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

MATHEMATICAL NOTES 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, 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 so that