Let $g$ be a metric tensor. The trace-free Ricci tensor for $g$ is defined by $T_{\mathrm{ij}}\=R_{\mathrm{ij}}-\frac{1}{4}{g}_{\mathrm{ij}}S$ , where $R_{\mathrm{ij}}$ is the Ricci tensor and $S\=g^{\mathrm{ij}}{R}_{\mathrm{ij}}$ the Ricci scalar of $g$.

The command RicciSpinor(s$\mathbf{\sigma }\mathbf{ }$) first computes the metric tensor $g$ defined by the solder form s. The trace-free Ricci tensor $T$ for $g$ is then computed and converted, using the solder form $\mathrm{\σ}$ to a rank 4 covariant spinor with index type $T_{\mathrm{ABA}\'B\'}$. (See convert/DGspinor.) Finally, a scalar factor of $-\frac{1}{2}$ is introduced according to standard conventions. See Stewart, page 85.

If the Ricci tensor $R$ for the metric $g$ has been previously computed, then the Ricci spinor will be computed more quickly using the second calling sequence RicciSpinor($\mathbf{\sigma }$, R).

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

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

$\mathrm{DGsetup}\left(\left[t\,x\,y\,z\right]\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{w1}\,\mathrm{w2}\right]\,M\right)$

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

$g≔\mathrm{evalDG}\left(\mathrm{dt}\&t\mathrm{dt}\+{\ⅇ}^x\mathrm{dt}\&s\mathrm{dz}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}\+\frac{1{\ⅇ}^{2x}\mathrm{dz}\&t\mathrm{dz}}{2}\right)$

$g:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,1\right]\,\left[\left[1\,4\right]\,\frac{{\ⅇ}^x}{2}\right]\,\left[\left[2\,2\right]\,\mathrm{-1}\right]\,\left[\left[3\,3\right]\,\mathrm{-1}\right]\,\left[\left[4\,1\right]\,\frac{{\ⅇ}^x}{2}\right]\,\left[\left[4\,4\right]\,\frac{{\ⅇ}^{2x}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,1\right]\,\left[\left[1\,4\right]\,\frac{{\ⅇ}^x}{2}\right]\,\left[\left[2\,2\right]\,\mathrm{-1}\right]\,\left[\left[3\,3\right]\,\mathrm{-1}\right]\,\left[\left[4\,1\right]\,\frac{{\ⅇ}^x}{2}\right]\,\left[\left[4\,4\right]\,\frac{{\ⅇ}^{}}{}\right]\right]\right]\right)$