CHAPTER XXIII POTENTIALS AND FORCES
ACCORDING to classical science, force may be defined as the product of mass by acceleration. When no force is acting, the motion of a body is always Galilean; only when a force is acting will the motion of a body become accelerated.[76]
Now it must be noted that when a body is subjected to some definite muscular pull the magnitude of this pull or force depends solely on the muscular effort which we are willing or able to produce. On the other hand, when a body is situated in the gravitational field of the earth, the gravitational force which acts upon it and which is commonly known as the weight of the body not only depends on the mass of the earth but likewise varies with the mass of the body. In order to remedy this indeterminateness, it is usual to specify that the body on which the force is acting is one of unit mass; it is then called a test-body. Under these circumstances, we can explore a field of force by placing our test-body in successive regions of space and determining the magnitude and direction of the force which is acting on the body. When we have mapped out the magnitude and direction of the force for every point of space, we are in a position to state that we have determined the lay of the field of force.
In physics we are often concerned with fields of force. The gravitational field surrounding the earth constitutes only one particular illustration. We may also consider the electric field surrounding an electron, or the magnetic field surrounding the poles of a magnet, or even combinations of these two fields, such as the electromagnetic fields produced by electric currents.
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When the direction and magnitude of the force are the same throughout space and do not vary with time, the field of force is called uniform. Uniform fields of force are very difficult to obtain; the simplest illustration we can think of would be that given by a river flowing with constant speed along a rectilinear course. The force exerted by the current would of course remain the same in magnitude and in direction at all points in the river; and for this reason the field of force produced by the current would constitute a uniform field.
Now, if we rest satisfied with this type of description of a field of force, we see that in the case of a uniform field, at any rate, there is no means of differentiating a region of the river or field that is upstream from one that is downstream. All parts of the river are identical. But it is obvious that this apparent identity between the different parts of the river is very superficial. It is a fact of common experience that whereas a boat will float downstream of its own accord, it will need the expenditure of a considerable amount of work to force it upstream again. Mathematicians were therefore compelled to take into consideration a new type of abstraction called a potential. Thanks to the introduction of this new concept, it became possible to differentiate between the various regions of the river (upstream or downstream) by stating that these regions differed in potential. The regions upstream were called the regions of higher potential, and the regions downstream were termed those of lower potential. Hence, it was possible to state, in the form of a general law, that bodies left to themselves would move of their own accord from regions of higher to regions of lower potential.[77]
So far we have been dealing with generalities, and it remains to be seen in what precise manner forces and potentials will be connected. The mathematical connection is expressed as follows:
Potentials are defined in such a way that the expression of the projection of the force along any given direction at any specific point of the field is given by the change in the magnitude of the potential when we pass from this point to the neighbouring point along this same direction. Just as an acceleration projected along a given direction is the time rate of change of the velocity projected along this same direction, so now a force is the space rate of change of the potential; and just as the acceleration is directed towards the increasing velocities, so a force is directed towards the lower potentials.
The existence of these potentials applies generally to all fields of force, regardless of whether these fields are of the uniform or non-uniform variety.[78] It is, as we have said, the variations in value of these potentials from point to point that are associated with the existence of forces. When this variation is uniform in one direction and no variation exists in directions at right angles, we are in the presence of a uniform field of force. When the variation is irregular, we have a non-uniform field of force. When the potential has the same constant value throughout space, so that it does not change in value from place to place, no forces can be present; there is then no field of force.
From this we see that a certain degree of indeterminateness surrounds the precise numerical value of a potential, since it is only its variation from place to place, and not its absolute value, that defines the force. This indeterminateness can be removed in the case of a non-uniform field of force by stipulating that in those regions of the field where the force vanishes (such as would be the case with the gravitational force at an infinite distance from matter), the [Pg 247] potential itself vanishes. A definite value having been attributed to the potential in one part of the field, we can determine its precise value from place to place in all other parts of the field.
Let us now give a few concrete illustrations of fields of force and of potentials in classical science. We have already mentioned the field of force surrounding matter. This was the gravitational or Newtonian field, and the potential at every point derived therefrom was called the Newtonian Potential at the point. The distribution of the field of force was known in a perfectly definite manner when the potential distribution was known. In fact Newton’s law, which tells us how the field of force is distributed around matter, can also be expressed in an equivalent form by Laplace’s Equation, in which it is the potential distribution, and no longer the force distribution, that is described.
Let us now consider another type of field of force known as the inertial field. In perfectly free space, far from matter, there exists no field of force so long as we refer our observations to a Galilean frame. But if now we step into an accelerated or a rotating frame, we shall experience the effects of a field of force which we will ascribe to forces of inertia. Not only our own human bodies, but, in addition, all free bodies, will be subjected to the actions of these forces. Thus, consider a disk that is rotating, and a ball rolling without friction on the disk’s surface. If the ball is sent from the centre of the disk to its periphery it will of course follow a straight line at constant speed with respect to the earth or to any other Galilean frame. But then, with respect to the rotating disk itself, its course can no longer be rectilinear and uniform. Instead of following one of the radii of the rotating disk, it will follow a curve as though it had been pulled sideways by some force.
This result is general. If we ourselves tried to advance along one of the radii of the rotating disk we should experience a real physical force pulling us sideways. To all intents and purposes, when we referred events to the rotating disk and not to the non-rotating earth (the rotation of the earth being so slow that we may neglect it as a first approximation), the physical existence of a field of force would have to be taken into consideration. The precise type of force we have mentioned is called the Coriolis force; in addition, there also exists another type of force, the better-known centrifugal force. Both these types of force are called forces of inertia. It is their ensemble which constitutes the field of inertial force existing in a rotating frame.
If, in place of a rotating frame, we had considered an accelerated railway compartment, the field of inertial force would have been disposed in a different manner. In the case of a train moving with constant acceleration along a straight line, the field of inertial force generated would have been of the uniform variety, disposed longitudinally through the train.
All these different illustrations show us that whereas, in a Galilean frame, no field of force exists, yet a field springs into existence automatically as soon as we place ourselves in any accelerated frame. [Pg 248] It is for this reason that accelerated frames can be distinguished physically from Galilean frames; and it is owing to this generation of physical fields of force that accelerated motion must be regarded as absolute, whereas velocity, giving rise to no such fields, yields us no means of distinguishing one velocity from another, hence is relative.
Now we have seen that fields of force are accompanied by potential distributions. Hence it follows that, whereas, with respect to a Galilean frame, the inertial potential is zero at every point, or at least maintains some constant value, in the case of accelerated frames this inertial potential must vary from place to place in the frame. The magnitude of the force at any point of an accelerated frame is given, therefore, by a mathematical expression involving the variations in value of the potential in the neighbourhood of the point considered.
All we have explained so far pertains to classical science, but it appeared necessary to mention these results briefly before proceeding to a more systematic study of Einstein’s theory. We shall now see how all these results find a natural place in the space-time theory.
Consider a Galilean frame and an object moving freely in this frame. Its path will of course be rectilinear and its speed uniform, hence it will possess no acceleration, in accordance with Newton’s law of inertia. If we interpret these results in terms of space-time, we see that the world-line of the body is a straight line in space-time as referred to our mesh-system of equal four-dimensional cubes. If we now examine the motion of this same free body from some definite accelerated frame, all we have to do is to change our four-dimensional space-time mesh-system in an appropriate way. This change of mesh-system does not affect the world-line of the body, since this line remains a straight line or geodesic through flat space-time; but, on the other hand, it will certainly alter the appearance of the straight line when we refer our measurements to our curvilinear mesh-system. The erstwhile straight world-line will now appear curved in a definite way, and its precise mathematical equation as referred to our new curved mesh-system can be obtained without difficulty.
The physical significance of this apparent curvature of the world-line of the body will be that the body will now appear to possess acceleration with respect to our new non-Galilean frame. The precise magnitude of this acceleration will therefore be given by the apparent curvature of the world-line followed by the body, and this curvature is of course expressed implicitly by the equation of the world-line when referred to our curvilinear mesh-system. But for a given test-body the acceleration is related to the force acting on the body. From this it follows that the mathematical expression of the force can be obtained immediately.
This mathematical expression of the curved world-line, hence of the force of inertia at any particular point, is seen to be built up with the variations in value of the ’s around this point. Were the ’s to remain constant in value throughout, as would be the case in a Galilean frame, this mathematical expression would vanish in [Pg 249] value. It follows that the ’s, of which this expression of the force is built, must correspond to the potentials of inertia. We have thus discovered the physical significance of the ’s of space-time: they define potentials.
We see, indeed, that this identification is legitimate in every respect. Thus, in a Galilean frame, there are no forces of inertia, so that the potentials of inertia must be constants; and we know that in a mesh-system of equal four-dimensional cubes, which corresponds to a Galilean system, the ’s are all constants and are given by , all other ’s being zero. Again, in an accelerated frame, a field of inertial forces appears; hence the potential must vary from place to place; and we know that in a curvilinear mesh-system (corresponding to an accelerated frame) the ’s lose their constant values and vary from place to place.
So far the reason for the existence of forces of inertia has been made apparent. They arise owing to the uneven spread of numbers which accompanies all curvilinear mesh-systems (accelerated frames).[79] When this occurs, the mathematical expression of the force of inertia assumes a definite numerical value at each point, whereas, when the ’s are constants, as in a Cartesian mesh-system, this expression of the force maintains a zero value.
From this we see that forces of inertia arise from an attempt on our part to cut up space-time with curvilinear mesh-systems instead of Cartesian ones, just as they arise from our substitution of accelerated frames for Galilean ones. We cannot help but feel, however, that having proceeded thus far, a further generalisation is required. To be more explicit, it should be understood that the scheme of physics we have developed has compelled us to attribute a fundamental rôle to space-time. But Newton’s great law of universal attraction is expressed in terms of the separate space and time of classical science. If space-time is indeed as fundamental as Einstein has led us to believe, it appears incredible that Newton’s law should remain outside its scope. Yet this it certainly does, for if Newton’s law were a space-time law, it would preserve the same form in spite of any change in our space-time mesh-system. And this it fails to do.
Einstein’s next attempt was therefore to weld Newton’s law into the general fabric, a result he achieved about 1914. The mathematical generalisation which would allow this result to be obtained seems almost obvious to-day, since (as will be explained in Chapter XXV) it reduces to assuming that in the neighbourhood of matter, four-dimensional space-time loses its flatness and becomes non-Euclidean or curved. However, it appears to have been through the medium of physical observation that Einstein was led to his superb generalisation, so we shall proceed to follow the historical order by explaining the significance of his postulate of equivalence.
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CHAPTER XXIV THE POSTULATE OF EQUIVALENCE
EINSTEIN’S Postulate of Equivalence consists 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 follows [Pg 251] that we are compelled to conceive of a real force, a force of gravitation pulling 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 the inertial mass of the body, and in a gravitational field the pull is proportional to the gravitational mass of 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 no a priori reason 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 units [Pg 252] suitably 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.
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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 operate in vacuo so 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.
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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 experiments conducted 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 value [Pg 255] remained 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 also all manner of experiments, will fail to detect the slightest difference. This is indeed Einstein’s attitude, and he upholds it in the postulate 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 the [Pg 256] existence 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 force appears to 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 these [Pg 257] fields 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 is not meant 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 is [Pg 258] no 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 termed semi-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 its [Pg 259] physical 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 the Einstein-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 not uniform gravitational fields generated 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 regions [Pg 260] of 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.
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CHAPTER XXV THE INCLUSION OF GRAVITATION IN THE MODEL OF SPACE-TIME
IN the preceding chapter the following results were established:
1. No essential difference exists between a field of gravitation produced by matter and a field of inertia produced by the acceleration of our frame of reference (postulate of equivalence).
2. Although these two types of fields of force are essentially of the same nature, yet the actual spread of the gravitational field around matter is never quite the same as that of the field generated by the acceleration of a frame in free space far from matter. As a result of this feature, whereas it is always possible to cancel a field of force entirely by changing the motion of our frame in a suitable way when the field contemplated exists in regions remote from matter, on the other hand, in the neighbourhood of matter, the field of force can be modified, and even cancelled in a small region, but never cancelled over any wide extension. It is like a protuberance which can be flattened out in one place, only to reappear with increased intensity elsewhere.
These are the results which we must weave into the general model of space-time, unless we reject Einstein’s postulate of equivalence, and maintain that fields of gravitation produced by matter have no more in common with fields of inertia than they have with electromagnetic fields, for instance. The postulate of equivalence, as we have seen, appears to be justified owing to the empirical fact that the two types of masses are identical. But we now propose to show that quite independently of this empirical fact, even if the postulate of equivalence had never been stated in a specific manner, a natural mathematical generalisation of the geometry of space-time would have led us indirectly to exactly the same identification of forces of inertia and gravitation. Accordingly, let us recall briefly the results we arrived at in Chapter XXIII when we examined the origin of forces of inertia in our space-time model.
We remember that space-time was assumed to be perfectly flat, so that its geodesics were Euclidean straight lines when referred to Cartesian mesh-systems of equal four-dimensional cubes. These Cartesian mesh-systems, splitting up space-time into mutually perpendicular rectilinear directions, constituted the geometrical representations of our methods of breaking up space-time into space and time for purposes of referring our measurements to Galilean systems of reference. In these Cartesian mesh-systems the ’s of space-time maintained constant values throughout, and the numerical values of these ’s represented the potentials of the field of inertial force. These potentials being constants, remaining unchanged in value from [Pg 262] point to point in our Cartesian systems, no field of force could be present; and this was in full accord with what we know of Galilean systems, since these systems are characterised by the total absence of all such fields. Furthermore, as all free bodies and light waves followed geodesics in space-time according to the requirements of Least Action, and as these geodesics when referred to Cartesian systems were straight lines, free bodies and light waves would always appear to follow straight lines with constant speeds when observed from Galilean frames. This was in accord with Newton’s law of inertia.
Subsequently we saw that accelerated frames moving in interstellar space would be represented by curvilinear or Gaussian mesh-systems. In these systems the values of the ’s of space-time were no longer constant, but varied from place to place. This variation in the values of the ’s connoted the variation in value of the potentials; and as a result the mathematical expression of a force (given as it was by relations between these variations) no longer vanished. These curvilinear mesh-systems represented, therefore, systems in which fields of force would be present. This again was in perfect accord with what we knew of accelerated or non-Galilean frames. Further, as referred to these curvilinear mesh-systems, the geodesics of space-time would appear to be curved. The physical significance of this fact would be to imply that the motion of free bodies and light waves would appear to be accelerated when viewed from accelerated frames. Once more the complete accord with experiment is apparent.
Now let us proceed to the natural generalisation of these results. So far, we have assumed space-time to be flat. Indeed, we could not escape this assumption if we were to admit that Galilean frames, free from all fields of force, could exist in space far from matter, and that as observed from these frames the geodesics of space-time would be Euclidean straight lines. Only thus would Newton’s law of the motion of free bodies in interstellar space be verified. But suppose now that, for some reason or other, space-time were to develop non-Euclideanism or curvature, and so depart from its flatness. How would such a condition manifest itself physically?
In the first place, it would be impossible to split up this curved space-time into the straight directions which define a Cartesian mesh-system. Physically speaking, this would imply that in those regions where space-time was curved no Galilean systems could exist. We might, of course, conceive of mesh-systems less curved than others; but even the straightest of these systems would at best be only approximately Cartesian within a restricted region. As we considered parts of the mesh-system farther and farther removed from this special region, the meshes would become more and more widely spaced or more and more closely gathered together.
An elementary illustration of this would be given by considering a curved surface of two dimensions, such as that of a sphere. On a sphere we can trace an infinite variety of mesh-systems, but none of these mesh-systems is made up of equal Euclidean squares. The nearest approach to a network of squares would be afforded by a mesh-system [Pg 263] of meridians and parallels. But even a mesh-system of this sort could be likened to a network of equal squares only in the immediate vicinity of the equator. As we examined our mesh-system at regions farther and farther removed from the equator, we should notice that the meridians had a tendency to run closer and closer together, so that the quadrilaterals bounded by the meridians and parallels would depart more and more in shape from those of Euclidean squares. In the general case, where we consider an arbitrarily curved surface and no longer a uniformly curved one, such as a sphere, it could be shown that the straightest type of mesh-system compatible with the surface was assimilable to one of squares only around a point, and not, as in the case of a sphere, along a line (such as the equator). The point could be selected arbitrarily, and our pseudo-straight mesh-system drawn around this point accordingly.
We see, therefore, that if space-time is unevenly curved or unevenly non-Euclidean, the nearest approach to a Cartesian mesh-system around any given point could be approximately Cartesian only in a restricted region around this point. As we moved away from this point in our mesh-system, its curvilinear characteristics would become more and more pronounced. So far as the ’s of space-time are concerned, this would imply that in a region where curvature was present these ’s could maintain constant values only in the more or less immediate neighbourhood of a point. Expressed in terms of fields of force, this assertion is equivalent to stating that in a region of space-time curvature it would be quite impossible to rid ourselves of a field of force throughout space. We might annul it at the point and in its immediate neighbourhood, but the field of force would reappear for more distant regions.
It is easy to see that these physical conditions are precisely those which we have found to exist in a gravitational field produced by matter. The straightest type of mesh-system around a point (namely, a semi-Cartesian mesh-system) would correspond to that of the observer in the falling elevator or in the hollow projectile, since for an observer in a frame of this type the force of gravitation disappears in the immediate interior of his frame, but reappears at more distant points. Further, space-time being curved, its geodesics would no longer be Euclidean straight lines. They would still appear approximately straight when referred to a semi-Cartesian mesh-system in the immediate neighbourhood of the point for which the mesh-system was approximately one of equal four-dimensional cubes; but their curvature would become more and more apparent when we surveyed them from this same mesh-system in more distant regions. The physical interpretation of this geometrical fact would be afforded by the seemingly uniform and rectilinear motion of free bodies and rays of light in the immediate neighbourhood of the falling elevator when viewed by an observer situated in this falling frame, and by the apparent acceleration of these bodies in regions more distant from the falling elevator.
It is therefore obvious that the existence of a non-Euclideanism of [Pg 264] space-time around matter would betray itself physically in exactly the same way as a field of gravitation produced by matter.[85] Assuming this generalisation to be correct, we see that no essential difference can exist between a field of gravitation and one of inertia. Both types of field arise from a relationship between the mesh-system adopted by the observer, and the fundamental structure of the space-time continuum. The only difference that exists is that in interstellar space far from matter, space-time being flat, it is always possible to select a Cartesian mesh-system corresponding to a Galilean system in which no field of force is present, whereas in the neighbourhood of matter, owing to the inherent curvature of space-time, the nearest approach to Galilean frames is given by semi-Cartesian or semi-Galilean frames (falling elevator). In such frames the field of force cannot vanish throughout the frame at more or less distant points.
It follows that these same ’s whose values in a given mesh-system represented the potentials of the field of inertia can now with equal justification be considered to yield the potentials of the field of gravitation. The mathematical expression comprising the variations of the ’s from point to point, which was formerly proved to represent the force of inertia, can now with equal justification be deemed to represent the force of gravitation. As was the case with the force of inertia, this mathematical expression of the force, depending as it does on the variations in the values of the ’s from point to point, varies with our choice of a mesh-system; in exactly the same way, the magnitude of a gravitational force in our frame of reference varies with the acceleration of our frame (as was established when we discussed the postulate of equivalence). Furthermore, the value of the mathematical expression of this force will vanish completely throughout a mesh-system when this mesh-system is Cartesian, whereas it will vanish only around a definite point when, owing to the curvature of space-time, a Cartesian mesh-system is impossible, so that we have to be content with a semi-Cartesian one. We see that the analogy is complete in all respects.
Here it must be noticed that, identifying as we are the ’s in a given mesh-system with the gravitational potentials in the frame corresponding to this mesh-system, we find ourselves in the presence of as many different gravitational potentials as there are separate ’s. That is, we have ten different potentials. Newton knew of only one of these, the Newtonian potential, which turns out to be —the one directed along the time-axis of the mesh-system. It is the presence of these other potentials ignored by Newton and discovered by Einstein which is responsible for the major differences distinguishing Newton’s law of planetary motions from that of Einstein. We see, therefore, that this natural generalisation of the geometry of space-time, by compelling us to recognise forces of gravitation and inertia as of the same nature, leads us to exactly the same conclusions as were obtained by Einstein when he started from the postulate of equivalence. So although we laid stress on the postulate of equivalence as a separate discovery in order to abide by the [Pg 265] historical sequence of Einstein’s investigations, we see that when we view the problem from a loftier standpoint the postulate does not add anything to what would normally have been discovered, or at least suspected, as a result of the more philosophical generalisation we have discussed.
Of course, if the postulate of equivalence should prove to be unjustified, as it would were experiment to detect a difference between inertial and gravitational mass, the generalisation would also be unjustified and gravitation could never be ascribed to a curvature of space-time. However, as the most refined experiments have invariably disclosed the complete identity of the two types of masses, the space-time generalisation appears to be in order.
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CHAPTER XXVI TENSORS AND THE LAWS OF NATURE
WE have now reached a point where further progress would appear to require at least a few elementary notions on tensors and the absolute calculus. We shall endeavour to present these matters in as clear a way as possible by considering, first of all, tensors from the standpoint of pure mathematics, without any reference to their applications in Einstein’s theory.
Let us suppose that we are dealing with the ordinary three-dimensional Euclidean space of classical science. We remember that it is often convenient to refer the positions of objects situated in space to some definite system of reference or system of co-ordinates. This procedure is equivalent to drawing an imaginary three-dimensional mesh-system with respect to which we locate the points of space. The precise shape of this mesh-system is arbitrary. It may split up space into equal cubes (in which case it is called Cartesian), or again it may be curvilinear and split up space into irregularly shaped volumes. In all cases, however, we shall assume that these mesh-systems remain fixed and do not vary in shape or orientation once they have been drawn.
Now, though we have stated that the precise shape of the co-ordinate system, or mesh-system, is arbitrary, there are mathematical reasons which render the use of Cartesian systems especially convenient. So long as we are dealing with Euclidean space, in which, as we know, Cartesian mesh-systems are always permissible, we need not be concerned with Gaussian co-ordinate systems. It was for this reason that classical science, when dealing with physical problems, confined itself to Cartesian co-ordinates; and by a change of co-ordinate system it meant a rotation or a displacement of the Cartesian frame of reference from one position to another.
Nevertheless, we always had to view the possibility of having to investigate problems in which Cartesian co-ordinate systems would be impracticable, owing to the curvature or non-Euclideanism of the continuum. Accordingly, even before the advent of the theory of relativity, mathematicians had endeavoured to perfect a form of calculus which would free them from all dependence on any particular type of co-ordinate system. This task was accomplished chiefly by Levi-Civita and Ricci, and the mathematical instrument evolved was called the absolute calculus. And so, when in the general theory Einstein was led to his conception of a curved or non-Euclidean space-time, in which Cartesian co-ordinate systems were impossible, the mathematical instrument he required was at hand. All he had to do was to put it to practical use.
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Although, in the Euclidean space of classical physics which we now propose to discuss, there was no need to consider Gaussian or curvilinear co-ordinate systems, yet, in view of the greater generality of the method, we shall treat the subject matter of this chapter from the standpoint of all systems of co-ordinates, whether Gaussian or Cartesian. The conclusions we shall reach must, however, be construed as holding for all manner of co-ordinate systems. If these preliminary points are understood, we may proceed to a more detailed discussion of the problem.
Let us consider a straight rod situated in this space. If we agree on a unit of measurement, we can assign some definite length to the rod. Obviously, this length, whatever it may be, will remain unaffected by a change in our co-ordinate system in space. Magnitudes of this sort, which transcend in every respect our choice of a co-ordinate system, are known as scalars, numbers or invariants.
But suppose now that we wish to determine the exact orientation of our rod. Orientation, in contrast to length, has no meaning until some mesh-system has been prescribed; for we can define a direction in empty space only by referring it to some accepted system of standard directions at each point. The meshes of our mesh-system at each point will, of course, define such standard directions for us. When, therefore, we attempt to determine the orientation of the rod, our procedure will be to project its length upon the three meshes that pass through one of its extremities. We thereby obtain what are known as the three components of the rod; and when these three components are known, the orientation and the length of the rod are fully determined.[86]
From this example we see that a considerable difference exists between the directed magnitude we are here considering and the invariant one which was given by the mere length of the rod irrespective of its orientation. Whereas the invariant magnitude possessed no components, the directed one has as many components as the space has dimensions, three in the present case. A directed magnitude of this sort is termed a vector.
Suppose now that we change our system of reference, rotate it slightly, for instance. We shall thereby obtain three new components for our vector in the new co-ordinate system, and these new components will differ in magnitude from our former ones. However, the change in the value of the components for a definite change of frame of reference is not arbitrary, but will be submitted to well-defined mathematical rules.[87] This fact enables us to discover a criterion of objectivity.
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For, suppose that in a first co-ordinate system certain components, , , , are revealed by measurement, and that in a second co-ordinate system new components , , are obtained. If under these circumstances we were to find that the two trios of components , , and , , were not linked together by the stringent mathematical rules we have referred to, we should have to recognise that we could not have been contemplating the same vector when we passed from the first co-ordinate system to the second. Conversely, if the mathematical rules were found to be satisfied by the components, we should realise that we were indeed dealing with the same vector. In this way the vector would no longer be some indeterminate magnitude. It would be a definite and precise one to which we might attribute an objective existence transcending our choice of a mesh-system of reference. The components themselves have no absolute existence; they do not transcend our mesh-system. They represent but partial aspects of the vector, mere modes or shadows, varying, as they do, with our system of reference. But to the vector itself an objective existence can be conceded.
Now, mathematical investigation soon showed that there existed two different types of vectors, distinguished by two different rules of mathematical transformation when we passed from one co-ordinate system to another. As these two rules of mathematical transformation, known as the contravariant and the covariant rule, respectively, appeared to be the only consistent ones, it was recognised that there existed only these two different types of vectors, the so-called contravariant vector and the covariant vector. In a certain sense this distinction is somewhat artificial, for we may always regard a given vector as contravariant or covariant, as we see fit. All that we mean to imply by this terminology is that we intend to subject the components of the vector to the contravariant or the covariant rule of transformation when we change our co-ordinate system.
So far we have considered two types of magnitudes, invariants or scalars, and vectors. Mathematicians discovered, however, that there existed a vast array of other types of magnitudes named tensors,[88] whose components likewise obeyed rigid rules of mathematical transformation when we passed from one co-ordinate system to another, and which for this reason could be credited, just as truly as vectors and invariants, with an objective existence transcending our mesh-system. Tensors differed from vectors owing to the greater number of their components. Whereas a vector possessed only as many components as the space had dimensions, a tensor could have a much greater [Pg 269] number. According to the number of their components in a space of a given number of dimensions, tensors were grouped into tensors of the second order, of the third order, of the th order.
In a three-dimensional space a tensor of the second order has nine components instead of three, as has a vector. A tensor of the third order has twenty-seven, and so on. Just as in the case of vectors, the components of a tensor situated at a point in space vary in value with the choice of our mesh-system, and, again, just as with vectors, the values of the components in one mesh-system are connected with their values in another mesh-system by rigid mathematical rules of transformation. In the present case, however, these rules of transformation are more complex. They are seen to be generalisations of the rules regulating the transformations of vector components. As an immediate sequel, tensors appear to be mere generalisations of the older known vectors.
In particular, a vector was found to be nothing else than a tensor of the first order, while an invariant appeared to be a tensor of zero order. The general name “tensor” expresses, therefore, all those types of mathematical magnitudes which possess an objective significance transcending our system of co-ordinates, although it will often be convenient to reserve the name “tensor” for tensors of the second and higher orders, retaining thereby the classical appellations “vector” and “invariant” for tensors of lower order.
Now we shall see that the distinction we have made between the two different types of vectors (contravariant and covariant) will extend automatically to tensors. Let us consider, for instance, a tensor of the second order. When we change our co-ordinate system, the nine components of this tensor will vary, and to obtain their values in the new co-ordinate system we apply a rule of transformation which is, so to speak, the vector rule taken twice in succession. As there are two vector rules, one contravariant and one covariant, we get, in the case of tensors of the second order, a twice contravariant rule, a twice covariant one or a mixed rule both contravariant and covariant. In this way we are led to distinguish three different types of second-order tensors: the twice contravariant, the twice covariant and the mixed variety.
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Here it must be stated that the two rules, the contravariant and the covariant, are in certain respects the antitheses of each other. What one rule does the other undoes. If, therefore, we combine any vector considered as contravariant with any vector considered as covariant, the result will be similar to what happens when a squirrel runs in a drum which is rotating backwards. The forward motion of the squirrel is counteracted by the backward motion of the drum, so that finally the squirrel does not move at all. The result of combining a contravariant and a covariant vector will be, therefore, to obtain a scalar or invariant. In the same way, a covariant tensor of the second order combined with two contravariant vectors, or a covariant tensor combined with a contravariant one of the same order, will each yield an invariant.
Let us now pass to one of the most important characteristics of tensors and vectors. The fact that the components of all contravariant vectors are submitted to the same rules of change when we pass from one co-ordinate system to another, proves that if two contravariant vectors situated at the same point of space are equal, that is, if the components of the one are equal to the components of the other in our co-ordinate system, this equality will inevitably endure in any other co-ordinate system. In other words, the equality of two vectors at a point of space constitutes an equality which a change in our co-ordinate system can never destroy; the equality is thus an absolute. Similar conclusions apply to the equality of two tensors at the same point of space.
It must be noted, however, that this equality endures only because the components of either vector or of either tensor on the two sides of the equation are submitted to the same rules of transformation when our co-ordinate system is changed. This implies that we are dealing with vectors of the same nature (covariant or contravariant), or with tensors of the same nature and order. If these conditions are satisfied, we see that equations between vectors or tensors, often called vector equations and tensor equations, exhibit the remarkable property of remaining unaffected by a change of mesh-system.
In a certain sense, then, vector equations and tensor equations constitute invariant or absolute equations. This last appellation would, in fact, be perfectly justified were we to argue in terms of the absolute vectors and tensors which exist independently of our mesh-system. But in practice it is very difficult to dispense with a mesh-system. As a result, what we come into contact with—what we measure in reality—is not the vector or the tensor itself, which is absolute, but its components in the mesh-system that we have selected. These components are relative, since they change in value when we change our mesh-system. True, these changes being the same on either side of the equation, the equality endures regardless of our choice of mesh-system. But, on the other hand, a definite numerical change has taken place in the value of either side of our equation. The form of the equation as a whole has remained unaltered, but the contents of the two sides has changed.
Let us explain this point more fully. Consider two vectors which are situated at the same point and which are equal to each other. In three-dimensional space, this equality is expressed by the respective equality of the three components of the first vector with the three components of the second vector in our co-ordinate system; hence we have three equations, such as
If, now, we change our mesh-system, the components of either vector change in exactly the same way, so that our equations subsist. In the new mesh-system, however, becomes and becomes , etc., so we now have
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Obviously, although the equations subsist, yet the expressions on either side of our three equations have varied; for is different from and so on. Thus we see that it would not be quite correct to state that our equations remained completely unchanged when we passed from one mesh-system to another.
For these reasons it is customary to speak of the covariance rather than of the invariance of vector or tensor equations. The word “covariance” expresses the fact that both sides have varied in exactly the same way. The appellation “invariance” would tend to make us believe that neither side had varied at all, which would, of course, be incorrect. In short, we must remember that it is the relationship of equality between the two sides, and not the individual terms themselves, that remain unaffected by a change of mesh-system.[89]
Now, there exists still another type of vector and of tensor equation. If, for instance, a vector or a tensor vanishes, this means that all the components vanish individually. A simple glance at the rules of transformation shows that in this case all the components will continue to vanish in any co-ordinate system, so that the equating of a tensor, a vector or an invariant to zero at a point of space will endure in all co-ordinate systems if it endures in any particular system.
Summarising, we find that there exist two types of covariant equations:
The first type is given by the equality of two vectors (the vectors being both covariant or both contravariant) or, more generally, by the equality of two tensors, these tensors being of the same order and same nature (i.e., covariant, mixed or contravariant). A limiting case would be given by the equality of two invariants; we should then have an invariant equation.
The second type is given by the vanishing of a vector or a tensor at a point in space. A limiting case would be given by the vanishing of an invariant.
All these deductions obtained mathematically are in a certain sense self-evident physically when we recall that a tensor, a vector and an invariant represent magnitudes to which an objective existence may be conceded. It is obvious, indeed, that if two such magnitudes are equal to each other at a point in space, a mere change of our co-ordinate system can never disturb this intrinsic equality. In the same way, if a concrete entity, such as a tensor, has vanished at a point in space, a mere change of our co-ordinate system is incapable of bringing it into existence. Once again, we realise the strict dependence which exists between physical representations of this sort and the fact that invariants, vectors and tensors must be recognised to have an objective existence transcending the particular point of view of our co-ordinate system.
If these mathematical considerations have been grasped, we may pass to physical applications of these purely mathematical discoveries. We will restrict ourselves for the present to the three-dimensional [Pg 272] Euclidean space of classical science; hence by a change of co-ordinate system we again mean a mere change in shape, position and orientation of our mesh-system. We do not mean the passage from one system to another in relative motion thereto.
In classical science we often come upon invariants and vectors. Examples of invariants are given by the mass and the density of mass of a body. Vectors are illustrated by a velocity, an acceleration, a force, a displacement; and we may say that in a general way a vector can be represented by an arrow. The simplest physical illustration of a tensor at each point of space is afforded by the state of compression and tension at each point of an elastic medium which has been distorted.
It is impossible to express this state of tension and compression at a point by means of three components in our three-dimensional space. Owing to lateral wrenching effects at each point, nine components are necessary; and we realise that we can no longer appeal to a vector. We are now in the presence of a tensor. The physical origin of the appellation “tensor,” coined by Willard Gibbs, thus becomes apparent. In the present case the tensor is a second-order tensor, and, furthermore, it is what is known as a symmetrical tensor. The characteristics of a symmetrical tensor are defined by the identity of certain of its components two by two; this reduces the number of independent components of a second-order tensor in three-dimensional space to six in place of nine.
It can be proved that the numbers which we have often mentioned constitute symmetrical second-order covariant tensors. The tensor nature of the ’s means nothing else than that the components of the ’s in our co-ordinate system vary according to the rules of tensors when we pass from one co-ordinate system to another. The ’s being second-order covariant tensors, we may infer that by combining them with two displacements, or vectors, of opposite nature (i.e., contravariant vectors), we shall obtain an invariant. It is for this reason that yields an invariant,[90] which, as we know, is nothing but the square of the length .
In three-dimensional space there are nine of the components, but, the tensor being symmetrical, this number is reduced to six. Thus . For this reason we may in the case of three-dimensional space write out as follows: instead of [91]
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When we come to examine the concrete significance of the ’s, we find them to be expressive of the angles between our co-ordinate meshes, and of the general shape of our mesh-system from place to place. Since, however, the shape of our mesh-system, though arbitrary in large measure, is nevertheless restricted to certain types compatible with the structure of the space in which we trace them, these ’s are also representative of the structure or geometry of our space. For this reason they are called structural or fundamental tensors.
Let us recall once more that we are not yet dealing with Einstein’s discoveries or with space-time; we are discussing three-dimensional space, and all we have said refers to pre-Einsteinian science.
Other illustrations of tensors are afforded by an angular velocity and by a magnetic force. The tensor nature of these two magnitudes is less obvious because it so happens that these second-order tensors are of a special type known as anti-symmetrical. In tensors of this type, certain components are always zero in all systems of co-ordinates, and the remaining ones are of the same magnitude but of opposite sign. The result is that in the case of three-dimensional space anti-symmetrical tensors of the second order possess only three independent components just like vectors, and for this reason are often represented by arrows and confused with vectors. As a matter of fact, this confusion need not arise; for it is obvious that a rotation, for example, is better represented by a corkscrew curve than by an arrow, and the same applies to a magnetic force when we realise that it is generated by a whirling of electrons. At all events, had it not been for the accidental fact that space had three dimensions, the confusion could never have occurred at all; for, in a four-dimensional space, anti-symmetrical tensors of the second order have a number of components greater than four (which is the number of components of a vector in a four-dimensional space). Still another illustration of an anti-symmetrical tensor of the second order is furnished by an element of a surface.
In short, we see that the magnitudes in which classical science was chiefly interested were either invariants, vectors or tensors. At this we need not be surprised, seeing that the magnitudes to which physical science pays greatest attention are always those to which we may concede an objective existence transcending our choice of a mesh-system. Thus, an object itself appears to be of greater [Pg 274] significance than the shape of its shadow; for the shadow will change as we alter the orientation and shape of the surface on which it falls, whereas the existence of the object will transcend any such changes.
If now we consider the laws of nature, we can anticipate, for two different reasons, that they must be vector or tensor equations. First, because all those physical magnitudes to which we attribute an objective existence and which enter into the expression of the laws of nature are invariants, vectors and tensors; and, secondly, because a law of nature must obviously remain unaffected by a mere change of orientation or shape of our frame of reference.
It is, indeed, evident that a mesh-system is a mere artifice of calculation. It can have no intrinsic significance, and, as a matter of fact, we might dispense with it altogether. Such an absolute condition of the world as the equality of two vectors at a point or the vanishing of a vector or of a tensor would endure even had mesh-systems never been invented by Descartes and Gauss. Suppose, then, that a law appeared to hold when we selected some particular mesh-system, but no longer when our meshes were squeezed together or rotated to some new position. Obviously our law would possess no generality; it could refer to nothing objective and would be no law at all.
We may summarise these discoveries by saying that all the laws of classical science were vector or tensor laws. They remained covariant or unchanged in form when we changed the orientation or shape of our mesh-system while maintaining it fixed in space. This last point is important; for in classical science, time being absolute and separate from space, it was solely from the point of view of a spatial change in the shape and orientation of our mesh-system that the laws remained covariant.
We may also mention that in addition to the covariance of all natural laws to a spatial change of mesh-system, there existed a second requirement which all natural laws had to obey. We refer to the test of dimensionality. A brief discussion will make this point clear.
The numerical values of all physical quantities depend on our choice of standard units for the measurement of distance, duration and mass. If, therefore, we change our units, if we adopt the minute instead of the second, or the foot instead of the metre, or the pound instead of the kilogramme, the numerical values of both sides of the equation representing the law will be affected accordingly. For the equation to endure in spite of the change in value of its two terms, it is necessary that a change in our unit of time or of length or of mass should affect the numerical values of both sides of the equation in exactly the same way. This condition is expressed by saying that both sides of the equation must possess the same dimensionality. It is perfectly obvious that a law of nature must constitute an equation of this type. It would be ridiculous to suppose that a change in our units from the avoirdupois to the metric system, for example, [Pg 275] could ever upset the validity of a law of nature. This test of dimensionality was often useful in determining the second side of the equation representing some unknown law. We might ignore the exact form of the second side, but we could always tell in advance that its dimensionality would have to be the same as that of the first side. In this way a certain amount of a priori information was often obtained.
[Pg 276]
CHAPTER XXVII THE PRINCIPLE OF GENERAL COVARIANCE, OR THE GENERAL PRINCIPLE OF RELATIVITY
WE have endeavoured to show that the great change in our views of the world brought about by the theory of relativity arises from the discovery that four-dimensional space-time and not the three-dimensional space and the independent time of classical science correspond to reality. It might seem but a natural generalisation to extend to space-time the conclusions we arrived at in the last chapter when discussing the covariance of the natural laws to a change of mesh-system in space. We should then expect the laws of nature to express relationships between vectors and tensors in space-time, hence to remain covariant to a change in our space-time mesh-system.
Although this generalisation would turn out to be perfectly legitimate, its validity is by no means self-evident a priori. We can accept it only after we have convinced ourselves that the laws of physics, such as experiment has revealed them to be, can indeed be regarded as expressing relationships between entities (vectors and tensors) in space-time. Now, from a purely formal standpoint, it is conceivable that such might not be the case. Suppose, for instance, that the four-dimensional spatio-temporal background were characterised by some finite invariant velocity greater than Maxwell’s constant , which is given, as we know, by the velocity of light in vacuo. In an extended sense we could still regard our universe as a four-dimensional space-time, since, the invariant velocity being finite, and not infinite, the fusion of space and time would hold, just as it did in Minkowski’s space-time.
But, on the other hand, owing to the discrepancy between and , the laws of electromagnetics would cease to be covariant to a change of Galilean frame, hence to a change of space-time mesh-system. And so we could not view these laws as expressing relationships between the absolute entities of the fundamental space-time continuum.
The situation changes when we find, as a result of experiment, that the laws of electromagnetics remain covariant to a change of Galilean frame. This discovery results, of course, from the negative experiments in electromagnetics and from others of a similar nature. The first consequence of this discovery is to prove that , the invariant velocity of space-time, and , Maxwell’s constant, must be one and the same, since only on this condition will the laws of electromagnetics remain covariant.
Einstein then extended this property of covariance to all physical laws. We thus obtain the special principle of relativity, according to which the laws of nature remain covariant when we refer them to one Galilean frame or another. In short, the special theory deals solely [Pg 277] with Galilean frames.
From the standpoint of mesh-systems, let us recall that Galilean frames are given by Cartesian mesh-systems in space-time, the differences in their orientations representing the relative velocities existing between the frames. When we pass from measurements performed in one of these Cartesian mesh-systems to those executed in some other one of these frames, we must apply the transformations corresponding to a rotation in space-time; and these are the Lorentz-Einstein transformations. We may say, therefore, that in the special theory the principle of covariance entails the covariance of all natural laws when submitted to the Lorentz-Einstein transformations.
Now the fact that the laws of nature maintain the same form regardless of the orientation of our Cartesian space-time mesh-system, or frame of reference, suggests that they must be the expressions of relationships between such entities as vectors and tensors in space-time. If this be the case, the laws should remain covariant not merely to a change of Cartesian mesh-system, but also to a change to any species of mesh-system, whether Cartesian or curvilinear. In other words, they should be expressible as vector or tensor equations in flat space-time.
This discovery permits us to advance a step farther. For let us suppose that this erstwhile flat continuum manifests curvature in certain regions. Einstein assumed that in this case its curvature would curve along with it the entities embedded in its substance. Under the circumstances, the covariance of the laws of nature should continue to hold regardless of the flatness or curvature of space-time, hence also of the species of mesh-system we might appeal to. It is this generalisation of Einstein’s which constitutes the principle of general covariance, or the general principle of relativity. We see that it represents an extension of the special principle of covariance, or of relativity, extending as it does to all regions of space-time, whether flat or curved, and to all mesh-systems, whether Cartesian or curvilinear, the covariance established originally only in a special case. Henceforth it becomes permissible to say that all the laws of nature express relationships between entities of space-time (vectors and tensors); and as a result we are able to extend to space-time the conclusions we arrived at when discussing space in the previous chapter. After this preliminary discussion, we may investigate the objectivity of our physical magnitudes.
In classical science these magnitudes were required to be invariants, vectors or tensors in space. In the light of the new discoveries, we must now substitute space-time for space. The result is that both gravitational force and potential energy turn out to be deprived of all objective significance, for by merely changing our mesh-system in space-time we can annihilate or bring into existence the components of these magnitudes at a point. Potential energy entering as it does into the expression of the law of conservation, it became apparent that this law of classical science was obviously of an artificial nature. We shall discuss the principles of conservation in greater detail in a later chapter.
Let us now examine the subject of the general laws of nature. In [Pg 278] classical science these laws had to remain covariant to a change in the shape and orientation of our mesh-system in three-dimensional space. The new requirement is that these laws shall remain covariant to a change of shape and orientation of our mesh-system in four-dimensional space-time. This new condition will obviously exert an additional restrictive influence on the formulation of our laws, for it comprises as a special case all the classical restrictions as to covariance in space, supplementing them with the added restriction of covariance in space-time. When, therefore, we consider the laws of classical science, we shall be able to accept them without modification only provided they happen to satisfy the restrictive condition imposed by the space-time theory.
It must be kept in mind, however, that the space-time tensor test cannot aspire to tell us with certainty what the laws of the real world will be. Its action is merely one of elimination; it tells us what these laws can never be. When it comes to deciding which one of the many possible laws is to be accepted as the true law, experiment must be our guide. We can also foresee that the classical laws, though they may appear impossible in the light of the new test, must nevertheless be approximately correct since classical science had considered them to be borne out by experiment. We must not be surprised to find, therefore, that from a practical point of view very slight differences exist between the new laws and the old ones when crude experiments at least are considered. But from a theoretical point of view the difference is of momentous importance. The new laws pass the test; some of the classical ones do not.
It was soon noted that the classical equations of electrodynamics satisfied the new theoretical requirements, for they could be expressed as tensor laws in space-time. Inasmuch as they were in perfect harmony with experiment, they were left undisturbed. On the other hand, the classical equations of mechanics and, with them, Newton’s law of gravitation were certainly incorrect, for they did not remain covariant to a change in our space-time mesh-system.
Accordingly, Einstein had no difficulty in writing out the correct laws for mechanics by modifying ever so slightly the classical ones so as to make them conform to the restrictive space-time condition. We have stated elsewhere that in this way mass was proved to be a relative, no longer an invariant, its value varying according to our space-time mesh-system. We have also mentioned that the most exacting experimental tests have proved the correctness of the new laws and the fallacy of the classical ones.
In the preceding chapter we stated that classical science never paid much heed to the tensor notation, because it so happened that three-dimensional space was assumed to be permanently Euclidean and flat, and hence a standard mesh-system of the Cartesian variety was applicable everywhere throughout space. In the special theory, space-time is also considered flat, so here again the tensor symbolism can be dispensed with. We have seen, indeed, that the special theory was concerned only with Galilean observers in space far from matter, [Pg 279] and we know that observers of this type always split up space-time into space and time through the medium of Cartesian mesh-systems. These mesh-systems differ from one another merely by their various orientations in space-time; and we pass from one mesh-system to another by means of the Lorentz-Einstein transformations.
On the other hand, when we come to consider accelerated observers, we are dealing with observers who split up space-time with curved mesh-systems. Again, when masses of attracting matter are present, space-time assumes an intrinsic curvature which renders the utilisation of Cartesian mesh-systems impossible. Certain particular observers, such as those who are falling freely in a gravitational field, adopt mesh-systems which are approximately Cartesian in the immediate neighbourhood of the observer; but were we to prolong these mesh-systems sufficiently, we should soon discover that they became curvilinear, hence varied in shape from place to place, owing to the intrinsic curvature of space-time. In much the same way, a network of parallels and meridians on the surface of a sphere can be assimilated to a Cartesian one of equal squares only in the neighbourhood of the equator, since the meshes taper together as we near the poles. Thus, in a general way, we see why it is that when we leave the special theory, where space-time is flat and where the observer, being Galilean, abides by Cartesian mesh-systems, and when we investigate the general theory, where the observer is accelerated or space-time is curved, the Lorentz-Einstein transformations must give place to a more general type. Under these circumstances the tensor calculus becomes a mathematical instrument of great power.
While it is true that in the early days of the special theory Einstein never made use of this form of calculus, with which he was probably unacquainted, most modern treatises on relativity appeal to it from the very start, since it brings out the beautiful unity of the entire theory and shows that the special theory is merely a particular limiting case of the general one. This allows us to treat both forms of the theory simultaneously. We lay stress on this point, because some non-mathematical critics are of the mistaken opinion that the tensor calculus applies only to the general theory. From this belief they derive the equally erroneous opinion that the two theories constitute separate creations, having little in common. We have seen that such is not the case.
In this presentation of Einstein’s theory, we have emphasised the absolute characteristics of space-time, showing how the covariance of the natural laws followed as a necessary consequence as soon as we regarded them as expressing relationships between space-time entities (tensors, vectors) transcending the mesh-system of the observer. Viewed in this way, the theory has sometimes been referred to as “the quest of the absolute.” But we must remember that it is only as a result of the principle of relativity, necessitated by the negative experiments, among others, that space-time and the laws of nature can be viewed in this light. Were it not for the principle, hence for the identity of , invariant velocity, and , Maxwell’s constant, [Pg 280] though there might still be an absolute four-dimensional world, the physical laws such as we know them could no longer be conceived of as expressing relations between its tensors and vectors. We might conceive of space-time entities, in this new type of space-time, as entering into laws that would remain covariant to any change of mesh-system, but these entities and laws would not be represented by the physical entities and laws that experiment has revealed. From the standpoint of the physicist, they would be mere fictions, foreign to the world of our experience. It would be as though the entities and laws revealed to us by experience differed from the laws that would normally be expected had the world a unity of structure and of content. In this event the great beauty of Einstein’s space-time theory would be lost, and with it the wonderful unity and harmony it has shown to exist in nature. Thus, although the absolute viewpoint is the one we have stressed in this chapter, we should bear in mind that it was the disclosures of relativity that rendered this absolute viewpoint possible.
Furthermore we may very easily confer a relativistic aspect on the theory if we recall that by selecting curvilinear or Gaussian mesh-systems in space-time, we are merely placing ourselves in the position of an accelerated observer, who refers his measurements to some frame of reference, squirming like an octopus and who times events with clocks running wild. So far as can be gathered, Einstein was guided originally by a desire to prove that all motion was relative; it was thus the relativistic view that preceded the absolute viewpoint in the elaboration of the theory. There is no cause to be surprised at this precedence, seeing that we are referring to days when space-time was as yet unknown. But when space-time was discovered by Minkowski, the theory received an added impetus (though we can realise full well that even before this discovery a theory which aspired to prove that certain permanencies endured regardless of the observer’s motion would obviously lead us to suspect the underlying presence of some absolute world in which these permanencies would be situated). This absolute world could not be one of separate space and time; for in the absolute space of classical science the laws changed in form when our motion was modified. Obviously something more was needed, and space-time was the answer.
It is imperative, however, not to misinterpret the significance of the covariance of natural laws. The mere fact that all observers, whatever their motion, will discover the same laws, does not mean that all observers are alike in every respect. Even in Einstein’s theory, a rotating observer will realise full well, owing to centrifugal forces, that he does not stand in a Galilean frame. The complete relativity of all motion from every point of view is not by any means embodied in the covariance of the laws; and we shall see that this complete relativity of motion, leading to Mach’s mechanics, is far from being accepted by all scientists. The discussion of this subject will have to be left aside for the present, as it is intimately connected with the difficult problem of the shape of the universe, concerning which we shall have something to say in a later chapter.
[Pg 281]
CHAPTER XXVIII THE DISCOVERY OF THE EINSTEINIAN LAW OF GRAVITATION
IN the two preceding chapters we have discussed in a very general way the subject of tensors and of the laws of nature. It now remains to be seen how Einstein, guided by the requirements that accompanied all general laws, succeeded in formulating with mathematical precision the true law of gravitation.
In order to understand what is to follow we must revert to Newton’s law. This law defines the intensity and distribution of the field of gravitational force in the empty space which surrounds matter. Also, it enables us in turn to ascertain how the field of force would be distributed in the interior of matter, in the interior of the earth, for instance. Thus, in its most general aspect, Newton’s law of gravitation defines the intensity and distribution of the gravitational field both inside and outside of matter.
Now we have seen that the presence of a field of force connotes the existence of an uneven distribution of potentials in the space where the field exists,[92] since any perfectly even and constant distribution of potentials would correspond to the complete vanishing of the field. Hence, Newton’s law of gravitation can also be written as a law of potential distribution inside and outside of matter. Expressed in this form, it is known as Poisson’s equation and is written In this celebrated equation, represents the potential at each point; a complicated mathematical expression linking the potential at a point and its value at all neighbouring points; expresses the density at each point of the matter producing the field, and is a constant of proportionality known as the constant of gravitation. And since the equation gives us the law of potential distribution at each point of space where the density of matter is , in regions outside matter we obviously have [Pg 282] since outside matter , is of course zero. Expressed in this latter form (), in which it gives us the potential distribution outside matter, the Newtonian law of gravitation is known as Laplace’s equation.
Thus, we may say that Newton’s law of gravitation inside and outside of matter is given by , in which is zero or has some definite non-vanishing value according to whether we are considering a region outside matter or in its interior. Now, this potential distribution in Newton’s law refers to the distribution through space of one same potential, an invariant in space, known as the Newtonian potential. But we know that if Einstein’s ideas are correct, if gravitation is to be attributed to space-time curvature, Newton’s single potential (an invariant) will have to be replaced at every point by the ten separate ’s of space-time. These, as we know, are no longer invariants, they are second-order tensors;[93] and Newton’s invariant potential turns out to be none other than one of the ’s, namely, the time one, .[94]
According to Einstein’s views, therefore, the expression of the gravitational potential distribution in the interior of matter will be given by an equation between some complicated expression connecting the ten ’s at every point and at neighbouring points, and some quantity representing the characteristics of matter at every point. This equation will then constitute Einstein’s law of gravitation in the interior of matter, and will represent the Einsteinian substitute for Poisson’s classical equation.
Now Poisson’s equation was an equation between two invariants in space. Einstein’s law, on the other hand, in accordance with the requirements that must be satisfied by all general laws in the space-time theory, will have to be an equation between invariants or tensors in space-time. By analogy with Poisson’s equation we see that the left-hand side of Einstein’s law will be a space-time invariant or tensor which will represent the potential distribution, or -distribution, from point to point, while the second side will be given by an invariant or a tensor representing matter.
[Pg 283]
A tensor representative of matter and describing it fully had already been discovered by Einstein. It was a second-order symmetrical tensor generally written and its ten separate components, as referred to any definite space-time mesh-system, defined such physical magnitudes as the density of matter at a point, its internal stresses, vis viva, and momentum in that mesh-system. Matter, as a source of gravitation, was obviously described much more completely by this tensor than was the case in classical science, where the mere mention of its density of mass was considered sufficient to determine its gravitational field. Thus, Einstein was in possession of the right-hand side of the equation of gravitation in the interior matter, that corresponding to in Poisson’s equation (neglecting constants).
It remained to discover the left-hand side of the equation. Owing to the condition of covariance, this left-hand side had to be represented by a tensor of the same order and nature as the right-hand side, hence, in the present case, by a symmetrical tensor of the second order. Further, it would have to be built up of the ten ’s or potentials, expressing a relationship between these ten ’s at every point and at the neighbouring points. It is here that Riemann’s discoveries fit in with Einstein’s requirements; for Riemann and his successors had discovered tensors built up from the fundamental structural tensors of a continuum, namely, from the ’s. All these tensors served to represent various possible types of structure, geometry or curvature.
In a previous chapter we mentioned these tensors. First, there was the Riemann-Christoffel tensor . It was a tensor of the fourth order, as is indicated by the four indices underlying the letter . Then there was a second-order symmetrical tensor . Lastly, there was an invariant of curvature , no longer a tensor in the ordinary sense of the word. By differentiating and combining these magnitudes, we could of course obtain others, but the ones we have mentioned are the simplest.
Einstein had still one further clue to guide him. Classical science had assumed that at an infinite distance from attracting matter the law of inertia would hold rigorously; that is to say, a body moving freely would follow a rectilinear course with constant speed when referred to a Galilean frame. Expressed in the language of space-time, this would imply that at an infinite distance from matter, space-time would become rigorously flat, as we had assumed it to be in the special theory. This condition necessitated the vanishing of the Riemann-Christoffel tensor , at infinity. If this condition is maintained, the curvature of space-time around matter will have to be of a type which is capable of dying down to perfect flatness at infinity, degenerating to .
Now the law of curvature was precisely of such a type; hence Einstein, in his original paper, found it natural to select the tensor for the left-side term of his gravitational equations. Accordingly he wrote:
Outside matter, this law would of course reduce to , since, outside matter, , representative of matter, would vanish. It is easy to see that these gravitational equations satisfy all the requirements we have mentioned. First, we are in the presence of covariant equations, since and are tensors of the same nature and order. Secondly, is built up from the ’s or potentials, as it should be. Thirdly, outside matter, our equations reduce to , and the curvature connoted by this relation is compatible with the existence of perfect flatness at infinity, i.e., with at infinity.
Nevertheless, Einstein soon recognised that his equations could not correspond to reality. The fact was that matter appeared to be conserved in nature. By this we mean that matter disappears from here only to reappear there.[95] It followed that the tensor of curvature [Pg 284] which would be identified with the presence of matter in the law of gravitation should likewise possess the attributes of conservation. Now the tensor possessed no such attributes; but a very similar tensor, also symmetrical and of the second order, namely, fulfilled all the previously mentioned requirements. Henceforth, the law of potential distribution, or of gravitation inside matter, would have to be (neglecting constant factors) (analogous to Poisson’s equation); and outside matter, as disappeared, there was left which reduces to (analogous to Laplace’s equation). From this we see that Einstein’s initial error affected the law of space-time curvature only in the interior of matter. Outside matter, we had, as before, . [96]
Let us recall that in all examples where tensor equations are discussed, the indices and (in the present case) are to be replaced by all manner of arrangements of the integers from 1 to 4 (since space-time has four dimensions). An equation such as or as represents therefore ten separate equations. As a matter of fact, sixteen equations would be obtained were we to substitute all possible arrangements; but as the tensors are symmetrical, only ten out of sixteen art independent, hence need be retained; the other six are mere repetitions We see, then, that in Einstein’s theory ten separate equations are necessary in order to define the distribution of the gravitational field in the interior and around matter, whereas in classical science one equation, i.e. Poisson’s equation (), was sufficient.
Here we may mention that if we abandon the restriction that the law of inertia holds rigorously at infinity, hence if we abandon the condition of perfect flatness at infinity, we may build up other [Pg 285] tensors satisfying all the remaining requirements. This can be done by adding a further tensor (where represents an arbitrary constant) to the left-hand side of our equation. The presence of this additional term will imply that even for very distant regions, space-time would still manifest a trace of residual spherical curvature so that the universe would close round on itself and be finite. In the first development of his theory, Einstein rejected this additional term, but the fact remains that so far as the mathematical requirements of the theory of relativity are concerned there is no reason to exclude it ab initio. At all events, we can assert that if the term exists at all, must be exceedingly minute; for even though finite, the universe is certainly of gigantic proportions, and its curvature therefore exceedingly slight. For the present we shall ignore the term since in any case its influence would be too insignificant to affect perceptibly such phenomena as those of planetary motions. Only when we come to consider the shape of the universe as a whole and its bearing on the Mach-Einstein principle of the relativity of all motion will the presence of assume a vast philosophical importance.
Now the methods whereby Einstein was led to the discovery of his law of gravitation might appear rather piecemeal, much like the fitting together of a jig-saw puzzle in which the structural tensors constituted the pieces. But this, in itself, can scarcely be regarded as a drawback to the theory. It is rather the reverse, since it proves that just as the pieces of a puzzle can only be fitted together in certain ways, so now the universe can scarcely be otherwise than it is. In this respect, there is a marked contrast between the present situation and that which endured in classical science, where practically any law of gravitation might have been expected.
Nevertheless, from the standpoint of mathematical elegance, something more was needed. When, therefore, Einstein had cleared the ground and his ideas were more fully understood, the greatest of the pure mathematicians, such as Klein, Hilbert, Weyl and subsequently Einstein himself, undertook to discover the law of gravitation by more general methods. They found that if we accepted the existence of space-time and if we agreed that the gravitational field around matter was due to a curvature of space-time, the Principle of Stationary Action showed that a large number of laws of gravitation could be entertained. If, however, it was specified that by analogy with Newton’s law, the law of curvature should contain no derivatives of the ’s or potentials to an order higher than the second, Einstein’s law of gravitation with or without the term was the only one possible. Furthermore, all these possible laws of gravitation were found to verify the mathematical condition expressing the conservation of matter; hence, by proceeding in this way through the principle of action, it was possible to obviate the rather artificial condition of conservation which would otherwise have had to be introduced as an additional hypothesis.
[Pg 286]
CHAPTER XXIX THE GENERAL LAWS OF MOTION
IN classical science, when we wished to describe a body’s motion, it was of course insufficient to describe its spatial course in our system of reference. It was necessary, also, to define its motion along this course. As soon, however, as we reason in terms of space-time instead of in terms of separate space and time, all the particularities of motion are fixed by the single space-time world track of the body. This opens up the possibility of describing the laws of motion of free bodies in a purely geometrical manner in four-dimensional space-time, so that kinematics becomes a four-dimensional statics. Let us now pass to a more precise determination of the problem.
According to Newtonian mechanics, free bodies in interstellar space far from matter, moving under no constraint, always described rectilinear courses with constant speeds. This law was known as Newton’s law of inertia. We remember, of course, that a law of motion of this type remained completely indeterminate until we had defined the frame of reference to which this motion would be referred, and we saw that any Galilean frame answered our requirements. Now it was assumed that Newton’s law of inertia was universally valid in the case of free bodies moving under no constraint; when, therefore, the law was found to be at fault in the neighbourhood of large masses of matter, the discrepancies were ascribed to a real force of gravitation emanating from matter and compelling the erstwhile free bodies to move under constraint. In other words, in a field of gravitation, bodies were no longer free; so the validity of the law of inertia, applying to perfectly free bodies, was not overthrown thereby. The motions actually followed by bodies in a gravitational field would thus result from a compromise between the courses the bodies would have followed had no gravitational field been present, and the perturbating influence due to the field. These laws of motion were expressed by Kepler’s three laws of planetary motion, according to which planets described ellipses round the sun (Kepler’s first law), with variable velocities determined by the law of areas (Kepler’s second law) and by the dimensions of the ellipses described (Kepler’s third law).
As in the case of the law of inertia, Kepler’s laws of planetary motion held only for a Galilean or inertial frame and, in particular, for the Galilean frame attached to the sun’s centre. Thus there existed a complete duality between the two types of law and motion, those for free bodies and those for bodies moving under gravitational constraint.
[Pg 287]
In Einstein’s space-time theory this duality is removed. There is now but one law of motion, namely: All free bodies pursue geodesics, or the straightest of paths, through space-time, regardless of whether we are considering regions in the neighbourhood of matter or remote regions in interstellar space. This law is in full accord with the Principle of Least Action, and may be regarded as a direct consequence of this fundamental principle.[97]
True, owing to the curvature of space-time around matter and its flatness in distant regions, the actual courses and motions will vary according to the more or less great proximity of matter; and we might be tempted to say that we had succeeded in removing a duality in the laws of motion only by introducing a new duality referring to the structure of space-time—curved around matter and flat for distant regions. But this line of argument would be incorrect. There is but one law of space-time curvature or gravitation; for this law, which ascribes a definite curvature to space-time in the presence of matter, tells us also how this curvature dies away to perfect flatness at infinity. The existence of flat space-time far from matter is thus comprised in the one statement of the law of space-time curvature or gravitation. Thus, a precise determination of the motions of bodies, whether near matter or far from matter, is given by the one law of space-time curvature together with the law of least action, which states that the space-time courses or world lines of the bodies will be geodesics.
Let us now examine another aspect of the question. We have seen that according to Einstein’s ideas it was no longer necessary to differentiate between free bodies (those moving in interstellar space) and bodies moving under constraint (those moving in the vicinity of attracting matter). Henceforth, a planet moving round the sun is just as much a free body as one moving in interstellar space. If it moves in a different way, in these two cases, it is simply because the structure of space-time is curved around matter, instead of being flat as in interstellar space. There is, therefore, no justification for invoking a physical pull or force of gravitation in the proximity of matter; this so-called physical pull is inexistent so long as the planet moves without any additional interference. In all cases the planets and bodies in interstellar space are guided through space-time by the intrinsic structure of the space-time wherein they are situated.
Now structure implies geometry, and space-time, like any other mathematical extension, is amorphous per se; hence this [Pg 288] structure of space-time must be attributed to a foreign cause, to the presence of the four-dimensional metrical field of space-time similar to a magnetic field. It is this four-dimensional field which acts as a guiding influence on bodies, constraining them along the geodesics of space-time. When, therefore, the observer partitions space-time into his space and time mesh-system, the effect of the guiding field is to direct bodies along certain paths in space and to determine their motions along these paths. Were there no guiding field, or metrical field, were space-time completely amorphous, there would be no space and time partitions, motion would be meaningless and inconceivable; even were it conceivable, the bodies would not know how to move.
It would appear, then, to be more correct to state that there can exist no such thing as freely moving bodies, since all bodies are subjected to the influence of the guiding field. We can, however, retain the usual terminology by remembering that by a freely moving body we mean one submitted to the action of the guiding field and to nothing else. The important point to understand is that as regards freedom of motion, all bodies in empty space, whether situated in a gravitational field or far from matter, are on the same footing. In all cases their actions are controlled by the guiding field, and by the guiding field alone. The sole difference is that in the neighbourhood of matter, the lay of the guiding field is curved slightly. It is incorrect, therefore, to invoke a real physical pull acting on the body when it moves near matter, and to deny the existence of such a pull when the body is moving in interstellar space.
And yet it might be said: “We experience a real physical pull when we stand on the earth, whereas we should experience no such pull were we floating in interstellar space.” Let us see why it is that this real physical pull appears to arise in one case and not in the other.
A freely moving body follows, as we have explained, a geodesic of space-time, regardless of whether space-time be curved by the proximity of matter or whether it be flat. So long as the body is not interfered with, it will continue to follow its geodesic, allowing itself to be guided by the space-time structure or field. But suppose now that the motion is interfered with, so that the body is torn away from the geodesic which it would normally have followed. The guiding field will immediately react and oppose this foreign interference. It is as though the geodesic were a groove from which the body could not easily be diverted.
Consider then a body moving uniformly in interstellar space, according to the law of inertia. If we interfere with this body’s motion, accelerate it along its straight line or compel it to describe a curve, we shall, in effect, be tearing it away from its space-time geodesic. The guiding field will react and we shall experience a physical pull which we call a force of inertia or a centrifugal pull. In exactly the same way we may explain the appearance of the physical pull which we ascribe to gravitation when we stand on the earth’s surface.
Thus, a body which is falling freely towards the earth is following a [Pg 289] geodesic in the curved space-time that surrounds the earth. But if, for some reason or other, the free fall of the body is interfered with, if, for instance, the rigidity of the earth’s crust prevents the body from falling farther along the geodesic, the geodesic will react and will strive to force the body through the earth’s crust. It was this force which we formerly ascribed to the pull of gravitation. But we see once again that this force has exactly the same origin as the forces of inertia we discussed previously, so that there exists no essential difference between a force of inertia and one of gravitation. They are both real physical forces in that they will produce physical effects; and they will both disappear in exactly the same way when we cease to interfere with the natural motion of the body. In short, they are both of them but manifestations of the four-dimensional metrical field or guiding field of space-time.
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CHAPTER XXX THE VERIFICATION OF EINSTEIN’S LAW
EINSTEIN’S law, as we have expressed it, is one of space-time curvature around matter and in the interior of matter. It is also the law of distribution of the ten potentials or ’s throughout space-time, or, again, the law of distribution of the four-dimensional metrical field or guiding field of space-time. But in order to study the implications of the law we may discuss it from yet another angle.
Being a law of space-time curvature or structure, we are able to derive from it the precise lay of the geodesics of space-time around matter. These geodesics, or paths of least resistance, are, we remember, the paths which planets, light rays and all free bodies will pursue. A knowledge of the lay of the geodesics around the sun will therefore allow us to determine the precise courses in space and the precise motions along these courses of all free bodies moving in a gravitational field. When these geodesics are submitted to mathematical analysis, it is found that the courses and motions of the planets should be very nearly identical with those required by Newton’s law.
There would be slight discrepancies, however, between the requirements of the two laws. In particular, Einstein’s law of gravitation would predict that the courses of the planets should be very nearly elliptical, but not exactly so; there should be a slight precession of the perihelion of the planet, increasing in magnitude with the velocity of the planet on its orbit. Now the most rapidly moving planet of the solar system is Mercury; and Leverrier had observed nearly a century ago that Mercury’s perihelion displayed a precessional motion which Newton’s law was unable to explain. Einstein’s law accounts for this precessional motion, not only qualitatively, but quantitatively as well. Hence, a first verification of Einstein’s law was obtained.
Here it must be noticed that, in the preceding example, Mercury’s curious motion was already known to astronomers. But we have said enough of Einstein’s methodology in his discovery of the law of space-time curvature to realise that he did not adjust his law of space-time so as to account for Mercury’s hitherto unexplained motion. We saw, indeed, that the law of gravitation, as formulated by him, was practically inevitable if the principle of action, the most fundamental pillar of science, was to endure. However, in the following verification, we shall see that Einstein’s law was able to foretell the existence of a phenomenon totally unsuspected by classical science. We refer to the deviation of a light ray in a gravitational field.
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Classical science had always assumed that a gravitational field, such as that of the sun, would attract material bodies, but that it would be without effect on light rays, which were considered to be immune from gravitational attractions. Had classical science been thoroughly convinced of the identity of the two types of mass, it would have realised that light waves must be possessed of gravitational mass, since they were known to manifest the characteristics of inertial mass. This inertial property of light rays had been foreseen theoretically by Maxwell and proved experimentally by Lebedew, who detected the pressure exerted by rays of light on a body on which they impinge. It is a pressure of this type which is supposed to be responsible for the apparent repulsion of comets’ tails away from the sun. It would have been, therefore, only a step to assume that light rays must manifest gravitational mass, and hence must be attracted by the sun.
Classical science, however, had never felt justified in taking this step. As a result, it was thought that light rays would proceed in straight lines through a gravitational field just as they would through free space far from matter. When, therefore, the first eclipse observations prompted by Einstein’s predictions revealed an undeniable bend in the course of a ray of light emitted by a star and grazing the sun’s limb, classical scientists did not consider it necessary to abandon Newton’s law on this account. They merely recognised that light was attracted by a gravitational field in common with all other forms of matter, and calculation showed that assuming Newton’s law to be correct, the angle of this deflection would be 0".87 for a ray grazing the sun’s limb. The first eclipse observations were conducted under very unfavourable conditions, and all that could be ascertained was that a certain amount of bending was present. The precise degree could not be determined in any reliable way, so that Newton’s law still had its defenders.
Let us now pass to Einstein’s attack. Of course, in Einstein’s theory, even prior to his discovery of the new law of gravitation, a ray of light would certainly have been bent in a gravitational field in virtue of the postulate of equivalence. It is easy to see how this would arise. Consider a chest, floating vertically in interstellar space, and a ray of light traversing it horizontally. If now the chest be accelerated upwards along the vertical, the floor of the chest will move with acceleration across the ray and, as a result, the light ray will follow a curved path through the chest. But in this case the postulate of equivalence allows us to regard the chest as unaccelerated but permeated by a gravitational field; we may infer, therefore, that the ray of light would bend downwards under the action of this field. Hence, the mere qualitative bending of a ray of light in the sun’s gravitational field would not vindicate Einstein’s law of gravitation and dethrone Newton’s; it would merely prove the correctness of the postulate of equivalence and the identity of the two species of masses. It is the precise quantitative bending of the ray, as determined by Einstein’s law of gravitation, that constitutes the crucial test of the correctness of Einstein’s law. This bending, as computed from [Pg 292] the lay of the geodesic which a light ray grazing the limb of the sun would follow, turns out to be 1".75; that is, just twice as great as the Newtonian deflection 0".87. When this precise angle predicted by Einstein was observed during the subsequent solar eclipse of 1919, interest in relativity became widespread; for it was recognised at last that the theory was not a mere mathematical dream.
But even this verification of Einstein’s law of gravitation would have been insufficient to establish the theory. As we shall see in the next chapter, other very curious phenomena were suggested by the new law of gravitation; and if these new phenomena should not be verified in turn, the theory, in spite of its wonderful achievements, would have to be abandoned.
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CHAPTER XXXI THE SEPARATION OF SPACE-TIME INTO SPACE AND TIME IN A GRAVITATIONAL FIELD
WE must remember that the one fundamental continuum of the world is space-time; space and time considered separately, varying as they do with the observer’s motion, have no absolute significance. The law of curvature, or of gravitation, which we have discussed is one of space-time curvature. It is the law giving the distribution of the four-dimensional metrical field of space-time in the interior of matter and around matter to infinity. It is true that we may be able to effect some sort of separation of space-time into the space and time of some particular observer; and this may enable us to represent the law of curvature of space-time as two separate laws of curvature, one affecting space and the other affecting time. But for the reasons we have just mentioned, no absolute significance could be attached to the curvatures we should obtain. Thus, in interstellar space, an observer situated in a Galilean frame would contend that neither space nor time manifested any trace of curvature, whereas an observer on the rim of a rotating disk would recognise that both space and time were curved. On the other hand, regardless of his frame of reference, either observer would always recognise that space-time possessed no trace of curvature.
It is because space-time transcends the observer that its curvature or flatness, as the case may be, can be considered absolute; whereas a curvature of space or of time alone can never represent anything but a mere relationship between the intrinsic condition of space-time and the observer’s motion or frame of reference.
Of course, if we wish to split up space-time into the particular space and time directions of some particular observer we may always attempt to do so. But in the majority of cases this procedure will be impossible.
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If these considerations are properly understood, there is no harm done in trying to separate the curved space-time which endures around the sun into the space and time of an observer at rest with respect to the sun and viewing the solar system as a whole. This separation, however, is possible only in the event that the masses of the moving planets can be ignored, so that the only field we have to consider is that of the sun, remaining unchanged as time passes. It is owing to this permanent condition of the gravitational field that the separation into space and time is possible. Fields of this type are called stationary. For non-stationary fields, such as those where the distribution of the field changes with time, the separation into space and time becomes impracticable.
We will assume, therefore, that the gravitational field of the sun is stationary; that is, that the masses and motions of the planets do not modify the field to any appreciable extent. We may then proceed to split up curved space-time into the space and time of an observer at rest with respect to the sun and non-rotating with respect to the stars. By so doing we shall be splitting up into two separate fields the metrical field of space-time, that four-dimensional field which pertains neither to space nor to time. In this way we shall obtain, first, a purely spatial metrical field, which will be responsible for the measurements of our rods when these are maintained at rest in our frame; and, secondly, a purely temporal metrical field, which will determine the behaviour of our clocks or vibrating atoms while these are at rest in various regions of the space around the sun.
When Levi-Civita and Schwarzschild had performed this separation, it was found that the curvature of the observer’s space along a plane section through the centre of the sun would be identical with that of the surface of a paraboloid of revolution disposed around the sun; and that the observer’s time direction would also be curved in a definite way. Both curvatures would die down gradually for regions of space farther and farther removed, to infinity. We must not forget, however, that this separation and the curvatures we have mentioned hold solely for the observer at rest with respect to the sun when surveying the solar system as a whole.
The advantage of this separation of space-time into space and time is that it allows us to express the purely formal results deduced from Einstein’s space-time gravitational equations in terms of our familiar notions of space and time. It thus becomes possible to describe the world lines of the planets under Einstein’s law by representing them as spatial orbits and motions along these orbits; hence we are enabled to check up Einstein’s previsions with astronomical observation. Furthermore, this splitting up of space-time enables us to study separately those effects which must be attributed to the respective curvatures of space and of time, considered by themselves and independently of one another.
We will first consider the curvature of time alone. Let us assume that for some reason or other the curvature of space were to disappear, so that all we should be concerned with would be a more or less curved time direction. In this case, it is found that the motions of the planets would be in perfect accord with Newton’s law. The precessional advance of Mercury’s perihelion would be non-existent, and the bending of a ray of light grazing the sun’s limb would be 0".87, as required by Newton’s law, and no longer twice that amount, as required by Einstein’s law. From this we see that it is the additional curvature of space around the sun (and around matter generally) that is the distinguishing feature of Einstein’s law of gravitation.
Here it might be asked: “Since a peculiar type of space-time curvature (curved time and flat space) would lead us to Newton’s law, why should Einstein have rejected the classical law a priori and urged his own more complicated type of space-time curvature in its place? [Pg 295] It could not have been for the purpose of accounting for the double bending of a ray of light, since this effect was unknown at the time. Was it for the sole purpose of explaining the motion of the planet Mercury?”
The answer must be decidedly in the negative. When Einstein was led to his law of space-time curvature , he never could have recognised in advance that this law would account for the motion of Mercury. He may have suspected the fact, convinced as he was that he was on the right track; but in order to establish the nature of the planetary motions when governed by this law, it was necessary to submit the law to a mathematical integration. It was only after the integration had been performed that the motion of Mercury, as required by the law, was found to be in agreement with previous astronomical observations. Hence we must discard all ideas of a law patched up for the sole purpose of harmonising with the planetary motions.
In order to understand Einstein’s procedure more fully, we must recall that the general principle of covariance requires that all natural laws be expressed in the form of space-time tensor equations. This criterion alone, however, does not permit us to decide a priori on any particular law of space-time curvature. But Einstein, as we remember, introduces a second restriction, according to which, in the empty space surrounding matter, none but the structural tensors of space-time, the ’s, should enter into the law of curvature.[98] This second restriction is a natural one to set, for were foreign tensors to be introduced at each point of space-time, the space could no longer be empty.
When these two restrictions are taken into consideration, we find that the law is practically the only one that can be constructed; at all events, it is the simplest. Theoretically, could also be countenanced; but this law, indicating perfect flatness, would connote the absence of gravitation. Then again, we might suggest , and this would lead to de Sitter’s finite universe, a possibility which has not yet been discarded.
But in any case, with the restrictions imposed, Newton’s law is barred, so that unless we wish to introduce foreign tensors, acting in the capacity of hypotheses ad hoc, it is hard to see how Einstein’s law can be escaped.[99] If we convince ourselves on this score, we can see that it is quite impossible to think that we may adjust the curvatures of space and of time to suit ourselves. These curvatures are determined in advance by the law of space-time curvature, itself imposed by considerations of a rational order.
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Let us revert to our separation of curved space-time into curved space and curved time. We have seen that if the curvature of space were non-existent, the curvature of time alone remaining, the motions of planets and free bodies would be in accord with Newton’s law. We may understand the reason for this by following a different line of thought.
If space is flat, the ’s which refer to space remain constant (in a Cartesian mesh-system at least).[100] The only , or potential, that can vary is therefore the time one . This last potential is therefore the only one that can be responsible for a field of force. It follows that by considering space to be flat, we are preserving only one of the ten potentials of space-time; and we find ourselves in exactly the same position as under Newton’s law, where only one potential was present, i.e., the Newtonian potential. This allows us to identify the time-potential with the Newtonian potential , and the proof of the statement we made in a previous chapter is given thereby.[101]
It now remains to be seen how this curvature of time may be detected directly. Here we must recall that the curvature of time is expressive of the repartition of the temporal metrical field obtained by splitting up the metrical field of space-time into one of space and one of time. The repartition of this temporal metrical field in the space and time separation we are discussing shows that its curvature will increase as we consider regions of space nearer and nearer the sun, indicating thereby a slowing down of time as we approach the sun. Now it is this temporal metrical field which controls the rate of evolution of any isolated periodic phenomenon from place to place, hence which defines practical time-congruence as given by perfect clocks. In the present case, therefore, the action of this field would be to slow down the temporal evolution of all phenomena when these have their spatial locus nearer and nearer the sun. This effect was anticipated by Einstein and is known as the Einstein shift-effect.
Of course, if we, as conscious beings, were to be situated at any particular point of the sun’s gravitational field, we should never appreciate this slowing down of time and of the evolution of all things in our immediate neighbourhood; for our consciousness would participate in the same slowing down that was active around us. It would only be when we viewed distant phenomena occurring in other parts of the field of gravitation that discrepancies in their speeds of evolution would [Pg 297] become apparent. Thus, although we should not be directly conscious of aging at various rates in the different parts of a gravitational field, we should discover eventually, either by direct vision or by meeting again, that our friend who had lived in the proximity of the sun had aged less than ourselves, who had lived farther away from its mass. Gravitation would thus preserve youth, and, more generally, two men living on two different stars of unequal mass would not age at the same rate.
It is to be noted that it is not the force of gravitation itself which is responsible for this slowing down of time as we near the sun. It is rather the decrease in the value of the potential; and in the present case we have seen that the only potential which enters into account is , or the Newtonian potential.
All these phenomena predicted by Einstein were entirely new to science, and it was most important to verify their correctness before accepting Einstein’s theory. The method suggested was that of exploring a field of gravitation with vibrating atoms, these being assumed to behave like perfect clocks marking correct time. For instance, if Einstein’s law of space-time curvature was correct, atoms of sodium placed in different parts of the solar field would appear to be vibrating faster or slower than the atom in our immediate neighbourhood, according to whether they were farther removed from or nearer the sun than we happened to be situated. These variations in the rates of vibrations of the atoms would be betrayed physically by variations in the colour of the light they emitted or absorbed. It would follow that while the atom of sodium by our side would always emit a yellow light in our estimation, those atoms nearer the sun would appear redder, and those farther away from the sun would appear greener. In particular, an atom of sodium situated in the sun’s atmosphere would beat more slowly and appear to us redder than an identical atom examined in our laboratory (the earth’s gravitational field being insignificant compared with that of the sun).[102]
For a number of years it was questionable whether this effect really [Pg 298] existed, because the difficulty in detecting so minute a displacement of the spectral lines was very great. Quite recently, however, the phenomenon has been observed on a star of enormous density, the companion of Sirius.[103] This is perhaps the most beautiful of all the verifications of Einstein’s theory, since the effect contemplated was totally unsuspected by classical science—even more so than the bending of light in a gravitational field.
Now the postulate of equivalence, by assimilating gravitational fields to fields of inertia, requires that similar phenomena should occur in an accelerated enclosure (allowing, of course, for the slight discrepancies in the actual field-distribution when produced by matter or by acceleration). Thus, consider a train pursuing a Galilean motion, and two identical sodium atoms or clocks placed on the engine and rear car, respectively. So long as the motion of the train remains Galilean, an observer at rest in any point of the train would detect no difference in the rates of vibration of the atoms,[104] hence in the colours which they emitted. On the other hand, were the train to be accelerated, moving faster and faster, a field of inertial force directed from engine to rear car would now be present; and as a result the atom on the rear car would appear to the train observer to be vibrating more slowly than the atom on the engine.
We must realise, therefore, that whereas in the special theory a slowing down of time was always due to relative motion, in the present case, the slowing down of time takes place in spite of the absence of relative motion between the atoms and the train observer. In the same way, there was no relative motion in the example of the Einstein shift-effect in the sun’s gravitational field, and yet we had to accept a slowing down of time. So we see that acceleration, even when unaccompanied by relative motion, will produce a variation in the flow of time; and the same variations would also ensue in the case of distance.
Instead of considering an accelerated enclosure pursuing a rectilinear path, we could of course consider a rotating enclosure, and the argument would be the same. Atoms situated at greater distances from the centre of rotation would appear to vibrate more and more slowly; and a parcel of radium, for instance, placed on the rim of a rotating disk, would be slowed down in its rate of disruption, so that the effect of the inertial field would be to decrease the radioactivity of the atom.
Now that we have discussed the effects due to the curvature of time and have seen that this type of curvature accounts for all the major effects of gravitation, let us consider the additional effects due to the curvature of space. The presence of this curvature of space, so [Pg 299] far as the motions of bodies are concerned, will modify the effects due to the curvature of time. It will therefore be responsible for the small discrepancies that differentiate Einstein’s law from Newton’s. However, so long as the motions of bodies in the gravitational field are small compared with that of light, the curvature of space will not affect their motions to any appreciable extent.
Among the planets, Mercury alone is animated with a speed (with respect to the inertial frame of the sun) sufficient to render observable the influence due to the curvature of space; and it is this influence which is responsible for the slight precessional advance of Mercury’s perihelion. In the case of light waves, which move with a speed many times greater than Mercury, the influence of the curvature of space will be marked to a still greater degree. It will cause an increased bending of the ray of light equal to that already produced by the curvature of time alone, which, as we know, would correspond to the bending required by Newton’s law. In this way the double bending is accounted for.
In order to exhaust this discussion, we must mention that there exists still another case, in which the curvature of space would exert a perceptible influence, even in the case of slowly moving bodies. This would be in the event of a very intense field of gravitation, many times greater than that of the sun. In such a field, even slowly moving bodies would deviate perceptibly from the course required by Newton’s law of gravitation.
It may appear strange that this curvature of space should seem to be of such minor importance as contrasted with that of time, and produce observable effects only for bodies moving with great velocities with respect to the sun. The solution of this puzzle will be found, however, when we consider some of the peculiarities of the space-time continuum of relativity. In an ordinary continuum like space, where all the dimensions are of exactly the same nature, we have no difficulty in specifying some unit of length which will apply in exactly the same way to measurements computed along the dimensions of height, length and breadth. In space-time, however, the situation is somewhat modified, since the fourth dimension (time) represents something different from the other three. Thus, whereas our measurements along the spatial dimensions will be conducted with material rods, our time measurements will require time-rods, or clocks. The question is, then, how are we to co-ordinate our measurements with clocks and our measurements with rods; or, in other words, how shall we be able to establish any sort of comparison of magnitude between these measurements so different in nature?
The answer to this question is that the variable which represents measurements along the time direction is not t but ct. This means that one second in time corresponds in our formulæ to the distance which light covers during one second, namely, 186,000 miles. It follows, therefore, that if we measure time in seconds we must, in order to co-ordinate results, measure space in terms of unit rods 186,000 miles long.
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Obviously there exists a vast discrepancy in actual practice between the magnitudes of our temporal and spatial units; for the great majority of bodies with which we come in contact cover far less than a unit of space (186,000 miles) in a unit of time (one second). Only in the case of bodies approaching the velocity of light in our frame are the two progressions, that in space and that in time, in any way comparable. Thus, consider our own earth. It moves in relation to the sun with a speed of some eighteen miles a second, which is equivalent to approximately one ten-thousandth part of a unit of space per unit of time. This emphasises the fact that to all intents and purposes the motions of bodies with which we are concerned reduce to a motion in time, the displacement in space being insignificant in comparison.
Now when we come to investigate the respective effects of the curvatures of time and space, taken separately, in the gravitational field of the sun, we must remember that for a curvature to manifest itself we must survey relatively large areas on the curved surface (large as compared with the intensity of the curvature). Thus, on the curved surface of the earth, for example, if we content ourselves with surveying restricted areas, no effects of its curvature will be apparent. It is only when we conduct measurements over large areas, say over a few hundred miles, that the direction of the vertical will change perceptibly and that geodesical surveys will render the curvature manifest. It is much the same with the curvatures of space and of time in a gravitational field. Only when we consider planets moving through the sun’s inertial frame with a speed approaching that of light (hence covering large distances in unit time) will the space-curvature produce effects comparable to those produced by the curvature of time. We understand, therefore, how it comes that the curvature of time alone is able to produce the major effect of Newtonian attraction, such as weight, whereas the effects due to the curvature of space remain imperceptible until we consider the motion of Mercury, which is moving fairly fast with respect to the inertial frame of the sun. When we consider the propagation of light rays, the effects due to the curvature of space reassert themselves fully, and as a result the double bending of a ray of light can be attributed in equal proportion to the curvature of time and to the curvature of space.
In short, we now understand why it is that Einstein’s law of gravitation deviates appreciably in its effects from Newton’s only when we consider bodies moving with velocities approximating to that of light. Once again we realise that if this invariant velocity of the universe had happened to be much smaller, the deviations between Einstein’s law of gravitation and that of Newton would have been far easier to detect; and, conversely, had the invariant velocity happened to be much greater than 186,000 miles a second, had it been infinite, for example, there would have been no discrepancies left between the two laws. Thus, once again we reach the same conclusions that we mentioned when discussing space-time. The fundamental difference between Einstein’s theory and classical science arises from the [Pg 301] discovery that the invariant velocity of the universe is finite and not infinite.
We may summarise all these results by stating that the major effects of gravitation are produced by the curvature of time, i.e., of the time component of the metrical field of space-time; the curvature of space exerts perceptible effects only when the velocities of the moving bodies are very great or when the gravitational field is of very great intensity. Hence, for weak fields such as that of the sun, and for slow motions, Einstein’s law of space-time curvature leads us to results practically identical with those necessitated by Newton’s law of gravitation.
It now remains to be seen how the curvature of space could be detected by direct measurement, without our having recourse to moving bodies. It can, of course, be detected indirectly, since we have seen that it is responsible for the advance of Mercury’s perihelion and for the double bending of a ray of light passing near the sun. But we now propose to detect it by direct spatial measurements.
We have said that our separation of space-time into space and time was equivalent to splitting up the metrical field of space-time into a metrical field of time and a metrical field of space. It is this metrical field of space with which we are now concerned. To this metrical field of space must be ascribed the structure of our space (curved in the present case). Were the metrical field non-existent, space would, of course, have no structure, no geometry whatsoever. As the geometry of a continuum is revealed by measurement, we see once again that it is the metrical field of space which will control the behaviour of our measuring rods at rest in our frame, hence which will determine practical congruence.
The obvious method of verifying the anticipated lay of this metrical field of space, or, what comes to the same thing, this curvature of space would be to explore space with measuring rods; we should, of course, obtain non-Euclidean results. Calculation shows that the results should be identical with those which would be obtained were we to apply rigid Euclidean rods on an appropriate paraboloid of revolution. The type of curvature, or non-Euclideanism, implied thereby is of the Riemann variety, since a paraboloid of revolution is a surface of positive curvature. Hence, we should expect the ratio of the circumference to its diameter to be smaller when some massive body such as the sun was situated at the centre of the circumference.
But here again we are met with the same difficulty. In order to detect a curvature of this sort, we should have to survey vast areas of space, much greater than that of the solar system; and it is scarcely probable that even in future years our measurements will be sufficiently refined to permit a proper test. The situation would have been different had the earth to which we refer our measurements been moving with great speed through the sun’s gravitational field. The inhabitants of the planet Mercury, for example, would be in a much better position to verify the curvature of space by direct measurement, owing to the greater velocity of their post of observation through the inertial frame of the sun. Once more we see how it comes that [Pg 302] the finite invariant velocity of the universe, though most important from a philosophical viewpoint, in practice loses much of its palpable influence by reason of its incredible magnitude. As a result, ultra-precise experimentation must always be resorted to when we wish to test the Einstein theory.
Incidentally, we are now in a position to understand another erroneous assertion that has often been upheld. As soon as Riemann’s discoveries were known, the possibility of interpreting gravitation in terms of a curvature of space had been suggested (chiefly by Clifford). But the great obstacle to this attitude was that the most refined spatial measurements obstinately refused to reveal the slightest trace of non-Euclideanism. Assuming, for the sake of argument, that it had been possible to interpret the curved orbits of the planets and of projectiles in terms of guided motions along curved grooves threading their way through a curved space,[105] even then, in view of the tremendous degree of non-Euclideanism that it would have been necessary to invoke, the scheme would have had to be abandoned; the chief reason being, of course, the incompatibility of this hypothesis with the Euclideanism given by measurements with rigid rods. It was only after Minkowski had discovered space-time that a modified form of Clifford’s premonitions could be vindicated; only then was it possible to reconcile the Euclideanism of our spatial measurements with that non-Euclideanism of space-time which was the source of all gravitational phenomena. In this respect it cannot be urged too strongly that the general theory and the special theory are but two chapters of one same theory. In so far as the general theory cannot work without space-time, any experiment which would conflict with the special theory would thereby conflict with the general theory.
Let us now examine another point. The velocity of light in a gravitational field is no longer an invariant, as it was in the special theory. However, if we were to measure the velocity of light from point to point, displacing ourselves from one point to the next, we should find that wherever we might be placed, this velocity would always appear the same everywhere, owing to the progressive modification in the behaviour of our rods and clocks. It would only be when viewing distant points that we should recognise variations in the velocity of light.
Thus we see that the principle of the invariant velocity of light is accurately true only in free space far from matter, and, even then, only when computed with reference to our Galilean frame. In an accelerated frame, as in the neighbourhood of matter, the cornerstone of the special theory of relativity no longer applies.
It follows that in the general theory the law of the invariant velocity of light must be replaced by some more general principle, comprising the invariant-velocity principle as a particular case. This more general principle can be expressed by saying that a ray of light moves [Pg 303] along a world-line which is a very particular kind of geodesic called a null-line or minimal geodesic. This holds regardless of whether space-time be flat or curved.
Thus far we have studied the curvatures of space and of time outside matter in those particularly simple stationary cases where it is feasible to split up space-time into a separate space and time. Under similar conditions we may investigate these curvatures in the interior of matter. Consider a homogeneous incompressible fluid at rest in our Galilean frame. Conditions being stationary, that is, not varying with time, we can again split up space-time into space and time and study the two types of curvatures separately. We find that the time direction is curved, and this curvature once more is connected with the gravitational forces which exist in the interior of the fluid. We next find that the three-dimensional space inside the fluid possesses a curvature at every point which is proportional to the density of the fluid at every point. Assuming this density to be the same throughout, we see that the curvature of space is constant, and we obtain a spherical space of three dimensions analogous to a cap on the two-dimensional surface of an ordinary sphere in three-dimensional Euclidean space. This curvature of space inside the sphere has nothing to do with the dimensions of our fluid sphere; the curvature is governed solely by the density of the fluid.
Now, in our ordinary three-dimensional Euclidean space, when the curvature or the radius of a sphere is given, its total area is thereby fixed. In a similar way, the density of the fluid predetermines the curvature of space and hence fixes the total volume of the spherical space. Obviously, it would be impossible for us to increase the volume of our fluid indefinitely, for there would come a time when there would be no more room for it to occupy in the spherical space which it had itself created. In the case of a fluid of the density of water, calculation shows that when the sphere of water had attained a radius of about 400 million kilometres, it would be impossible to increase its size further by adding more water to it. When this critical volume was reached, the fluid would fill the spherical space entirely. Moreover, calculation shows that the flow of time would be arrested altogether at the surface of the fluid, so here again we have another reason which would prevent the sphere from growing any larger. Reverting to our two-dimensional analogy, it would be as though the entire surface of the two-dimensional sphere were occupied by the water.
As a matter of fact, this critical volume could never be reached; for even before it could be attained the pressure at the centre of the sphere would have become infinite, and calculation shows that this condition, too, would be accompanied by an arrest in the flow of time. In fine, we discover that the greatest possible volume our fluid sphere could ever occupy would be about of the volume mentioned previously.
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It follows that the fluid could never fill completely the spherical volume its density had created in space. Using a two-dimensional analogy, it would be as though on the surface of an ordinary sphere the entire surface could not be covered; an uncovered portion like a cap would always remain. On this account it is sometimes said that space would not close round on itself, since part of the spherical space created by the fluid could never be filled.
Just as a fluid of given density can never increase in volume beyond a certain point, so, conversely, a fluid of given mass could never be compressed into a volume which would cause its density to increase beyond a certain limit. The argument would be the same. The pressure would become infinite, time would stop, and nothing further could happen. A sphere of fluid possessing the same total mass as the earth could never be compressed into a volume less than that of a thimble.
The numerical values mentioned show that there is only a very slim chance of putting these anticipations of the theory to an experimental test. Even in the case of a giant star such as Betelgeuse, its density is so small that the critical volume is well beyond the actual size of the star. Nevertheless, from a theoretical point of view, it is interesting to note that there must exist a definite limit to the size of a star of given density. If these critical limits should ever be reached, time would be arrested, and it is difficult to foresee what the consequences might be. Einstein has called these critical conditions the “catastrophes.” When pressed to give his opinion as to what would happen were they ever to be reached, he is said to have replied that very possibly we should witness the disruption of matter into radiant or electromagnetic energy. However, he preferred not to venture an opinion, as anything he might say would be little better than a guess.
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CHAPTER XXXII THE PRINCIPLES OF CONSERVATION
CLASSICAL science had assumed that the mass of a body was an invariant, which would not change in value when the relative motion of the body was changed. It is true that highly refined experiments on electrons moving at enormous speeds (Bucherer’s experiment) had shown that the mass of an electron increased with its velocity; but it was assumed that this increase in mass was due to other causes, either to a modification in the electromagnetic field of the electron, or possibly to the mass of the ether which the electron in its rapid motion would drag along with it.
One of the first triumphs of the special theory of relativity was to prove that mass, just like time and space, must be a relative; and that a body in motion with respect to the observer would suffer an increase in mass in all ways identical with that disclosed by Bucherer’s experiment. This increase could no longer be attributed to the drag of the ether, since in the special theory of relativity exactly the same increase would have to be expected, regardless of whether it was the observer or the electron that was in motion through the ether. As in all previous examples, the essential factor was the relative motion between observer and observed. The ether played no part.
Calculation then showed that the mass of a body would become infinite when the velocity of the body reached that of light; for this reason no material body could ever move faster than light, since it would require an infinite force to increase its velocity any further.
When Einstein had discovered this influence of relative velocity over mass, it was suggested that the curious motion of the planet Mercury might be explained without our having to abandon Newton’s law of gravitation, but merely by taking into consideration the variations of the mass of this fast-moving planet at different times of the year. Calculation proved that under the circumstances a perihelial precession in Mercury’s orbit would indeed be in order, but that it would be considerably smaller than that actually observed. Only when the general theory compelled Einstein to abandon Newton’s law was the precise motion of Mercury accounted for, as has been explained in a previous chapter.
On the other hand, this variation of mass with velocity was soon to lead to the discovery of very important phenomena in the realm of intra-atomic motions. The atom, we must remember, is a miniature solar system, with electrons revolving round a central nucleus just as the planets revolve round the sun. Sommerfeld, by taking into consideration the relativistic variation of mass with velocity, proved [Pg 306] that if Bohr’s conception of the atom was correct, the spectral lines emitted from an incandescent atom should be in the nature of bundles of very fine lines closely packed together. Very refined optical tests soon proved that these anticipations were correct; the thick spectral lines formerly observed turned out, upon more refined investigation, to be formed by a number of fine lines in the precise manner predicted by Sommerfeld. This discovery had therefore a twofold effect. It vindicated Bohr’s theory of the structure of the atom, while at the same time it supported Einstein’s special theory of relativity.
By affording a further empirical proof of the correctness of Einstein’s views, these experiments, in conjunction with Bucherer’s, proved conclusively that the mass of a body was a relative and not an invariant, so that the mass of a billiard ball in motion through our frame of reference would have to be considered greater than its mass when at rest. Nevertheless, as was shown by Einstein, it was still possible to adhere to the classical belief in the principle of the conservation of mass, provided that by mass we understood the relative mass as now formulated, and not the mass of the body at rest. Imagine, for instance, two billiard balls colliding and then rebounding in the space defined by our Galilean frame. After their rebound the masses of the two individual balls will have varied, since their velocities with respect to the observer will have changed; but the relativity theory proves that the sum total of both these masses will remain the same for all Galilean observers, after as before the impact. This is what is meant by the conservation of mass.
Were we to conceive of only one billiard ball, first at rest in our Galilean frame and then in motion, its mass, of course, would vary, according to relativity. But it must be realised that in order to set the body in motion we should have to submit it to the action of a force, so that we should be introducing a foreign influence which would have to be taken into consideration.
Now, this expression of the mass of a body, varying as it did with the relative motion of the body, was proved by Einstein to be equal to the mass of the body at rest, plus a certain mathematical expression which for slow velocities became identical with the vis viva, or classical energy of motion, of the body in our frame. Accordingly, the energy of motion of a body could be regarded as identical with the increase of its mass in motion over its mass at rest. It became, then, highly probable, for a number of reasons, that what we called the mass of a body at rest was itself due to the contained energy of the body. In a general way, therefore, a body, when heated or electrified or compressed, would increase in mass.
This identification of the mass of a body at rest with its contained energy, and of its mass when in motion with its contained energy plus its generalised vis viva, or energy of motion, allows us to assert that the principle of the conservation of mass is none other than the principle of the conservation of energy. Hence, Einstein’s special theory is also in harmony with the latter principle. In [Pg 307] a similar way, the principle of the conservation of momentum, or Newton’s law of action and reaction, would be found to stand in full harmony with Einstein’s special theory, provided that by momentum we meant relative mass multiplied by velocity, instead of mass at rest multiplied by velocity.
In short, the principles of conservation endured only when we enlarged our understanding of mass, momentum and energy; and Newtonian mechanics had to be amended as a result of Einstein’s discoveries in the special theory of relativity.
Let us now study the problems of conservation in the general theory when we make use not merely of a Galilean frame of reference, but of any frame whatever, hence when our space is permeated with a field of inertial forces or, again, with a field of gravitation. We shall discover once more that conservation is required by the general theory; though, as we shall see, it will not be the type of conservation to which we were accustomed in classical science.
In order to make these points clearer we must revert to Einstein’s law of gravitation. We remember that assuming the correctness of Einstein’s views of gravitation as being due to space-time curvature, the principle of stationary action led us to the law of gravitation:
Now the tensor on the left possesses the remarkable mathematical property of expressing a form of curvature which is preserved regardless of our mesh-system. It follows that in virtue of the equations of gravitation written above, the same conservation must hold for , hence for mass, energy and momentum.
The mathematical identity that expresses this property, let it be noted, does not depend on Einstein’s theory; it is a purely mathematical property between the ’s of a continuum, which had been discovered by the mathematician Ricci before the theory of relativity was born. But in view of the physical relationships which Einstein had established between the curvatures of the space-time continuum and the presence of matter, it now followed as a necessary consequence of the theory that the right-hand side of the gravitational equations must present the same characters of conservation as the left-hand side. If we interpret this result physically, we see that it implies the conservation of mass, momentum and energy.
We thus reach the remarkable conclusion that the principles of conservation, which in classical science were of an empirical nature and were totally unconnected with the law of gravitation, now appear as immediate and necessary consequences of Einstein’s law of gravitation. In other words, granting the correctness of the law of gravitation, conservation cannot help but exist in this universe. To many thinkers this is one of the most beautiful aspects of Einstein’s theory, showing as it does the wonderful unity and relatedness of nature.
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Nevertheless we must be careful to understand the deeper significance of conservation in relativity. We shall see that the type of conservation we are now discussing is somewhat different from the classical conception of conservation, under which matter, for instance, disappeared from here only to reappear there. In the present case conservation is more ethereal and abstract. We may explain these points better by examining conservation in classical science. Here, when fields of force are present, the principles of the conservation of energy and momentum would appear at first sight to be contradicted by experiment. Thus, a body falling freely towards the earth can certainly not be said to possess a constant vis viva, or energy, since, originally starting from rest, it acquires an increasing speed as it falls towards the ground. Rather than recognise the breakdown of the principle of the conservation of energy, classical science preferred to assume that the body possessed two types of energy: a kinetic energy given by , and a potential energy depending on the position of the body in the field of gravitation. It was the sum total of these two types of energy that remained constant and was therefore conserved during the fall of the body.
It must be noted that the necessity for introducing potential energy, as an additional requirement in order to ensure conservation, exists only when we are considering a phenomenon occurring in a gravitational field or in an inertial field of force, such as would be present were we to take our stand in an accelerated frame. In the absence of such a field, potential energy would become completely meaningless, and conservation of energy would endure without it.
If now we take into consideration Einstein’s discoveries, in the special theory where mass is identified with energy, we see that in the general case of a gravitational or inertial field being present, conservation of both mass and energy can be upheld, but only provided we introduce the concept of potential energy. It remains to be seen whether potential energy possesses any intrinsic significance in the world, or whether it represents a mere mathematical artifice introduced for the purpose of saving the principles of conservation. The general theory proves that this last alternative is the correct one and that conservation in the classical sense must be abandoned. The reason is that when we express this space-time equation of conservation in a semi-Cartesian mesh-system of space and time (a strictly Cartesian one being impossible, owing to the intrinsic curvature of space-time round matter), we find that it expresses true conservation in space and time, in the sense of the special theory, without the adjunction of potential energy; but when we select an arbitrary curvilinear mesh-system of space and time, the mathematical expression we are studying splits up into two terms, and the conservation of this expression implies, therefore, the conservation of the sum total of these two terms. The first term refers to the mass, vis viva and momentum of the matter, while the second term refers to the energy and momentum of the gravitational or inertial field. It would thus appear that conservation could be maintained provided we extended it so as to include the [Pg 309] energy and momentum of the gravitational field as well as that of the matter. This, indeed, was Einstein’s original stand. Later, he seems to have abandoned it for the following reasons:
The second term, which represents gravitational or potential energy in our equations of conservation, is given by a mathematical expression which is not a tensor. It expresses, therefore, merely a relationship between the motion of the observer and the structure of space-time; it does not describe the structure of absolute space-time itself. We must realise, then, that potential energy represents nothing inherent in the absolute world, and we must exclude it as a fundamental entity in the formulation of our laws.
As soon, however, as we refuse to take gravitational or potential energy into consideration, conservation breaks down in a field of force. Hence, it follows that conservation of mass, energy and momentum holds only for Galilean observers. Conservation is not, therefore, a true law of nature, valid for all observers. In the light of these discoveries, the permanent nature of mass and matter can no longer be upheld. Lavoisier’s principle of the conservation of mass must be abandoned, and the conception of matter as a substance is shown to be fundamentally incorrect.
But then if the ordinary laws of conservation are discarded, by what shall we replace them? Obviously, they will now give way to the more general principle of relativistic conservation, a sort of space-time principle. This type of conservation cannot in the general case be expressed in terms of the conservation of our habitual physical quantities in space and time. Its significance will be deeper, referring as it will to space-time uniformities existing in the absolute world of space-time.
Only when the observer adopts a Galilean or a semi-Galilean system of reference will physical conservation endure; only then will it be possible for a man to conceive of a universe of permanent spatial entities. As Eddington remarks, it is only when these Cartesian or semi-Cartesian mesh-systems are used that the separation of space-time into four directions yields the habitual space and time of men. Thus there appears to be a relationship, between the natural separation of time and space by our consciousness, and the insistent demands of the mind to seek out the permanent rather than the transient in order to construct an edifice of knowledge.
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CHAPTER XXXIII OTHER ASPECTS OF THE GRAVITATIONAL EQUATIONS
WE remember that in classical science the Newtonian law of gravitation was expressed by Poisson’s equation, where represented the density of the matter exciting the field at each point of space. It followed that it was solely the mass of a body which was active in creating a gravitational field. Likewise, it was the mass of a body which was the sole cause of its being attracted in a gravitational field, so that the weight of a body at a given point near the earth was fully determined when we knew the mass of the body and the mass of the earth.
Now, in Einstein’s theory, the gravitational equations become where and refer to the curvatures of space-time, or, if we prefer, to the potential or -distribution, while represents the characteristics of the matter exciting the field. There are ten of these gravitational equations corresponding to the various values from 1 to 4 which we must assign to the indices i and k (space-time being four-dimensional). And here we must note that the tensor represents matter much more fully than does the of classical science, for the various components of refer to such additional characteristics as the momentum, the vis viva and the conditions of stress existing in the interior of matter. We may infer, therefore, that whereas in classical science the mass of the matter was solely responsible for the creation of a gravitational field, in the present case all the additional characteristics mentioned above contribute their share.
Similar considerations would hold for the gravitational field outside matter, and as a result, were the sun to be rotating with greater speed on its axis, the gravitational field it would produce would be slightly different. In the case of the planet Jupiter and its moons, it would appear possible to submit to an observational test this additional disturbance caused by the rotation of the central mass on the satellites moving around it. As yet, however, no checking up of the theory has been possible on this point.
Thus far we have considered the gravitational field, or space-time curvature, produced by matter, but it should be apparent that owing [Pg 311] to the identification of mass and energy, space-time curvature is a necessary consequence of the presence of energy. The electromagnetic field should thus be susceptible of giving rise to a gravitational field. Briefly, we may present the problem as follows:
Minkowski and Einstein had succeeded in establishing that electric and magnetic forces were the components of the same tensor . From this tensor it was possible to build up another tensor , representing the energy and characteristics of the electromagnetic field, just as represented the energy and characteristics of matter. These two types of energy, which are not identical, are designated as loose energy and bound energy, respectively. If, now, we wish to discover the precise type of space-time curvature, or gravitational field, which the presence of an electromagnetic field would produce, all we have to do is to replace by in the gravitational equations. We thus obtain We can see at a glance that the type of curvature generated by an electromagnetic field must differ from one due to matter, since and are different magnitudes. A more conspicuous difference is illustrated when we manipulate the gravitational equations according to the permissible mathematical rules. We then find that whereas the gravitational equations for matter yield where represents the density of mass of the matter, the gravitational equations for electromagnetic fields yield for happens to be a quantity which vanishes in all electromagnetic fields. Hence, we realise that in regions of space where electromagnetic fields are present, but all matter is absent, the Gaussian curvature of space-time always vanishes; whereas in regions occupied by matter it is equal to the density of the matter, and does not vanish.
Before proceeding to compare these various types of curvature, let us recall that there exist three types of tensors of curvature of especial importance: the Riemann-Christoffel tensor , the contracted tensor and the invariant . If we write these three magnitudes in line: , , , we find that the vanishing of over a region signifies the complete flatness of the continuum over this region. As a result, both and will certainly vanish everywhere within the region. If it is that vanishes, will automatically vanish but will not necessarily vanish. Hence , while comprising perfect flatness as an extreme case, permits a certain degree of curvature. Finally, if it is that vanishes, [Pg 312] need not vanish; and if does not vanish, neither will ; whereas if vanishes, we are thrown back on the previous case, i.e., may or may not vanish. Hence, , while compatible with perfect flatness as a special case, connotes a law of curvature still less stringent than .
We may now contrast the various types of space-time curvature produced by matter and electromagnetic energy. By writing the gravitational equations in a slightly different though equivalent way, we obtain, in the interior of matter, whence Thus, in the interior of matter, none of the curvatures need vanish, and hence no limitation is imposed on the curvature of space-time.
In regions where there are electromagnetic fields but no matter, we have whence The curvature is thus less thorough, since vanishes.
In regions outside matter, where no electromagnetic fields are present, the curvature is still less pronounced, for there , however, vanishes only for points infinitely distant from matter (in the hypothesis of the infinite universe).
In absolutely free space, remote from matter and where there is no electromagnetic energy, all the tensors and invariants vanish.
We thus find that there exists a progressive curvature of space-time as we pass from empty space-time far removed from all matter and energy to empty space-time in the neighbourhood of matter, then to regions where there are electromagnetic fields but no matter, and finally to regions filled with matter or electrons. If we regard the successive gradations of curvature of space-time as corresponding to successive grades of materialisation, starting from emptiness, we must place electromagnetic fields, or loose energy, at a stage of materialisation preceding that of matter proper, or bound energy.
Still another discovery which Einstein was able to deduce from his gravitational equations refers to the propagation of a gravitational force. Classical science had held to the view that gravitation must be propagated with a speed much greater than light, perhaps even with infinite speed. But Einstein, by integrating the equations of gravitation, proved that in a weak field gravitational waves would be propagated with the speed of light. In a strong field, of course, the presence of the field itself would influence the speed of propagation of the waves, and the problem would assume much greater complexity.
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