When working with tensors in spaces where the covariant and contravariant tensors' components have a different value (the underlying metric is not Euclidean) one frequently wants to express formulations with some or all of the tensors's indices expressed either in covariant or contravariant form. In previous Maple releases, also in Maple 2021, you can raise or lower free indices multiplying by the metric and performing the contraction. That, however, involves a whole simplification process not always desired, and does not result in flipping the character of repeated indices. The SubstituteTensorIndices is also useful for that purpose but requires changing the indices one by one. Instead, to handle the whole manipulation operation, you can use ToCovariant and ToContravariant.

Several options are available to adjust the operation in different ways, as explained in the Options section above. Perhaps two more relevant ones are changecharacteroffreeindices (default value is false), that can be used to receive an expression where you get all free indices flipping their character, and onlytheseindices = ... to restrict the operation to only some of the indices.

Note that closely related to ToCovariant and ToContravariant, the Physics package includes a SubstituteTensorIndices command.

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

$\mathrm{Setup}\left(\mathrm{coordinates}=\mathrm{Cartesian}\,\mathrm{tensors}=A\left[\mathrm{\mu }\right]\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(x\,y\,z\,t\right)\right\}$

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

$\left[\mathrm{coordinatesystems}=\left\{X\right\}\,\mathrm{tensors}=\left\{A_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}\right]$

$\mathrm{CompactDisplay}\left(A\left(X\right)\right)$

$A\left(x\,y\,z\,t\right)\mathrm{will\; now\; be\; displayed\; as}A$

$F\left[\mathrm{\mu },\mathrm{\nu }\right]=\mathrm{d\_}\left[\mathrm{\mu }\right]\left(A\left[\mathrm{\nu }\right]\left(X\right)\right)-\mathrm{d\_}\left[\mathrm{\nu }\right]\left(A\left[\mathrm{\mu }\right]\left(X\right)\right)$

$F_{\mathrm{\mu },\mathrm{\nu }}={\partial }_{\mathrm{\mu }}\left(A_{\mathrm{\nu }}\right)-{\partial }_{\mathrm{\nu }}\left(A_{\mathrm{\mu }}\right)$

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

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

$\left\{A_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,F_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}$

$F\left[\right]$

$F_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}0& {\left(A_2\right)}_x-{\left(A_1\right)}_y& {\left(A_3\right)}_x-{\left(A_1\right)}_z& {\left(A_4\right)}_x-{\stackrel{\mathbf{.}}{A}}_1\\ {\left(A_1\right)}_y-{\left(A_2\right)}_x& 0& {\left(A_3\right)}_y-{\left(A_2\right)}_z& {\left(A_4\right)}_y-{\stackrel{\mathbf{.}}{A}}_2\\ {\left(A_1\right)}_z-{\left(A_3\right)}_x& {\left(A_2\right)}_z-{\left(A_3\right)}_y& 0& {\left(A_4\right)}_z-{\stackrel{\mathbf{.}}{A}}_3\\ {\stackrel{\mathbf{.}}{A}}_1-{\left(A_4\right)}_x& {\stackrel{\mathbf{.}}{A}}_2-{\left(A_4\right)}_y& {\stackrel{\mathbf{.}}{A}}_3-{\left(A_4\right)}_z& 0\end{array}\right]$

$F\left[\mathrm{`~`}\right]$

$F_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}0& -{\left(A_{\phantom{}\phantom{2}}^{\phantom{}2}\right)}_x+{\left(A_{\phantom{}\phantom{1}}^{\phantom{}1}\right)}_y& -{\left(A_{\phantom{}\phantom{3}}^{\phantom{}3}\right)}_x+{\left(A_{\phantom{}\phantom{1}}^{\phantom{}1}\right)}_z& -{\left(A_{\phantom{}\phantom{4}}^{\phantom{}4}\right)}_x-{\stackrel{\mathbf{.}}{A}}_{\phantom{}\phantom{1}}^{\phantom{}1}\\ -{\left(A_{\phantom{}\phantom{1}}^{\phantom{}1}\right)}_y+{\left(A_{\phantom{}\phantom{2}}^{\phantom{}2}\right)}_x& 0& -{\left(A_{\phantom{}\phantom{3}}^{\phantom{}3}\right)}_y+{\left(A_{\phantom{}\phantom{2}}^{\phantom{}2}\right)}_z& -{\left(A_{\phantom{}\phantom{4}}^{\phantom{}4}\right)}_y-{\stackrel{\mathbf{.}}{A}}_{\phantom{}\phantom{2}}^{\phantom{}2}\\ -{\left(A_{\phantom{}\phantom{1}}^{\phantom{}1}\right)}_z+{\left(A_{\phantom{}\phantom{3}}^{\phantom{}3}\right)}_x& -{\left(A_{\phantom{}\phantom{2}}^{\phantom{}2}\right)}_z+{\left(A_{\phantom{}\phantom{3}}^{\phantom{}3}\right)}_y& 0& -{\left(A_{\phantom{}\phantom{4}}^{\phantom{}4}\right)}_z-{\stackrel{\mathbf{.}}{A}}_{\phantom{}\phantom{3}}^{\phantom{}3}\\ {\stackrel{\mathbf{.}}{A}}_{\phantom{}\phantom{1}}^{\phantom{}1}+{\left(A_{\phantom{}\phantom{4}}^{\phantom{}4}\right)}_x& {\stackrel{\mathbf{.}}{A}}_{\phantom{}\phantom{2}}^{\phantom{}2}+{\left(A_{\phantom{}\phantom{4}}^{\phantom{}4}\right)}_y& {\stackrel{\mathbf{.}}{A}}_{\phantom{}\phantom{3}}^{\phantom{}3}+{\left(A_{\phantom{}\phantom{4}}^{\phantom{}4}\right)}_z& 0\end{array}\right]$

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

$g_{\mathrm{\alpha },\mathrm{\mu }}{g}_{\mathrm{\beta },\mathrm{\nu }}{F}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}=g_{\mathrm{\alpha },\mathrm{\mu }}{g}_{\mathrm{\beta },\mathrm{\nu }}{\partial }_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(A_{\phantom{}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\beta }}\right)-g_{\mathrm{\alpha },\mathrm{\nu }}{g}_{\mathrm{\beta },\mathrm{\mu }}{\partial }_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(A_{\phantom{}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\beta }}\right)$

$\mathrm{indets}\left(\,\mathrm{Or}\left(\mathrm{specindex}\left(F\right)\,\mathrm{specfunc}\left(A\right)\right)\right)$

$\left\{F_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta }}\,A_{\phantom{}\phantom{\mathrm{\beta }}}^{\phantom{}\mathrm{\beta }}\right\}$

$\mathrm{ToContravariant}\left(\,\mathrm{only}=\mathrm{\mu }\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{only}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{onlytheseindices}\text{'}$

$g_{\mathrm{\alpha },\mathrm{\mu }}{F}_{\phantom{}\phantom{\mathrm{\alpha }}\mathrm{\nu }}^{\phantom{}\mathrm{\alpha }\phantom{\mathrm{\nu }}}=g_{\mathrm{\alpha },\mathrm{\mu }}{\partial }_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\left(A_{\mathrm{\nu }}\right)-g_{\mathrm{\alpha },\mathrm{\mu }}{\partial }_{\mathrm{\nu }}\left(A_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right)$

$\mathrm{indets}\left(\,\mathrm{Or}\left(\mathrm{specindex}\left(F\right)\,\mathrm{specfunc}\left(A\right)\right)\right)$

$\left\{F_{\mathrm{\nu }\phantom{\mathrm{\alpha }}}^{\phantom{\mathrm{\nu }}\mathrm{\alpha }}\,A_{\mathrm{\nu }}\,A_{\phantom{}\phantom{\mathrm{\alpha }}}^{\phantom{}\mathrm{\alpha }}\right\}$

$\mathrm{ToContravariant}\left(\,\mathrm{only}=\mathrm{\mu }\,\mathrm{changecharacter}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{only}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{onlytheseindices}\text{'}$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{changecharacter}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{changecharacteroffreeindices}\text{'}$

$F_{\phantom{}\phantom{\mathrm{\mu }}\mathrm{\nu }}^{\phantom{}\mathrm{\mu }\phantom{\mathrm{\nu }}}={\partial }_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(A_{\mathrm{\nu }}\right)-{\partial }_{\mathrm{\nu }}\left(A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)$

$\mathrm{indets}\left(\,\mathrm{Or}\left(\mathrm{specindex}\left(F\right)\,\mathrm{specfunc}\left(A\right)\right)\right)$

$\left\{F_{\mathrm{\nu }\phantom{\mathrm{\mu }}}^{\phantom{\mathrm{\nu }}\mathrm{\mu }}\,A_{\mathrm{\nu }}\,A_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right\}$

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

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

$\left\{B\,G\,A_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,F_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}$

$A\left[\mathrm{\alpha }\right]B\left[\mathrm{\beta }\right]F\left[\mathrm{\mu },\mathrm{\nu }\right]G\left[\mathrm{\nu },\mathrm{\alpha }\right]+A\left[\mathrm{\beta }\right]B\left[\mathrm{\alpha }\right]F\left[\mathrm{\mu },\mathrm{\rho }\right]G\left[\mathrm{\rho },\mathrm{\alpha }\right]$

$A_{\mathrm{\alpha }}{B}_{\mathrm{\beta }}{F}_{\mathrm{\mu },\mathrm{\nu }}{G}_{\mathrm{\nu },\mathrm{\alpha }}+A_{\mathrm{\beta }}{B}_{\mathrm{\alpha }}{F}_{\mathrm{\mu },\mathrm{\rho }}{G}_{\mathrm{\rho },\mathrm{\alpha }}$

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

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

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

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

$A_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}{g}_{\mathrm{\beta },\mathrm{\lambda }}{B}_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}{g}_{\mathrm{\mu },\mathrm{\sigma }}{F}_{\phantom{}\phantom{\mathrm{\sigma },\mathrm{\tau }}}^{\phantom{}\mathrm{\sigma },\mathrm{\tau }}{G}_{\phantom{}\phantom{\mathrm{\upsilon },\mathrm{\chi }}}^{\phantom{}\mathrm{\upsilon },\mathrm{\chi }}{g}_{\mathrm{\tau },\mathrm{\upsilon }}{g}_{\mathrm{\kappa },\mathrm{\chi }}+g_{\mathrm{\beta },\mathrm{\nu }}{A}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{B}_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}{g}_{\mathrm{\lambda },\mathrm{\mu }}{F}_{\phantom{}\phantom{\mathrm{\lambda },\mathrm{\sigma }}}^{\phantom{}\mathrm{\lambda },\mathrm{\sigma }}{G}_{\phantom{}\phantom{\mathrm{\tau },\mathrm{\upsilon }}}^{\phantom{}\mathrm{\tau },\mathrm{\upsilon }}{g}_{\mathrm{\sigma },\mathrm{\tau }}{g}_{\mathrm{\kappa },\mathrm{\upsilon }}$

$\mathrm{indets}\left(\,\mathrm{specindex}\left(\left[A\,B\,F\,G\right]\right)\right)$

$\left\{A_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}\,A_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\,B_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}\,B_{\phantom{}\phantom{\mathrm{\lambda }}}^{\phantom{}\mathrm{\lambda }}\,F_{\phantom{}\phantom{\mathrm{\lambda },\mathrm{\sigma }}}^{\phantom{}\mathrm{\lambda },\mathrm{\sigma }}\,F_{\phantom{}\phantom{\mathrm{\sigma },\mathrm{\tau }}}^{\phantom{}\mathrm{\sigma },\mathrm{\tau }}\,G_{\phantom{}\phantom{\mathrm{\tau },\mathrm{\upsilon }}}^{\phantom{}\mathrm{\tau },\mathrm{\upsilon }}\,G_{\phantom{}\phantom{\mathrm{\upsilon },\mathrm{\chi }}}^{\phantom{}\mathrm{\upsilon },\mathrm{\chi }}\right\}$

$\mathrm{Simplify}\left(-\right)$

$0$

$\mathrm{ToContravariant}\left(\,\mathrm{changerepeatedindices}=\mathrm{false}\right)$

$A_{\mathrm{\alpha }}{g}_{\mathrm{\beta },\mathrm{\kappa }}{B}_{\phantom{}\phantom{\mathrm{\kappa }}}^{\phantom{}\mathrm{\kappa }}{g}_{\mathrm{\lambda },\mathrm{\mu }}{F}_{\phantom{}\phantom{\mathrm{\lambda }}\mathrm{\nu }}^{\phantom{}\mathrm{\lambda }\phantom{\mathrm{\nu }}}{G}_{\mathrm{\nu },\mathrm{\alpha }}+g_{\mathrm{\beta },\mathrm{\nu }}{A}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{B}_{\mathrm{\alpha }}{g}_{\mathrm{\kappa },\mathrm{\mu }}{F}_{\phantom{}\phantom{\mathrm{\kappa }}\mathrm{\rho }}^{\phantom{}\mathrm{\kappa }\phantom{\mathrm{\rho }}}{G}_{\mathrm{\rho },\mathrm{\alpha }}$

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