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 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.
[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]
Curvilinear co-ordinates, [65]