APPENDIX IIITHE GRAVITATIONAL EQUATIONS
FROM a purely mathematical point of view, without any regard to physical applications, we may always select a co-ordinate system, or mesh-system, in a continuum regardless of whether the continuum be Euclidean or non-Euclidean. As referred to this co-ordinate system, the position of every point of the continuum is defined by its co-ordinates,....There are as many of these as the continuum has dimensions.
In addition to these co-ordinates of every point, there exist certain structural tensors (the’s), their number beingat every point, whererepresents the dimensionality of the continuum. The values both of the’s and the’s at a point will depend on the co-ordinate system selected; but the’s are remarkable in that they depend not solely on the co-ordinate system, but also on the structure (curved or flat) of the continuum which they define. Once, then, a particular co-ordinate system is selected in a continuum whose structure is known from place to place, the values of the’s in this co-ordinate system can be determined at all points of the continuum. We then find that the square of the distance between two infinitely close points of co-ordinatesand(hence whose co-ordinates differ by), is given bywhererepresents summation for all values attributable toand.Note thatandcan receive all whole values from 1 to,whereis the dimensionality of the continuum.
It is a remarkable fact that when, and only when, the continuum is flat, this expression forcan break up into a sum or difference of squares. When it breaks up into a sum of squares, the continuum is called Euclidean; when into sums and differences of squares, it is called semi-Euclidean, and is said to have positive and imaginary dimensions; but in all such cases it remains flat. (Space-time is an illustration of this latter species of continuum.)
And now suppose we wish to discover the equations of geodesics, that is to say, of the straightest of lines compatible with the structure of the continuum. We obtain a geodesic between two pointsandby expressing the fact that the line stretching betweenandis a maximum or a minimum. Mathematically, this means that by summating the value of,expressed above, along the line joining the two pointsand,this value must be greater or less than that of all other lines stretching between these two points. The calculus of variations enables us to solve the problem. As a result, we obtain thefollowing equations, in which the co-ordinates (,,...) of each and every point of the geodesic, as referred to our co-ordinate system, are given byThese equations are written for short:where,,must be given all whole values from 1 to;being the dimensionality of the continuum.
It follows that, having selected a co-ordinate system, all we have to do is to determine the values of the’s in this co-ordinate system, and then substitute these values in the expression ofand of the geodesics; and the entire geometry of the continuum is thereby established.
Thus far we have been reasoning as though the continuum possessed a structure; but in a purely amorphous continuum no such structure exists, and the mathematician must postulate it. When, however, we are dealing with physical space, we find that the laws of dispositions of material rods, as also the laws of moving bodies (law of inertia), suggest that space is three-dimensional and Euclidean. This implies that it is possible to obtain a particular co-ordinate system, called a Cartesian one, in which the’s will have constant values equal to unity. Einstein’s and Minkowski’s great achievement was to prove that this conception of the world was incompatible with the verdict of the negative experiments. Instead it appeared necessary to assume that the fundamental continuum was four-dimensional, possessing one imaginary dimension, which was none other than time. In this four-dimensional space-time continuum, experiment proved thathad the following form when a Galilean frame of reference was resorted to:We have reversed the sign of,but this is of no particular importance. This formula proves that the’s have the following values when computed in a Cartesian space-time co-ordinate system:all other’s vanishing. Now, while leaving the time axis unaltered, we might also have split up space, no longer with parallel planes, but with concentric spheres and with lines radiating from their centre. We then obtain a polar co-ordinate system. In this case, neglecting one of the spatial dimensions for purposes of simplicity, and callingthe distance from the centre, andan angle, we getso thatall other’s vanishing. It may be noted that this change in the value of the’s is due solely to our change of co-ordinate system; it is in no wise attributable to any change in the structure of space-time. This remains flat, as before.
Now we know that in a gravitational field space-time is no longer flat; it becomes curved, and its law of curvature outside matter is given by.Butis what is known as a differential equation in the’s. By this we mean that it contains in its expression the differences in the values of the’s at neighbouring points. What we wish to discover, however, are the precise values of the’s, and not their mere differences in value. For this purpose the differential equation must be manipulated in a certain way, or, as mathematicians would say, integrated. Only after the integration has been accomplished can solutions be obtained. Now all differential equations have a character in common in that they yield classes of solutions. In order to single out that particular solution which answers to the problem we may be considering, it will be necessary to impose certain conditions upon it. In the present case, the differential equations of the gravitational field are of the second order, which means that two separate conditions will have to be imposed.
Schwarzschild succeeded in integrating these equations for the particular case of the radially symmetrical field of gravitation produced by a mass-particle. The flatness of space-time at infinity, or, in other words, the degeneration oftoat infinity, yielded a first condition; and the second condition was obtained by introducing into the solution the numerical value of the massexciting the gravitational field. In this way the solution became determinate. Schwarzschild then found that for a polar co-ordinate system the expression ofassumed the following form:or, in other words,all other’s vanishing. We can see that when,the mass of the particle, vanishes, or whenbecomes infinite, we obtain the values of flat space-time; this is as it should be.
Now that we are in possession of these values of the’s, all we have to do is to introduce them into the equations of the geodesics, and automatically we shall obtain the equations of free motion of particles moving in our system of co-ordinates under the influence ofa central mass.It is these equations which yield us the double bending of light and the motion of the planet Mercury.
We may note that if we neglect to consider the influence due to the variations in value ofasincreases, that is, if we assign tothe value of -1, which it had in flat space-time, we obtain approximately Newton’s law. It is then the deviation offrom the value -1 which is the distinguishing feature of Einstein’s law of gravitation. But it is obvious thatwas not adjusted for the mere satisfaction of accounting for Mercury’s motion, since its value follows mathematically from the integration of.
We may also add that the gravitational equationsare not linear. In ordinary language, this means that the resultant field produced by two masses cannot be obtained by superposing the two separate fields; additional cross-terms must also be taken into consideration. In the case of weak fields, however, these cross-terms may be neglected, so that in practice a superposition of separate fields is possible. Nevertheless, owing to this non-linearity of the gravitational equations, the problem of two bodies (when neither of the masses may be neglected) presents insurmountable mathematical difficulties. And so it happens that the problem of double stars, so easy to solve in Newtonian science, has as yet failed to receive a solution in the case of Einstein’s law of gravitation.
One last point is worth noting. We have seen that there were ten gravitational equations, and that there were ten’s whose values were to be determined. Owing to the fact that there were as many equations as unknowns, it might be thought that one single set of values for the ten’s would be obtained. But inasmuch as the values of the’s deduced frommust necessarily differ as we choose one mesh-system or another, it would appear as though some definite mesh-system were imposed on us—a fact incompatible with the arbitrariness of mesh-system demanded by the general theory of relativity. According to relativity, there should be, therefore, a fourfold arbitrariness in the values of the’s of space-time corresponding to the four arbitrarily chosen meshes of our space-time mesh-system. Mathematically, this would imply that four of our gravitational equations should be superfluous, hence linked to one another by relations of identity. Now we know that four such identical relations actually exist. Physically, they express the conservation principles of momentum and energy. We are thus led to the remarkable conclusion that the relativity of motion entails the existence of permanent entities in the world. In Eddington’s words:
“The argument is so general that we can even assert that corresponding to anyabsoluteproperty of a volume of a world of four dimensions (in this case,curvature), there must be fourrelativeproperties which are conserved. This might be made the starting-point of a general inquiry into the necessary qualities of a permanent perceptual world,i.e., a world whose substance is conserved.”[165]