 find a spinor dyad which transforms the Weyl spinor to normal form - Maple Programming Help

Calling Sequences

Parameters

W         - a symmetric rank 4 covariant spinor

PT        - the Petrov type of the spinor $W$

options   - one or more of the keyword arguments method and output

Description

 • Let ${W}_{\mathrm{ABCD}}$ be a rank 4 symmetric spinor and let  be a solder form on a 4-dimensional spacetime of signature [1, -1, -1, -1]. Then the rank 4 tensor

=

enjoys all the symmetries of the Weyl tensor. It is skew-symmetric in the indices $\mathrm{ab}$ and satisfies the cyclic identity on $\mathrm{bcd},$ and is trace-free with respect to the metric defined by the solder form $\mathrm{σ}$.

 • If is a spinor dyad (a pair of rank-2 spinors with ) then the spinor can be expressed as

.    (*)

The coefficients coincide with the Newman-Penrose Weyl scalars for the Weyl tensor constructed from the null tetrad defined by the spinor dyad

 • The Petrov type of the spinor ${W}_{\mathrm{ABCD}}$ coincides with the Petrov type of the Weyl tensor or the Petrov type of the spacetime determined by the Newman-Penrose Weyl scalars. See NPCurvatureScalars, NullTetrad, PetrovType, SolderForm, WeylSpinor.
 • The normal forms for the Newman-Penrose coefficients are as follows.

Type I.

Type II. ${\mathrm{Ψ}}_{0}$

Type III.

Type D.

Type N.

Type O.

See Penrose and Rindle Vol. 2, Section 8.3.

Thus, for example, if the Petrov type of the Weyl spinor is $\mathrm{D}$, then there is a spinor dyad such that the Weyl spinor takes the form . Such a spinor dyad is said to be an adapted spinor dyad.

 • The command AdaptedSpinorDyad returns a contravariant spinor dyad which will put the given Weyl spinor in the above normal form. If the Petrov type is I, then the values of are also returned. If the Petrov type is II or D, the value of is given.
 • The adapted spinor dyads need not be unique; multiple adapted spinor dyads may be returned. All computed dyads will be returned with the keyword argument output = $"all"$. If the type is III or N, then a quasi-adapted dyad (where the 1 non-zero Newman-Penrose coefficient is not normalized to 1) will be calculated if method = "unnormalized".
 • The command AdaptedSpinorDyad is part of the DifferentialGeometry:-Tensor package. It can be used in the form AdaptedSpinorDyad(...) only after executing the commands with(DifferentialGeometry) and with(Tensor), but can always be used by executing DifferentialGeometry:-Tensor:-AdaptedSpinorDyad(...).

Examples

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

Set the global environment variable _EnvExplicit to true to insure that our results are free of expressions.

 Spin > $\mathrm{_EnvExplicit}≔\mathrm{true}:$

We give examples of Weyl spinors of each Petrov type and calculate an adapted spinor dyad. We check that the spinor dyad has the desired properties.

First create the spinor bundle over a 4 dimensional spacetime.

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

In order to construct the Weyl spinors for our examples, we need a basis for the vector space of symmetric rank 4 spinors. This we obtain from the GenerateSymmetricTensors command.

 Spin > $S≔\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{dz1},\mathrm{dz2}\right],4\right)$
 ${S}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}\frac{{1}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{6}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{4}}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{4}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.2)

Example 1. Type I

Define a rank 4 spinor ${W}_{1}.$

 Spin > $\mathrm{W1}≔\mathrm{DGzip}\left(\left[6,12,30,24,6\right],S,"plus"\right)$
 ${\mathrm{W1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{6}\right]{,}\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{5}\right]{,}{3}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}{5}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{6}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{5}\right]{,}{6}\right]{,}\left[\left[{6}{,}{6}{,}{6}{,}{6}\right]{,}{6}\right]\right]\right]\right)$ (2.3)

Calculate the Newman-Penrose coefficients for ${W}_{1}$ with respect to the initial dyad basis .

 Spin > $\mathrm{NP1}≔\mathrm{NPCurvatureScalars}\left(\mathrm{W1},\left[\mathrm{dz1},\mathrm{dz2}\right]\right)$
 ${\mathrm{NP1}}{:=}{\mathrm{table}}\left(\left[{"Psi2"}{=}{5}{,}{"Psi0"}{=}{6}{,}{"Psi1"}{=}{-}{6}{,}{"Psi3"}{=}{-}{3}{,}{"Psi4"}{=}{6}\right]\right)$ (2.4)

Use these coefficients to find the Petrov type of ${W}_{1}.$

 Spin > $\mathrm{PetrovType}\left(\mathrm{NP1}\right)$
 ${"I"}$ (2.5)

 Spin > $\mathrm{SpinorDyadCon1},\mathrm{η1},\mathrm{χ1}≔\mathrm{AdaptedSpinorDyad}\left(\mathrm{W1},"I"\right)$
 ${\mathrm{SpinorDyadCon1}}{,}{\mathrm{η1}}{,}{\mathrm{χ1}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{Spin}}{,}\left[\right]\right]{,}\left[\left[\left[{6}\right]{,}{1}\right]\right]\right]\right)\right]{,}{1}{,}{4}$ (2.6)

Here is the covariant form of the spinor dyad.

 Spin > $\mathrm{Dyad1}≔\mathrm{map}\left(\mathrm{RaiseLowerSpinorIndices},\mathrm{SpinorDyadCon1},\left[1\right]\right)$
 ${\mathrm{Dyad1}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]{,}\left[\left[{6}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}\right]{,}{-1}\right]\right]\right]\right)\right]$ (2.7)

We check that this answer is correct in two ways. First we can re-calculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.

 Spin > $\mathrm{NPCurvatureScalars}\left(\mathrm{W1},\mathrm{Dyad1}\right)$
 ${\mathrm{table}}\left(\left[{"Psi2"}{=}{-}{1}{,}{"Psi0"}{=}{6}{,}{"Psi1"}{=}{0}{,}{"Psi3"}{=}{0}{,}{"Psi4"}{=}{6}\right]\right)$ (2.8)

This is the correct normal form since and

Second, we can calculate a type I Weyl spinor from this spinor dyad using the command WeylSpinor and check that the result coincides with the original spinor ${W}_{1}$.

 Spin > $\mathrm{W1Check}≔\mathrm{WeylSpinor}\left(\mathrm{Dyad1},"I",1,4\right)$
 ${\mathrm{W1Check}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{\mathrm{Spin}}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{6}\right]{,}\left[\left[{5}{,}{5}{,}{5}{,}{6}\right]{,}{3}\right]{,}\left[\left[{5}{,}{5}{,}{6}{,}{5}\right]\right]\right]\right]\right)$