The e_[a, mu] and eta[a, b] commands respectively represent the tetrad (also vierbein; by default, this is an orthonormal tetrad) and the tetrad metric, that is, the metric of the local frame - which by default is inertial, of Minkowski type. These two tensors are defined in terms of each other by ${𝔢}_{a,\mathrm{\mu }}{𝔢}_b^{\mathrm{\mu }}={\mathrm{\eta }}_{a,b}$.

Both e_ and eta_ accept the keywords accepted by the other tensors of the Physics package, these are definition, matrix and nonzero, that can be given with or without indices. If given with indices, the corresponding output takes their character (covariant or contravariant) into account. In the case of e_, you can also use the keyword nullvectors, to see the null vectors corresponding to a given tetrad. Note anyway that these null vectors are available as commands of the Tetrads package; these are the l_, n_, m_ and mb_ commands.

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

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

$\mathrm{Setting}\mathrm{lowercaselatin\_ah}\mathrm{letters\; to\; represent}\mathrm{tetrad}\mathrm{indices}$

$\mathrm{Defined\; as\; tetrad\; tensors}\left(\mathrm{see\; ?Physics,tetrads}\right),{𝔢}_{a,\mathrm{\mu }},{\mathrm{\eta }}_{a,b},{\mathrm{\gamma }}_{a,b,c},{\mathrm{\lambda }}_{a,b,c}$

$\mathrm{Defined\; as\; spacetime\; tensors\; representing\; the\; NP\; null\; vectors\; of\; the\; tetrad\; formalism}\left(\mathrm{see\; ?Physics,tetrads}\right),l_{\mathrm{\mu }},n_{\mathrm{\mu }},m_{\mathrm{\mu }},{\stackrel{\&conjugate0;}{m}}_{\mathrm{\mu }}$

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

$\left[\mathrm{IsTetrad}\,\mathrm{NullTetrad}\,\mathrm{OrthonormalTetrad}\,\mathrm{PetrovType}\,\mathrm{SegreType}\,\mathrm{TransformTetrad}\,\mathrm{WeylScalars}\,\mathrm{e\_}\,\mathrm{eta\_}\,\mathrm{gamma\_}\,\mathrm{l\_}\,\mathrm{lambda\_}\,\mathrm{m\_}\,\mathrm{mb\_}\,\mathrm{n\_}\right]$

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

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

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{eta\_}\left[\right]$

${\mathrm{\eta }}_{a,b}=\left[\right]$

$\mathrm{e\_}\left[\right]$

${𝔢}_{a,\mathrm{\mu }}=\left[\right]$

$\mathrm{g\_}\left[\left[13\,7\,5\right]\right]$

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

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

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

$\mathrm{The}\mathrm{metric\; in\; coordinates}\left[x\,y\,z\,t\right]$

$\mathrm{Parameters:}\left[\mathrm{\epsilon }\,A\left(t\right)\,B\left(t\right)\,\mathrm{A1}\right]$

$\mathrm{Comments:}\mathrm{\ⅇpsilon=1\; or\; \ⅇpsilon=-1}$

$\mathrm{Domain:}\left[0<t\right]$

$\mathrm{Assumptions:}\left[A\left(t\right)::\mathrm{real}\&comma;B\left(t\right)::\mathrm{real}\&comma;0<\mathrm{A1}\&comma;0<x\&comma;0<A\left(t\right)\&comma;0<B\left(t\right)\right]$

$\mathrm{Resetting\; the\; signature\; of\; spacetime\; from}\left(\mathrm{-\; -\; -\; +}\right)\mathrm{to}\left(\mathrm{+\; +\; +\; -}\right)\mathrm{in\; order\; to\; match\; the\; signature\; in\; the\; database\; of\; metrics}$

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

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{PDEtools}:-\mathrm{declare}\left(A\left(t\right)\&comma;B\left(t\right)\right)$

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

$B\left(t\right)\mathrm{will\; now\; be\; displayed\; as}B$

$\mathrm{e\_}\left[\right]$

${𝔢}_{a,\mathrm{\mu }}=\left[\right]$

$\mathrm{e\_}\left[\mathrm{nullvectors}\right]$

$l_{\mathrm{\mu }}=\left[\right],n_{\mathrm{\mu }}=\left[\right],m_{\mathrm{\mu }}=\left[\right],{\stackrel{\&conjugate0;}{m}}_{\mathrm{\mu }}=\left[\right]$

${\mathrm{l\_}\left[\mathrm{\mu }\right]}^2,\mathrm{l\_}\left[\mathrm{\mu }\right]\mathrm{n\_}\left[\mathrm{\mu }\right],\mathrm{l\_}\left[\mathrm{\mu }\right]\mathrm{m\_}\left[\mathrm{\mu }\right],\mathrm{l\_}\left[\mathrm{\mu }\right]\mathrm{mb\_}\left[\mathrm{\mu }\right]$

$l_{\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }},l_{\mathrm{\mu }}{n}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }},l_{\mathrm{\mu }}{m}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }},{\stackrel{\&conjugate0;}{m}}_{\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}$

$\mathrm{map}\left(u\mapsto u=\mathrm{SumOverRepeatedIndices}\left(u\right)\&comma;\left[\right]\right)$

$\left[l_{\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=0\&comma;l_{\mathrm{\mu }}{n}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=\mathrm{-1}\&comma;l_{\mathrm{\mu }}{m}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=0\&comma;{\stackrel{\&conjugate0;}{m}}_{\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=0\right]$

$\mathrm{m\_}\left[\mathrm{definition}\right]$

$m_{\mathrm{\mu }}{l}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=0,m_{\mathrm{\mu }}{n}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=0,m_{\mathrm{\mu }}{m}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=0,m_{\mathrm{\mu }}{\stackrel{\&conjugate0;}{m}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}=1,g_{\mathrm{\mu },\mathrm{\nu }}=-l_{\mathrm{\mu }}{n}_{\mathrm{\nu }}-l_{\mathrm{\nu }}{n}_{\mathrm{\mu }}+m_{\mathrm{\mu }}{\stackrel{\&conjugate0;}{m}}_{\mathrm{\nu }}+m_{\mathrm{\nu }}{\stackrel{\&conjugate0;}{m}}_{\mathrm{\mu }}$

$\mathrm{eta\_}\left[\mathrm{definition}\right]$

${\mathrm{\eta }}_{a,b}={𝔢}_{a,\mathrm{\mu }}{𝔢}_{b\phantom{\mathrm{\mu }}}^{\phantom{b}\mathrm{\mu }}$

$\mathrm{e\_}\left[\mathrm{definition}\right]$

${𝔢}_{a,\mathrm{\mu }}{𝔢}_{b\phantom{\mathrm{\mu }}}^{\phantom{b}\mathrm{\mu }}={\mathrm{\eta }}_{a,b}$

$\mathrm{TensorArray}\left(\&comma;\mathrm{simplifier}=\mathrm{simplify}\right)$

The Physics[Tetrads][e_] and Physics[Tetrads][eta_] commands were introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.