Page:Eddington A. Space Time and Gravitation. 1920.djvu/225

when the actual deflection of the ray is only 10^-7, or 0″.02. In the case of a star seen behind the sun the added divergence has no time to take effect; but when the light has to travel a stellar distance after the divergence is produced, it becomes weakened by it. Generally in stellar phenomena the weakening of the light should be more prominent than the actual deflection.

Note 13 (p. 141).

The relations are (Report, § 39) $G_{\mu \nu }^{\nu }={\tfrac {1}{2}}{\frac {\partial G}{\partial x_{\mu }}}$($\mu$ = 1, 2, 3, 4),</math> where $G_{\mu \nu }^{\nu }$ is the (contracted) covariant derivative of $G_{\mu }^{\nu }$, or $g^{\nu a}G_{\mu a}$.

I doubt whether anyone has performed the laborious task of verifying these identities by straightforward algebra.

Note 14 (p. 158).

The modified law for spherical space-time is in empty space $G_{\mu \nu }=\lambda g_{\mu \nu }$.

In cylindrical space-time, matter is essential. The law in space occupied by matter is $G_{\mu \nu }-{\tfrac {1}{2}}g_{\mu \nu }\left(G-2\lambda \right)=-8\pi T_{\mu \nu }$, the term $2\lambda$ being the only modification. Spherical space-time of radius $R$ is given by $\lambda =3/R^{2}$; cylindrical space-time by $\lambda =1/R^{2}$ provided matter of average density $\rho =1/4\pi R^{2}$ is present. (See Report, §§ 50, 51.) The total mass of matter in the cylindrical world is ${\tfrac {1}{2}}\pi R$. This must be enormous, seeing that the sun s mass is only 1 1/2 kilometres.

Note 15 (p. 174).

Weyl's theory is given in Berlin. Sitzungsberichte, 30 May, 1918; Annalen der Physik, Bd. 59 (1919), p. 101.

Note 16 (p. 177).

The argument is rather more complicated than appears in the text, where the distinction between action-density and action in a region, curvature and total curvature in a region, has not been elaborated. Taking a definitely marked out region in space and time, its measured volume will be increased 16-fold