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)\:$

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

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

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

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

$F≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D\_t}\,\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]\,g\,\mathrm{signature}\=\left[1\,-1\,-1\,-1\right]\right)assumingx::\mathrm{real}$

$F:=\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,\mathrm{I}\right]\,\left[\left[4\right]\,-2\mathrm{I}{\ⅇ}^{-x}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,\mathrm{I}\right]\,\left[\left[4\right]\,-2\mathrm{I}{\ⅇ}^{-x}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,\mathrm{I}\right]\,\left[\left[4\right]\,-2\mathrm{I}{\ⅇ}^{-x}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,\mathrm{I}\right]\,\left[\left[4\right]\,-2\mathrm{I}{\ⅇ}^{-x}\right]\right]\right]\right)\right]$

$\mathrm{\σ}≔\mathrm{SolderForm}\left(F\right)$

$\mathrm{\σ}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''con\_vrt''\,''con\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,5\,7\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[1\,6\,8\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[2\,5\,8\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[2\,6\,7\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[3\,5\,8\right]\,-\frac{\mathrm{I}}{2}\sqrt{2}\right]\,\left[\left[3\,6\,7\right]\,\frac{\mathrm{I}}{2}\sqrt{2}\right]\,\left[\left[4\,5\,7\right]\,\frac{\mathrm{I}{\ⅇ}^x\sqrt{2}}{4}+\frac{{\ⅇ}^x\sqrt{2}}{4}\right]\,\left[\left[4\,6\,8\right]\,-\frac{\mathrm{I}{\ⅇ}^x\sqrt{2}}{4}+\frac{{\ⅇ}^x\sqrt{2}}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''con\_vrt''\,''con\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,5\,7\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[1\,6\,8\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[2\,5\,8\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[2\,6\,7\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[3\,5\,8\right]\,-\frac{\mathrm{I}}{2}\sqrt{2}\right]\,\left[\left[3\,6\,7\right]\,\frac{\mathrm{I}}{2}\sqrt{2}\right]\,\left[\left[4\,5\,7\right]\,\frac{\mathrm{I}{\ⅇ}^x\sqrt{2}}{4}+\frac{{\ⅇ}^x\sqrt{2}}{4}\right]\,\left[\left[4\,6\,8\right]\,-\frac{\mathrm{I}{\ⅇ}^x\sqrt{2}}{4}+\frac{{\ⅇ}^x\sqrt{2}}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''con\_vrt''\,''con\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,5\,7\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[1\,6\,8\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[2\,5\,8\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[2\,6\,7\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[3\,5\,8\right]\,-\frac{\mathrm{I}}{2}\sqrt{2}\right]\,\left[\left[3\,6\,7\right]\,\frac{\mathrm{I}}{2}\sqrt{2}\right]\,\left[\left[4\,5\,7\right]\,\frac{\mathrm{I}{\ⅇ}^x\sqrt{2}}{4}+\frac{{\ⅇ}^x\sqrt{2}}{4}\right]\,\left[\left[4\,6\,8\right]\,-\frac{\mathrm{I}{\ⅇ}^x\sqrt{2}}{4}+\frac{{\ⅇ}^x\sqrt{2}}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''con\_vrt''\,''con\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,5\,7\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[1\,6\,8\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[2\,5\,8\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[2\,6\,7\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[3\,5\,8\right]\,-\frac{\mathrm{I}}{2}\sqrt{2}\right]\,\left[\left[3\,6\,7\right]\,\frac{\mathrm{I}}{2}\sqrt{2}\right]\,\left[\left[4\,5\,7\right]\,\frac{\mathrm{I}{\ⅇ}^x\sqrt{2}}{4}+\frac{{\ⅇ}^x\sqrt{2}}{4}\right]\,\left[\left[4\,6\,8\right]\,-\frac{\mathrm{I}{\ⅇ}^x\sqrt{2}}{4}+\frac{{\ⅇ}^x\sqrt{2}}{4}\right]\right]\right]\right)$

$\mathrm{\Φ1}≔\mathrm{RicciSpinor}\left(\mathrm{\σ}\right)$

$\mathrm{\Φ1}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,5\,7\,7\right]\,\frac{1}{2}\right]\,\left[\left[5\,5\,8\,8\right]\,\frac{3}{8}\right]\,\left[\left[5\,6\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[5\,6\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,6\,7\,7\right]\,\frac{3}{8}\right]\,\left[\left[6\,6\,8\,8\right]\,\frac{1}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,5\,7\,7\right]\,\frac{1}{2}\right]\,\left[\left[5\,5\,8\,8\right]\,\frac{3}{8}\right]\,\left[\left[5\,6\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[5\,6\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,6\,7\,7\right]\,\frac{3}{8}\right]\,\left[\left[6\,6\,8\,8\right]\,\frac{1}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,5\,7\,7\right]\,\frac{1}{2}\right]\,\left[\left[5\,5\,8\,8\right]\,\frac{3}{8}\right]\,\left[\left[5\,6\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[5\,6\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,6\,7\,7\right]\,\frac{3}{8}\right]\,\left[\left[6\,6\,8\,8\right]\,\frac{1}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,5\,7\,7\right]\,\frac{1}{2}\right]\,\left[\left[5\,5\,8\,8\right]\,\frac{3}{8}\right]\,\left[\left[5\,6\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[5\,6\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,6\,7\,7\right]\,\frac{3}{8}\right]\,\left[\left[6\,6\,8\,8\right]\,\frac{1}{2}\right]\right]\right]\right)$

$\mathrm{R2}≔\mathrm{RicciTensor}\left(g\right)$

$\mathrm{R2}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{1}{2}\right]\,\left[\left[1\,4\right]\,-\frac{{\ⅇ}^x}{4}\right]\,\left[\left[2\,2\right]\,-\frac{3}{2}\right]\,\left[\left[4\,1\right]\,-\frac{{\ⅇ}^x}{4}\right]\,\left[\left[4\,4\right]\,\frac{{\ⅇ}^{2x}}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{1}{2}\right]\,\left[\left[1\,4\right]\,-\frac{{\ⅇ}^x}{4}\right]\,\left[\left[2\,2\right]\,-\frac{3}{2}\right]\,\left[\left[4\,1\right]\,-\frac{{\ⅇ}^x}{4}\right]\,\left[\left[4\,4\right]\,\frac{{\ⅇ}^{2x}}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{1}{2}\right]\,\left[\left[1\,4\right]\,-\frac{{\ⅇ}^x}{4}\right]\,\left[\left[2\,2\right]\,-\frac{3}{2}\right]\,\left[\left[4\,1\right]\,-\frac{{\ⅇ}^x}{4}\right]\,\left[\left[4\,4\right]\,\frac{{\ⅇ}^{2x}}{4}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{1}{2}\right]\,\left[\left[1\,4\right]\,-\frac{{\ⅇ}^x}{4}\right]\,\left[\left[2\,2\right]\,-\frac{3}{2}\right]\,\left[\left[4\,1\right]\,-\frac{{\ⅇ}^x}{4}\right]\,\left[\left[4\,4\right]\,\frac{{\ⅇ}^{2x}}{4}\right]\right]\right]\right)$

$\mathrm{\Φ2}≔\mathrm{RicciSpinor}\left(\mathrm{\σ}\,\mathrm{R2}\right)$

$\mathrm{\Φ2}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,5\,7\,7\right]\,\frac{1}{2}\right]\,\left[\left[5\,5\,8\,8\right]\,\frac{3}{8}\right]\,\left[\left[5\,6\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[5\,6\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,6\,7\,7\right]\,\frac{3}{8}\right]\,\left[\left[6\,6\,8\,8\right]\,\frac{1}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,5\,7\,7\right]\,\frac{1}{2}\right]\,\left[\left[5\,5\,8\,8\right]\,\frac{3}{8}\right]\,\left[\left[5\,6\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[5\,6\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,6\,7\,7\right]\,\frac{3}{8}\right]\,\left[\left[6\,6\,8\,8\right]\,\frac{1}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,5\,7\,7\right]\,\frac{1}{2}\right]\,\left[\left[5\,5\,8\,8\right]\,\frac{3}{8}\right]\,\left[\left[5\,6\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[5\,6\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,6\,7\,7\right]\,\frac{3}{8}\right]\,\left[\left[6\,6\,8\,8\right]\,\frac{1}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,5\,7\,7\right]\,\frac{1}{2}\right]\,\left[\left[5\,5\,8\,8\right]\,\frac{3}{8}\right]\,\left[\left[5\,6\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[5\,6\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,7\,8\right]\,\frac{1}{16}\right]\,\left[\left[6\,5\,8\,7\right]\,\frac{1}{16}\right]\,\left[\left[6\,6\,7\,7\right]\,\frac{3}{8}\right]\,\left[\left[6\,6\,8\,8\right]\,\frac{1}{2}\right]\right]\right]\right)$

$\mathrm{g3}≔\mathrm{SpinorInnerProduct}\left(\mathrm{\σ}\,\mathrm{\σ}\right)$

$\mathrm{g3}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,1\right]\,\left[\left[1\,4\right]\,\frac{{\ⅇ}^x}{2}\right]\,\left[\left[2\,2\right]\,\mathrm{-1}\right]\,\left[\left[3\,3\right]\,\mathrm{-1}\right]\,\left[\left[4\,1\right]\,\frac{{\ⅇ}^x}{2}\right]\,\left[\left[4\,4\right]\,\frac{{\ⅇ}^{2x}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,1\right]\,\left[\left[1\,4\right]\,\frac{{\ⅇ}^x}{2}\right]\,\left[\left[2\,2\right]\,\mathrm{-1}\right]\,\left[\left[3\,3\right]\,\mathrm{-1}\right]\,\left[\left[4\,1\right]\,\frac{{\ⅇ}^x}{2}\right]\,\left[\left[4\,4\right]\,\frac{{\ⅇ}^{2x}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,1\right]\,\left[\left[1\,4\right]\,\frac{{\ⅇ}^x}{2}\right]\,\left[\left[2\,2\right]\,\mathrm{-1}\right]\,\left[\left[3\,3\right]\,\mathrm{-1}\right]\,\left[\left[4\,1\right]\,\frac{{\ⅇ}^x}{2}\right]\,\left[\left[4\,4\right]\,\frac{{\ⅇ}^{2x}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,1\right]\,\left[\left[1\,4\right]\,\frac{{\ⅇ}^x}{2}\right]\,\left[\left[2\,2\right]\,\mathrm{-1}\right]\,\left[\left[3\,3\right]\,\mathrm{-1}\right]\,\left[\left[4\,1\right]\,\frac{{\ⅇ}^x}{2}\right]\,\left[\left[4\,4\right]\,\frac{{\ⅇ}^{2x}}{2}\right]\right]\right]\right)$

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

$C:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,4\right]\,-\frac{{\ⅇ}^x}{8}\right]\,\left[\left[1\,1\,4\,1\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[1\,2\,1\,2\right]\,\frac{1}{4}\right]\,\left[\left[1\,2\,2\,1\right]\,-\frac{1}{4}\right]\,\left[\left[1\,2\,2\,4\right]\,-{\ⅇ}^x\right]\,\left[\left[1\,2\,4\,2\right]\,{\ⅇ}^x\right]\,\left[\left[1\,4\,1\,4\right]\,-\frac{{\ⅇ}^{2x}}{8}\right]\,\left[\left[1\,4\,4\,1\right]\,\frac{{\ⅇ}^{2x}}{8}\right]\,\left[\left[2\,1\,1\,2\right]\,\frac{1}{4}\right]\,\left[\left[2\,1\,2\,1\right]\,-\frac{1}{4}\right]\,\left[\left[2\,1\,2\,4\right]\,-\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,1\,4\,2\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,4\,1\,2\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,4\,2\,1\right]\,-\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,4\,2\,4\right]\,\frac{3{\ⅇ}^{2x}}{8}\right]\,\left[\left[2\,4\,4\,2\right]\,-\frac{3{\ⅇ}^{2x}}{8}\right]\,\left[\left[4\,1\,1\,4\right]\,\frac{1}{4}\right]\,\left[\left[4\,1\,4\,1\right]\,-\frac{1}{4}\right]\,\left[\left[4\,2\,2\,4\right]\,\frac{7}{4}\right]\,\left[\left[4\,2\,4\,2\right]\,-\frac{7}{4}\right]\,\left[\left[4\,4\,1\,4\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[4\,4\,4\,1\right]\,-\frac{{\ⅇ}^x}{8}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,4\right]\,-\frac{{\ⅇ}^x}{8}\right]\,\left[\left[1\,1\,4\,1\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[1\,2\,1\,2\right]\,\frac{1}{4}\right]\,\left[\left[1\,2\,2\,1\right]\,-\frac{1}{4}\right]\,\left[\left[1\,2\,2\,4\right]\,-{\ⅇ}^x\right]\,\left[\left[1\,2\,4\,2\right]\,{\ⅇ}^x\right]\,\left[\left[1\,4\,1\,4\right]\,-\frac{{\ⅇ}^{2x}}{8}\right]\,\left[\left[1\,4\,4\,1\right]\,\frac{{\ⅇ}^{2x}}{8}\right]\,\left[\left[2\,1\,1\,2\right]\,\frac{1}{4}\right]\,\left[\left[2\,1\,2\,1\right]\,-\frac{1}{4}\right]\,\left[\left[2\,1\,2\,4\right]\,-\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,1\,4\,2\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,4\,1\,2\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,4\,2\,1\right]\,-\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,4\,2\,4\right]\,\frac{3{\ⅇ}^{2x}}{8}\right]\,\left[\left[2\,4\,4\,2\right]\,-\frac{3{\ⅇ}^{2x}}{8}\right]\,\left[\left[4\,1\,1\,4\right]\,\frac{1}{4}\right]\,\left[\left[4\,1\,4\,1\right]\,-\frac{1}{4}\right]\,\left[\left[4\,2\,2\,4\right]\,\frac{7}{4}\right]\,\left[\left[4\,2\,4\,2\right]\,-\frac{7}{4}\right]\,\left[\left[4\,4\,1\,4\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[4\,4\,4\,1\right]\,-\frac{{\ⅇ}^x}{8}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,4\right]\,-\frac{{\ⅇ}^x}{8}\right]\,\left[\left[1\,1\,4\,1\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[1\,2\,1\,2\right]\,\frac{1}{4}\right]\,\left[\left[1\,2\,2\,1\right]\,-\frac{1}{4}\right]\,\left[\left[1\,2\,2\,4\right]\,-{\ⅇ}^x\right]\,\left[\left[1\,2\,4\,2\right]\,{\ⅇ}^x\right]\,\left[\left[1\,4\,1\,4\right]\,-\frac{{\ⅇ}^{2x}}{8}\right]\,\left[\left[1\,4\,4\,1\right]\,\frac{{\ⅇ}^{2x}}{8}\right]\,\left[\left[2\,1\,1\,2\right]\,\frac{1}{4}\right]\,\left[\left[2\,1\,2\,1\right]\,-\frac{1}{4}\right]\,\left[\left[2\,1\,2\,4\right]\,-\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,1\,4\,2\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,4\,1\,2\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,4\,2\,1\right]\,-\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,4\,2\,4\right]\,\frac{3{\ⅇ}^{2x}}{8}\right]\,\left[\left[2\,4\,4\,2\right]\,-\frac{3{\ⅇ}^{2x}}{8}\right]\,\left[\left[4\,1\,1\,4\right]\,\frac{1}{4}\right]\,\left[\left[4\,1\,4\,1\right]\,-\frac{1}{4}\right]\,\left[\left[4\,2\,2\,4\right]\,\frac{7}{4}\right]\,\left[\left[4\,2\,4\,2\right]\,-\frac{7}{4}\right]\,\left[\left[4\,4\,1\,4\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[4\,4\,4\,1\right]\,-\frac{{\ⅇ}^x}{8}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\,4\right]\,-\frac{{\ⅇ}^x}{8}\right]\,\left[\left[1\,1\,4\,1\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[1\,2\,1\,2\right]\,\frac{1}{4}\right]\,\left[\left[1\,2\,2\,1\right]\,-\frac{1}{4}\right]\,\left[\left[1\,2\,2\,4\right]\,-{\ⅇ}^x\right]\,\left[\left[1\,2\,4\,2\right]\,{\ⅇ}^x\right]\,\left[\left[1\,4\,1\,4\right]\,-\frac{{\ⅇ}^{2x}}{8}\right]\,\left[\left[1\,4\,4\,1\right]\,\frac{{\ⅇ}^{2x}}{8}\right]\,\left[\left[2\,1\,1\,2\right]\,\frac{1}{4}\right]\,\left[\left[2\,1\,2\,1\right]\,-\frac{1}{4}\right]\,\left[\left[2\,1\,2\,4\right]\,-\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,1\,4\,2\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,4\,1\,2\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,4\,2\,1\right]\,-\frac{{\ⅇ}^x}{8}\right]\,\left[\left[2\,4\,2\,4\right]\,\frac{3{\ⅇ}^{2x}}{8}\right]\,\left[\left[2\,4\,4\,2\right]\,-\frac{3{\ⅇ}^{2x}}{8}\right]\,\left[\left[4\,1\,1\,4\right]\,\frac{1}{4}\right]\,\left[\left[4\,1\,4\,1\right]\,-\frac{1}{4}\right]\,\left[\left[4\,2\,2\,4\right]\,\frac{7}{4}\right]\,\left[\left[4\,2\,4\,2\right]\,-\frac{7}{4}\right]\,\left[\left[4\,4\,1\,4\right]\,\frac{{\ⅇ}^x}{8}\right]\,\left[\left[4\,4\,4\,1\right]\,-\frac{{\ⅇ}^x}{8}\right]\right]\right]\right)$

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

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

$S≔\mathrm{RicciScalar}\left(g\,C\right)$

$S:=\frac{5}{2}$

$T≔\mathrm{evalDG}\left(R-\frac{1gS}{4}\right)$

$T:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{9}{8}\right]\,\left[\left[1\,4\right]\,-\frac{9{\ⅇ}^x}{16}\right]\,\left[\left[2\,2\right]\,-\frac{7}{8}\right]\,\left[\left[3\,3\right]\,\frac{5}{8}\right]\,\left[\left[4\,1\right]\,-\frac{9{\ⅇ}^x}{16}\right]\,\left[\left[4\,4\right]\,-\frac{{\ⅇ}^{2x}}{16}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{9}{8}\right]\,\left[\left[1\,4\right]\,-\frac{9{\ⅇ}^x}{16}\right]\,\left[\left[2\,2\right]\,-\frac{7}{8}\right]\,\left[\left[3\,3\right]\,\frac{5}{8}\right]\,\left[\left[4\,1\right]\,-\frac{9{\ⅇ}^x}{16}\right]\,\left[\left[4\,4\right]\,-\frac{{\ⅇ}^{2x}}{16}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{9}{8}\right]\,\left[\left[1\,4\right]\,-\frac{9{\ⅇ}^x}{16}\right]\,\left[\left[2\,2\right]\,-\frac{7}{8}\right]\,\left[\left[3\,3\right]\,\frac{5}{8}\right]\,\left[\left[4\,1\right]\,-\frac{9{\ⅇ}^x}{16}\right]\,\left[\left[4\,4\right]\,-\frac{{\ⅇ}^{2x}}{16}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,-\frac{9}{8}\right]\,\left[\left[1\,4\right]\,-\frac{9{\ⅇ}^x}{16}\right]\,\left[\left[2\,2\right]\,-\frac{7}{8}\right]\,\left[\left[3\,3\right]\,\frac{5}{8}\right]\,\left[\left[4\,1\right]\,-\frac{9{\ⅇ}^x}{16}\right]\,\left[\left[4\,4\right]\,-\frac{{\ⅇ}^{2x}}{16}\right]\right]\right]\right)$

$U≔\mathrm{convert}\left(T\,\mathrm{DGspinor}\,\mathrm{\σ}\,\left[1\,2\right]\right)$

$U:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,7\,5\,7\right]\,\mathrm{-1}\right]\,\left[\left[5\,7\,6\,8\right]\,-\frac{1}{8}\right]\,\left[\left[5\,8\,5\,8\right]\,-\frac{3}{4}\right]\,\left[\left[5\,8\,6\,7\right]\,-\frac{1}{8}\right]\,\left[\left[6\,7\,5\,8\right]\,-\frac{1}{8}\right]\,\left[\left[6\,7\,6\,7\right]\,-\frac{3}{4}\right]\,\left[\left[6\,8\,5\,7\right]\,-\frac{1}{8}\right]\,\left[\left[6\,8\,6\,8\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,7\,5\,7\right]\,\mathrm{-1}\right]\,\left[\left[5\,7\,6\,8\right]\,-\frac{1}{8}\right]\,\left[\left[5\,8\,5\,8\right]\,-\frac{3}{4}\right]\,\left[\left[5\,8\,6\,7\right]\,-\frac{1}{8}\right]\,\left[\left[6\,7\,5\,8\right]\,-\frac{1}{8}\right]\,\left[\left[6\,7\,6\,7\right]\,-\frac{3}{4}\right]\,\left[\left[6\,8\,5\,7\right]\,-\frac{1}{8}\right]\,\left[\left[6\,8\,6\,8\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,7\,5\,7\right]\,\mathrm{-1}\right]\,\left[\left[5\,7\,6\,8\right]\,-\frac{1}{8}\right]\,\left[\left[5\,8\,5\,8\right]\,-\frac{3}{4}\right]\,\left[\left[5\,8\,6\,7\right]\,-\frac{1}{8}\right]\,\left[\left[6\,7\,5\,8\right]\,-\frac{1}{8}\right]\,\left[\left[6\,7\,6\,7\right]\,-\frac{3}{4}\right]\,\left[\left[6\,8\,5\,7\right]\,-\frac{1}{8}\right]\,\left[\left[6\,8\,6\,8\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\,''cov\_vrt''\right]\,\left[\right]\right]\right]\,\left[\left[\left[5\,7\,5\,7\right]\,\mathrm{-1}\right]\,\left[\left[5\,7\,6\,8\right]\,-\frac{1}{8}\right]\,\left[\left[5\,8\,5\,8\right]\,-\frac{3}{4}\right]\,\left[\left[5\,8\,6\,7\right]\,-\frac{1}{8}\right]\,\left[\left[6\,7\,5\,8\right]\,-\frac{1}{8}\right]\,\left[\left[6\,7\,6\,7\right]\,-\frac{3}{4}\right]\,\left[\left[6\,8\,5\,7\right]\,-\frac{1}{8}\right]\,\left[\left[6\,8\,6\,8\right]\,\mathrm{-1}\right]\right]\right]\right)$