Page:Eddington A. Space Time and Gravitation. 1920.djvu/221

where $v_{r}$ is the velocity of the charge in the direction of $r$, $C$ the velocity of light, and the square bracket signifies antedated values. To the first order of $v_{r}/C$, the denominator is equal to the present distance $r$, so the expression reduces to $e/r$ in spite of the time of propagation. The foregoing formula for the potential was found by Liénard and Wiechert.

Note 7 (p. 97).

It is found that the following scheme of potentials rigorously satisfies the equations $G_{\mu \nu }=0$, according to the values of $G_{\mu \nu }$ in Note 5, ${\begin{matrix}-1/\gamma &0&0&0\\&-{x_{1}}^{2}&0&0\\&&-{x_{1}}^{2}\sin ^{2}{x_{2}}^{2}&0\\&&&\gamma \end{matrix}}$ where $\gamma =1-\kappa /x_{1}$ and $\kappa$ is any constant (see Report, § 28). Hence these potentials describe a kind of space-time which can occur in nature referred to a possible mesh-system. If $\kappa =0$, the potentials reduce to those for flat space-time referred to polar coordinates; and, since in the applications required $\kappa$ will always be extremely small, our coordinates can scarcely be distinguished from polar coordinates. We can therefore use the familiar symbols $r$, $\theta$, $\phi$, $t$, instead of $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$. It must, however, be remembered that the identification with polar coordinates is only approximate; and, for example, an equally good approximation is obtained if we write $x_{1}=r+{\tfrac {1}{2}}\kappa$, a substitution often used instead of $x_{1}=r$ since it has the advantage of making the coordinate-velocity of light more symmetrical.

We next work out analytically all the mechanical and optical properties of this kind of space-time, and find that they agree observationally with those existing round a particle at rest at the origin with gravitational mass ${\tfrac {1}{2}}\kappa$. The conclusion is that the gravitational field here described is produced by a particle of mass ${\tfrac {1}{2}}\kappa$—or, if preferred, a particle of matter at rest is produced by the kind of space-time here described.

Note 8 (p. 98).

Setting the gravitational constant equal to unity, we have for a circular orbit $m/r^{2}=v^{2}/r$, so that $m=v^{2}r$.