8 8 6 A P P E N D I X H space. The size of space depends on that term, and if the mean density is known, one can calculate this quantity. Since we do not know it, we can only say that it should be approxi- mately that of our neighborhood. The main thing is that the properties of space are given by matter, and the reciprocity is complete, since space gives the law of localization and matter gives the properties of space. This is more natural than to admit that the laws of space are given a priori (for example, the laws of the motion of bodies before they exist) we would thus have something both abso- lute and without cause: that is to say, we would have an incomplete causality. There is another point on which the theory of relativity is incomplete. In the gravitational equations there is a first term (see the summary of the second lecture) that represents the gravitational potential and which can be obtained exactly through purely theoretical proce- dures, while the second term contains a density obtained empirically. From Coulomb to Maxwell, the electromagnetic field has been constructed by inductive means. Based on what we already know, it is certain that the mathematical nature of this field is that of an antisymmetric tensor , and in the special theory of relativity it appears as such in Max- well’s equations when written in covariant notation. These equations have been written in a variational form that is very pretty from a mathematical standpoint. The quantities to vary are the and the nevertheless, there are things that do not satisfy me here. There is a problem that already existed in Maxwell’s electrodynamics. We know what protons and electrons are. These two elementary bodies must be solutions of field equations “and they aren’t.” There is no way to counteract the force of repulsion between the particles the elec- trons consist of. Poincaré introduced a pressure, but this is not natural and it is rather only a means of calculation. If we do away with the electron and deal only with the field, we are left with the gravitational and electromagnetic fields, that is, with two things totally inde- pendent of one another from the logical point of view. We thus face a dualistic system, something which one gets recurring to the variational form . If this integral is invariant, the equations obtained satisfy the principle of general relativ- ity. The function H is known for gravitational phenomena and, according to Maxwell, it can be calculated for electromagnetic phenomena, but one thus arrives at two logically indepen- dent things whose sum defines space. It would be preferable to have one theory in which the function from which all things are derived might have a single structure and not be com- posed of the sum of two independent things. This is a mathematical problem: to construct a function that is not composed of logically independent members, and what was said of H we must say of the and , which express the properties of space. The problem turns out to be very difficult because we try to satisfy a strive for unity without having any physical fact on which to rely, and so the question is reduced to the problem of mathematical simplicity. If it is resolved, we should start looking for observable consequences. fik gik fik var( Hdν) 0= fik gik