The SumOverRepeatedIndices performs the summation over the repeated indices of expression implied when using the Einstein summation convention. The summation takes into account the covariant and contravariant character of each contracted index.

The summation is performed from 1 to the dimension of spacetime, and optionally you indicate the indices (not their range) over which the summation is to be performed. The summation indices are indicated in sequence after expression. If no indices are indicated then summation is performed over all the repeated indices of expression.

To check and determine the free and repeated indices of an expression use Check.

By default, the summation is performed without simplifying the result; to have the result simplified before returning, indicate the simplifier on the right-hand-side of the optional argument simplifier = ...

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{g\_}\left[\mathrm{sc}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right\}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right\}$

$\mathrm{Setting}\mathrm{lowercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{space}\mathrm{indices}$

$\mathrm{The\; Schwarzschild\; metric\; in\; coordinates}\left[r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]$

$\mathrm{Parameters:}\left[m\right]$

$\mathrm{Signature:}\left(\mathrm{-\; -\; -\; +}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\frac{r}{2m-r}& 0& 0& 0\\ 0& -r^2& 0& 0\\ 0& 0& -r^2{\sin \left(\mathrm{\theta }\right)}^2& 0\\ 0& 0& 0& \frac{r-2m}{r}\end{array}\right]$

$\mathrm{R2}≔\frac{1}{2}\mathrm{LeviCivita}\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\rho },\mathrm{\sigma }\right]\mathrm{Riemann}\left[\mathrm{~rho},\mathrm{~sigma},\mathrm{\mu },\mathrm{\nu }\right]$

$\mathrm{R2}≔\frac{{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\rho },\mathrm{\sigma }}{R}_{\phantom{}\phantom{\mathrm{\rho },\mathrm{\sigma }}\mathrm{\mu },\mathrm{\nu }}^{\phantom{}\mathrm{\rho },\mathrm{\sigma }\phantom{\mathrm{\mu },\mathrm{\nu }}}}{2}$

$\mathrm{R2}\mathrm{Riemann}\left[\mathrm{~alpha},\mathrm{~beta},\mathrm{~mu},\mathrm{~nu}\right]$

$\frac{{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\rho },\mathrm{\sigma }}{R}_{\phantom{}\phantom{\mathrm{\rho },\mathrm{\sigma }}\mathrm{\mu },\mathrm{\nu }}^{\phantom{}\mathrm{\rho },\mathrm{\sigma }\phantom{\mathrm{\mu },\mathrm{\nu }}}{R}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}}{2}$

$\mathrm{Check}\left(\,\mathrm{all}\right)$

$\mathrm{The\; repeated\; indices\; per\; term\; are:}\left[\left\{\mathrm{...}\right\}\,\left\{\mathrm{...}\right\}\,\mathrm{...}\right]\mathrm{,\; the\; free\; indices\; are:}\left\{\mathrm{...}\right\}$

$\left[\left\{\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\mu }\,\mathrm{\nu }\,\mathrm{\rho }\,\mathrm{\sigma }\right\}\right],\varnothing$

$\mathrm{SumOverRepeatedIndices}\left(\right)$

$0$

$\mathrm{Riemann}\left[\mathrm{scalars}\right]$

$S_1=\frac{m^2}{r^6},S_2=-\frac{m^3}{r^9}$

$\mathrm{Riemann}\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }\right]\mathrm{Riemann}\left[\mathrm{~alpha},\mathrm{~beta},\mathrm{~mu},\mathrm{~nu}\right]$

$R_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}{R}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{SumOverRepeatedIndices}\left(\,'\mathrm{simplifier}'=\mathrm{simplify}\right)$

$\frac{48m^2}{r^6}$

$\mathrm{Riemann}\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }\right]\mathrm{Riemann}\left[\mathrm{~alpha},\mathrm{~beta},\mathrm{~rho},\mathrm{~sigma}\right]\mathrm{Riemann}\left[\mathrm{\rho },\mathrm{\sigma },\mathrm{~mu},\mathrm{~nu}\right]$

$R_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}{R}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta },\mathrm{\rho },\mathrm{\sigma }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta },\mathrm{\rho },\mathrm{\sigma }}{R}_{\mathrm{\rho },\mathrm{\sigma }\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{\mathrm{\rho }}\phantom{,\mathrm{\sigma }}\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{SumOverRepeatedIndices}\left(\,'\mathrm{simplifier}'=\mathrm{simplify}\right)$

$-\frac{96m^3}{r^9}$

Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.