DifferentialGeometry/Tensor/RicciSpinor - Maple Help

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 > $g≔\mathrm{evalDG}\left(\mathrm{dt}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dt}+\mathrm{exp}\left(x\right)\mathrm{dt}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}-\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}-\mathrm{dy}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+\frac{1}{2}\mathrm{exp}\left(2x\right)\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
 ${g}{:=}{\mathrm{dt}}{}{\mathrm{dt}}{+}\frac{{{ⅇ}}^{{x}}}{{2}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{+}\frac{{{ⅇ}}^{{x}}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}\frac{{{ⅇ}}^{{2}{}{x}}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.2)

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

 M > $F≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D_t},\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right],g,\mathrm{signature}=\left[1,-1,-1,-1\right]\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}x::\mathrm{real}$
 ${F}{:=}\left[{\mathrm{D_t}}{,}{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{I}{}{\mathrm{D_t}}{-}{2}{}{I}{}{{ⅇ}}^{{-}{x}}{}{\mathrm{D_z}}\right]$ (2.3)

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

 M > $\mathrm{\sigma }≔\mathrm{SolderForm}\left(F\right)$
 ${\mathrm{σ}}{:=}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{\sqrt{{2}}}{{2}}{}{\mathrm{dx}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{\mathrm{dy}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\left(\frac{{I}{}{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}{+}\frac{{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}\right){}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\left({-}\frac{{I}{}{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}{+}\frac{{{ⅇ}}^{{x}}{}\sqrt{{2}}}{{4}}\right){}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.4)

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

 M > $\mathrm{Φ1}≔\mathrm{RicciSpinor}\left(\mathrm{\sigma }\right)$
 ${\mathrm{Φ1}}{:=}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{}{\mathrm{dw1}}{+}\frac{{3}}{{8}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{}{\mathrm{dw2}}{+}\frac{{1}}{{16}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{}{\mathrm{dw2}}{+}\frac{{1}}{{16}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}{}{\mathrm{dw1}}{+}\frac{{1}}{{16}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{}{\mathrm{dw2}}{+}\frac{{1}}{{16}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{}{\mathrm{dw1}}{+}\frac{{3}}{{8}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{}{\mathrm{dw1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}{}{\mathrm{dw2}}$ (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 > $\mathrm{R2}≔\mathrm{RicciTensor}\left(g\right)$
 ${\mathrm{R2}}{:=}{-}\frac{{1}}{{2}}{}{\mathrm{dt}}{}{\mathrm{dt}}{-}\frac{{{ⅇ}}^{{x}}}{{4}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}\frac{{3}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}\frac{{{ⅇ}}^{{x}}}{{4}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}\frac{{{ⅇ}}^{{2}{}{x}}}{{4}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.6)
 M > $\mathrm{Φ2}≔\mathrm{RicciSpinor}\left(\mathrm{\sigma },\mathrm{R2}\right)$
 ${\mathrm{Φ2}}{:=}\frac{{1}}{{2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{}{\mathrm{dw1}}{+}\frac{{3}}{{8}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{}{\mathrm{dw2}}{+}\frac{{1}}{{16}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{}{\mathrm{dw2}}{+}\frac{{1}}{{16}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}{}{\mathrm{dw1}}{+}\frac{{1}}{{16}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{}{\mathrm{dw2}}{+}\frac{{1}}{{16}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{}{\mathrm{dw1}}{+}\frac{{3}}{{8}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{}{\mathrm{dw1}}{+}\frac{{1}}{{2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}{}{\mathrm{dw2}}$ (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 > $\mathrm{g3}≔\mathrm{SpinorInnerProduct}\left(\mathrm{\sigma },\mathrm{\sigma }\right)$
 ${\mathrm{g3}}{:=}{\mathrm{dt}}{}{\mathrm{dt}}{+}\frac{{{ⅇ}}^{{x}}}{{2}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}{\mathrm{dx}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{dy}}{+}\frac{{{ⅇ}}^{{x}}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}\frac{{{ⅇ}}^{{2}{}{x}}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.8)

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

 M > $C≔\mathrm{CurvatureTensor}\left(g\right)$
 ${C}{:=}{-}\frac{{{ⅇ}}^{{x}}}{{8}}{}{\mathrm{D_t}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dz}}{+}\frac{{{ⅇ}}^{{x}}}{{8}}{}{\mathrm{D_t}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}\frac{{1}}{{4}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dt}}{}{\mathrm{dx}}{-}\frac{{1}}{{4}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}{{ⅇ}}^{{x}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}{{ⅇ}}^{{x}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dx}}{-}\frac{{{ⅇ}}^{{2}{}{x}}}{{8}}{}{\mathrm{D_t}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}{+}\frac{{{ⅇ}}^{{2}{}{x}}}{{8}}{}{\mathrm{D_t}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}\frac{{1}}{{4}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dx}}{-}\frac{{1}}{{4}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dt}}{-}\frac{{{ⅇ}}^{{x}}}{{8}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}\frac{{{ⅇ}}^{{x}}}{{8}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}\frac{{{ⅇ}}^{{x}}}{{8}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dx}}{-}\frac{{{ⅇ}}^{{x}}}{{8}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dt}}{+}\frac{{3}{}{{ⅇ}}^{{2}{}{x}}}{{8}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}\frac{{3}{}{{ⅇ}}^{{2}{}{x}}}{{8}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}\frac{{1}}{{4}}{}{\mathrm{D_z}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}\frac{{1}}{{4}}{}{\mathrm{D_z}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}\frac{{7}}{{4}}{}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dx}}{}{\mathrm{dz}}{-}\frac{{7}}{{4}}{}{\mathrm{D_z}}{}{\mathrm{dx}}{}{\mathrm{dz}}{}{\mathrm{dx}}{+}\frac{{{ⅇ}}^{{x}}}{{8}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}\frac{{{ⅇ}}^{{x}}}{{8}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dt}}$ (2.9)
 M > $R≔\mathrm{RicciTensor}\left(C\right)$
 ${R}{:=}{-}\frac{{1}}{{2}}{}{\mathrm{dt}}{}{\mathrm{dt}}{-}\frac{{{ⅇ}}^{{x}}}{{4}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}\frac{{3}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}\frac{{{ⅇ}}^{{x}}}{{4}}{}{\mathrm{dz}}{}{\mathrm{dt}}{+}\frac{{{ⅇ}}^{{2}{}{x}}}{{4}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.10)
 M > $S≔\mathrm{RicciScalar}\left(g,C\right)$
 ${S}{:=}\frac{{5}}{{2}}$ (2.11)

Calculate the trace-free Ricci tensor $T$.

 M > $T≔\mathrm{evalDG}\left(R-\frac{1}{4}gS\right)$
 ${T}{:=}{-}\frac{{9}}{{8}}{}{\mathrm{dt}}{}{\mathrm{dt}}{-}\frac{{9}{}{{ⅇ}}^{{x}}}{{16}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}\frac{{7}}{{8}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}\frac{{5}}{{8}}{}{\mathrm{dy}}{}{\mathrm{dy}}{-}\frac{{9}{}{{ⅇ}}^{{x}}}{{16}}{}{\mathrm{dz}}{}{\mathrm{dt}}{-}\frac{{{ⅇ}}^{{2}{}{x}}}{{16}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.12)

Convert $T$ to a spinor $U$.

 M > $U≔\mathrm{convert}\left(T,\mathrm{DGspinor},\mathrm{\sigma },\left[1,2\right]\right)$
 ${U}{:=}{-}{\mathrm{dz1}}{}{\mathrm{dw1}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{-}\frac{{1}}{{8}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}{-}\frac{{3}}{{4}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{-}\frac{{1}}{{8}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{-}\frac{{1}}{{8}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{}{\mathrm{dz1}}{}{\mathrm{dw2}}{-}\frac{{3}}{{4}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{}{\mathrm{dz2}}{}{\mathrm{dw1}}{-}\frac{{1}}{{8}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}{}{\mathrm{dz1}}{}{\mathrm{dw1}}{-}{\mathrm{dz2}}{}{\mathrm{dw2}}{}{\mathrm{dz2}}{}{\mathrm{dw2}}$ (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 > $\mathrm{Φ1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{RearrangeIndices}\left(\left(-\frac{1}{2}\right)U,\left[\left[2,3\right]\right]\right)$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.14)