tensor(deprecated)/Ricci - Help

tensor

 Ricci
 compute the covariant Ricci tensor

 Calling Sequence Ricci(ginv, Rmn)

Parameters

 ginv - rank two tensor_type of character [1,1] representing the contravariant metric tensor; specifically, ${{\mathrm{ginv}}_{\mathrm{compts}}}_{i,j}≔{g}^{\left\{\mathrm{ij}\right\}}$ Rmn - rank four tensor_type of character [-1,-1,-1,-1] representing the covariant Riemann curvature tensor; specifically, ${\mathrm{Rmn}}_{\mathrm{compts}}[i,j,k,l]:=\mathrm{R_}\left\{\mathrm{ijkl}\right\}$

Description

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

 • The resultant tensor_type of this routine is the covariant Ricci tensor: a covariant rank 2 tensor that is symmetric in its indices (the component array of the result uses the Maple symmetric indexing function).
 • ginv should be indexed using the symmetric indexing function. Rmn should be indexed using the cov_riemann indexing function provided by the package.  It is recommended that tensor[invert] and tensor[Riemann] be used to compute these quantities.
 • Simplification:  This routine uses the tensor/Ricci/simp routine for simplification purposes.  The simplification routine is applied to each component of result after it is computed.  By default, tensor/Ricci/simp is initialized to the tensor/simp routine.  It is recommended that the tensor/Ricci/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 Ricci(..) only after performing the command with(tensor) or with(tensor, Ricci).  The function can always be accessed in the long form tensor[Ricci](..).

Examples

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

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

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

 > $\mathrm{coord}≔\left[t,r,\mathrm{θ},\mathrm{φ}\right]:$
 > $\mathrm{g_compts}≔\mathrm{array}\left(\mathrm{symmetric},\mathrm{sparse},1..4,1..4\right):$
 > ${\mathrm{g_compts}}_{1,1}≔1-\frac{2m}{r}:$${\mathrm{g_compts}}_{2,2}≔-\frac{1}{{\mathrm{g_compts}}_{1,1}}:$
 > ${\mathrm{g_compts}}_{3,3}≔-{r}^{2}:$${\mathrm{g_compts}}_{4,4}≔-{r}^{2}{\mathrm{sin}\left(\mathrm{θ}\right)}^{2}:$
 > $g≔\mathrm{create}\left(\left[-1,-1\right],\mathrm{eval}\left(\mathrm{g_compts}\right)\right)$
 ${g}{:=}{\mathrm{table}}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{cccc}{1}{-}\frac{{2}{}{m}}{{r}}& {0}& {0}& {0}\\ {0}& {-}\frac{{1}}{{1}{-}\frac{{2}{}{m}}{{r}}}& {0}& {0}\\ {0}& {0}& {-}{{r}}^{{2}}& {0}\\ {0}& {0}& {0}& {-}{{r}}^{{2}}{}{{\mathrm{sin}}{}\left({\mathrm{θ}}\right)}^{{2}}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-}{1}{,}{-}{1}\right]\right]\right)$ (1)

Compute the Ricci tensor.

 > $\mathrm{ginv}≔\mathrm{invert}\left(g,'\mathrm{detg}'\right):$
 > $\mathrm{D1g}≔\mathrm{d1metric}\left(g,\mathrm{coord}\right):$$\mathrm{D2g}≔\mathrm{d2metric}\left(\mathrm{D1g},\mathrm{coord}\right):$
 > $\mathrm{Cf1}≔\mathrm{Christoffel1}\left(\mathrm{D1g}\right):$
 > $\mathrm{RMN}≔\mathrm{Riemann}\left(\mathrm{ginv},\mathrm{D2g},\mathrm{Cf1}\right):$
 > $\mathrm{RICCI}≔\mathrm{Ricci}\left(\mathrm{ginv},\mathrm{RMN}\right)$
 ${\mathrm{RICCI}}{:=}{\mathrm{table}}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-}{1}{,}{-}{1}\right]\right]\right)$ (2)

You can also view the result using the tensor package function displayGR.