The function tensorsGR(coord, cov_metric, 'contra_metric', 'det_met', 'C1', 'C2', 'Rm', 'Rc', 'R', 'G', 'C') calculates the following objects given the coordinates, coord, and covariant metric tensor, cov_metric:

contravariant metric tensor, returned through contra_metric

determinant of the metric tensor components, returned through det_met

Christoffel symbols of the first kind, returned through C1

Christoffel symbols of the second kind, returned through C2

covariant Riemann tensor, returned through Rm

covariant Ricci tensor, returned through Rc

Ricci scalar, returned through R

covariant Einstein tensor, returned through G

covariant Weyl tensor, returned through C

The calculated quantities are returned via the third through eleventh parameters.  Since these are output parameters, they must be passed as unassigned names.  The return value is NULL.

The last parameter, print_flag, is optional directive to display the calculated results (using tensor[display_allGR]) after they have been calculated. If used, it must be passed with the value print.  Other values will result in an error.

The calculations are made simply by making the appropriate calls to the following procedures: tensor[invert], tensor[partial_diff], tensor[Christoffel1], tensor[Christoffel2], tensor[Riemann], tensor[Ricci], tensor[Ricciscalar], tensor[Einstein], and tensor[Weyl].

Note that this procedure is not strictly necessary.  However, it provides a convenient way to calculate all of the important general relativity curvature quantities in the natural basis.  The print_flag option provides the further convenience of displaying the results automatically once they are computed.

Simplification:  Since this routine computes all of the quantities by calling the appropriate routines from the package, simplification is done according to the simplification methods of each individual routine.

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

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

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

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

$g\left[1,1\right]≔1-\frac{2m}{r}\:$$g\left[2,2\right]≔-\frac{1}{g\left[1,1\right]}\:$$g\left[3,3\right]≔-r^2\:$$g\left[4,4\right]≔-r^2{\sin \left(\mathrm{th}\right)}^2\:$

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

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

$\mathrm{tensorsGR}\left(\mathrm{coords}\,\mathrm{metric}\,\mathrm{contra\_metric}\,\mathrm{det\_met}\,\mathrm{C1}\,\mathrm{C2}\,\mathrm{Rm}\,\mathrm{Rc}\,R\,G\,C\right)$

$\mathrm{displayGR}\left(\mathrm{Einstein}\,G\right)$

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

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

$\mathrm{None}$

$\mathrm{None}$

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

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

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

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

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

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

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

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

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

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

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