Part 3 — 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 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, 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.