The SubstituteTensorIndices substitutes indices in tensors - the ones displayed when you enter Define(); see Define to define one. Nowhere else are the indices substituted, and the substitution can be performed on a covariant index, the corresponding contravariant one, or on both.

The first argument is a substitution equation with an index on the left-hand side, or a set or list of them. All the substitution equations are expected in this first argument. The second argument is the target, where the substitutions are to be performed.

The tensors where indices are substituted are re-evaluated after substitution; this re-evaluation can optionally be suppressed giving the argument evaluatetensor = false. The expression where these re-evaluated tensors are introduce is by default not re-evaluated; you can change that passing the optional argument evaluateexpression = true.

By default both covariant and contravariant indices are substituted, even if the substitution equation has only one of them on the left-hand side; this behavior is convenient both when substituting contracted or free indices. To substitute only the kind of index found on the left-hand side of the substitution equation use covariantandcontravariant = false.

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

When substituting indices by numerical values, contravariant values are prefixed with ~, say as in ~1; otherwise, say as in 1, the value is interpreted as covariant. Hence, the equation ~nu = 1 will also transform ~nu into covariant. Substituting the natural way, say as in nu = 1, will automatically substitute both nu = 1 and also ~nu = ~1 keeping the covariant/contravariant character unchanged.

When the right-hand side of a substitution equation is a contravariant index, so suffixed with ~, say as in ~nu = ~1, note the separation between = and ~; that separation is important because ~ is also the element-wise operator and so =~ has other specific meaning.

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

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

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

$\mathrm{Define}\left(A\,B\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A\,B\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]A\left[\mathrm{~mu}\right]\mathrm{g\_}\left[\mathrm{~alpha},\mathrm{~nu}\right]B\left[\mathrm{\nu },\mathrm{\sigma },\mathrm{~rho}\right]$

$g_{\mathrm{\alpha },\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\nu }}{B}_{\mathrm{\nu },\mathrm{\sigma }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\nu }}\phantom{,\mathrm{\sigma }}\mathrm{\rho }}$

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

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

$\left[\left\{\mathrm{\alpha }\,\mathrm{\mu }\,\mathrm{\nu }\right\}\right],\left\{\mathrm{\sigma }\,\mathrm{~\ρ}\right\}$

$\mathrm{subs}\left(\mathrm{\alpha }=\mathrm{\beta }\,\right)$

$g_{\mathrm{\beta },\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\nu }}{B}_{\mathrm{\nu },\mathrm{\sigma }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\nu }}\phantom{,\mathrm{\sigma }}\mathrm{\rho }}$

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

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

$\left[\left\{\mathrm{\mu }\,\mathrm{\nu }\right\}\right],\left\{\mathrm{\beta }\,\mathrm{\sigma }\,\mathrm{~\α}\,\mathrm{~\ρ}\right\}$

$\mathrm{SubstituteTensorIndices}\left(\mathrm{\alpha }=\mathrm{\beta }\,\right)$

$g_{\mathrm{\beta },\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\beta },\mathrm{\nu }}}^{\phantom{}\mathrm{\beta },\mathrm{\nu }}{B}_{\mathrm{\nu },\mathrm{\sigma }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\nu }}\phantom{,\mathrm{\sigma }}\mathrm{\rho }}$

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

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

$\left[\left\{\mathrm{\beta }\,\mathrm{\mu }\,\mathrm{\nu }\right\}\right],\left\{\mathrm{\sigma }\,\mathrm{~\ρ}\right\}$

$\mathrm{subs}\left(\mathrm{\rho }=\mathrm{\beta }\,\right)$

$g_{\mathrm{\alpha },\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\nu }}{B}_{\mathrm{\nu },\mathrm{\sigma }\phantom{\mathrm{\rho }}}^{\phantom{\mathrm{\nu }}\phantom{,\mathrm{\sigma }}\mathrm{\rho }}$

$\mathrm{SubstituteTensorIndices}\left(\mathrm{\rho }=\mathrm{\beta }\,\right)$

$g_{\mathrm{\alpha },\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\nu }}{B}_{\mathrm{\nu },\mathrm{\sigma }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\nu }}\phantom{,\mathrm{\sigma }}\mathrm{\beta }}$

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

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

$\left[\left\{\mathrm{\alpha }\,\mathrm{\mu }\,\mathrm{\nu }\right\}\right],\left\{\mathrm{\sigma }\,\mathrm{~\β}\right\}$

$\mathrm{SubstituteTensorIndices}\left(\left[\mathrm{\rho }=\mathrm{\beta }\,\mathrm{\alpha }=\mathrm{\gamma }\right]\,\right)$

$g_{\mathrm{\gamma },\mathrm{\mu }}{A}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\gamma },\mathrm{\nu }}}^{\phantom{}\mathrm{\gamma },\mathrm{\nu }}{B}_{\mathrm{\nu },\mathrm{\sigma }\phantom{\mathrm{\beta }}}^{\phantom{\mathrm{\nu }}\phantom{,\mathrm{\sigma }}\mathrm{\beta }}$

$\mathrm{g\_}\left[\mathrm{\mu },\mathrm{\nu }\right]$

$g_{\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{SubstituteTensorIndices}\left(\mathrm{\nu }=1\,\right)$

$g_{1,\mathrm{\mu }}$

$\mathrm{SubstituteTensorIndices}\left(\mathrm{\nu }=1\,\,\mathrm{evaluatetensor}=\mathrm{false}\right)$

$g_{\mathrm{\mu },1}$

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