The Bel-Robinson tensor $B_{\mathrm{ijhk}}$ is a covariant rank 4 tensor defined in terms of the Weyl tensor $W_{\mathrm{ijhk}}$ on a 4-dimensional manifold by (see, for example, Penrose and Rindler Vol. 1)

The keyword argument indexlist = ind allows the user to specify the index structure for the Bel-Robinson tensor. For example, with indexlist = ["con", "con", "con", "con"], the contravariant form $B^{\mathrm{ijhk}}$ is returned. The default output is the purely covariant form (as above).

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form BelRobinson(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-BelRobinson.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\,u\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$g≔\mathrm{evalDG}\left(\exp \left(z\right)\mathrm{dx}\&t\mathrm{dx}+\exp \left(-2z\right)\left(\mathrm{dy}\&t\mathrm{dy}+\mathrm{dx}\&s\mathrm{du}\right)-\frac{3}{\mathrm{\Lambda }}\mathrm{dz}\&t\mathrm{dz}\right)$

$g:={\ⅇ}^z\mathrm{dx}\mathrm{dx}+\frac{{\ⅇ}^{-2z}}{2}\mathrm{dx}\mathrm{du}+{\ⅇ}^{-2z}\mathrm{dy}\mathrm{dy}-\frac{3}{\mathrm{\Lambda }}\mathrm{dz}\mathrm{dz}+\frac{{\ⅇ}^{-2z}}{2}\mathrm{du}\mathrm{dx}$

$B≔\mathrm{BelRobinson}\left(g\right)$

$B:=\frac{{\mathrm{\Lambda }}^2{\ⅇ}^{2z}}{4}\mathrm{dx}\mathrm{dx}\mathrm{dx}\mathrm{dx}$

$\mathrm{B1}≔\mathrm{BelRobinson}\left(g\,\mathrm{indexlist}=\left[''con''\,''con''\,''con''\,''con''\right]\right)$

$\mathrm{B1}:=4{\ⅇ}^{10z}{\mathrm{\Lambda }}^2\mathrm{D\_u}\mathrm{D\_u}\mathrm{D\_u}\mathrm{D\_u}$

$h≔\mathrm{InverseMetric}\left(g\right)$

$h:=2{\ⅇ}^{2z}\mathrm{D\_x}\mathrm{D\_u}+{\ⅇ}^{2z}\mathrm{D\_y}\mathrm{D\_y}-\frac{\mathrm{\Lambda }}{3}\mathrm{D\_z}\mathrm{D\_z}+2{\ⅇ}^{2z}\mathrm{D\_u}\mathrm{D\_x}-4{\ⅇ}^{5z}\mathrm{D\_u}\mathrm{D\_u}$

$\mathrm{ContractIndices}\left(h\,B\,\left[\left[1\,1\right]\,\left[2\,2\right]\right]\right)$

$0\mathrm{dx}\mathrm{dx}$

$C≔\mathrm{Christoffel}\left(g\right)$

$C:=-\mathrm{D\_x}\mathrm{dx}\mathrm{dz}-\mathrm{D\_x}\mathrm{dz}\mathrm{dx}-\mathrm{D\_y}\mathrm{dy}\mathrm{dz}-\mathrm{D\_y}\mathrm{dz}\mathrm{dy}+\frac{\mathrm{\Lambda }{\ⅇ}^z}{6}\mathrm{D\_z}\mathrm{dx}\mathrm{dx}-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{6}\mathrm{D\_z}\mathrm{dx}\mathrm{du}-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{3}\mathrm{D\_z}\mathrm{dy}\mathrm{dy}-\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{6}\mathrm{D\_z}\mathrm{du}\mathrm{dx}+3{\ⅇ}^{3z}\mathrm{D\_u}\mathrm{dx}\mathrm{dz}+3{\ⅇ}^{3z}\mathrm{D\_u}\mathrm{dz}\mathrm{dx}-\mathrm{D\_u}\mathrm{dz}\mathrm{du}-\mathrm{D\_u}\mathrm{du}\mathrm{dz}$

$\mathrm{nablaB1}≔\mathrm{CovariantDerivative}\left(\mathrm{B1}\,C\right)$

$\mathrm{nablaB1}:=-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\mathrm{D\_z}\mathrm{D\_u}\mathrm{D\_u}\mathrm{D\_u}\mathrm{dx}-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\mathrm{D\_u}\mathrm{D\_z}\mathrm{D\_u}\mathrm{D\_u}\mathrm{dx}-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\mathrm{D\_u}\mathrm{D\_u}\mathrm{D\_z}\mathrm{D\_u}\mathrm{dx}-\frac{2{\mathrm{\Lambda }}^3{\ⅇ}^{8z}}{3}\mathrm{D\_u}\mathrm{D\_u}\mathrm{D\_u}\mathrm{D\_z}\mathrm{dx}+24{\ⅇ}^{10z}{\mathrm{\Lambda }}^2\mathrm{D\_u}\mathrm{D\_u}\mathrm{D\_u}\mathrm{D\_u}\mathrm{dz}$

$\mathrm{Divergence}≔\mathrm{ContractIndices}\left(\mathrm{nablaB1}\,\left[\left[1\,5\right]\right]\right)$

$\mathrm{Divergence}:=0\mathrm{D\_x}\mathrm{D\_x}\mathrm{D\_x}$

$R≔\mathrm{RicciTensor}\left(g\right)$

$R:=\mathrm{\Lambda }{\ⅇ}^z\mathrm{dx}\mathrm{dx}+\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{2}\mathrm{dx}\mathrm{du}+\mathrm{\Lambda }{\ⅇ}^{-2z}\mathrm{dy}\mathrm{dy}-3\mathrm{dz}\mathrm{dz}+\frac{\mathrm{\Lambda }{\ⅇ}^{-2z}}{2}\mathrm{du}\mathrm{dx}$

$\mathrm{evalDG}\left(R-\mathrm{\Lambda }g\right)$

$0\mathrm{dx}\mathrm{dx}$

$W≔\mathrm{WeylTensor}\left(g\right)$

$W:=-\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\mathrm{dx}\mathrm{dy}\mathrm{dx}\mathrm{dy}+\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\mathrm{dx}\mathrm{dy}\mathrm{dy}\mathrm{dx}-\frac{3{\ⅇ}^z}{2}\mathrm{dx}\mathrm{dz}\mathrm{dx}\mathrm{dz}+\frac{3{\ⅇ}^z}{2}\mathrm{dx}\mathrm{dz}\mathrm{dz}\mathrm{dx}+\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\mathrm{dy}\mathrm{dx}\mathrm{dx}\mathrm{dy}-\frac{\mathrm{\Lambda }{\ⅇ}^{-z}}{2}\mathrm{dy}\mathrm{dx}\mathrm{dy}\mathrm{dx}+\frac{3{\ⅇ}^z}{2}\mathrm{dz}\mathrm{dx}\mathrm{dx}\mathrm{dz}-\frac{3{\ⅇ}^z}{2}\mathrm{dz}\mathrm{dx}\mathrm{dz}\mathrm{dx}$

$\mathrm{BelRobinson}\left(g\,W\right)$

$\frac{{\mathrm{\Lambda }}^2{\ⅇ}^{2z}}{4}\mathrm{dx}\mathrm{dx}\mathrm{dx}\mathrm{dx}$