 WeylSpinor - Maple Help

Tensor[WeylSpinor] - calculate the spinor form of the Weyl tensor

Calling Sequences

WeylSpinor(${\mathbf{σ}}$, W)

WeylSpinor(dyad, PT,${\mathbf{η}}$, ${\mathbf{χ}}$)

Parameters

$\mathrm{σ}$      - a solder form

W      - (optional) the Weyl tensor for the metric determined by the solder form sigma

dyad   - a list of 2 independent, rank 1 covariant two-component spinors

NP     - a table, with indices "Psi0", "Psi1", "Psi2", "Psi3", "Psi4" and specifying the 5 Newman-Penrose coefficients for the Weyl spinor to be constructed

PT     - the Petrov type of the Weyl spinor to be constructed

$\mathrm{η}$,   - the complex numbers used to construct the Penrose normal form of the Weyl spinor Description

 • Let be the metric tensor defined by the solder form $\mathrm{σ}$ and let $W$ be the Weyl tensor for $g$. Then the spinor form of $W$ is a covariant rank 8 Hermitian spinor which, because of the algebraic properties of W, can be decomposed as

The symmetric rank 4 spinor ${\mathrm{\Psi }}_{\mathrm{ABCD}}$ is called the Weyl spinor. Ifis a spinor dyad (a pair of rank-2 spinors with ) then the spinor ${\mathrm{Ψ}}_{\mathrm{ABCD}}$

can be expressed as

The complex scalars  are called the Newman-Penrose coefficients for the Weyl tensor. Every Weyl spinor can be transformed by a change of dyad to a certain canonical form depending on the Petrov type of the WeylTensor. See AdaptedSpinorDyad, convert/DGspinor,  NPCurvatureScalars, PetrovType, SolderForm, WeylTensor.

 • If the Weyl tensor for the metric has been previously computed, then the Weyl spinor will be computed more quickly using the calling sequence WeylSpinor(${\mathbf{σ}}$, W).
 • In the second calling sequence the Weyl spinor is calculated directly from the a spinor dyadand a set of Newman-Penrose coefficients using equation (2).
 • The third calling sequence also uses equation (2), but the Newman-Penrose coefficients are calculated from the Petrov type according to the following normal forms rules:

Type I.

Type II. ${\mathrm{Ψ}}_{0}$

Type III.

Type D.

Type N.

Type O.

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

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

Example 1.

First create a vector bundle over with base coordinates and fiber coordinates.

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

Define a metric $g$ on $M$. For this example we use a pure radiation metric of Petrov type N. (See (22.70) in Stephani, Kramer et al.) Note that we have changed the sign of the metric to conform to the signature convention [1, -1, -1, -1] used by the spinor formalism in DifferentialGeometry.

 M > $g≔\mathrm{evalDG}\left(\mathrm{exp}\left(2k\left(t-\mathrm{\rho }\right)\right)\left(\mathrm{dt}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dt}-\mathrm{drho}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{drho}\right)-{\mathrm{\rho }}^{2}\mathrm{dphi}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dphi}-\mathrm{dz}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}\right)$
 ${g}{:=}{{ⅇ}}^{{-}{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{\mathrm{dt}}{}{\mathrm{dt}}{-}{{ⅇ}}^{{-}{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{\mathrm{drho}}{}{\mathrm{drho}}{-}{{\mathrm{ρ}}}^{{2}}{}{\mathrm{dphi}}{}{\mathrm{dphi}}{-}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.2)

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

 M > $F≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D_t},\mathrm{D_rho},\mathrm{D_phi},\mathrm{D_z}\right],g,\mathrm{signature}=\left[1,-1,-1,-1\right]\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}k::\mathrm{real},t::\mathrm{real},0<\mathrm{\rho }$
 ${F}{:=}\left[{{ⅇ}}^{{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{\mathrm{D_t}}{,}{{ⅇ}}^{{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{\mathrm{D_rho}}{,}\frac{{\mathrm{D_phi}}}{{\mathrm{ρ}}}{,}{\mathrm{D_z}}\right]$ (2.3)

Use SolderForm to compute the solder form sigma from the frame F.

 M > $\mathrm{\sigma }≔\mathrm{SolderForm}\left(F\right)$
 ${\mathrm{σ}}{:=}\frac{{1}}{{2}}{}{{ⅇ}}^{{-}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}{{ⅇ}}^{{-}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}\sqrt{{2}}{}{\mathrm{dt}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}{{ⅇ}}^{{-}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}\sqrt{{2}}{}{\mathrm{drho}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}{{ⅇ}}^{{-}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}\sqrt{{2}}{}{\mathrm{drho}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{2}}{}{I}{}{\mathrm{ρ}}{}\sqrt{{2}}{}{\mathrm{dphi}}{}{\mathrm{D_z1}}{}{\mathrm{D_w2}}{+}\frac{{1}}{{2}}{}{I}{}{\mathrm{ρ}}{}\sqrt{{2}}{}{\mathrm{dphi}}{}{\mathrm{D_z2}}{}{\mathrm{D_w1}}{+}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z1}}{}{\mathrm{D_w1}}{-}\frac{{1}}{{2}}{}\sqrt{{2}}{}{\mathrm{dz}}{}{\mathrm{D_z2}}{}{\mathrm{D_w2}}$ (2.4)

Calculate the Weyl spinor from the solder form sigma.

 M > $\mathrm{Ψ1}≔\mathrm{WeylSpinor}\left(\mathrm{\sigma }\right)$
 ${\mathrm{Ψ1}}{:=}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}$ (2.5)

Example 2.

We check that the same answer for the Weyl spinor Psi1 is obtained if we first calculate the Weyl tensor of the metric $g$ defined by $\mathrm{σ}$.

 M > $W≔\mathrm{WeylTensor}\left(g\right)$
 ${W}{:=}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{dt}}{}{\mathrm{dphi}}{}{\mathrm{dt}}{}{\mathrm{dphi}}{-}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{dt}}{}{\mathrm{dphi}}{}{\mathrm{drho}}{}{\mathrm{dphi}}{-}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{dt}}{}{\mathrm{dphi}}{}{\mathrm{dphi}}{}{\mathrm{dt}}{+}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{dt}}{}{\mathrm{dphi}}{}{\mathrm{dphi}}{}{\mathrm{drho}}{-}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{drho}}{}{\mathrm{dz}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dt}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{drho}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{drho}}{}{\mathrm{dphi}}{}{\mathrm{dt}}{}{\mathrm{dphi}}{+}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{drho}}{}{\mathrm{dphi}}{}{\mathrm{drho}}{}{\mathrm{dphi}}{+}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{drho}}{}{\mathrm{dphi}}{}{\mathrm{dphi}}{}{\mathrm{dt}}{-}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{drho}}{}{\mathrm{dphi}}{}{\mathrm{dphi}}{}{\mathrm{drho}}{+}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{drho}}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{drho}}{}{\mathrm{dz}}{}{\mathrm{drho}}{}{\mathrm{dz}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{drho}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{dt}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{drho}}{}{\mathrm{dz}}{}{\mathrm{dz}}{}{\mathrm{drho}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{dphi}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dphi}}{+}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{dphi}}{}{\mathrm{dt}}{}{\mathrm{drho}}{}{\mathrm{dphi}}{+}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{dphi}}{}{\mathrm{dt}}{}{\mathrm{dphi}}{}{\mathrm{dt}}{-}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{dphi}}{}{\mathrm{dt}}{}{\mathrm{dphi}}{}{\mathrm{drho}}{+}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{dphi}}{}{\mathrm{drho}}{}{\mathrm{dt}}{}{\mathrm{dphi}}{-}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{dphi}}{}{\mathrm{drho}}{}{\mathrm{drho}}{}{\mathrm{dphi}}{-}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{dphi}}{}{\mathrm{drho}}{}{\mathrm{dphi}}{}{\mathrm{dt}}{+}\frac{{1}}{{2}}{}{k}{}{\mathrm{ρ}}{}{\mathrm{dphi}}{}{\mathrm{drho}}{}{\mathrm{dphi}}{}{\mathrm{drho}}{+}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dt}}{}{\mathrm{dz}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{drho}}{}{\mathrm{dz}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{dt}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{dz}}{}{\mathrm{dt}}{}{\mathrm{dz}}{}{\mathrm{drho}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{dz}}{}{\mathrm{drho}}{}{\mathrm{dt}}{}{\mathrm{dz}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{dz}}{}{\mathrm{drho}}{}{\mathrm{drho}}{}{\mathrm{dz}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{dz}}{}{\mathrm{drho}}{}{\mathrm{dz}}{}{\mathrm{dt}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{2}}{}\frac{{k}{}{\mathrm{dz}}{}{\mathrm{drho}}{}{\mathrm{dz}}{}{\mathrm{drho}}}{{\mathrm{ρ}}}$ (2.6)
 M > $\mathrm{Ψ2}≔\mathrm{WeylSpinor}\left(\mathrm{\sigma },W\right)$
 ${\mathrm{Ψ2}}{:=}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}$ (2.7)
 M > $\mathrm{Ψ1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{Ψ2}$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.8)

Example 3.

We see that the rank 4 spinor Psi1 calculated by WeylSpinor factors as the 4-th tensor product of a rank 1 spinor psi. This affirms the fact that the metric g has Petrov type N.

 M > $\mathrm{\alpha }≔-\frac{\frac{1}{4}\cdot 1}{\mathrm{\rho }}\mathrm{exp}\left(2k\left(-t+\mathrm{\rho }\right)\right)k$
 ${\mathrm{α}}{:=}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}}{{\mathrm{ρ}}}$ (2.9)
 M > $\mathrm{\psi }≔{\mathrm{\alpha }}^{\frac{1}{4}}\mathrm{evalDG}\left(\mathrm{dz1}-\mathrm{dz2}\right)$
 ${\mathrm{ψ}}{:=}{\left({-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}}{{\mathrm{ρ}}}\right)}^{{1}{/}{4}}{}\left({\mathrm{dz1}}{-}{\mathrm{dz2}}\right)$ (2.10)
 M > $\mathrm{ψ4}≔\mathrm{evalDG}\left(\left(\left(\mathrm{\psi }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\psi }\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\psi }\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{\psi }\right)$
 ${\mathrm{ψ4}}{:=}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}{+}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}}{{\mathrm{ρ}}}{-}\frac{{1}}{{4}}{}\frac{{{ⅇ}}^{{2}{}{k}{}\left({-}{t}{+}{\mathrm{ρ}}\right)}{}{k}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}}{{\mathrm{ρ}}}$ (2.11)
 M > $\mathrm{Ψ1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&minus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{ψ4}$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.12)

Example 4.

We check that the Weyl spinor Psi1 satisfies the decomposition (1) given above. From the Weyl tensor W calculated in Example 2, we find the left-hand side LHS of (1).  (The intermediate expressions, even in this simple example, are too long to display.)

 M > $\mathrm{WS}≔\mathrm{convert}\left(W,\mathrm{DGspinor},\mathrm{\sigma },\left[1,2,3,4\right]\right):$

We re-arrange the indices to place all the unbarred (unprimed) indices first and all the barred (primed) indices last.

 M > $\mathrm{LHS}≔\mathrm{RearrangeIndices}\left(\mathrm{WS},\left[1,5,2,6,3,7,4,8\right]\right):$

We calculate the first terms on the right-hand side of (1) as RHS1.

 M > $\mathrm{barE}≔\mathrm{EpsilonSpinor}\left("cov","barspinor"\right)$
 ${\mathrm{barE}}{:=}{\mathrm{dw1}}{}{\mathrm{dw2}}{-}{\mathrm{dw2}}{}{\mathrm{dw1}}$ (2.13)
 M > $\mathrm{RHS1}≔\left(\mathrm{Ψ1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&tensor\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{barE}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&tensor\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{barE}:$

We use the command ConjugateSpinor to find the complex conjugate of Psi1. Then we calculate the second terms on the right-hand side of (1) as RHS2.

 M > $\mathrm{barPsi1}≔\mathrm{ConjugateSpinor}\left(\mathrm{Ψ1}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}k::\mathrm{real}:$
 M > $E≔\mathrm{EpsilonSpinor}\left("cov","spinor"\right)$
 ${E}{:=}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{\mathrm{dz2}}{}{\mathrm{dz1}}$ (2.14)
 M > $\mathrm{RHS2}≔\left(E\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&tensor\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}E\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&tensor\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{barPsi1}:$

We check that the left-hand side and right-hand side of (1) are the same.

 M > $\mathrm{evalDG}\left(\mathrm{LHS}-\left(\mathrm{RHS1}+\mathrm{RHS2}\right)\right)$
 ${0}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.15)

Example 5.

We use the second calling sequence to calculate a Weyl spinor from a spinor dyad and a set of Newman-Penrose coefficients.

 M > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.16)
 > $\mathrm{dyad}≔\left[\mathrm{dz1},\mathrm{dz2}\right]$
 ${\mathrm{dyad}}{:=}\left[{\mathrm{dz1}}{,}{\mathrm{dz2}}\right]$ (2.17)
 > $\mathrm{NP}≔\mathrm{table}\left(\left["Psi0"=0,"Psi1"=0,"Psi2"=0,"Psi3"=z,"Psi4"=t\right]\right)$
 ${\mathrm{NP}}{:=}{\mathrm{table}}\left(\left[{"Psi1"}{=}{0}{,}{"Psi2"}{=}{0}{,}{"Psi0"}{=}{0}{,}{"Psi4"}{=}{t}{,}{"Psi3"}{=}{z}\right]\right)$ (2.18)
 > $\mathrm{WeylSpinor}\left(\mathrm{dyad},\mathrm{NP}\right)$
 ${t}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{z}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{-}{z}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{-}{z}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{-}{z}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.19)

Example 6.

We use the third calling sequence to calculate a Weyl spinor in adapted normal form.

 M > $\mathrm{dyad}≔\left[\mathrm{dz1},\mathrm{dz2}\right]$
 ${\mathrm{dyad}}{:=}\left[{\mathrm{dz1}}{,}{\mathrm{dz2}}\right]$ (2.20)
 M > $\mathrm{WeylSpinor}\left(\mathrm{dyad},"N"\right)$
 ${\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.21)
 M > $\mathrm{WeylSpinor}\left(\mathrm{dyad},"II",A\right)$
 ${6}{}{A}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{+}{A}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{+}{A}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{A}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{A}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{+}{A}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{+}{A}{}{\mathrm{dz2}}{}{\mathrm{dz2}}{}{\mathrm{dz1}}{}{\mathrm{dz1}}$ (2.22) See Also