By default, the dimension of the spacetime when you load the Physics package is 4 = 3 + 1, and the signature is `-`.

$\mathrm{Setup}\left(\mathrm{dimension}\,\mathrm{signature}\right)$

$\left[\mathrm{dimension}=4\,\mathrm{signature}=\mathrm{-\; -\; -\; +}\right]$

So the trace of the metric g_ is equal to 4.

${\mathrm{g\_}}_{\mathrm{\ν}\,\mathrm{\μ}}^2$

$g_{\mathrm{\mu },\mathrm{\nu }}{g}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}$

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

$4$

The metric is used to 'raise and lower' indices in other tensors, as shown below.

${\mathrm{g\_}}_{\mathrm{\ν}\,\mathrm{\μ}}{\mathrm{g\_}}_{\mathrm{\ν}\,\mathrm{\ρ}}$

$g_{\mathrm{\mu },\mathrm{\nu }}{\mathrm{g\_}}_{\mathrm{~nu}\,\mathrm{\ρ}}$

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

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

Define $A$ as an object having tensorial properties; that is, the summation convention for repeated indices in products should be taken into account.

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

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

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

So the metric can now have indices contracted with $A$.

${\mathrm{g\_}}_{\mathrm{\ν}\,\mathrm{\μ}}{\mathrm{g\_}}_{\mathrm{\ρ}\,\mathrm{\σ}}{A}_{\mathrm{\ν}\,\mathrm{\μ}\,\mathrm{\ρ}\,\mathrm{\σ}}$

$A_{\mathrm{\nu },\mathrm{\mu },\mathrm{\rho },\mathrm{\sigma }}{g}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}{g}_{\phantom{}\phantom{\mathrm{\rho },\mathrm{\sigma }}}^{\phantom{}\mathrm{\rho },\mathrm{\sigma }}$

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

$A_{\mathrm{\nu }\phantom{\mathrm{\nu }}\mathrm{\sigma }\phantom{\mathrm{\sigma }}}^{\phantom{\mathrm{\nu }}\mathrm{\nu }\phantom{\mathrm{\sigma }}\mathrm{\sigma }}$

The metric is totally symmetric with respect to interchange of positions of its indices, while the LeviCivita symbol, in the Maple worksheet displayed as epsilon, is totally antisymmetric. So the contraction of their respective indices is equal to zero.

${\mathrm{g\_}}_{\mathrm{\ν}\,\mathrm{\μ}}{\mathrm{LeviCivita}}_{\mathrm{\ν}\,\mathrm{\μ}\,\mathrm{\σ}\,\mathrm{\ρ}}$

${\mathrm{\epsilon }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\rho },\mathrm{\sigma }}{g}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}$

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

$0$

For the same reason, the contraction of any two of the indices of the LeviCivita symbol is also zero.

${\mathrm{LeviCivita}}_{\mathrm{\ν}\,\mathrm{\ν}\,\mathrm{\ρ}\,\mathrm{\σ}}$

$0$

As is the product of the LeviCivita symbol, where the same spacetime vector appears two times, contracting different indices of epsilon. To illustrate this case, first Define $q$ to represent this generic tensor.

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

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

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

${\mathrm{LeviCivita}}_{\mathrm{\ν}\,\mathrm{\μ}\,\mathrm{\σ}\,\mathrm{\ρ}}{q}_{\mathrm{\μ}}{q}_{\mathrm{\ν}}$

${\mathrm{\epsilon }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\rho },\mathrm{\sigma }}{q}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{q}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}$

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

$0$

When defining an object to have tensorial properties, you can define the symmetry properties of the indices of the object as well. The following Defines $B$ and $C$ as totally symmetric and totally antisymmetric, respectively.

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

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

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

$\mathrm{Define}\left(C\,\mathrm{antisymmetric}\right)$

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

$\left\{B\,C\,A_{\mathrm{\nu },\mathrm{\mu },\mathrm{\rho },\mathrm{\sigma }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,q_{\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

$B_{\mathrm{\μ}\,\mathrm{\ν}}{\mathrm{LeviCivita}}_{\mathrm{\ν}\,\mathrm{\μ}\,\mathrm{\σ}\,\mathrm{\ρ}}$

${\mathrm{\epsilon }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\rho },\mathrm{\sigma }}{B}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}$

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

$0$

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

$\mathrm{Totally\; symmetric\; tensor,\; structured\; as}\left[\mathrm{name}\,\mathrm{indices}\,\mathrm{variables}\right]:$

$\left[B\,\left[2\,0\,0\right]\,0\right]$

$C_{\mathrm{\μ}\,\mathrm{\ν}}{\mathrm{g\_}}_{\mathrm{\ν}\,\mathrm{\μ}}$

$g_{\mathrm{\mu },\mathrm{\nu }}{C}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}$

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

$0$

$\mathrm{Define}\left(C\,\mathrm{query}\right)$

$\mathrm{Totally\; antisymmetric\; tensor,\; structured\; as}\left[\mathrm{name}\,\mathrm{indices}\,\mathrm{variables}\right]:$

$\left[C\,\left[2\,0\,0\right]\,0\right]$

$B_{\mathrm{\μ}\,\mathrm{\ν}}{C}_{\mathrm{\μ}\,\mathrm{\ν}}$

$B_{\mathrm{\mu },\mathrm{\nu }}{C}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}$

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

$0$

The number of indices of the LeviCivita symbol depends on the dimension of spacetime. For any dimension and signature, the contracted product of two LeviCivita symbols can be expressed as a sum of products involving the metric g_. Note the use of Check to tell which indices are repeated (contracted) and which are free at any point.

${\mathrm{LeviCivita}}_{\mathrm{\ν}\,\mathrm{\μ}\,\mathrm{\α}\,\mathrm{\β}}{\mathrm{LeviCivita}}_{\mathrm{\μ}\,\mathrm{\σ}\,\mathrm{\ρ}\,\mathrm{\τ}}$

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

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

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

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

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

$\left(-g_{\mathrm{\alpha },\mathrm{\rho }}{g}_{\mathrm{\beta },\mathrm{\sigma }}+g_{\mathrm{\alpha },\mathrm{\sigma }}{g}_{\mathrm{\beta },\mathrm{\rho }}\right)g_{\mathrm{\nu },\mathrm{\tau }}-g_{\mathrm{\alpha },\mathrm{\sigma }}{g}_{\mathrm{\beta },\mathrm{\tau }}{g}_{\mathrm{\nu },\mathrm{\rho }}+g_{\mathrm{\alpha },\mathrm{\tau }}{g}_{\mathrm{\beta },\mathrm{\sigma }}{g}_{\mathrm{\nu },\mathrm{\rho }}-g_{\mathrm{\alpha },\mathrm{\tau }}{g}_{\mathrm{\beta },\mathrm{\rho }}{g}_{\mathrm{\nu },\mathrm{\sigma }}+g_{\mathrm{\alpha },\mathrm{\rho }}{g}_{\mathrm{\beta },\mathrm{\tau }}{g}_{\mathrm{\nu },\mathrm{\sigma }}$

${\mathrm{LeviCivita}}_{\mathrm{\ν}\,\mathrm{\μ}\,\mathrm{\σ}\,\mathrm{\ρ}}{\mathrm{LeviCivita}}_{\mathrm{\ν}\,\mathrm{\μ}\,\mathrm{\α}\,\mathrm{\β}}$

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

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

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

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

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

$2g_{\mathrm{\alpha },\mathrm{\rho }}{g}_{\mathrm{\beta },\mathrm{\sigma }}-2g_{\mathrm{\alpha },\mathrm{\sigma }}{g}_{\mathrm{\beta },\mathrm{\rho }}$

${\mathrm{LeviCivita}}_{\mathrm{\ν}\,\mathrm{\μ}\,\mathrm{\α}\,\mathrm{\σ}}{\mathrm{LeviCivita}}_{\mathrm{\ν}\,\mathrm{\μ}\,\mathrm{\α}\,\mathrm{\β}}$

${\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\mu },\mathrm{\nu },\mathrm{\sigma }}{\mathrm{\epsilon }}_{\phantom{}\phantom{\mathrm{\alpha }}\mathrm{\beta }\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha }\phantom{\mathrm{\beta }}\mathrm{\mu },\mathrm{\nu }}$

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

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

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

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

$-6g_{\mathrm{\beta },\mathrm{\sigma }}$

${\mathrm{LeviCivita}}_{\mathrm{\ν}\,\mathrm{\μ}\,\mathrm{\α}\,\mathrm{\β}}^2$

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

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

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

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

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

$\mathrm{-24}$

Note that the results above are different if the dimension or signature of spacetime are different from 4 and `-`, respectively.

During normal computations, a frequent occurrence is when two products have tensors with the same contracted indices, but in each product the contracted indices are represented by different letters, thus obscuring the fact that the two products are mathematically equal.

$\mathrm{Define}\left(F\,\'\mathrm{quiet}\'\right)$

$\left\{F\,A_{\mathrm{\nu },\mathrm{\mu },\mathrm{\rho },\mathrm{\sigma }}\,B_{\mathrm{\mu },\mathrm{\nu }}\,C_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,q_{\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

$F_{\mathrm{\μ}\,\mathrm{\ν}}{C}_{\mathrm{\ν}\,\mathrm{\ρ}}\+F_{\mathrm{\μ}\,\mathrm{\α}}{C}_{\mathrm{\α}\,\mathrm{\ρ}}$

$C_{\mathrm{\alpha },\mathrm{\rho }}{F}_{\mathrm{\mu }\phantom{\mathrm{\alpha }}}^{\phantom{\mathrm{\mu }}\mathrm{\alpha }}+C_{\mathrm{\nu },\mathrm{\rho }}{F}_{\mathrm{\mu }\phantom{\mathrm{\nu }}}^{\phantom{\mathrm{\mu }}\mathrm{\nu }}$

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

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

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

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

$2C_{\mathrm{\alpha },\mathrm{\rho }}{F}_{\mathrm{\mu }\phantom{\mathrm{\alpha }}}^{\phantom{\mathrm{\mu }}\mathrm{\alpha }}$

The following example would be a little trickier to tell.

$\mathrm{Define}\left(A_{\mathrm{\μ}}\,C_{\mathrm{\μ}}\,\'\mathrm{redo}\'\,\'\mathrm{quiet}\'\right)$

$\left\{A_{\mathrm{\mu }}\,B_{\mathrm{\mu },\mathrm{\nu }}\,C_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,F_{\mathrm{\mu }\phantom{\mathrm{\alpha }}}^{\phantom{\mathrm{\mu }}\mathrm{\alpha }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,q_{\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

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

$\mathrm{Tensors\; structured\; as}\left[\mathrm{name}\,\mathrm{indices}\,\mathrm{variables}\right]:$

$\left[A\,\left[1\,0\,0\right]\,0\right],\left[C\,\left[1\,0\,0\right]\,0\right]$

${\mathrm{LeviCivita}}_{\mathrm{\ν}\,\mathrm{\μ}\,\mathrm{\α}\,\mathrm{\σ}}{\mathrm{LeviCivita}}_{\mathrm{\ν}\,\mathrm{\μ}\,\mathrm{\α}\,\mathrm{\β}}\left(A_{\mathrm{\β}}{C}_{\mathrm{\σ}}-A_{\mathrm{\σ}}{C}_{\mathrm{\β}}\right)$

$\left(A_{\mathrm{\beta }}{C}_{\mathrm{\sigma }}-A_{\mathrm{\sigma }}{C}_{\mathrm{\beta }}\right){\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\mu },\mathrm{\nu }\phantom{\mathrm{\sigma }}}^{\phantom{\mathrm{\alpha }}\phantom{,\mathrm{\mu },\mathrm{\nu }}\mathrm{\sigma }}{\mathrm{\epsilon }}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}$

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

$0$