Page:Eddington A. Space Time and Gravitation. 1920.djvu/220

204 Note 4 (p. 81).

The condition for flat space in two dimensions is

$$ \frac{\partial}{\partial x_1} \left (\frac{g_{12}}            {g_{11}\sqrt{\left (g_{11}g_{22}-{g_{12}}^2 \right )}}        \frac{\partial g_{11}}             {\partial x_2}      - \frac{1}             {\sqrt{\left (g_{11}g_{22} - {g_{12}}^2 \right )}}       \frac{\partial g_{22}}             {\delta x_1} \right )$$ $ + \frac{\partial}{\partial x_2} \left (\frac{2}              {\sqrt{\left (g_{11}g_{22} - {g_{12}}^2 \right )}}         \frac{\partial g_{12}}{\partial x_1}        - \frac{1}{\sqrt{\left (g_{11} g_{22} - {g_{12}}^2 \right )}}          \frac{\partial g_{11}}{\partial x_2}   \right. $|undefined $ \left. - \frac{g_{12}}       {g_{11} \sqrt{\left ( g_{12}g_{22} - {g_{12}}^2 \right )}}   \frac{\partial g_{11}}{\partial x_1} \right ) = 0$.|undefined

Note 5 (p. 89).

Let $$g$$ be the determinant of four rows and columns formed with the elements $$g_{\mu\nu}$$.

Let $$g^{\mu\nu}$$ be the minor of $$g_{\mu\nu}$$, divided by $$g$$. Let the "3-index symbol" {$$\mu\nu$$, $$\lambda$$} denote summed for values of $$a$$ from 1 to 4. There will be 40 different 3-index symbols.

Then the Riemann-Christoffel tensor is the terms containing $$\epsilon$$ being summed for values of $$\epsilon$$ from 1 to 4.

The "contracted" Riemann-Christoffel tensor $$G_{\mu\nu}$$ can be reduced to

$$ G_{\mu\nu} = - \frac{\partial}{\partial x_a} \{ \mu\nu, a \} + \{ \mu a, \beta \} \{ \nu\beta , a \}$$ $ + \frac {\partial^2}{\partial x_\mu \partial x_\nu} \log \sqrt{-g} - \{ \mu\nu, a \} \frac{\partial}{\partial x_a} \log \sqrt{-g} $, where in accordance with a general convention in this subject, each term containing a suffix twice over ($$a$$ and $$\beta$$) must be summed for the values 1, 2, 3, 4 of that suffix.

The curvature $$G = g^{\mu\nu}G_{\mu\nu}$$, summed in accordance with the foregoing convention.

Note 6 (p. 94).

The electric potential due to a charge $$e$$ is