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tensor

 Weyl
 compute the covariant Weyl tensor

 Calling Sequence Weyl( g, Rmn, Ricci, R)

Parameters

 g - rank two tensor_type of character [-1,-1] representing the covariant metric tensor; specifically, ${g}_{\mathrm{compts}}[i,j]:=\mathrm{g_}\left\{\mathrm{ij}\right\}$; the g component array should use the symmetric indexing function Rmn - rank four tensor_type of character [-1,-1,-1,-1] representing the covariant Riemann curvature tensor; specifically, ${\mathrm{Rmn}}_{\mathrm{compts}}[i,j,k,l]:=\mathrm{R_}\left\{\mathrm{ijkl}\right\}$; the Rmn component array should use the cov_riemann indexing function provided by the tensor package Ricci - rank two tensor_type of character [-1,-1] representing the covariant Ricci tensor; specifically, ${\mathrm{Ricci}}_{\mathrm{compts}}[i,j]:=\mathrm{R_}\left\{\mathrm{ij}\right\}$; the Ricci component array should use the symmetric indexing function R - rank zero tensor_type of character [], representing the Ricci scalar (note it is recognized as a zeroth rank tensor_type in the tensor package).

Description

Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[Tensor][WeylTensor] and Physics[Weyl] instead.

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

${\left({\mathrm{WEYL}}_{\mathrm{compts}}\right)}_{i,j,k,l}≔{C}_{ijkl}$

 • 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](..).

Examples

Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[Tensor][WeylTensor] and Physics[Weyl] instead.

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

Define the coordinate variables and the covariant components of the Schwarzchild metric.

 > $\mathrm{coord}≔\left[t,r,\mathrm{th},\mathrm{ph}\right]:$
 > $\mathrm{g_compts}≔\mathrm{array}\left(\mathrm{symmetric},\mathrm{sparse},1..4,1..4\right):$
 > $\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}{\mathrm{sin}\left(\mathrm{th}\right)}^{2}:$
 > $g≔\mathrm{create}\left(\left[-1,-1\right],\mathrm{eval}\left(\mathrm{g_compts}\right)\right)$
 ${g}{≔}{table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{cccc}{1}{-}\frac{{2}{}{m}}{{r}}& {0}& {0}& {0}\\ {0}& {-}\frac{{1}}{{1}{-}\frac{{2}{}{m}}{{r}}}& {0}& {0}\\ {0}& {0}& {-}{{r}}^{{2}}& {0}\\ {0}& {0}& {0}& {-}{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{th}}\right)}^{{2}}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-1}{,}{-1}\right]\right]\right)$ (1)

Now compute all of the necessary quantities for the computation of the Weyl tensor components, and then compute the components themselves:

 > $\mathrm{ginv}≔\mathrm{invert}\left(g,'\mathrm{detg}'\right):$
 > $\mathrm{D1g}≔\mathrm{d1metric}\left(g,\mathrm{coord}\right):$$\mathrm{D2g}≔\mathrm{d2metric}\left(\mathrm{D1g},\mathrm{coord}\right):$
 > $\mathrm{Cf1}≔\mathrm{Christoffel1}\left(\mathrm{D1g}\right):$
 > $\mathrm{RMN}≔\mathrm{Riemann}\left(\mathrm{ginv},\mathrm{D2g},\mathrm{Cf1}\right):$
 > $\mathrm{RICCI}≔\mathrm{Ricci}\left(\mathrm{ginv},\mathrm{RMN}\right):$
 > $\mathrm{RS}≔\mathrm{Ricciscalar}\left(\mathrm{ginv},\mathrm{RICCI}\right):$
 > $\mathrm{WEYL}≔\mathrm{Weyl}\left(g,\mathrm{RMN},\mathrm{RICCI},\mathrm{RS}\right):$

Show the nonzero components.

 > $\mathrm{displayGR}\left('\mathrm{Weyl}',\mathrm{WEYL}\right)$
 ${}$
 ${\mathrm{The Weyl Tensor}}$
 ${\mathrm{non-zero components :}}$
 ${\mathrm{C1212}}{=}\frac{{2}{}{m}}{{{r}}^{{3}}}$
 ${\mathrm{C1313}}{=}\frac{\left({-}{r}{+}{2}{}{m}\right){}{m}}{{{r}}^{{2}}}$
 ${\mathrm{C1414}}{=}\frac{\left({-}{r}{+}{2}{}{m}\right){}{m}{}{{\mathrm{sin}}{}\left({\mathrm{th}}\right)}^{{2}}}{{{r}}^{{2}}}$
 ${\mathrm{C2323}}{=}{-}\frac{{m}}{{-}{r}{+}{2}{}{m}}$
 ${\mathrm{C2424}}{=}{-}\frac{{m}{}{{\mathrm{sin}}{}\left({\mathrm{th}}\right)}^{{2}}}{{-}{r}{+}{2}{}{m}}$
 ${\mathrm{C3434}}{=}{-}{2}{}{r}{}{m}{}{{\mathrm{sin}}{}\left({\mathrm{th}}\right)}^{{2}}$
 ${\mathrm{character : \left[-1, -1, -1, -1\right]}}$ (2)