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

Calling Sequences

     WeylSpinor(σ, W)

     WeylSpinor(dyad, NP)

     WeylSpinor(dyad, PT,η, χ)

Parameters

   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

   η,χ    - the complex numbers used to construct the Penrose normal form of the Weyl spinor

 

Description

Examples

See Also

Description

• 

Let g be the metric tensor defined by the solder form 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

WAA'BB'CC'DD= ΨABCD ϵA'B'ϵC'D' + ΨA'B'C'D' ϵAB ϵCD .    1

 The symmetric rank 4 spinor ΨABCD is called the Weyl spinor. IfιA, οA is a spinor dyad (a pair of rank-2 spinors with ιAοA =1) then the spinor ΨABCD

 can be expressed as

ΨABCD= Ψ0 ιAιBιCιD 4 Ψ1 ι(AιBιCοD)+ 6 Ψ2 ι(AιBοCοD) 4 Ψ3 ι(AοBοCοD) + Ψ4 οAοBοCοD .     2

The complex scalars Ψ0 , Ψ1,Ψ2,Ψ3,Ψ4 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 g has been previously computed, then the Weyl spinor will be computed more quickly using the calling sequence WeylSpinor(σ, W).

• 

In the second calling sequence the Weyl spinor is calculated directly from the a spinor dyad ιA, οA and 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. Ψ0= 32 η χ , Ψ1 = 0, Ψ2=12η2  χ, Ψ3 =0, Ψ4= 32η χ .

Type II. Ψ0 = 0, Ψ1 = 0, Ψ2=η, Ψ3 =0, Ψ4= 6 η.

Type III. Ψ0= 0, Ψ1 = 0, Ψ2 =0,Ψ3 =1, Ψ4 = 0.

Type D. Ψ0= 0, Ψ1 = 0, Ψ2=η, Ψ3 = 0, Ψ4 = 0.

Type N. Ψ0 = 0, Ψ1 =0, Ψ2= 0, Ψ3 = 0, Ψ4= 1.

Type O. Ψ0= 0, Ψ1 = 0, Ψ2=0, Ψ3 =0, Ψ4 = 0.

• 

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

withDifferentialGeometry:withTensor:

 

Example 1.

First create a vector bundle over M with base coordinates t, ρ, φ , z and fiber coordinatesz1 ,z2, w1,w2.

DGsetupt,ρ,φ,z,z1,z2,w1,w2,M

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 > 

gevalDGexp2ktρdt&tdtdrho&tdrhoρ2dphi&tdphidz&tdz

g:=ⅇ2kt+ρdtdtⅇ2kt+ρdrhodrhoρ2dphidphidzdz

(2.2)

 

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

M > 

FDGGramSchmidtD_t&comma;D_rho&comma;D_phi&comma;D_z&comma;g&comma;signature=1&comma;1&comma;1&comma;1assumingk::real,t::real,0<ρ

F:=&ExponentialE;kt&plus;&rho;D_t&comma;&ExponentialE;kt&plus;&rho;D_rho&comma;D_phi&rho;&comma;D_z

(2.3)

 

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

M > 

σSolderFormF

&sigma;:=12&ExponentialE;kt&plus;&rho;2dtD_z1D_w1&plus;12&ExponentialE;kt&plus;&rho;2dtD_z2D_w2&plus;12&ExponentialE;kt&plus;&rho;2drhoD_z1D_w2&plus;12&ExponentialE;kt&plus;&rho;2drhoD_z2D_w112I&rho;2dphiD_z1D_w2&plus;12I&rho;2dphiD_z2D_w1&plus;122dzD_z1D_w1122dzD_z2D_w2

(2.4)

 

Calculate the Weyl spinor from the solder form sigma.

M > 

&Psi;1WeylSpinorσ

&Psi;1:=14&ExponentialE;2kt&plus;&rho;kdz1dz1dz1dz1&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz1dz1dz1dz2&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz1dz1dz2dz1&rho;14&ExponentialE;2kt&plus;&rho;kdz1dz1dz2dz2&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz1dz2dz1dz1&rho;14&ExponentialE;2kt&plus;&rho;kdz1dz2dz1dz2&rho;14&ExponentialE;2kt&plus;&rho;kdz1dz2dz2dz1&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz1dz2dz2dz2&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz2dz1dz1dz1&rho;14&ExponentialE;2kt&plus;&rho;kdz2dz1dz1dz2&rho;14&ExponentialE;2kt&plus;&rho;kdz2dz1dz2dz1&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz2dz1dz2dz2&rho;14&ExponentialE;2kt&plus;&rho;kdz2dz2dz1dz1&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz2dz2dz1dz2&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz2dz2dz2dz1&rho;14&ExponentialE;2kt&plus;&rho;kdz2dz2dz2dz2&rho;

(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 &sigma;.

M > 

WWeylTensorg

W:=12k&rho;dtdphidtdphi12k&rho;dtdphidrhodphi12k&rho;dtdphidphidt&plus;12k&rho;dtdphidphidrho12kdtdzdtdz&rho;&plus;12kdtdzdrhodz&rho;&plus;12kdtdzdzdt&rho;12kdtdzdzdrho&rho;12k&rho;drhodphidtdphi&plus;12k&rho;drhodphidrhodphi&plus;12k&rho;drhodphidphidt12k&rho;drhodphidphidrho&plus;12kdrhodzdtdz&rho;12kdrhodzdrhodz&rho;12kdrhodzdzdt&rho;&plus;12kdrhodzdzdrho&rho;12k&rho;dphidtdtdphi&plus;12k&rho;dphidtdrhodphi&plus;12k&rho;dphidtdphidt12k&rho;dphidtdphidrho&plus;12k&rho;dphidrhodtdphi12k&rho;dphidrhodrhodphi12k&rho;dphidrhodphidt&plus;12k&rho;dphidrhodphidrho&plus;12kdzdtdtdz&rho;12kdzdtdrhodz&rho;12kdzdtdzdt&rho;&plus;12kdzdtdzdrho&rho;12kdzdrhodtdz&rho;&plus;12kdzdrhodrhodz&rho;&plus;12kdzdrhodzdt&rho;12kdzdrhodzdrho&rho;

(2.6)
M > 

&Psi;2WeylSpinorσ&comma;W

&Psi;2:=14&ExponentialE;2kt&plus;&rho;kdz1dz1dz1dz1&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz1dz1dz1dz2&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz1dz1dz2dz1&rho;14&ExponentialE;2kt&plus;&rho;kdz1dz1dz2dz2&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz1dz2dz1dz1&rho;14&ExponentialE;2kt&plus;&rho;kdz1dz2dz1dz2&rho;14&ExponentialE;2kt&plus;&rho;kdz1dz2dz2dz1&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz1dz2dz2dz2&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz2dz1dz1dz1&rho;14&ExponentialE;2kt&plus;&rho;kdz2dz1dz1dz2&rho;14&ExponentialE;2kt&plus;&rho;kdz2dz1dz2dz1&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz2dz1dz2dz2&rho;14&ExponentialE;2kt&plus;&rho;kdz2dz2dz1dz1&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz2dz2dz1dz2&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz2dz2dz2dz1&rho;14&ExponentialE;2kt&plus;&rho;kdz2dz2dz2dz2&rho;

(2.7)
M > 

&Psi;1&minus&Psi;2

0dz1dz1dz1dz1

(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 > 

α141ρexp2kt+ρk

&alpha;:=14&ExponentialE;2kt&plus;&rho;k&rho;

(2.9)
M > 

ψα14evalDGdz1dz2

&psi;:=14&ExponentialE;2kt&plus;&rho;k&rho;1&sol;4dz1dz2

(2.10)
M > 

&psi;4evalDGψ&tψ&tψ&tψ

&psi;4:=14&ExponentialE;2kt&plus;&rho;kdz1dz1dz1dz1&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz1dz1dz1dz2&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz1dz1dz2dz1&rho;14&ExponentialE;2kt&plus;&rho;kdz1dz1dz2dz2&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz1dz2dz1dz1&rho;14&ExponentialE;2kt&plus;&rho;kdz1dz2dz1dz2&rho;14&ExponentialE;2kt&plus;&rho;kdz1dz2dz2dz1&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz1dz2dz2dz2&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz2dz1dz1dz1&rho;14&ExponentialE;2kt&plus;&rho;kdz2dz1dz1dz2&rho;14&ExponentialE;2kt&plus;&rho;kdz2dz1dz2dz1&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz2dz1dz2dz2&rho;14&ExponentialE;2kt&plus;&rho;kdz2dz2dz1dz1&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz2dz2dz1dz2&rho;&plus;14&ExponentialE;2kt&plus;&rho;kdz2dz2dz2dz1&rho;14&ExponentialE;2kt&plus;&rho;kdz2dz2dz2dz2&rho;

(2.11)
M > 

&Psi;1&minus&psi;4

0dz1dz1dz1dz1

(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 > 

WSconvertW&comma;DGspinor&comma;σ&comma;1&comma;2&comma;3&comma;4&colon;

 

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

M > 

LHSRearrangeIndicesWS&comma;1&comma;5&comma;2&comma;6&comma;3&comma;7&comma;4&comma;8&colon;

 

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

M > 

barEEpsilonSpinorcov&comma;barspinor

barE:=dw1dw2dw2dw1

(2.13)
M > 

RHS1&Psi;1&tensorbarE&tensorbarE&colon;

 

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 > 

barPsi1ConjugateSpinor&Psi;1assumingk::real&colon;

M > 

EEpsilonSpinorcov&comma;spinor

E:=dz1dz2dz2dz1

(2.14)
M > 

RHS2E&tensorE&tensorbarPsi1&colon;

 

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

M > 

evalDGLHSRHS1+RHS2

0dz1dz1dz1dz1dz1dz1dz1dz1

(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 > 

DGsetupt&comma;x&comma;y&comma;z&comma;z1&comma;z2&comma;w1&comma;w2&comma;M

frame name: M

(2.16)

dyaddz1&comma;dz2

dyad:=dz1&comma;dz2

(2.17)

NPtablePsi0=0&comma;Psi1=0&comma;Psi2=0&comma;Psi3=z&comma;Psi4=t

NP:=tablePsi1&equals;0&comma;Psi2&equals;0&comma;Psi0&equals;0&comma;Psi4&equals;t&comma;Psi3&equals;z

(2.18)

WeylSpinordyad&comma;NP

tdz1dz1dz1dz1zdz1dz1dz1dz2zdz1dz1dz2dz1zdz1dz2dz1dz1zdz2dz1dz1dz1

(2.19)

 

Example 6.

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

M > 

dyaddz1&comma;dz2

dyad:=dz1&comma;dz2

(2.20)
M > 

WeylSpinordyad&comma;N

dz1dz1dz1dz1

(2.21)
M > 

WeylSpinordyad&comma;II&comma;A

6Adz1dz1dz1dz1&plus;Adz1dz1dz2dz2&plus;Adz1dz2dz1dz2&plus;Adz1dz2dz2dz1&plus;Adz2dz1dz1dz2&plus;Adz2dz1dz2dz1&plus;Adz2dz2dz1dz1

(2.22)

See Also

DifferentialGeometry, Tensor, AdaptedSpinorDyad, AdaptedNullTetrad, ConjugateSpinor, DGGramSchmidt, NPCurvatureScalars, Physics[Riemann], PetrovType, RicciSpinor, Physics[Ricci], SolderForm, WeylTensor, Physics[Weyl]