The TensorArray receives a tensorial expression having n free indices, typically involving sums and products, and returns a corresponding n dimensional Array, which can be indexed as a single object to return the values of the tensorial expression for given values of its free indices.

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

The returned Array is constructed taking into account the covariant and contravariant character of each free index in expression. To compute the values of expression you index this array giving values between 1 and the spacetime dimension to the indices.

Pauli, Dirac and Gell-Mann matrices, respectively represented by the Psigma Dgamma and StandardModel:-Glambda commands, are implemented as spacetime and SU(3) vectors, where each component represents a matrix. Thus, when computing tensor arrays of components, one can optionally visualize the matrices behind these tensor representations and rewrite in matrix form, or additionally perform the matrix operations involved if any. For this purpose, pass the optional arguments rewriteinmatrixform or performmatrixoperations.

By default, in the returned result, summation is explicitly performed over all the repeated indices found in expression, taking into account the covariant/contravariant character of each index. To avoid performing this summation and keep repeated indices not summed pass the optional argument performsumoverrepeatedindices = false.

By default, the Array is constructed without simplifying its components; to have them simplified indicate the simplifier on the right-hand-side of the optional argument simplifier = .... A frequently convenient simplification is achieved with simplifier = simplify.

When expression involves sums and products of tensors having free indices, the ordering of the free indices in the returned Array is arbitrary. To indicated a desired ordering of these free indices, use the option freeindices = [...] where the list [...] indicates the desired ordering.

The array of components of a tensorial expression with at most 2 free indices is all visible on the screen as the output TensorArray. Things are not so simple when that number is 3 or larger, when only a slice of the array of components is shown. To visualize the expression's components in these cases you can use the explore option, which launches an applet that allows for giving values to all but two of the free indices, and shows the 2x2 corresponding slice for the other two, with the ordering shown.

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

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

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

${\mathrm{g\_}}_{\mathrm{sc}}$

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

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

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

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

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

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

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

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

${\mathrm{g\_}}_{\mathrm{\μ}\,\mathrm{\ν}}\=\left(\left[\begin{array}{cccc}\frac{r}{2m-r}& 0& 0& 0\\ 0& -r^2& 0& 0\\ 0& 0& -r^2{\sin \left(\mathrm{\θ}\right)}^2& 0\\ 0& 0& 0& \frac{r-2m}{r}\end{array}\right]\right)$

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

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

$\left\{A\,{▿}_{\mathrm{\mu }}\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,R_{\mathrm{\mu },\mathrm{\nu }}\,R_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,C_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha },\mathrm{\beta }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\gamma }}_{i,j}\,{\mathrm{\Gamma }}_{\mathrm{\mu },\mathrm{\nu },\mathrm{\alpha }}\,G_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\,X_{\mathrm{\mu }}\right\}$

${\mathrm{g\_}}_{\mathrm{\μ}\,\mathrm{\ρ}}{A}_{\mathrm{`~rho`}}{A}_{\mathrm{\ν}}$

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

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

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

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

$T≔\mathrm{TensorArray}\left(\right)$

$\left[\begin{array}{cccc}\frac{A_1{A}_{\mathrm{~1}}r}{2m-r}& \frac{A_2{A}_{\mathrm{~1}}r}{2m-r}& \frac{A_3{A}_{\mathrm{~1}}r}{2m-r}& \frac{A_4{A}_{\mathrm{~1}}r}{2m-r}\\ -A_1{r}^2{A}_{\mathrm{~2}}& -A_2{r}^2{A}_{\mathrm{~2}}& -A_3{r}^2{A}_{\mathrm{~2}}& -A_4{r}^2{A}_{\mathrm{~2}}\\ -A_1{A}_{\mathrm{~3}}{r}^2{\sin \left(\mathrm{\θ}\right)}^2& -A_2{A}_{\mathrm{~3}}{r}^2{\sin \left(\mathrm{\θ}\right)}^2& -A_3{A}_{\mathrm{~3}}{r}^2{\sin \left(\mathrm{\θ}\right)}^2& -A_4{A}_{\mathrm{~3}}{r}^2{\sin \left(\mathrm{\θ}\right)}^2\\ \frac{A_1{A}_{\mathrm{~4}}\left(r-2m\right)}{r}& \frac{A_2{A}_{\mathrm{~4}}\left(r-2m\right)}{r}& \frac{A_3{A}_{\mathrm{~4}}\left(r-2m\right)}{r}& \frac{A_4{A}_{\mathrm{~4}}\left(r-2m\right)}{r}\end{array}\right]$

$\mathrm{TensorArray}\left(\,\mathrm{performsumoverrepeatedindices}\=\mathrm{false}\right)$

$\left[\begin{array}{cccc}{\mathrm{g\_}}_{1\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_1& {\mathrm{g\_}}_{1\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_2& {\mathrm{g\_}}_{1\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_3& {\mathrm{g\_}}_{1\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_4\\ {\mathrm{g\_}}_{2\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_1& {\mathrm{g\_}}_{2\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_2& {\mathrm{g\_}}_{2\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_3& {\mathrm{g\_}}_{2\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_4\\ {\mathrm{g\_}}_{3\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_1& {\mathrm{g\_}}_{3\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_2& {\mathrm{g\_}}_{3\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_3& {\mathrm{g\_}}_{3\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_4\\ {\mathrm{g\_}}_{4\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_1& {\mathrm{g\_}}_{4\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_2& {\mathrm{g\_}}_{4\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_3& {\mathrm{g\_}}_{4\,\mathrm{\ρ}}{A}_{\mathrm{~rho}}{A}_4\end{array}\right]$

$T≔\mathrm{TensorArray}\left(\,\mathrm{freeindices}\=\left[\mathrm{\ν}\,\mathrm{\μ}\right]\right)$

$\left[\begin{array}{cccc}\frac{A_1{A}_{\mathrm{~1}}r}{2m-r}& -A_1{r}^2{A}_{\mathrm{~2}}& -A_1{A}_{\mathrm{~3}}{r}^2{\sin \left(\mathrm{\θ}\right)}^2& \frac{A_1{A}_{\mathrm{~4}}\left(r-2m\right)}{r}\\ \frac{A_2{A}_{\mathrm{~1}}r}{2m-r}& -A_2{r}^2{A}_{\mathrm{~2}}& -A_2{A}_{\mathrm{~3}}{r}^2{\sin \left(\mathrm{\θ}\right)}^2& \frac{A_2{A}_{\mathrm{~4}}\left(r-2m\right)}{r}\\ \frac{A_3{A}_{\mathrm{~1}}r}{2m-r}& -A_3{r}^2{A}_{\mathrm{~2}}& -A_3{A}_{\mathrm{~3}}{r}^2{\sin \left(\mathrm{\θ}\right)}^2& \frac{A_3{A}_{\mathrm{~4}}\left(r-2m\right)}{r}\\ \frac{A_4{A}_{\mathrm{~1}}r}{2m-r}& -A_4{r}^2{A}_{\mathrm{~2}}& -A_4{A}_{\mathrm{~3}}{r}^2{\sin \left(\mathrm{\θ}\right)}^2& \frac{A_4{A}_{\mathrm{~4}}\left(r-2m\right)}{r}\end{array}\right]$

$T≔\mathrm{TensorArray}\left(\,\mathrm{freeindices}\=\left[\mathrm{\ν}\,\mathrm{\μ}\right]\,\mathrm{output}\=\mathrm{setofequations}\right)$

$T≔\left\{\frac{A_1{A}_{\phantom{}\phantom{1}}^{\phantom{}1}r}{2m-r}=0\,\frac{A_1{A}_{\phantom{}\phantom{4}}^{\phantom{}4}\left(r-2m\right)}{r}=0\,\frac{A_2{A}_{\phantom{}\phantom{1}}^{\phantom{}1}r}{2m-r}=0\,\frac{A_2{A}_{\phantom{}\phantom{4}}^{\phantom{}4}\left(r-2m\right)}{r}=0\,\frac{A_3{A}_{\phantom{}\phantom{1}}^{\phantom{}1}r}{2m-r}=0\,\frac{A_3{A}_{\phantom{}\phantom{4}}^{\phantom{}4}\left(r-2m\right)}{r}=0\,\frac{A_4{A}_{\phantom{}\phantom{1}}^{\phantom{}1}r}{2m-r}=0\,\frac{A_4{A}_{\phantom{}\phantom{4}}^{\phantom{}4}\left(r-2m\right)}{r}=0\,-A_1{r}^2{A}_{\phantom{}\phantom{2}}^{\phantom{}2}=0\,-A_2{r}^2{A}_{\phantom{}\phantom{2}}^{\phantom{}2}=0\,-A_3{r}^2{A}_{\phantom{}\phantom{2}}^{\phantom{}2}=0\,-{}_{}\right\}$