Page:Eddington A. Space Time and Gravitation. 1920.djvu/107

V] geometrical scheme hitherto considered, so that force is not purely relative, and Newton's super-observer exists.

Perhaps the best survey of the meaning of our theory can be obtained from the standpoint of a ten-dimensional Euclidean continuum, in which space-time is conceived as a particular four-dimensional surface. It has to be remarked that in ten dimensions there are gradations intermediate between a flat surface and a fully curved surface, which we shall speak of as curved in the "first degree" or "second degree ." The distinction is something like that of curves in ordinary space, which may be curved like a circle, or twisted like a helix; but the analogy is not very close. The full "curvature" of a surface is a single quantity called $$G$$, built up out of the various terms $$G_{\mu\nu}$$ in somewhat the same way as these are built up out of $$B^\rho_{\mu\nu\sigma}$$.The following conclusions can be stated.

If $$B^\rho_{\mu\nu\sigma} = 0$$ (20 conditions) space-time is flat. This is the state of the world at an infinite distance from all matter and all forms of energy.

If $$G_{\mu\nu} = 0$$ (6 conditions) space-time is curved in the first degree. This is the state of the world in an empty region—not containing matter, light or electromagnetic fields, but in the neighbourhood of these forms of energy.

If $$G = 0$$ (1 condition) space-time is curved in the second degree. This is the state of the world in a region not containing matter or electrons (bound energy), but containing light or electromagnetic fields (free energy).

If $$G \text{ is not zero}$$

space-time is fully curved. This is the state of the world in a region containing continuous matter.

According to current physical theory continuous matter does not exist, so that strictly speaking the last case never arises. Matter is built of electrons or other nuclei. The regions lying between the electrons are not fully curved, whilst the regions inside the electrons must be cut out of space-time altogether. We cannot imagine ourselves exploring the inside of an electron