the gravitational field has vanished; it is solely that the field of gravity has been counteracted by an equal and opposite field of inertia generated by the elevator’s accelerated motion. The equality in the intensities of these two opposing fields is due once more to the equality of the two masses. Thus, according to classical science, there existed a very decided difference between the two types of fields of force, and it was always assumed that this difference would be detected when we performed non-mechanical experiments such as optical or electromagnetic ones.
It is this classical interpretation which Einstein challenges. He maintains that the field of force generated in an enclosure is of exactly the same nature regardless of whether it has been generated by acceleration or by gravitation. Also, in the case of the falling elevator, we should not say that the field of gravitation had been counteracted; we should say that it had vanished. Thus interpreted, the miraculous equality of the two types of masses is no miracle at all. There are no two types of masses which react to two types of forces. Since there is but one type of force, there is but one type of mass, to which classical science had erroneously given two different names, and then marvelled at the fact that whatever name was given, its valueremained the same. In this way Einstein succeeded in accounting for a miraculous equality which had baffled classical science.
But, of course, Einstein’s idea was susceptible of experimental proof or disproof. For if we assume that the two types of fields of force are all one, we must assume that not alone mechanical experiments, but alsoall manner of experiments, will fail to detect the slightest difference. This is indeed Einstein’s attitude, and he upholds it in thepostulate of equivalence.
This postulate states that there can exist no difference in the conditions prevailing in the interior of an enclosure, regardless of whether the field of force experienced be generated by acceleration or by gravitation.
Now, when all is said and done, it might appear that when Einstein stated his postulate of equivalence he was taking a leap in the dark. The equality of the two masses proved that so far as mechanical experiments were concerned, conditions existing in a falling elevator would be identical with those enduring in an enclosure floating in free space far from matter. These anticipations were based on experience. But what justification was there for generalising these results and extending them without hesitation to all manner of non-mechanical experiments? At first sight the generalisation might appear somewhat hasty, and most certainly such generalisations should not be encouraged at the hands of those who do not possess any profound scientific knowledge. But with Einstein the case is different. On this occasion, as on many others, his generalisations are those of a man who seems to be able to read the secrets of nature as no one else has ever done before. It is not his knowledge of mathematics or of physics that causes admiration; it is his insight into the philosophy of nature, which is stupendous. As a matter of fact, a similar generalisation on his part was encountered in the special theory. There he generalised from electrodynamics to mechanics; here it is the reverse. But, in either case, a desire to find unity in nature appears to have been his guiding motive.
There is a great analogy between Einstein’s interpretation of the problem we are discussing and his explanation of Michelson’s experiment. In the latter, we remember that Galilean motion through the ether appeared to generate no ether drift. Lorentz explained this absence of an ether drift by assuming that a physical drift was really present, but that it was automatically cancelled in a miraculous way by the trickiness of nature working through the compensating influences of a real physical contraction of bodies moving through the ether and of a real physical slowing down of the duration of events. Einstein, as we have seen, preferred to assume that if no ether drift could be detected, it was because no ether drift was present. Classical science demanded the ether drift and attempted to explain why in spite of its expectations none could be detected. Einstein, by refusing to patch up classical science so as to explain the miraculous, was led to declare that classical science was wrong in attaching any credence to theexistence of an ether drift, and that its belief in this ether drift was attributable to mistaken premises.
If we now pass to the problem of gravitation we see that the situation is very similar. Gravitational force vanishes in a falling elevator; this is the physical fact. Classical science took the stand that the force did not really vanish, but that it was compensated. Einstein prefers to assume that if the gravitational forceappearsto vanish, it is because it really does vanish.
With these new views, the force of gravitation acting at a point loses its attributes of absoluteness. Just like a force of inertia, a force of gravitation betrays a relationship existing between the frame of reference selected and the surrounding conditions of the world. Just as a force of inertia can be annulled by changing the motion of our frame, so now can a force of gravitation be annulled. This does not mean that a force of gravitation is unreal. It is perfectly real, since it can be detected and measured. But it is no longer an absolute; it is a relative, like a force of inertia, for its value varies with our choice of a frame of reference. Henceforth no intrinsic difference exists between a force of gravitation and a force of inertia; these forces are of the same essence. When, therefore, we are in the presence of a force acting on mass, we may refer to it as a force without specifying whether it has been generated by the acceleration of our frame (field of inertia) or by the proximity of matter (field of gravitation).[82]
Einstein maintains the appellation of a gravitational field of force. But he applies it indifferently to fields which classical science would have considered due to the acceleration of the frame, and to the fields which classical science would have recognised as generated by the proximity of matter. By classing both types of fields as gravitational, his object is to differentiate these identical types of fields from fields of other types, such as fields of electric, magnetic or electromagnetic forces.
However, it is well to caution the beginner against a misunderstanding which might cause him some trouble. In classical science the word “gravitational” was always associated with the attraction caused by matter. In Einstein’s theory, when we identify an inertial field with a gravitational one and call both these species of fields gravitational, it is not meant to imply that an exact replica of the inertial field could be reproduced by disposing matter in a suitable way with respect to our frame of reference. Inversely, a field of force produced by matter cannot be duplicated in every detail by communicating a suitable accelerated motion to our frame of reference in free space far from matter. In spite of the complete identification of forces of gravitation and forces of inertia, there still exists a difference in the spatial distribution of the field of force, according to whether it is produced by the proximity of matter or by the acceleration of our frame. In consequence, although all fields of force may be called gravitational fields, regardless of whether they be generated by matter or by acceleration, we must remember that the actual lay of thesefields through space will vary with their origin.
A few precise illustrations will make this point clearer. Consider a hollow chest pulled by some unseen hand through interstellar space far from matter. If we suppose that the chest is rising vertically with constant acceleration, we know that as a result of this constant acceleration a uniform field of inertial force will be present in the interior of the chest. The postulate of equivalence consists in asserting that the observer in the interior of the chest might with equal justification consider the chest to be unaccelerated or at rest in space, while the field of force he perceives would be assimilated to a field of gravitation. All that it is meant here to imply is that the physical nature of fields of gravity and of inertia is one and the same. It isnotmeant to imply that any actual distribution of matter could ever produce a perfectly uniform field of force in the chest’s interior such as existed under the influence of constant acceleration. We know, indeed, that a distribution of matter under the chest would produce a somewhat similar field of force, but the field would be non-uniform, the magnitude of the force being smaller at the top than at the base of the chest. Thus, although the physical nature of both types of fields would be the same, the precise distribution of these fields through space would be different.
These, at least, are the conclusions by which we must abide at the present stage of the theory. We shall see that when we come to consider the universe as a whole, there may be grounds for modifying our opinions. But at the present stage of the discussion we must admit that fields of force produced by matter can never be distributed in exactly the same way as fields produced by acceleration, and vice versa.
Another example is afforded by the field of force generated on a rotating disk. This field of force can be split up for purposes of analysis into two separate fields: a centrifugal field constituted by forces directed away from the centre of the disk, and a Coriolis field pulling bodies sideways as they approach or move away from the centre. Now, no distribution of matter around the disk can be conceived of at this stage which would produce a field of gravitation on the disk distributed in exactly the same way as this field of inertia. Nevertheless, the inertial field on the disk can be called a field of gravitation for the reasons previously set forth in this chapter.
Again, just as matter cannot produce a field of force distributed in exactly the same way as a field generated by acceleration, so also is it impossible to produce by the sole virtue of acceleration a field of force distributed in exactly the same way as the gravitational field around matter. From this it follows that it is impossible, by merely selecting some suitably accelerated frame, to cancel in its entirety a field of force produced by matter. This statement may appear to be in conflict with the example of the falling elevator, in which it was explained that owing to the motion of the elevator the field of force produced by the earth vanished in its interior. However, there isno conflict between the two statements, if we are careful to express ourselves with precision.
In the falling elevator no field of force is experienced in the interior of the elevator,provided the extension of the elevator is small in comparison with the size of the earth. If the elevator were very large, if, for instance, it were a gigantic box completely surrounding the earth, no conceivable motion of the box could ever cause the gravitational field of the earth to vanish everywhere in the box’s interior. Suppose, for instance, that the roof of the box were accelerated towards the North Pole, its floor would then be moving with accelerated motion away from the South Pole. Hence, whereas an observer attached to the roof of the box would perceive no force of gravitation, an observer attached to the floor would experience a force of doubled intensity. In other words, it would be impossible to banish the force of gravitation in all parts of the box. Only when the box or elevator is small in comparison with the earth, so that within its confines the field produced by our planet is practically uniform, is it possible to cancel the earth’s field in the whole interior of the box. Even then, however, this cancellation would be rigorous only at a point, and would decrease in thoroughness as we moved away from this point.
It follows that a falling elevator cannot be assimilated in all rigour to a Galilean frame, since in a Galilean frame, wherever we might be stationed, no forces would be experienced. Nevertheless, as in practice, the frames we consider are not supposed to extend for great distances in all directions, conditions enduring in a falling elevator would be identical to a very high order of precision with those existing in a Galilean frame. For this reason, rigid frames of small dimensions falling freely in a gravitational field are termedsemi-Galilean.
If we summarise the conclusions reached in this chapter, we may say that the postulate of equivalence has allowed us to identify forces of inertia with forces of gravitation. But this identification applies solely to the nature of the forces, not to their spatial distribution. And so it is not correct to say that it would be impossible for us to ascertain whether the field of force experienced in our enclosure was due to the acceleration of the enclosure or to the proximity of gravitating masses. Even without peering out and discovering whether large masses were present, we could always, at least in theory, by a mere exploration of the field distribution, ascertain the true conditions.
We see, then, that whereas velocity was relative in that it was quite impossible for us to ascertain the absolute velocity of our enclosure, acceleration still remains absolute in spite of the postulate, since (theoretically at least) it can always be detected and its effects separated from those due to matter. In fact, as we shall see later, the complete relativity of all motion can be established only if the universe is finite. Under the circumstances, we must be careful not to overestimate the philosophical significance of the postulate in its bearing on the relativity of motion.
Now it might be thought that owing to this fundamental difference in the spatial distribution of fields of force (inertial and gravitational), the postulate would not be of much use in itsphysical applications. But this view would be erroneous. From a purely qualitative standpoint the postulate permits us to assert that any phenomenon whose behaviour should be affected by the acceleration of the enclosure, must also be sensitive to the presence of a gravitational field due to matter. Inasmuch as it is often easy to see that the acceleration of our frame of reference must inevitably modify the observed behaviour of a phenomenon, we are able to infer therefrom that the same phenomenon will also be affected by a gravitational field generated by matter.
It was by following this method that Einstein was able to anticipate a number of gravitational effects which classical science had never even suspected. Chief among these are the bending of a ray of light in a gravitational field, and theEinstein-shift effect, since observed on the companion of Sirius.
We may understand without difficulty how the bending of a ray of light was anticipated. For consider an enclosure floating in empty space far from matter. A wave of light enters by a crack in the wall and travels through the enclosure along a line parallel to the floor. But if now the enclosure be uniformly accelerated, the floor advances with accelerated motion across the ray. As referred to the enclosure, the ray of light will thus assume a bent path, just as the course of a bullet would appear bent under similar circumstances.[83]Then the postulate of equivalence allows us to assert that exactly the same results would have ensued had the enclosure been at rest in a uniform gravitational field; so that regardless of whether or notuniform gravitational fieldsgenerated by matter can exist, we are led to the general conclusion that a gravitational field will bend the course of light waves and modify their velocity.
It is this qualitative discovery that is of importance and that constitutes a fact which was unknown to classical science. What the precise course of a light ray will be in a non-uniform gravitational field generated by matter, is a question of another order.
The postulate, however, enables us to answer this question, provided we know how the gravitational field is spatially distributed around matter. We may understand this point as follows: Owing to its gradual variations from place to place, even a non-uniform field may be regarded as uniform if we restrict our attention to very small regionsof space.[84]It follows that we can decompose a non-uniform field into a series of contiguous volumes of space, each one of which may be assimilated to the spaces in enclosures possessing appropriate uniform accelerations. If, therefore, the spatial distribution of a non-uniform gravitational field is given, we may deduce (according to the methods of differential geometry) the path of a ray of light from place to place over contiguous infinitesimal distances. In this way, assuming Newton’s law to be correct, Einstein deduced the curvature of a ray of light in the sun’s gravitational field. He found that for a ray grazing the sun’s limb the deflection would be 0".87, just one-half of what he was to establish subsequently, after he had recognised that Newton’s law could not be correct.
In other cases, we are not concerned with the study of a phenomenon extending over large areas; its behaviour in a small area, say in our room, suffices. In this case, of course, the earth’s gravitational field may always be regarded as uniform; so that by considering the effect arising from the introduction of a constant acceleration of suitable magnitude, the action of gravitation on our phenomenon can be anticipated.
In a general way, the postulate of equivalence led Einstein to the following conclusions: Inertial mass and gravitational mass being one and the same thing, all forms of existence which possess inertial mass must also manifest weight in a gravitational field. Hence, as all forms of energy possess inertial mass, we see that energy has weight. A hot brick weighs more than a cold one, an electrified body more than a neutral one; possibly, also, a living animal more than a dead one, and so on. Conversely, all forms of energy must develop a gravitational field.
In the following pages we shall show that the postulate of equivalence, which in the present chapter was derived as a generalisation from the equality of the two masses, can also be deduced in a purely rational way from the existence of space-time. We shall thus be in a position to understand how all these discoveries dealing with forces and gravitation can be woven into the common space-time fabric.