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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)


   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




See Also



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(...).




Example 1.

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


frame name: M



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 > 

gevalDGⅇ2ktρdt &t dtdrho &t drhoρ2dphi &t dphidz &t dz




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

M > 





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

M > 





Calculate the Weyl spinor from the solder form sigma.

M > 





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 > 



M > 



M > 

&Psi;1 &minus &Psi;2




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 > 



M > 



M > 

&psi;4evalDG&psi; &t &psi; &t &psi; &t &psi;


M > 

&Psi;1 &minus &psi;4




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 > 



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

M > 



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

M > 



M > 

RHS1&Psi;1 &tensor barE &tensor barE&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 > 


M > 



M > 

RHS2E &tensor E &tensor barPsi1&colon;


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

M > 





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 > 


frame name: M












Example 6.

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

M > 



M > 



M > 




See Also

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