The Decompose receives a tensorial expression having n free indices, 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 the n free indices. When there are no free indices, the returned output is not an Array but of the same type of expression.

In the Array returned by Decompose, the 4D indices can assume the values 1 and 2 where, when the position of the time-like-component in the signature is 1 (e.g. the signature is $\left(\mathrm{-\; +\; +\; +}\right)$) , the value 1 returns the time component and the value 2 returns the 3D (space part) of the 4D tensorial expression, expressed using 3D space indices. If the position of the time-like-component in the signature is equal to the spacetime dimension (e.g. the signature is $\left(\mathrm{+\; +\; +\; -}\right)$), then the values 1 and 2 of the Array indices get reversed.

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

By default, in the returned result, summation is explicitly performed over all the repeated indices found in expression by splitting 4D indices into 3D + 1, so for example $A_{\mathrm{\mu }}{A}^{\mathrm{\mu }}=A_0{A}_{}^0+A_i{A}^i$, and the space indices are not summed up. To avoid this decomposition of 4D repeated indices pass the optional argument decomposerepeatedindices = 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 just simplifier = simplify.

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

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\,\mathrm{spaceindices}=\mathrm{lowercase\_is}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\,\mathrm{spaceindices}=\mathrm{lowercaselatin\_is}\right]$

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

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

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

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

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

$\mathrm{Decompose}\left(A\left[\mathrm{~mu}\right]\right)$

$\left[\begin{array}{cc}{A}_{\phantom{}\phantom{i}}^{\phantom{}i}& A_{\phantom{}\phantom{0}}^{\phantom{}0}\end{array}\right]$

$\mathrm{Decompose}\left(\mathrm{\%g\_}\left[\mathrm{\mu },\mathrm{\nu }\right]\right)$

$\left[\begin{array}{cc}{g}_{i,j}& g_{i,0}\\ g_{0,j}& g_{0,0}\end{array}\right]$

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

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\begin{array}{cccc}\mathrm{-1}& 0& 0& 0\\ 0& \mathrm{-1}& 0& 0\\ 0& 0& \mathrm{-1}& 0\\ 0& 0& 0& 1\end{array}\right]$

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

$\left[\begin{array}{cc}{g}_{i,j}& 0\\ 0& 1\end{array}\right]$

$\mathrm{gamma\_}\left[\mathrm{`~`},\mathrm{definition}\right]$

${\mathrm{\gamma }}_{\phantom{}\phantom{i,j}}^{\phantom{}i,j}=-g_{\phantom{}\phantom{i,j}}^{\phantom{}i,j}$

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

${\mathrm{\gamma }}_{i,j}=-g_{i,j}+\frac{g_{0,i}{g}_{0,j}}{g_{0,0}}$

$\mathrm{\%g\_}\left[\mathrm{~alpha},\mathrm{~mu}\right]\mathrm{\%g\_}\left[\mathrm{\mu },\mathrm{\beta }\right]=\mathrm{g\_}\left[\mathrm{~alpha},\mathrm{\beta }\right]$

$g_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\mu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\mu }}{g}_{\mathrm{\mu },\mathrm{\beta }}={\mathrm{\delta }}_{\mathrm{\beta }\phantom{\mathrm{\alpha }}}^{\phantom{\mathrm{\beta }}\mathrm{\alpha }}$

$\mathrm{Eq}≔\mathrm{Decompose}\left(\,\mathrm{repeatedindices}=\mathrm{false}\right)$

$\mathrm{Eq}≔\left[\begin{array}{cc}{g}_{\phantom{}\phantom{i,\mathrm{\mu }}}^{\phantom{}i,\mathrm{\mu }}{g}_{\mathrm{\mu },j}={\mathrm{\delta }}_{j\phantom{i}}^{\phantom{j}i}& g_{\phantom{}\phantom{i,\mathrm{\mu }}}^{\phantom{}i,\mathrm{\mu }}{g}_{\mathrm{\mu },0}=0\\ g_{\phantom{}\phantom{0,\mathrm{\mu }}}^{\phantom{}0,\mathrm{\mu }}{g}_{\mathrm{\mu },j}=0& g_{\phantom{}\phantom{0,\mathrm{\mu }}}^{\phantom{}0,\mathrm{\mu }}{g}_{\mathrm{\mu },0}=1\end{array}\right]$

$\mathrm{Eq}≔\mathrm{Decompose}\left(\,\mathrm{freeindices}=\mathrm{false}\right)$

$\mathrm{Eq}≔g_{0,\mathrm{\beta }}{g}_{\phantom{}\phantom{\mathrm{\alpha },0}}^{\phantom{}\mathrm{\alpha },0}+g_{i,\mathrm{\beta }}{g}_{\phantom{}\phantom{\mathrm{\alpha },i}}^{\phantom{}\mathrm{\alpha },i}={\mathrm{\delta }}_{\mathrm{\beta }\phantom{\mathrm{\alpha }}}^{\phantom{\mathrm{\beta }}\mathrm{\alpha }}$

$\mathrm{Eq}≔\mathrm{Decompose}\left(\right)$

$\mathrm{Eq}≔\left[\begin{array}{cc}{g}_{0,k}{g}_{\phantom{}\phantom{j,0}}^{\phantom{}j,0}+g_{i,k}{g}_{\phantom{}\phantom{j,i}}^{\phantom{}j,i}={\mathrm{\delta }}_{k\phantom{j}}^{\phantom{k}j}& g_{0,0}{g}_{\phantom{}\phantom{j,0}}^{\phantom{}j,0}+g_{i,0}{g}_{\phantom{}\phantom{j,i}}^{\phantom{}j,i}=0\\ g_{0,k}{g}_{\phantom{}\phantom{0,0}}^{\phantom{}0,0}+g_{i,k}{g}_{\phantom{}\phantom{0,i}}^{\phantom{}0,i}=0& g_{0,0}{g}_{\phantom{}\phantom{0,0}}^{\phantom{}0,0}+g_{i,0}{g}_{\phantom{}\phantom{0,i}}^{\phantom{}0,i}=1\end{array}\right]$

$\mathrm{isolate}\left(\mathrm{Eq}\left[1,2\right]\,\mathrm{\%g\_}\left[\mathrm{~j},\mathrm{~0}\right]\right)$

$g_{\phantom{}\phantom{j,0}}^{\phantom{}j,0}=-\frac{g_{i,0}{g}_{\phantom{}\phantom{j,i}}^{\phantom{}j,i}}{g_{0,0}}$

$\mathrm{subs}\left(\,\mathrm{Eq}\left[1,1\right]\right)$

$-\frac{g_{0,k}{g}_{i,0}{g}_{\phantom{}\phantom{j,i}}^{\phantom{}j,i}}{g_{0,0}}+g_{i,k}{g}_{\phantom{}\phantom{j,i}}^{\phantom{}j,i}={\mathrm{\delta }}_{k\phantom{j}}^{\phantom{k}j}$

$\mathrm{collect}\left(\,\mathrm{\%g\_}\left[\mathrm{~j},\mathrm{~i}\right]\right)$

$\left(-\frac{g_{0,k}{g}_{i,0}}{g_{0,0}}+g_{i,k}\right)g_{\phantom{}\phantom{j,i}}^{\phantom{}j,i}={\mathrm{\delta }}_{k\phantom{j}}^{\phantom{k}j}$

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

${\mathrm{\gamma }}_{i,j}=-g_{i,j}+\frac{g_{0,i}{g}_{0,j}}{g_{0,0}}$

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