The result of this routine is the Ricci scalar (note it is represented by a rank zero tensor_type).

Both ginv and Ricci should use the Maple symmetric indexing function for their components.

Simplification:  This routine uses the `tensor/Ricciscalar/simp` routine for simplification purposes.  The simplification routine is applied to each component of result after it is computed.  By default, `tensor/Ricciscalar/simp` is initialized to the `tensor/simp` routine.  It is recommended that the `tensor/Ricciscalar/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 Ricciscalar(..) only after performing the command with(tensor) or with(tensor, Ricciscalar).  The function can always be accessed in the long form tensor[Ricciscalar](..).

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

$\mathrm{coord}≔\left[t\,r\,\mathrm{\theta }\,\mathrm{\phi }\right]\:$

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

$\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{\theta }\right)}^2\:$

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

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

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

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

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

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

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

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

$\mathrm{RS}≔table\left(\left[\mathrm{index\_char}=\left[\right]\,\mathrm{compts}=-\frac{4}{r^2}\right]\right)$