pressure is due to the momentum of the moving light energy. It has also been proved, as we shall see, that light has weight. A material particle may be regarded as a very condensed localization of energy. Matter is no longer regarded as permanent in the old sense. A cooling body is losing mass. We have seen how this theory explains the stability of the helium nucleus, and also how the variation of mass with velocity explains the "fine structure" of the hydrogen lines. CHAPTER X GEOMETRY AND PHYSICS: PERHAPS the most important aspect of the new co-ordination of phenomena given by Einstein's theory is the rôle played by geometry in that theory. This is the aspect that Minkowski was the first to grasp clearly. Certain effects, which, on the old outlook, would have been ascribed to special " laws of nature," are now regarded as due to the geometrical properties of the space-time continuum. It has been shown, in fact, that geometry and laws of nature are not independent of one another. As long as it was assumed that all spatial measurements must necessarily conform to one particular geometry -Euclid's geometry-science had only, as it were, one variable to play with, namely, laws of nature. The mathematical description of phenomena proceeded on the assumption that all measurements necessarily conformed to Euclid's geometry. A triangle formed by straight lines, for instance, must necessarily contain two right angles. An instance of the way in which laws of nature are conditioned by such assumptions has been given by Poincaré. Suppose it were found that a triangle formed by a star and by the earth in two positions on its orbit had its interior angles less than two right angles. This would not necessarily be taken to indicate that Euclidean geometry could not be applied to stellar space. Such an assumption would explain the observations, but they can also be explained by assuming different laws of nature. For it is assumed, in measuring this stellar triangle, that light is propagated in straight lines. If we wish to preserve Euclidean geometry, therefore, we can do so by assuming a new law of nature, namely, that light is not propagated in straight lines. We have two variables in terms of which to describe any phenomenon, geometry and laws of nature. They are not independent. The choice we make in one region influences the choice we make in the other. The criterion appears to be " simplicity.' Thus, in the supposed case of the stellar triangle, retention of Euclidean geometry means a farreaching alteration in our physics. It might prove to be so extensive an alteration that Euclidean geometry would hardly seem worth the price paid for it. An altered geometry, on the other hand, might lead to all sorts of complications. The advantages and disadvantages of each course would have to be balanced up. Up to the time that the theory of relativity appeared no geometry other than Euclid's had ever been assumed as the basis of a scientific theory. Indeed, until the invention, about a hundred years ago, of non-Euclidean geometries, the geometry of Euclid was generally regarded as a necessity of thought. The axioms on which Euclid's geometry is based were regarded as unescapable. There was some uneasiness, it is true, about one of the axioms, the axiom concerning parallel lines, but this uneasiness attached chiefly to the clumsy and apparently arbitrary form of the axiom. It was not seriously questioned that the axiom was true. Attempts were made, for something like a thousand years, to deduce this axiom from the other axioms of Euclid. None of these attempts were successful. Early in the eighteenth century a very able logician, Saccheri, tried the effect of constructing a geometry that denied Euclid's parallel axiom. Saccheri expected, in this way, to be led into self-contradiction but, although he tried very hard, he never succeeded in contradicting himself. What he actually did, although he never realised it, was to construct the first non-Euclidean geometry. The first serious doubts of the validity of Euclid's axiom seem to have occurred to Gauss, but he was afraid to publish his discoveries. The first non-Euclidean geometry to be published was discovered early in the nineteenth century by two men independently, a Russian, Lobachevsky, and a Hungarian, Bolyai. Each of these men constructed a perfectly self-consistent geometry which denied Euclid's parallel axiom. Another non-Euclidean geometry was later constructed by Riemann, and we now know that an infinite number of such geometries is possible. Geometrical axioms have, in fact, acquired an entirely new logical status. We now know that they are not necessities of thought. What were supposed to be logical necessities are really the outcome of limited experience and deeply rooted habits of mind. We can invent what geometrical axioms we like, provided that they are consistent with one another, and the logical consequences of these axioms form a system of geometry. With this discovery the whole task of giving a mathematical description of nature assumes a new aspect. We need not assume that physical space, the space in which events happen, is necessarily Euclidean, any more than we need assume that the music of the spheres, should we ever hear it, must be in the diatonic scale. The geometry of space must be determined by experiment. We must observe the behaviour of our measuring apparatus without any a priori ideas as to how that apparatus must behave. The geometry of actual physical space can only be determined by actual physical apparatus. As long as Euclid's geometry was regarded as a logical necessity it was obvious that if space had any mathematical properties at all they must be consistent with Euclidean geometry. Otherwise the attempt to formulate a mathematical description of nature would have to be given up. But now that we know that there are an infinite number of geometries all on the same footing, all logically possible, it becomes an open question which of these geometries is most suited to describe natural phenomena. Thus the whole of the Newtonian picture of the world becomes open to examination. Newton assumed, as a necessity of thought, that the geometry of space was Euclidean ; |