, which is equivalent to
. If now we integrate
round a point-centre, assuming space-time to be flat at infinity, a certain undetermined constant of integration must be introduced into our solution. This is a purely mathematical necessity, having nothing to do with the physical significance of our equations. Now, the constant being undetermined, we may assign any value to it we please. According to the positive or negative value we might ascribe to this constant, we should obtain a field of force either attractive or repulsive around our particle; and the law of force distribution would be approximately that of the inverse square. Only if we attributed a vanishing value to the constant would no field of force be present. This solution, however, would be scarcely permissible. Thus, we have shown that the space-time theory, independently of any empirical study of nature, suggests the existence of permanent entities surrounded by fields of force. In actual practice, we identify the undetermined constant with the “mass” of the central particle; and since mass is always positive in the world of our experience, the constant is always a positive number, and attraction is the outcome. In this way the integrated form of Einstein’s law of gravitation is obtained.
[157] Or else so profoundly modified as to constitute, to all intents and purposes, a new theory.
[158] From the standpoint of scientific method, it is most instructive to contrast with Einstein’s theory one which fails to satisfy the demands of science. We refer to Dr. Whitehead’s attempt to interpret the phenomenon of gravitation in terms of Minkowski’s flat space-time. In the last chapter we noted that prior to Einstein’s sweeping generalisation a solution of gravitation on the basis of flat space-time had been worked out by Poincaré, Nordström and others. Dr. Whitehead wishes to revert to this type of solution. One of its advantages, he claims, is that it obviates the idea that matter influences the structure of space-time. Whether this is an advantage or not is a mere question of feeling, but in science it is a highly dangerous procedure to reject a simple co-ordination of the facts of experience in favour of some a priori preference which offers no means of being put to a test. Dr. Whitehead would probably claim that in his case this criticism does not hold, for he has given definite arguments in support of his contentions. But what are these arguments? The first consists in his writing: “The only possible structure is that of planes and straight lines.” Yet, as no mathematician would agree with this contention, it would appear that Whitehead was merely restating his a priori belief in the inevitable homogeneity of the space-time structure without justifying his views in any way. Elsewhere he writes: “Are these material bodies really the ultimate data of perception, incapable of further analysis? If they are, I at once surrender.” Now, in point of fact, it is scarcely probable that any scientist would agree with Whitehead’s hypothetical opponent in maintaining that material objects do constitute the ultimate data of perception. But what conceivable connection is there between this problem and the totally different one pertaining to a possible action of matter on the structure of space-time? Obviously, Whitehead is implicitly assuming that we know enough of space-time a priori to assert that it can never be affected by whatever is not an ultimate datum of perception. Hence his second argument, just like the first, reduces to a mere restatement of his a priori convictions. Let us proceed.
Having satisfied himself that material bodies are not what his hypothetical opponent claims them to be, he considers his point proved. Material bodies are henceforth referred to as “a certain coherence of sense-objects such as colours, sounds and touches.” And these sense-objects “at once proclaim themselves to be adjectives of events.” This, according to Dr. Whitehead, precludes matter from exerting any effect on space-time, so that Einstein’s general theory must be discarded.
It is hard to see how a solution of the problems of nature will ever be advanced by such loose arguments. At any rate, having assumed space-time to be flat, or at least homogeneous, everywhere and everywhen, Dr. Whitehead is compelled to account in terms of forces generated by matter for the gravitational effects predicted by Einstein. He explains the occurrence of the shift-effect by tracing it “to the combination of two causes, one being the change in the apparent mass due to the gravitational potential and the other being the change in the electric cohesive forces of the molecule due to the gravitational field.” We shall assume on his authority that the first of the two causes he mentions can be deduced from his gravitational equations, and shall concern ourselves solely with the second cause. Now, unless we get our information from text-books of physics, written more than thirty years ago, we shall never labour under the impression that the spectral lines have anything to do with the constitution of the molecule. Molecular vibrations may cause a broadening of the lines owing to a Doppler effect, and are recognised as giving rise to the diffuse band spectra, but modern experiment has traced the visible line spectra to occurrences within the atom. It follows that these molecular adjustments which Dr. Whitehead is seeking to introduce would appear to have no effect one way or another on the shift he proposes to account for. However, it is not from the standpoint of scientific accuracy that we wish to criticise Dr. Whitehead’s statements, but solely from that of method.