Note 156 — Chapter XXXIX · THE CO-ORDINATION OF PHYSICAL FACTS ACCOMPLISHED BY RELATIVITY

A tensor expression built up of the ’s, and presenting the property of conservation, is given by . We may identify this expression, therefore, with the presence of a permanent entity such as matter. Outside these permanent entities the space-time structure would be given by , 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.