Observing that the field equation (96) may be written in the form we get from (96) the equations, From this follows If the universe is quasi-Euclidean, and its radius of curvature therefore infinite, then σ would vanish. But it is improbable that [Pg 113] the mean density of matter in the universe is actually zero; this is our third argument against the assumption that the universe is quasi-Euclidean. Nor does it seem possible that our hypothetical pressure can vanish; the physical nature of this pressure can be appreciated only after we have a better theoretical knowledge of the electromagnetic field. According to the second of equations (123) the radius, , of the universe is determined in terms of the total mass, , of matter, by the equation The complete dependence of the geometrical upon the physical properties becomes clearly apparent by means of this equation. Thus we may present the following arguments against the conception of a space-infinite, and for the conception of a space-bounded, universe:— 1. From the standpoint of the theory of relativity, the condition for a closed surface is very much simpler than the corresponding boundary condition at infinity of the quasi-Euclidean structure of the universe. 2. The idea that Mach expressed, that inertia depends upon the mutual action of bodies, is contained, to a first approximation, in the equations of the theory of relativity; it follows from these equations that inertia depends, at least in part, upon mutual actions between masses. As it is an unsatisfactory assumption to make that inertia depends in part upon mutual actions, and in part upon an independent property of space, Mach's idea gains in probability. But this idea of Mach's corresponds only to a finite universe, bounded in space, and not to a quasi-Euclidean, infinite universe. From the standpoint of epistemology [Pg 114] it is more satisfying to have the mechanical properties of space completely determined by matter, and this is the case only in a space-bounded universe. 3. An infinite universe is possible only if the mean density of matter in the universe vanishes. Although such an assumption is logically possible, it is less probable than the assumption that there is a finite mean density of matter in the universe. [Pg 115] INDEX A Accelerated masses, inductive action of, 108 Addition and subtraction of tensors, 14 —theorem of velocities, 38 B Biot-Savart force, 44 C Centrifugal force, 64 Clocks, moving, 38 Compressible viscous fluid, 22 Concept of space, 3 —time, 28 Conditions of orthogonality, 7 Congruence, theorems of, 3 Conservation principles, 54 Continuum, four-dimensional, 31 Contraction of tensors, 14 Contra-variant vectors, 69 —tensors, 71 Co-ordinates, preferred systems of, 8 Co-variance of equation of continuity, 21 Co-variant, 12 et seq. —vector, 68 Criticism of principle of inertia, 62 Criticisms of theory of relativity, 29