The function displayGR(Einstein, G) displays the nonzero components and the index character of the tensor_type G using the symmetries of the Einstein tensor to reduce the number of components shown. G is expected to be a tensor of rank 2 with the proper symmetries. It can be calculated using tensor[Einstein].

The function display_allGR(coord, cov_met, cont_met, det_met, C1, C2, Rm, Rc, R, G, C) displays all of the GR-related objects passed in the parameter list. They must be passed in the following order: the coordinates list, the covariant metric tensor ([-1,-1]), the contravariant metric tensor ([1,1]), the determinant of the metric tensor components (algebraic type), the Christoffel symbols of the 1st kind ([-1,-1,-1]), the Christoffel symbols of the 2nd kind ([1,-1,-1]), the Riemann tensor, the Ricci tensor, the Ricciscalar ([]), the Einstein tensor, the Weyl tensor.  Each of the quantities is displayed by making the appropriate call to tensor[displayGR]. See tensor[tensorsGR] for the calculation of these quantities.

For displaying general tensor_type objects not listed above, use the 'display' option of tensor[act].

These functions are part of the tensor package, and so can be used in the form displayGR(..) / display_allGR(..) only after performing the command with(tensor), or with(tensor, displayGR) / with(tensor, display_allGR).

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

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

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

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

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

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

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

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

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

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

$\mathrm{None}$

$\mathrm{None}$

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

$\mathrm{displayGR}\left(\mathrm{Christoffel1}\,\mathrm{C1}\right)$

$\mathrm{The\; Christoffel\; Symbols\; of\; the\; First\; Kind}$

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

$\mathrm{[11,2]}=-\frac{m}{r^2}$

$\mathrm{[12,1]}=\frac{m}{r^2}$

$\mathrm{[22,2]}=\frac{m}{{\left(-r+2m\right)}^2}$

$\mathrm{[23,3]}=-r$

$\mathrm{[24,4]}=-r{\sin \left(\mathrm{\theta }\right)}^2$

$\mathrm{[33,2]}=r$

$\mathrm{[34,4]}=-\frac{\sin \left(2\mathrm{\theta }\right)r^2}{2}$

$\mathrm{[44,2]}=r{\sin \left(\mathrm{\theta }\right)}^2$

$\mathrm{[44,3]}=\frac{\sin \left(2\mathrm{\theta }\right)r^2}{2}$

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

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{The\; coordinates\; variables\; are\; :}$

$\mathrm{x1}=t$

$\mathrm{x2}=r$

$\mathrm{x3}=\mathrm{\theta }$

$\mathrm{x4}=\mathrm{\phi }$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{The\; Covariant\; Metric}$

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

$\mathrm{cov\_g11}=1-\frac{2m}{r}$

$\mathrm{cov\_g22}=-\frac{1}{1-\frac{2m}{r}}$

$\mathrm{cov\_g33}=-r^2$

$\mathrm{cov\_g44}=-r^2{\sin \left(\mathrm{\theta }\right)}^2$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Determinant\; of\; the\; covariant\; metric\; tensor\; :}$

$\mathrm{detg}=-r^4{\sin \left(\mathrm{\theta }\right)}^2$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{The\; Contravariant\; Metric}$

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

$\mathrm{contra\_g11}=-\frac{r}{-r+2m}$

$\mathrm{contra\_g22}=\frac{-r+2m}{r}$

$\mathrm{contra\_g33}=-\frac{1}{r^2}$

$\mathrm{contra\_g44}=-\frac{{\csc \left(\mathrm{\theta }\right)}^2}{r^2}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{The\; Christoffel\; Symbols\; of\; the\; First\; Kind}$

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

$\mathrm{[11,2]}=-\frac{m}{r^2}$

$\mathrm{[12,1]}=\frac{m}{r^2}$

$\mathrm{[22,2]}=\frac{m}{{\left(-r+2m\right)}^2}$

$\mathrm{[23,3]}=-r$

$\mathrm{[24,4]}=-r{\sin \left(\mathrm{\theta }\right)}^2$

$\mathrm{[33,2]}=r$

$\mathrm{[34,4]}=-\frac{\sin \left(2\mathrm{\theta }\right)r^2}{2}$

$\mathrm{[44,2]}=r{\sin \left(\mathrm{\theta }\right)}^2$

$\mathrm{[44,3]}=\frac{\sin \left(2\mathrm{\theta }\right)r^2}{2}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{The\; Christoffel\; Symbols\; of\; the\; Second\; Kind}$

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

$\mathrm{\{1,12\}}=-\frac{m}{r\left(-r+2m\right)}$

$\mathrm{\{2,11\}}=-\frac{\left(-r+2m\right)m}{r^3}$

$\mathrm{\{2,22\}}=\frac{m}{r\left(-r+2m\right)}$

$\mathrm{\{2,33\}}=-r+2m$

$\mathrm{\{2,44\}}=\left(-r+2m\right){\sin \left(\mathrm{\theta }\right)}^2$

$\mathrm{\{3,23\}}=\frac{1}{r}$

$\mathrm{\{3,44\}}=-\frac{\sin \left(2\mathrm{\theta }\right)}{2}$

$\mathrm{\{4,24\}}=\frac{1}{r}$

$\mathrm{\{4,34\}}=\cot \left(\mathrm{\theta }\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

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

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

$\mathrm{R1212}=\frac{2m}{r^3}$

$\mathrm{R1313}=\frac{\left(-r+2m\right)m}{r^2}$

$\mathrm{R1414}=\frac{\left(-r+2m\right)m{\sin \left(\mathrm{\theta }\right)}^2}{r^2}$

$\mathrm{R2323}=-\frac{m}{-r+2m}$

$\mathrm{R2424}=-\frac{m{\sin \left(\mathrm{\theta }\right)}^2}{-r+2m}$

$\mathrm{R3434}=-2{\sin \left(\mathrm{\theta }\right)}^{2}mr$

$\mathrm{R3434}=-2{\sin \left(\mathrm{\theta }\right)}^{2}mr$

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

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{The\; Ricci\; tensor}$

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

$\mathrm{None}$

$\mathrm{None}$

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

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{The\; Ricci\; Scalar}$

$R=0$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

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

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

$\mathrm{None}$

$\mathrm{None}$

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

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$

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

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

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

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

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

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

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

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

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

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_}$