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Tensor[RicciSpinor] - compute the spinor form of the trace-free Ricci tensor

Calling Sequences

RicciSpinor(${\mathbf{σ}}$, R)

Parameters

$\mathrm{σ}$   - a solder form

R   - (optional) the Ricci tensor for the metric determined by the solder form $\mathrm{σ}$

Description

 • 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) 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 . (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.

Examples

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

Example 1.

First create a vector bundle $M$ with base coordinates$\left(t,x,y,z\right)$ and fiber coordinates $\left(\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right)$.

 M > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)

Define a metric $g$ on the base. For this example we use the Godel metric. (See (12.26) in Exact Solutions to Einstein's Field Equations.) Note that we have adjusted the metric to conform to the signature conventions $\left[1,-1,-1,-1\right]$ used by the spinor formalism in the DifferentialGeometry package. See SpacetimeConventions.

 M > $g≔\mathrm{evalDG}\left(\mathrm{dt}&t\mathrm{dt}+{ⅇ}^{x}\mathrm{dt}&s\mathrm{dz}-\mathrm{dx}&t\mathrm{dx}-\mathrm{dy}&t\mathrm{dy}+\frac{1{ⅇ}^{2x}\mathrm{dz}&t\mathrm{dz}}{2}\right)$
 ${g}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{2}}\right]\right]\right]\right)$ (2.2)

Use DGGramSchmidt to calculate an orthonormal frame $F$ for the metric $g$.

 M > $F≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D_t},\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right],g,\mathrm{signature}=\left[1,-1,-1,-1\right]\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}x::\mathrm{real}$
 ${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]{,}{I}\right]{,}\left[\left[{4}\right]{,}{-}{2}{}{I}{}{{ⅇ}}^{{-}{x}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{I}\right]{,}\left[\left[{4}\right]{,}{-}{2}{}{I}{}{{ⅇ}}^{{-}{x}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{I}\right]{,}\left[\left[{4}\right]{,}{-}{2}{}{I}{}{{ⅇ}}^{{-}{x}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{I}\right]{,}\left[\left[{4}\right]{,}{-}{2}{}{I}{}{{ⅇ}}^{{-}{x}}\right]\right]\right]\right)\right]$ (2.3)

Use SolderForm to compute the solder form $\mathrm{σ}$ from the frame $F$.

 M > $\mathrm{σ}≔\mathrm{SolderForm}\left(F\right)$
 ${\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{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{7}\right]{,}\frac{{I}{}{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}{+}\frac{{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}\right]{,}\left[\left[{4}{,}{6}{,}{8}\right]{,}{-}\frac{{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{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{7}\right]{,}\frac{{I}{}{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}{+}\frac{{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}\right]{,}\left[\left[{4}{,}{6}{,}{8}\right]{,}{-}\frac{{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{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{7}\right]{,}\frac{{I}{}{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}{+}\frac{{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}\right]{,}\left[\left[{4}{,}{6}{,}{8}\right]{,}{-}\frac{{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{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{7}\right]{,}\frac{{I}{}{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}{+}\frac{{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}\right]{,}\left[\left[{4}{,}{6}{,}{8}\right]{,}{-}\frac{{I}{}{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}{+}\frac{{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}\right]\right]\right]\right)$ (2.4)

Calculate the Ricci spinor from the solder form $\mathrm{σ}$.

 M > $\mathrm{Φ1}≔\mathrm{RicciSpinor}\left(\mathrm{σ}\right)$
 ${\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)$ (2.5)

Example 2.

In this example we first calculate the Ricci tensor of the metric $g$ and then use the second calling sequence for RicciSpinor.

 M > $\mathrm{R2}≔\mathrm{RicciTensor}\left(g\right)$
 ${\mathrm{R2}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{1}{,}{4}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}\frac{{3}}{{2}}\right]{,}\left[\left[{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{1}{,}{4}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}\frac{{3}}{{2}}\right]{,}\left[\left[{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{1}{,}{4}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}\frac{{3}}{{2}}\right]{,}\left[\left[{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{1}{,}{4}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}\frac{{3}}{{2}}\right]{,}\left[\left[{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{4}}\right]\right]\right]\right)$ (2.6)
 M > $\mathrm{Φ2}≔\mathrm{RicciSpinor}\left(\mathrm{σ},\mathrm{R2}\right)$
 ${\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)$ (2.7)

Example 3.

We can check the result of Example 1 by direct computation, starting from the solder form $\mathrm{σ}$. First use the command SpinorInnerProduct to calculate the metric $\mathrm{g3}$ from $\mathrm{σ}$. (Note that $\mathrm{g3}$ coincides with the original metric $g$.)

 M > $\mathrm{g3}≔\mathrm{SpinorInnerProduct}\left(\mathrm{σ},\mathrm{σ}\right)$
 ${\mathrm{g3}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{1}\right]{,}\left[\left[{1}{,}{4}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-1}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{1}\right]{,}\frac{{{ⅇ}}^{{x}}}{{2}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{2}}\right]\right]\right]\right)$ (2.8)

Second, calculate the curvature tensor $C$, the Ricci tensor $R$, and the Ricci scalar $S$.

 M > $C≔\mathrm{CurvatureTensor}\left(g\right)$
 ${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}}^{{2}{}{x}}}{{8}}\right]{,}\left[\left[{1}{,}{4}{,}{4}{,}{1}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{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}}^{{2}{}{x}}}{{8}}\right]{,}\left[\left[{2}{,}{4}{,}{4}{,}{2}\right]{,}{-}\frac{{3}{}{{ⅇ}}^{{2}{}{x}}}{{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}}^{{2}{}{x}}}{{8}}\right]{,}\left[\left[{1}{,}{4}{,}{4}{,}{1}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{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}}^{{2}{}{x}}}{{8}}\right]{,}\left[\left[{2}{,}{4}{,}{4}{,}{2}\right]{,}{-}\frac{{3}{}{{ⅇ}}^{{2}{}{x}}}{{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}}^{{2}{}{x}}}{{8}}\right]{,}\left[\left[{1}{,}{4}{,}{4}{,}{1}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{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}}^{{2}{}{x}}}{{8}}\right]{,}\left[\left[{2}{,}{4}{,}{4}{,}{2}\right]{,}{-}\frac{{3}{}{{ⅇ}}^{{2}{}{x}}}{{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}}^{{2}{}{x}}}{{8}}\right]{,}\left[\left[{1}{,}{4}{,}{4}{,}{1}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{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}}^{{2}{}{x}}}{{8}}\right]{,}\left[\left[{2}{,}{4}{,}{4}{,}{2}\right]{,}{-}\frac{{3}{}{{ⅇ}}^{{2}{}{x}}}{{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)$ (2.9)
 M > $R≔\mathrm{RicciTensor}\left(C\right)$
 ${R}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{1}{,}{4}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}\frac{{3}}{{2}}\right]{,}\left[\left[{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{1}{,}{4}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}\frac{{3}}{{2}}\right]{,}\left[\left[{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{1}{,}{4}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}\frac{{3}}{{2}}\right]{,}\left[\left[{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{-}\frac{{1}}{{2}}\right]{,}\left[\left[{1}{,}{4}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}\frac{{3}}{{2}}\right]{,}\left[\left[{4}{,}{1}\right]{,}{-}\frac{{{ⅇ}}^{{x}}}{{4}}\right]{,}\left[\left[{4}{,}{4}\right]{,}\frac{{{ⅇ}}^{{2}{}{x}}}{{4}}\right]\right]\right]\right)$ (2.10)
 M > $S≔\mathrm{RicciScalar}\left(g,C\right)$
 ${S}{:=}\frac{{5}}{{2}}$ (2.11)

Calculate the trace-free Ricci tensor $T$.

 M > $T≔\mathrm{evalDG}\left(R-\frac{1gS}{4}\right)$
 ${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}}^{{2}{}{x}}}{{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}}^{{2}{}{x}}}{{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}}^{{2}{}{x}}}{{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}}^{{2}{}{x}}}{{16}}\right]\right]\right]\right)$ (2.12)

Convert $T$ to a spinor $U$.

 M > $U≔\mathrm{convert}\left(T,\mathrm{DGspinor},\mathrm{σ},\left[1,2\right]\right)$
 ${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]{,}{-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]{,}{-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]{,}{-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]{,}{-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]{,}{-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]{,}{-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]{,}{-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]{,}{-1}\right]\right]\right]\right)$ (2.13)

Rearrange the indices of $U$ and scale by $-\frac{1}{2}$ to arrive at the Ricci spinor $\mathrm{Φ1}$ (or $\mathrm{Φ2}$).

 M > $\mathrm{Φ1}&minus\left(\mathrm{RearrangeIndices}\left(-\frac{1}{2}U,\left[\left[2,3\right]\right]\right)\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right)$ (2.14)