Let $g$be a metric tensor with associated Ricci tensor R and Ricci scalar S. The trace-free Ricci tensor P is the symmetric, rank 2 covariant tensor with components $P_{\mathrm{ij}}\=R_{\mathrm{ij} }- \frac{1}{n} g_{\mathrm{ij}} S$, where $n$ is the dimension of the underlying manifold. It is trace-free with respect to the metric $g$ in the sense that $g^{\mathrm{ij}} P_{\mathrm{ij}} \= 0\,$where $g^{\mathrm{ij}}$are the components of the inverse metric.

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

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

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

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

$g≔\mathrm{evalDG}\left(y\mathrm{dx}\&t\mathrm{dx}+z\mathrm{dy}\&t\mathrm{dy}+\mathrm{dz}\&t\mathrm{dz}\right)$

$g:=y\mathrm{dx}\mathrm{dx}\+z\mathrm{dy}\mathrm{dy}\+\mathrm{dz}\mathrm{dz}$

$P≔\mathrm{TraceFreeRicciTensor}\left(g\right)$

$P:=-\frac{1}{12}\frac{\left(2y^2-z\right)\mathrm{dx}\mathrm{dx}}{z^{2}y}\+\frac{1}{12}\frac{\left(y^2\+z\right)\mathrm{dy}\mathrm{dy}}{z{y}^2}\+\frac{1}{4}\frac{\mathrm{dy}\mathrm{dz}}{yz}\+\frac{1}{4}\frac{\mathrm{dz}\mathrm{dy}}{yz}\+\frac{1}{12}\frac{\left(y^2-2z\right)\mathrm{dz}\mathrm{dz}}{z^2{y}^2}$

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

$h:=\frac{\mathrm{D\_x}\mathrm{D\_x}}{y}\+\frac{\mathrm{D\_y}\mathrm{D\_y}}{z}\+\mathrm{D\_z}\mathrm{D\_z}$

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

$0$

$\mathrm{TensorInnerProduct}\left(g\,g\,P\right)$

$0$

$\mathrm{A1}≔\mathrm{evalDG}\left(\mathrm{dx}\&s\mathrm{dy}\right)$

$\mathrm{A1}:=\mathrm{dx}\mathrm{dy}\+\mathrm{dy}\mathrm{dx}$

$\mathrm{A2}≔g$

$\mathrm{A2}:=y\mathrm{dx}\mathrm{dx}\+z\mathrm{dy}\mathrm{dy}\+\mathrm{dz}\mathrm{dz}$

$\mathrm{A3}≔\mathrm{evalDG}\left(\mathrm{dx}\&t\mathrm{dx}\right)$

$\mathrm{A3}:=\mathrm{dx}\mathrm{dx}$

$\mathrm{TraceFreeRicciTensor}\left(g\,\mathrm{A1}\right)$

$\mathrm{dx}\mathrm{dy}\+\mathrm{dy}\mathrm{dx}$

$\mathrm{TraceFreeRicciTensor}\left(g\,\mathrm{A2}\right)$

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

$T≔\mathrm{TraceFreeRicciTensor}\left(g\,\mathrm{A3}\right)$

$T:=\frac{2}{3}\mathrm{dx}\mathrm{dx}-\frac{1}{3}\frac{z\mathrm{dy}\mathrm{dy}}{y}-\frac{1}{3}\frac{\mathrm{dz}\mathrm{dz}}{y}$

$\mathrm{TensorInnerProduct}\left(g\,g\,T\right)$

$0$