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

Note 4 (p. 81).

The condition for flat space in two dimensions is

${\frac {\partial }{\partial x_{1}}}\left({\frac {g_{12}}{g_{11}{\sqrt {\left(g_{11}g_{22}-{g_{12}}^{2}\right)}}}}{\frac {\partial g_{11}}{\partial x_{2}}}-{\frac {1}{\sqrt {\left(g_{11}g_{22}-{g_{12}}^{2}\right)}}}{\frac {\partial g_{22}}{\delta x_{1}}}\right)$

$+{\frac {\partial }{\partial x_{2}}}\left({\frac {2}{\sqrt {\left(g_{11}g_{22}-{g_{12}}^{2}\right)}}}{\frac {\partial g_{12}}{\partial x_{1}}}-{\frac {1}{\sqrt {\left(g_{11}g_{22}-{g_{12}}^{2}\right)}}}{\frac {\partial g_{11}}{\partial x_{2}}}\right.$

⁠⁠⁠⁠$\left.-{\frac {g_{12}}{g_{11}{\sqrt {\left(g_{12}g_{22}-{g_{12}}^{2}\right)}}}}{\frac {\partial g_{11}}{\partial x_{1}}}\right)=0$.

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 ${\tfrac {1}{2}}g^{\lambda a}\left({\frac {\partial g_{\mu a}}{\partial x_{\nu }}}+{\frac {\partial g_{\nu a}}{\partial x_{\mu }}}-{\frac {\partial g_{\mu \nu }}{\partial x_{a}}}\right)$ summed for values of $a$ from 1 to 4. There will be 40 different 3-index symbols.

Then the Riemann-Christoffel tensor is $B_{\mu \nu \sigma }^{\rho }=\{\mu \sigma ,\epsilon \}\{\epsilon \nu ,\rho \}-\{\mu \nu ,\epsilon \}\{\epsilon \sigma ,\rho \}+{\frac {\partial }{\partial x_{\nu }}}\{\mu \sigma ,\rho \}-{\frac {\partial }{\partial x_{\sigma }}}\{\mu \nu ,\rho \}$, 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 $\phi ={\frac {e}{\left[r\left(1-v_{r}/C\right)\right]}}$,