The resultant tensor_type, WEYL say, of this routine is the covariant Weyl tensor, indexed as shown below:

Indexing Function:  Because the covariant Weyl tensor components have the same symmetrical properties as those of the Riemann tensor, the component array of the result uses the package's cov_riemann indexing function.

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

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

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

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

$\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\:$

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

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

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

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

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

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

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

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

$\mathrm{WEYL}≔\mathrm{Weyl}\left(g\,\mathrm{RMN}\,\mathrm{RICCI}\,\mathrm{RS}\right)\:$

$\mathrm{displayGR}\left('\mathrm{Weyl}'\,\mathrm{WEYL}\right)$

$\mathrm{The\; Weyl\; Tensor}$

$\mathrm{non-zero\; components\; :}$

$\mathrm{C1212}=\frac{2m}{r^3}$

$\mathrm{C1313}=\frac{\left(-r+2m\right)m}{r^2}$

$\mathrm{C1414}=\frac{\left(-r+2m\right)m{\sin \left(\mathrm{th}\right)}^2}{r^2}$

$\mathrm{C2323}=-\frac{m}{-r+2m}$

$\mathrm{C2424}=-\frac{m{\sin \left(\mathrm{th}\right)}^2}{-r+2m}$

$\mathrm{C3434}=-2{\sin \left(\mathrm{th}\right)}^{2}mr$

$\mathrm{character\; :\; [-1,\; -1,\; -1,\; -1]}$