Page:Eddington A. Space Time and Gravitation. 1920.djvu/224

It may be mentioned that the line-element $ds^{2}=-dr^{2}-r^{2}d\theta ^{2}+\left(1-2m/r\right)dt^{2}$, gives one-half the observed deflection of light, and one-third the motion of perihelion of Mercury. As both these can be obtained on older theories, taking account of the variation of mass with velocity, the coefficient $\gamma ^{-1}$ of $dr^{2}$ is the essentially novel point in Einstein's theory.

Note 11 (p. 131).

It is often supposed that by the Principle of Equivalence any invariant property which holds outside a gravitational field also holds in a gravitational field; but there is necessarily some limitation on this equivalence. Consider for instance the two invariant equations $ds^{2}=1$, $ds^{2}\left(1+k^{4}B_{\mu \nu \sigma }^{\rho }B_{\rho }^{\mu \nu \sigma }\right)=1$, where $k$ is some constant having the dimensions of a length. Since $B_{\mu \nu \sigma }^{\rho }$ vanishes outside a gravitational field, if one of these equations is true the other will be. But they cannot both hold in a gravitational field, since there $B_{\mu \nu \sigma }^{\rho }B_{\rho }^{\mu \nu \sigma }$ does not vanish, and is in fact equal to $24m^{2}/r^{6}$. (I believe that the numerical factor 24 is correct; but there are 65,536 terms in the expression, and the terms which do not vanish have to be picked out.

This ambiguity of the Principle of Equivalence is referred to in Report, §§ 14, 27; and an enunciation is given which makes it definite. The enunciation however is merely an explicit statement, and not a defence, of the assumptions commonly made in applying the principle.

So far as general reasoning goes there seems no ground for choosing $ds^{2}$ rather than $ds^{2}(1+24k^{4}m^{2}/r^{6})$, or any similar expression, as the constant character in the vibration of an atom.

Note 12 (p. 134).

Let two rays diverging from a point at a distance $R$ pass at distances $r$ and $r+dr$ from a star of mass $m$. The deflection being $4m/r$, their divergence will be increased by $4mdr/r^{2}$. This increase will be equal to the original divergence $dr/R$ if $r={\sqrt {4mR}}$. Take for instance $4m=10$ km., $R=10^{1}5$ km., then $r=10^{8}$ km. So that the divergence of the light will be doubled,