The resultant tensor_type, Estn say, of this routine is the COVARIANT Einstein tensor, indexed as shown below:

The component arrays of both g and Ricci should use the Maple symmetric indexing function.  They can be computed using the appropriate routines from the package.  The component array of the result uses Maple's symmetric indexing function.

Simplification:  This routine uses the `tensor/Einstein/simp` routine for simplification purposes.  The simplification routine is applied to each component of result after it is computed.  By default, `tensor/Einstein/simp` is initialized to the `tensor/simp` routine.  It is recommended that the `tensor/Einstein/simp` routine be customized to suit the needs of the particular problem.

This function is part of the tensor package, and so can be used in the form Einstein(..) only after performing the command with(tensor) or with(tensor, Einstein).  The function can always be accessed in the long form tensor[Einstein](..).

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

$\mathrm{coord}≔\left[t\,r\,\mathrm{th}\,\mathrm{ph}\right]\:$

$\mathrm{g\_compts}≔\mathrm{array}\left(\mathrm{symmetric}\,\mathrm{sparse}\,1..4\,1..4\right)\:$

$\mathrm{g\_compts}\left[1,1\right]≔1-\frac{2m}{r}\:$$\mathrm{g\_compts}\left[2,2\right]≔-\frac{1}{\mathrm{g\_compts}\left[1,1\right]}\:$

$\mathrm{g\_compts}\left[3,3\right]≔-r^2\:$$\mathrm{g\_compts}\left[4,4\right]≔-r^2{\sin \left(\mathrm{th}\right)}^2\:$

$g≔\mathrm{create}\left(\left[-1\,-1\right]\,\mathrm{eval}\left(\mathrm{g\_compts}\right)\right)$

$g≔table\left(\left[\mathrm{index\_char}=\left[\mathrm{-1}\,\mathrm{-1}\right]\,\mathrm{compts}=\left[\begin{array}{cccc}1-\frac{2m}{r}& 0& 0& 0\\ 0& -\frac{1}{1-\frac{2m}{r}}& 0& 0\\ 0& 0& -r^2& 0\\ 0& 0& 0& -r^2{\sin \left(\mathrm{th}\right)}^2\end{array}\right]\right]\right)$

$\mathrm{ginv}≔\mathrm{invert}\left(g\,'\mathrm{detg}'\right)\:$

$\mathrm{D1g}≔\mathrm{d1metric}\left(g\,\mathrm{coord}\right)\:$$\mathrm{D2g}≔\mathrm{d2metric}\left(\mathrm{D1g}\,\mathrm{coord}\right)\:$

$\mathrm{Cf1}≔\mathrm{Christoffel1}\left(\mathrm{D1g}\right)\:$

$\mathrm{RMN}≔\mathrm{Riemann}\left(\mathrm{ginv}\,\mathrm{D2g}\,\mathrm{Cf1}\right)\:$

$\mathrm{RICCI}≔\mathrm{Ricci}\left(\mathrm{ginv}\,\mathrm{RMN}\right)\:$

$\mathrm{RS}≔\mathrm{Ricciscalar}\left(\mathrm{ginv}\,\mathrm{RICCI}\right)\:$

$\mathrm{Estn}≔\mathrm{Einstein}\left(g\,\mathrm{RICCI}\,\mathrm{RS}\right)$

$\mathrm{Estn}≔table\left(\left[\mathrm{index\_char}=\left[\mathrm{-1}\,\mathrm{-1}\right]\,\mathrm{compts}=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\right]\right)$