APPENDIX IVSPACE, GRAVITATION AND SPACE-TIME
A NUMBER of philosophers have expressed the view that the special theory is too paradoxical to be accepted, but that the general theory meets with their approval. Statements of this sort are indefensible; they arise from too meagre an understanding of mechanics and geometry. Nevertheless, it may be of interest to investigate the problem more fully.
The special theory is but a particular limiting case of the general theory, one in which space-time remains flat instead of manifesting curvature. Now it is space-time which is the source of all the paradoxes of feeling (trip to the star; relativity of simultaneity, etc.). And since this new continuum is as essential to the general as to the special theory, we may be quite certain that all the paradoxes of the special theory will subsist in the generalised case, aggravated, however, by an additional element of complication produced by curvature. In short, the rejection of space-time would entail the downfall of both the general and special theories. In much the same way, if there were no omelet there might still be eggs, but if there had been no eggs in the first place there could certainly be no omelet. Indeed, we may summarise Einstein’s work by saying that in the special theory space-time was discovered, and that in the general theory its possibilities were investigated mathematically.
However, we may consider the problem under a different aspect. The main achievement of the general theory has been to account for gravitation in a purely geometrical way in terms of the curvatures of the fundamental continuum (space-time in the present case). But we may examine whether similar developments might not have been pursued had we been content to abide by the classical picture of a separate space and time. Gravitation would then be accounted for in terms of space-curvature alone, and space-time would be obviated together with the paradoxes of feeling it entails. However, we shall see that there are a number of reasons which would render success along these lines impossible. But it is not by a mere line of philosophical talk that the situation can be made clear; hence a short digression seems necessary.
When Galileo and Newton initiated their epochal discoveries in mechanics, they assumed that space was three-dimensional and Euclidean. But when we state that space is Euclidean, what is it we wish to imply? We mean that if ideally accurate measurements could be performed, the numerical results of pure Euclidean geometry would be obtained. If, therefore, on performing measurements with material rods, slight discrepancies are noted, we shall assume that owing to contingentinfluences of one sort or another, our rods are not true to standard.
Now it is obvious that by proclaiming space to be Euclideana prioriand then laying the blame on our physical measurements if oura prioriassumption is not borne out in practice, we are placing ourselves in a position which is impregnable but useless. We are professing to know everything before we start, and hence are depriving ourselves of the aid of experiment in our study of nature.
It might therefore appear strange to find Newton, the great empiricist, following ana prioriprocedure which he was never weary of combating in science. But here we must remember that in Newton’s day Euclidean space was the only type known; hence it was regarded as axiomatic, not alone by Kant, but, more important still, by the most competent mathematicians of the seventeenth and eighteenth centuries. In view of this supposed inevitableness of Euclidean geometry, it was assumed that if the geometry of space was to convey any meaning at all, it could not help but be Euclidean. And so the danger of ana prioriattitude towards the geometry of space failed to impress itself.
But the entire situation changed when non-Euclidean geometry was discovered. For since it was now established that various types of spaces could exist, it was impossible to anticipatea prioriwhich one of these possible types would be realised in nature. Euclidean geometry was thus deprived of its position of inevitableness. Henceforth, our sole means of ascertaining the geometry of space was to appeal to physical measurements; and the geometry of space would be that of light rays and of material bodies maintained under constant conditions of temperature and pressure.
It is, however, permissible to question whether such physical measurements could ever teach us anything about the geometry of space; it might be claimed that they merely yielded information as to the behaviour of light rays and material bodies. This was Poincaré’s attitude. But we have seen that if the aid of physical measurements is denied us, the geometry of space escapes us completely; and there is nothing further to argue about. In view of the various types that are rationally possible, any species we might select would be in the nature of an arbitrary definition, hence would be purely conventional. It would be as useless to try and prove that our definition was incorrect as it would be to state that the system of numbers should be decimal, or binary, or duodecimal;conveniencealone could guide us in such cases.
But when we analyse what is implied by the word “convenience,” we see that in the final analysis it is often based on the results of physical measurements. Thus, were all physical measurements to yield non-Euclidean results, were all the laws of nature to be expressed more simply in terms of the same non-Euclidean geometry, it would certainly be more convenient to attribute a non-Euclidean structure to space. For the same reason it is more convenient to attribute three dimensions to space, since this hypothesis permits us to account very simply for our inability to get out of a closed room without opening the dooror window. The reason the word “convenient” is still adhered to in this case is because it might still be possible to account for the observed facts in terms of, say, a four-dimensional space, provided we were willing to vary our fundamental ordering relation and with it our understanding of sameness.
We may summarise these statements by saying that “reality” for the scientist reduces to the simplest co-ordination of experimental facts and that in view of the various possible types of spaces suggested by non-Euclidean geometry, physical measurements constitute our only means of determining which is the simplest, hence which is the real solution. This explains why Gauss (who as far back as 1804 had mastered in secret the essentials of non-Euclidean geometry) endeavoured to establish the geometry of space by means of light-ray triangulations. But in Newton’s day the fusion of geometry and physics was not contemplated, since Euclidean geometry was thought to constitute the only possible type. Furthermore, the greater the care in performing measurements, the nearer did results approximate to those of pure Euclidean geometry; and so it appeared reasonable enough to assume that space was Euclidean.
We have deemed it necessary to recall these fundamental notions in view of their importance in the discussions which are to follow. But for the present we shall revert to the problem as it stood in the days of Newton, when non-Euclidean geometry was unknown and space was assumed to be Euclidean. In particular, we shall examine briefly the foundations of rational mechanics.
In all problems of motion it is essential to specify the frame of reference in which the motions of bodies are to be computed since the velocities and accelerations that are measurable vary with our choice of this frame. Following Copernicus’ suggestion, scientists defined the standard frame of reference as one in which the stars appeared to be fixed, or at least non-rotating. A frame of this sort is termedinertialorGalilean.
In addition to the choice of a frame, the measurement of a motion entails measurements of both space and time. The Euclideanism of space gave significance to the equality of two spatial distances measured in our Galilean frame. To a high degree of approximation, equal spatial distances were assumed to be defined by the displacement of the same material rod maintained under standard conditions. As for measurements in time, they were assumed to be given by clocks regulated ultimately by the earth’s rotation.
The operations of measurement being thus defined physically, both Galileo and Newton considered that on the strength of the empirical evidence, it was permissible to assert that a free body in motion, unsolicited by forces, would pursue a straight course with constant speed. This statement constitutes thelaw of inertia. Nevertheless, the empirical methods whence Galileo and Newton had derived the law of inertia were extremely crude. Not only were more or less inaccurate physical measurements involved, but, worse still, perfectly free bodies could never be contemplated, since theearth’s gravitational effect and frictional influences could never be eliminated. It follows that the law of inertia could lay claim to nodirectempirical justification.
But, on the other hand, if mechanics were to be developed along mathematical lines, it was essential that certain rigorous premises be accepted. Newton could not content himself with the statement that free bodies pursued more or less straight courses with more or less constant speeds. Hence the necessity of elevating the law of inertia to the position of a principle. Thus the principle, though originally adduced as a generalisation from experience, now assumed the position of the definition of the motion of afree body. When, therefore, a body appeared to deviate from the dictates of the principle, it was agreed that the body could not be free; just as, when rods did not yield perfectly Euclidean results, it was assumed that they could not be perfectly rigid (since, in classical science, the Euclideanism of space had been accepted as a principle).
The principle of inertia having been posited in this way, we see that it yields us an ideal definition of time, for, by definition, a perfectly free body describes equal Euclidean distances in rigorously equal intervals of time. Thanks to the principle of inertia, Newton thus gave an accurate (though ideal) definition of time.
In addition to this first principle, it was necessary to consider two others, also obtained as generalisations from experience; namely, the principle of force and acceleration, and the principle of action and reaction. These three basic principles constitute the foundations of rational mechanics.
Henceforth mechanics, originally an empirical science, becomes rationalised and is placed on a mathematical basis. It can now be developed without further appeal to experience by purely mathematical methods.[166]For this reason it is calledrational mechanics. If, however, we consider the mechanics of the solar system, further empirical information must be obtained. The observational data were furnished by Tycho Brahe and Kepler. From this information Newton deduced the law of gravitation. When, therefore, the mutual force acting between bodies is of the inverse-square variety, we obtain a special branch of mathematical mechanics, calledcelestial mechanics.
From what has been said, we realise that the solutions of these problems of rational and celestial mechanics are of a purely mathematical nature, involving technical difficulties of their own. And so we need not be surprised to find the names of pure mathematicians on every page of the treatises consecrated to these arduous sciences. Thus, in rational mechanics we find the great names of Newton, Euler, d’Alembert, Lagrange and Hamilton, whereas in celestial mechanics we come upon the equally great names of Laplace, Jacobi and Poincaré. We have stressed these points in order to show how mechanics, originally,an empirical science, has since become mathematised.
Now, theoretically at least, it is of no interest to the pure mathematician to know whether these mathematical speculations correspond in any way to the workings of the real world; and regardless of whether Newton’s law is correct or at fault, the solution of the problem of the n bodies acting under Newton’s law of the inverse square would still present considerable interest. But, on the other hand, it would be a grave mistake to assume that these abstruse mathematical problems constitute mere intellectual pastimes, cross-word puzzles of a higher sort. Rational and celestial mechanics claim to correspond to the real workings of the world. The only way to verify the justice of this claim will be continually to check the anticipations of theory by the results of observation.
If this checking up proves harmony to prevail between theory and observation, we may assume that our fundamental principles are correct. If not, three courses are open: Either we may recognise that our fundamental principles are at fault, and attempt to replace them by others; or else we may retain our fundamental principles, and adjust theory to observation by introducing suitable disturbing influences or hypothesesad hoc; finally, we may assume that nature is irrational, hence is not amenable to mathematical investigation.
It is impossible to give any golden rule in this respect. The hypothesis of Neptune was an illustration of the second procedure, where disturbing influences were introduced while the basic principles were maintained. The Einstein theory is an example of the first procedure, where the fundamental principles have been abandoned. As for the third procedure, it would be adopted only as a last resort, since it would be equivalent to throwing up the sponge. Nevertheless, it is far from certain that we may not have to resign ourselves to this attitude eventually, when our observations deal with phenomena on a microscopic scale.
Reverting to Newtonian science, we may say that until quite recently the fundamental principles of rational mechanics appeared to be in no danger. The most accurate astronomical observations yielded results in full conformity with theoretical previsions, when account was taken of a minimum of disturbing influences and of the inevitable inaccuracy in our knowledge of the planetary masses. We understand, therefore, whence arises the justification for the law of inertia, hence, incidentally, for the classical definitions of time and space.
It was not because stones thrown on the ice appeared to follow straight courses with constant speeds that the law of inertia was deemed to be established, for stones eventually come to a stop. It was because the numerous consequences of the law appeared to be verified indirectly to a high degree of accuracy. Hence, although it is undoubtedly correct to say that the fundamental principles of mechanics were nothing but hazardous generalisations from exceedingly crude observations, their justification appeared to have been establisheda posteriori, at least until quite recently.
From the principles of mechanics it is possible to deduce a certain very general principle. We refer to the principle ofLeastAction, alternative forms of which are given by Hamilton’s principle, and by Gauss’ principle of minimum effort. Any one of these general principles comprises all the principles of mechanics in a highly condensed form. Least action enables us to anticipate the following result: Free bodies unsolicited by forces invariably follow geodesics with constant speeds. Even when only partially free, as, for instance, when constrained to move over a fixed surface (without friction), the bodies will invariably follow the geodesics of the surface, also with constant speed.
Here a word of explanation may be necessary. The geodesics of three-dimensional Euclidean space are the familiar Euclidean straight lines; and our previous statement is but a presentation of the law of inertia. But when we consider a curved surface, the straight lines of space cannot lie on its surface. The geodesics of the curved surface are then no longer straight lines in the Euclidean sense, but they are the nearest approach to such lines,i.e., the least curved of all lines traceable on the curved surface. In the case of the sphere, geodesics are great circles. All surfaces have their geodesics more or less capriciously distorted in shape according to the variation in curvature from place to place of the surface. Again, as in the case of three-dimensional space, a geodesic is the shortest line between two points of the surface (when the distance is computed along the surface).[167]
An inextensible thread stretched between two points over a curved surface lies along a geodesic. Of course, in practice, it would be difficult to obtain adherence of the thread to the surface, were the surface concave. But this difficulty can be remedied if we consider a second surface fitting over the original one. The thread would then be stretched in the interstice between the two surfaces; and a body moving over the surface would be given by a small ball rolling without friction in the interstice between the two surfaces. In this way the ball would always remain in contact with the surface.
We may say, then, that in the absence of forces—for instance, in the interior of a falling elevator—the ball, if started with some initial speed, would describe that geodesic of the surface which lay tangentially to its initial velocity. In addition, the ball’s speed along the geodesic would remain constant (in the absence of friction).
And so, were we to discover one of the geodesics of the surface by stretching a string, and were we to mark the lay of this geodesic with a line of ink, we should find that the ball, when constrained to move without friction over the surface, would follow this line provided its initial velocity were directed tangentially to it. To all intents and purposes, geodesics act, therefore, as grooves guiding the progressof free bodies; and the free bodies follow these grooves with whatever initial speed we may have given them. All these considerations apply solely when no external forces are acting on the moving body. The presence of a force tears the body away from the geodesic it would normally follow, and also accelerates its speed.
We now come to Newton’s law of gravitation. It was a remarkable fact that although, according to the law of inertia, the motions of free bodies should be directed along the straight geodesics of three-dimensional Euclidean space, yet the courses and motions of the planets round the sun and of projectiles near the earth’s surface deviated widely from these geodesic paths. In the case of projectiles, a cause for this deviation was easy to discover. It was attributed to the disturbing action of the force of gravitation exerting itself in the earth’s neighbourhood. But in the case of the planets the discrepancy was harder to explain. Newton solved the problem by extending the scope of gravitation; namely, by assuming that a gravitational force varying inversely as the square of the distance, and directed towards the sun, was acting on the planets. He was thus able to account for the elliptical orbits of the planets, and also for their motions along these orbits, as given by Kepler’s laws.
Thus, in classical science, we were faced with straight geodesics and with forces emanating from matter and compelling bodies to depart from their normal geodesic courses when moving in the proximity of matter.
But we, who, in contradistinction to Newton, know of non-Euclidean geometry, might be tempted to offer another solution of gravitation. Viewed from the standpoint of non-Euclidean geometry, the motion of the point over the surface (discussed previously) can be interpreted as the free motion of a point in a two-dimensional non-Euclidean space (the space of the surface). Hence we may say that whether space be Euclidean or non-Euclidean, least action demands that free bodies follow its geodesics with constant speed. This holds regardless of the dimensionality of the space, whence we may conclude that were our three-dimensional space non-Euclidean instead of Euclidean, Newton’s law of inertia would have to be revised. It would no longer be correct to state that free bodies describedEuclidean straight lineswith constant speeds. Instead we should say: “Free bodies pursue thegeodesics of spacewith constant speed.”
Thanks to this generalisation of the law of inertia, we are enabled to extend classical mechanics to spaces of all kinds. Only in the particular case where space is Euclidean does our generalised law of inertia coincide with that given by Newton.
From an observational standpoint, what would be the result of space manifesting itself as non-Euclidean? The geodesics would be curved, as contrasted with Euclidean ones, and free bodies unsolicited by forces would appear to follow curves with constant speeds. Under the circumstances, it would seem to be possible to account for the curved paths of planets and projectiles without having to introduce a disturbing force of gravitation. All we should do would be to assumethat space was suitably curved around large masses, such as the sun and earth; and as a result the geodesics, hence the paths of free bodies, would be curved in turn. In particular, planets would now circle round the sun, not because the attraction of the sun compelled them to do so, tearing them away from their straight geodesics, but because the space around the sun being now curved, its geodesics would automatically become curved lines.
Ideas of this sort were advanced tentatively by Lobatchewski, Riemann and Clifford; but, as we shall proceed to explain, a solution along these lines was physically and mathematically impossible so long as we restricted our investigations to the separate space and time of classical science.
A first difficulty would be encountered when we contrasted an exploration of the metrics of space by the method of rod measurements, with that of free bodies following geodesics. We remember that when discussing the free motion of a ball constrained to move on a curved surface, we said that the ball would describe the shortest course between two points on the surface, in other words, a geodesic; that had we stretched a string over the surface between these two same points, the string in turn would have placed itself along the same geodesic (it was in this sense that the geodesic could be considered as defining the shortest line between the two points over the surface). In short, measurements with rods or motions of bodies yielded the same geometry, the same metrics, for the two-dimensional surface. Similar considerations would apply to a three-dimensional space.
But with our tentative theory of gravitation this accord would disappear and we should be faced with an insufferable dualism. Consider, for instance, the case of a stone thrown into the air; it describes a parabola. According to our present views, we should have to assume that this parabola represented a geodesic of three-dimensional space, and therefore the shortest distance between two points on the trajectory. But obviously, were a string to be stretched between the starting point and the point of fall of the stone, it would never coincide with the parabolic trajectory. In the same way, the distance between two points on the earth’s orbit—say, at six months’ interval—is not given by the line of the orbit. Thus the geometry of space established by freely moving bodies would be incompatible with that established by rod measurements. In the first case, intense non-Euclideanism would manifest itself, whereas, in the second case, space would appear to be Euclidean.
Under such conditions, it would be difficult to see what argument we could invoke for maintaining that the curved course of the parabola was a geodesic, hence constituted the shortest spatial distance between two points on the trajectory. In short, we should be faced with a dualism subsisting as between rod measurements and explorations with moving bodies. Inasmuch as it is the constant endeavour of science to obviate dualism wherever possible and to establish unity, we may anticipate that an interpretation of gravitation along the preceding lines wouldhave had no chance of being accepted by scientists. Rather would they have retained Newton’s Euclidean space, which would account perfectly for the results of rod measurements and light-ray triangulations, and then appealed to an additional force of gravitation whose action would be to tear bodies away from the straight geodesics they would normally have followed.
Furthermore, there is still another aspect to the problem. If we assume that moving points describe geodesics in a non-Euclidean space which presents a high degree of non-Euclideanism, we must recognise that our material bodies (which behave Euclideanly) must diverge widely from standard when displaced. If, then, we consider the translational motion of a large non-rotating mass, all its molecules should describe geodesics. But this would be impossible, since, owing to the enormous variations in the shape of the body, as contrasted with the geometry of space, the majority of the molecules would be torn from their natural courses. Enormous forces of reaction would therefore arise, so that the body would be either torn apart, crushed or unable to move at all.
With space-time, however, all these drawbacks are removed, since no dualism arises as between measurements with rods and the courses of free bodies. In Einstein’s theory it is the geodesics of space-time,not those of space, which are followed by free bodies. This curvature of space-time is sufficient to account for the motions of projectiles and planets; and when space-time is split up into space and time, we find that over limited areas the geodesics of space remain Euclidean straight lines (owing to the enormous value of), as in classical science. For this reason, it is fully in order that in spite of thespace-time curvature, measurements with rods and stretched strings should yield Euclidean results.
Thus we see that in Einstein’s theory the metrics of space-time permits us to account both for gravitation and for the results of ordinary measurements with rods. It also enables us to co-ordinate temporal processes as exhibited in the Einstein shift-effect. In short, with the rejection of space-time, the unity of nature revealed by the relativity theory would be lost.
Let us now consider a second objection to the space-curvature hypothesis. The law of motion for falling bodies is not concerned solely with spatial courses. For instance, projectiles describe parabolas, but their motions along these trajectories are just as characteristic as are the shapes of the trajectories themselves. Again, in the case of the planets, we are telling only half the tale when we state that they describe ellipses round the sun, situated at one of the foci. It was not from an incomplete statement of this sort that Newton could ever have obtained his law of the inverse square. Kepler’s second and third laws complete the description of the planetary motions by defining the nature of the motions and velocities of the planets along their elliptical orbits. Only when all three of Kepler’s laws are taken into consideration can Newton’s law of gravitation be deduced without ambiguity. For example, if the planets pursued their elliptical orbitswith constant speeds, Newton’s law would be unable to account for their motions.
Turning, then, to our tentative scheme of gravitation in terms of a curved space, we see that there is nothing in it to suggest the peculiar accelerated motions of planets and projectiles, so that we are unable to account for all the effects that Newton ascribed to gravitation. Worse still, with a theory of space-curvature the motions of bodies along their trajectories would be uniform, since geodesics are described with constant velocity. And this would be in utter conflict with the accelerated motions of falling bodies and of the planets.
We might, however, seek an avenue of escape by claiming that the curvature of space would account for the accelerated appearance of these motions, much as the apparent velocity of a body depends on its orientation with respect to our line of vision. To be sure, a curvature of space would entail a visual distortion of this sort, since the light rays emanating from a luminous source would now reach us along curved routes. As a result, we should misjudge the position of the source, much as occurs in the case of refraction. This effect has to be taken into consideration in Einstein’s cylindrical universe, for example. The distances of stars, as deduced from a measurement of their parallaxes, have to be decreased somewhat, owing to the spherical curvature of space. It follows that a star leaving us radially with uniform speed would appear to possess an increasing acceleration as it moved away from us, as though repelled from the point where we happened to be standing.[168]
But it is obvious that if separate space and time are retained, our belief in the accelerated motions of bodies falling in a gravitational field can in no wise be attributed to effects due to optical distortions varying with the relative position of the percipient. These accelerations are objective in that they exist to the same degree for all observers in the objective space and time world of science. To take a simple case, if we allow a stone to fall on our foot, its velocity of impact will vary with the height whence it was released. We cannot say that these variations in velocity are to be ascribed to a mere optical distortion of the light rays reflected from the stone. We know, indeed, that quite aside from any visual image of motion, the pain we experience will tell the tale. Added to all these considerations, it would be quite impossible to account for the precise accelerations of falling bodies and of the planets, even were we to interpret them by means of optical distortions.
And so we see that the best that could be hoped from a curvature ofspace alone would be an explanation of the precise paths of falling bodies, but not of their motions along these paths. Hence we should not have succeeded in interpreting gravitation, since, once more, the motion along the paths is as essential as the spatial shape of the paths. Newton’s law, on the other hand, accounts both for paths and for motions.
But we may consider another means of escape. We might assume that the spatial geodesics which free bodies follow act not only as grooves, but more like arteries which propel the blood along by their expansions and contractions. We might then assume that these pulsations of the geodesics would give the planets and projectiles those precise variations in motion which are observed. Needless to say, the hypothesis would be absurd and useless, and, so far as the writer is aware, has fortunately never been suggested. At all events, even with this hypothesisad hoc, we should have failed to account for gravitation in terms of space-curvature, since we should have been compelled to introduce this additional influence. Furthermore, as in the absence of attracting masses free bodies are not accelerated, we should have to assume that in the absence of attracting masses these pulsations of the geodesics would cease; hence it would be the action of matter which would produce the pulsations. In other words, what we formerly called the force of gravitation would now be called the force of pulsation, and we should be thrown back on the Newtonian law of gravitation under a new name, in spite of our absurd hypothesisad hoc.
With space-time, again, all these difficulties are obviated, for the geodesics of space-time, in contradistinction to those of space, account both for the paths of bodies and for their precise motions along these paths. This is due to the fact that a space-time geodesic deals with both time and space.
Thus far we have seen that one of the main obstacles to a space-curvature theory of gravitation resided in its inability to account for the precise motions of bodies along their orbits or trajectories. Yet, on the other hand, were it not for the other difficulties mentioned previously in this chapter, it might not have appeared impossible to account for the spatial shapes of the orbits of bodies in a gravitational field. We now propose to show that, quite independently of the reasons already given, even this partial success can never be realised. Here, however, we are compelled to consider a more mathematical aspect of the problem.
First, let us consider ordinary three-dimensional Euclidean space. If we select a pointat random, we know that there exists a triple infinity of straight lines passing through.But if we impose the restriction that these lines shall all be perpendicular to some arbitrary direction,the possible straight lines are decreased in number. In their aggregate they now constitute a plane, the plane passing throughand perpendicular to.
Similar reasonings may be applied to any-dimensional space, whether Euclidean or non-Euclidean. But if the space is non-Euclidean our straight lines, or geodesics, will be curved from the Euclidean standpoint. Hence we see that the geodesics passing throughandperpendicular atto the directionwill now form a curved surface, no longer a plane as in Euclidean space. A surface of this sort formed by geodesics is called ageodesic surface; and the one we are considering will be the geodesic surface passing throughand perpendicular to the directionat.We see, then, that in a Euclidean space the geodesic surfaces are planes, just as the geodesics are straight lines; whereas in a non-Euclidean space the geodesic surfaces are curved, just as the geodesics are curves.
If, now, we assume that the directionis pivoted round,we shall obtain a new geodesic surface for each new orientation of.In the case of Euclidean space these surfaces will always be planes, but in non-Euclidean space the curvatures of the various surfaces atmay vary with the orientation of,and also with the position of the point.Riemann then defined the curvature of the space at the pointin the directionby the curvature of the geodesic surface passing throughand perpendicular toat.Thus we see that whereas a two-dimensional surface has but one curvature at every point, a three-dimensional space has various curvatures at a point, depending on the direction in which the curvature is computed. If for all orientations of,and for all points,the curvature of the geodesic surface remains the same, we have a space of constant curvature. When the geodesic surfaces are spheres of the same radius, hence are surfaces of the same positive curvature, we have a spherical space; when pseudospheres, hence surfaces of constant negative curvature, we have a Lobatchewskian space, and when planes, hence surfaces of constant zero curvature, we have a Euclidean space.
It remains to be said that, as Riemann discovered, these curvatures of a space at a point can be described fully only in terms of the Riemann-Christoffel tensorat each point. For this reason the Riemann-Christoffel tensor is the only one which enables us to define the nature of the space in a complete way.
And now let us revert to the space-curvature theory of gravitation. Inasmuch as the orbits of free bodies do not present a constant curvature, we should be in the presence of a space of variable curvature from place to place. This in itself is no drawback. But here comes the difficulty: Consider a pointabove the earth’s surface, and letbe directed along a vertical. Then the geodesic surface perpendicular toatcontains all the geodesics passing throughand proceeding initially in a horizontal direction parallel to the earth’s surface. Let us consider, in particular, one of these geodesics, say the one proceeding from east to west. There exists one and only one such geodesic passing throughand starting out horizontally from east to west. According to our tentative theory of gravitation, this would imply that there existed but one path which a body could follow when projected horizontally fromin a westerly direction. But we know that this is not true, since by varying the initial velocity of a body we obtain an infinite variety of parabolas; whence we must conclude that it is impossible to identifythe spatial courses of bodies shot out fromwith the geodesics of a curved three-dimensional space. But we have seen that unless we are to introduce forces, least action requires that free bodies should describe geodesics. Furthermore, geodesics are the only lines that possess any absolute structural significance. If, then, bodies moving in a gravitational field do not follow them, we cannot attribute the courses of these bodies to the intrinsic geometry of the space. Once more we are compelled to appeal to a force of gravitation, just as Newton did, and our entire theory collapses. In order to succeed in an identification of the courses of bodies in a gravitational field with the geodesics of the continuum, a fourth non-spatial dimension is unavoidable. Only thus can we obtain the infinite number of geodesics necessitated by the infinite number of paths that bodies are known to follow. Here again, space-time solves all our difficulties.
Summarising, we may say that the great merit of Einstein’s general theory resides in the fact that his law of space-time curvature,(practically the only one possible) yields us not only the motions of bodies, hence the law of gravitation, but also the metrical properties of the fundamental continuum. By this we mean that the law of space-time curvature permits us to anticipate the precise results of measurements with rods and the precise rates of temporal evolution of processes situated in various parts of the gravitational field. Thanks to the law of curvature, we know that an atom must beat faster here than there, hence that its light must be modified in colour. This is, of course, the celebrated Einstein shift-effect since observed by astronomers. The importance Einstein attributes to this phenomenon is plainly illustrated when, writing in 1918, before the shift was observed, he tells us: “If the displacement of spectral lines towards the red by the gravitational potential does not exist, then the general theory of relativity will be untenable.”
Thus we see that this fusion of geometry and physics, this linking together of mechanics and gravitation with the geometry, or metrics, of space and of time, constitutes one of the most beautiful aspects of the general theory. It emphasises a degree of unity in nature heretofore undreamt of. Were it not for this superb unity, were it not that measurements with rods and clocks were in perfect harmony with the motions of bodies, the theory would be abandoned. At all events, the passage previously quoted expresses Einstein’s personal attitude on the subject.
Finally, we may consider still another drawback of the space-curvature hypothesis. Whatever the law of curvature, it would consist in some three-dimensional tensor law—say,,restricted to space alone. But a tensor law of this sort, while remaining covariant for any change in the orientation of our spatial frame of reference, would cease to manifest this property when referred to a frame in motion. It is admitted that this in itself is in no sense a decisive argument against the space-curvature attitude, since the covariance of natural laws for all observers, regardless of motion, is not a condition which is by any means imposed on usa priori. Classical science, for instance, did not anticipate any such covariance. Nevertheless, it must be admitted that Einstein’stheory, by permitting us to establish this irrelevance of the laws of nature to the observer’s motion, has achieved a result which, from a philosophical point of view, is compelling. And again this covariance entails space-time.
We have thus presented a number of arguments establishing the impossibility of accounting for gravitation in terms of space-curvature. With the exception of the last argument, it is safe to say that any of the preceding ones would prove the point to the satisfaction of scientists. We might have mentioned a number of other reasons, but those discussed seem sufficient. We see, then, why it was that the ideas of Riemann and Clifford were incapable of realisation in their day, when space-time was as yet unknown. And so Newton’s conception of a force emanating from matter and operating in a three-dimensional Euclidean space was in no danger of being overthrown.