This discovery of the inertia of energy created an entirely new starting-point for erecting the structure of mechanics. Classical mechanics regards the inertial mass of a body as an absolute, invariable, characteristic quantity. The special theory of relativity, it is true, makes no direct mention of the inertial mass associated with matter, but it tells us that every kind of energy has also inertia. But, as every kind of matter has at all times a probably enormous amount of latent energy, its inertia is composed of two components; the inertia of the matter and the inertia of its contained energy, which consequently alters with the amount of the energy-content. This view leads us naturally to ascribe the phenomenon of inertia in bodies to their energy-content altogether.
Thus, there arose the important task of absorbing these new discoveries concerning the nature of inert mass into the principles of mechanics. A difficulty hereby arose which, in a certain sense, pointed out the limits of achievement of the special theory of relativity. One of the fundamental facts of mechanics is the equality of the inertial and gravitational mass of a body. It is on the supposition that this is true that we determine the mass of a body by measuring its weight. The weight of a body is, however, only defined with reference to a gravitational field ([Note 18]): in our case, with reference to the earth. The idea of inertial mass of a body is, however, introduced as an attribute of matter without any reference whatsoever to physical conditions external to the body. How does the mysterious coincidence in the values of the inertial and gravitational mass of a body come about?
Nor does the special theory of relativity provide an answer to this question. The special theory of relativity does not even preserve the equality in the values of inertia and gravitational mass; a fact which is to be reckoned amongst the most firmly established facts in the whole of physics. For, although the special theory of relativity makes allowance for an inertia of energy, it makes none for a gravitation of energy. Consequently, a body which absorbs energy in any way will register a gain of inertia but not of weight, thereby transgressing the principle of the equality of inertial and gravitational mass; for this purpose a theory of gravitational phenomena, a theory of gravitation, is required. The special theory of relativity can, therefore, be regarded only as a stepping-stone to a more general principle, which orders gravitational phenomena satisfactorily into the principles of mechanics.
This is the point where Einstein's researches towards establishing a general theory of relativity set in. He has discovered that, by extending the application of the relativity-principle to accelerated motions, and by introducing gravitational phenomena into the consideration of the fundamental principles of mechanics, a new foundation for mechanics is made possible, in which all the difficulties occurring up to the present are solved. Although this theory represents a consistent development of the knowledge gathered by means of the special theory of relativity, it is so deeply rooted in the substructure of our principles of knowing, in their application to physical phenomena, that it is possible thoroughly to grasp the new theory only by clearly understanding its attitude toward these guiding lines provided by the theory of knowledge.
I shall, therefore, commence the account of his theory by discussing two general postulates, which should be fulfilled by every physical law, but neither of which is satisfied in classical mechanics: whereas their strict fulfilment is a characteristic feature of the new theory. Here we have thus a suitable point of entry into the essential outlines of the general theory of relativity.
§ 2
TWO FUNDAMENTAL POSTULATES IN THE MATHEMATICAL FORMULATION OF PHYSICAL LAWS
NEWTON had established the simple and fruitful law that two bodies, even when they are not visibly connected with one another, as in the case of the heavenly bodies, exert a mutual influence, attracting one another with a force directly proportional to the product of their masses, and inversely proportional to the square of the distance between them. But Huygens and Leibniz refused to acknowledge the validity of this law, on the ground that it did not satisfy a fundamental condition to which every physical law is subject, viz. that of continuity (continuity in the transmission of force, action "by contact" in contradistinction to action "at a distance"). How were two bodies to exert an influence upon one another without a medium between them to transmit the action? The demand for a satisfactory answer to this question became, in fact, so imperative that finally, in order to satisfy it, the existence of a substance which pervaded the whole of cosmic space and permeated all matter—the "luminiferous ether"—was assumed, although this substance seemed to be condemned to remain intangible and invisible (i.e. imperceptible to the senses for all time) and had to be endowed with all sorts of contradictory properties. In the course of time, however, there arose in opposition to such assumptions the more and more definite demand that, in the formulation of physical laws, only those things were to be regarded as being in causal connection which were capable of being actually observed: a demand which doubtless originates from the same instinct in the search for knowledge as that of continuity, and which really gives the law of causality the true character of an empirical law, i.e. one of actual experience.
The consistent fulfilment of these two postulates combined together is, I believe, the mainspring of Einstein's method of investigation; this imbues his results with their far-reaching importance in the construction of a physical picture of the world. In this respect his endeavours will probably not encounter any opposition in the matter of principle on the part of scientists. For both postulates—(1) that of continuity and (2) that of causal relationship between only such things as lie within the realm of observation—are of an inherent nature, i.e. contained in the very nature of the problem. The only question that might be raised is whether it is expedient to abandon such useful working hypotheses as "forces at a distance."
The principle of continuity requires that all physical laws allow of formulation as differential laws, i.e. physical laws must be expressible in a form such that the physical state at any point is completely determined by that of the point in its immediate neighbourhood. Consequently, the distances between points, which are at finite distances from one another, must not occur in these laws, but only those between points infinitely near to one another. The law of attraction of Newton given above, inasmuch as it involves "action at a distance," disobeys the first postulate.
The second postulate, that of a stricter form of expression for causality in its occurrence in physical laws, is intimately connected with a general theory of relativity of motions. Such a general principle of relativity requires that all possible systems of reference in nature be equivalent for the description of physical phenomena, and hence it avoids the introduction of the very questionable conception of absolute space which, for reasons we know (see [§ 4]), could not be circumvented by Newtonian mechanics. A general theory of relativity would, in excluding the fictitious quantity "absolute space," reduce the laws of mechanics to motions of bodies relative to one another, which are actually and exclusively what we observe. Thus, its laws would be founded on observed facts more completely than are those of classical mechanics.