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
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' + Ψ‾A'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(...).
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
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.
g≔evalDGexp2kt−ρdt&tdt−drho&tdrho−ρ2dphi&tdphi−dz&tdz
g:=ⅇ−2k−t+ρdtdt−ⅇ−2k−t+ρdrhodrho−ρ2dphidphi−dzdz
Use DGGramSchmidt to calculate an orthonormal frame F for the metric g.
F≔DGGramSchmidtD_t,D_rho,D_phi,D_z,g,signature=1,−1,−1,−1assumingk::real,t::real,0<ρ
F:=ⅇk−t+ρD_t,ⅇk−t+ρD_rho,D_phiρ,D_z
Use SolderForm to compute the solder form sigma from the frame F.
σ≔SolderFormF
σ:=12ⅇ−k−t+ρ2dtD_z1D_w1+12ⅇ−k−t+ρ2dtD_z2D_w2+12ⅇ−k−t+ρ2drhoD_z1D_w2+12ⅇ−k−t+ρ2drhoD_z2D_w1−12Iρ2dphiD_z1D_w2+12Iρ2dphiD_z2D_w1+122dzD_z1D_w1−122dzD_z2D_w2
Calculate the Weyl spinor from the solder form sigma.
Ψ1≔WeylSpinorσ
Ψ1:=−14ⅇ2k−t+ρkdz1dz1dz1dz1ρ+14ⅇ2k−t+ρkdz1dz1dz1dz2ρ+14ⅇ2k−t+ρkdz1dz1dz2dz1ρ−14ⅇ2k−t+ρkdz1dz1dz2dz2ρ+14ⅇ2k−t+ρkdz1dz2dz1dz1ρ−14ⅇ2k−t+ρkdz1dz2dz1dz2ρ−14ⅇ2k−t+ρkdz1dz2dz2dz1ρ+14ⅇ2k−t+ρkdz1dz2dz2dz2ρ+14ⅇ2k−t+ρkdz2dz1dz1dz1ρ−14ⅇ2k−t+ρkdz2dz1dz1dz2ρ−14ⅇ2k−t+ρkdz2dz1dz2dz1ρ+14ⅇ2k−t+ρkdz2dz1dz2dz2ρ−14ⅇ2k−t+ρkdz2dz2dz1dz1ρ+14ⅇ2k−t+ρkdz2dz2dz1dz2ρ+14ⅇ2k−t+ρkdz2dz2dz2dz1ρ−14ⅇ2k−t+ρkdz2dz2dz2dz2ρ
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 σ.
W≔WeylTensorg
W:=12kρdtdphidtdphi−12kρdtdphidrhodphi−12kρdtdphidphidt+12kρdtdphidphidrho−12kdtdzdtdzρ+12kdtdzdrhodzρ+12kdtdzdzdtρ−12kdtdzdzdrhoρ−12kρdrhodphidtdphi+12kρdrhodphidrhodphi+12kρdrhodphidphidt−12kρdrhodphidphidrho+12kdrhodzdtdzρ−12kdrhodzdrhodzρ−12kdrhodzdzdtρ+12kdrhodzdzdrhoρ−12kρdphidtdtdphi+12kρdphidtdrhodphi+12kρdphidtdphidt−12kρdphidtdphidrho+12kρdphidrhodtdphi−12kρdphidrhodrhodphi−12kρdphidrhodphidt+12kρdphidrhodphidrho+12kdzdtdtdzρ−12kdzdtdrhodzρ−12kdzdtdzdtρ+12kdzdtdzdrhoρ−12kdzdrhodtdzρ+12kdzdrhodrhodzρ+12kdzdrhodzdtρ−12kdzdrhodzdrhoρ
Ψ2≔WeylSpinorσ,W
Ψ2:=−14ⅇ2k−t+ρkdz1dz1dz1dz1ρ+14ⅇ2k−t+ρkdz1dz1dz1dz2ρ+14ⅇ2k−t+ρkdz1dz1dz2dz1ρ−14ⅇ2k−t+ρkdz1dz1dz2dz2ρ+14ⅇ2k−t+ρkdz1dz2dz1dz1ρ−14ⅇ2k−t+ρkdz1dz2dz1dz2ρ−14ⅇ2k−t+ρkdz1dz2dz2dz1ρ+14ⅇ2k−t+ρkdz1dz2dz2dz2ρ+14ⅇ2k−t+ρkdz2dz1dz1dz1ρ−14ⅇ2k−t+ρkdz2dz1dz1dz2ρ−14ⅇ2k−t+ρkdz2dz1dz2dz1ρ+14ⅇ2k−t+ρkdz2dz1dz2dz2ρ−14ⅇ2k−t+ρkdz2dz2dz1dz1ρ+14ⅇ2k−t+ρkdz2dz2dz1dz2ρ+14ⅇ2k−t+ρkdz2dz2dz2dz1ρ−14ⅇ2k−t+ρkdz2dz2dz2dz2ρ
Ψ1&minusΨ2
0dz1dz1dz1dz1
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.
α≔−14⋅1ρexp2k−t+ρk
α:=−14ⅇ2k−t+ρkρ
ψ≔α14evalDGdz1−dz2
ψ:=−14ⅇ2k−t+ρkρ1/4dz1−dz2
ψ4≔evalDGψ&tψ&tψ&tψ
ψ4:=−14ⅇ2k−t+ρkdz1dz1dz1dz1ρ+14ⅇ2k−t+ρkdz1dz1dz1dz2ρ+14ⅇ2k−t+ρkdz1dz1dz2dz1ρ−14ⅇ2k−t+ρkdz1dz1dz2dz2ρ+14ⅇ2k−t+ρkdz1dz2dz1dz1ρ−14ⅇ2k−t+ρkdz1dz2dz1dz2ρ−14ⅇ2k−t+ρkdz1dz2dz2dz1ρ+14ⅇ2k−t+ρkdz1dz2dz2dz2ρ+14ⅇ2k−t+ρkdz2dz1dz1dz1ρ−14ⅇ2k−t+ρkdz2dz1dz1dz2ρ−14ⅇ2k−t+ρkdz2dz1dz2dz1ρ+14ⅇ2k−t+ρkdz2dz1dz2dz2ρ−14ⅇ2k−t+ρkdz2dz2dz1dz1ρ+14ⅇ2k−t+ρkdz2dz2dz1dz2ρ+14ⅇ2k−t+ρkdz2dz2dz2dz1ρ−14ⅇ2k−t+ρkdz2dz2dz2dz2ρ
Ψ1&minusψ4
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.)
WS≔convertW,DGspinor,σ,1,2,3,4:
We re-arrange the indices to place all the unbarred (unprimed) indices first and all the barred (primed) indices last.
LHS≔RearrangeIndicesWS,1,5,2,6,3,7,4,8:
We calculate the first terms on the right-hand side of (1) as RHS1.
barE≔EpsilonSpinorcov,barspinor
barE:=dw1dw2−dw2dw1
RHS1≔Ψ1&tensorbarE&tensorbarE:
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.
barPsi1≔ConjugateSpinorΨ1assumingk::real:
E≔EpsilonSpinorcov,spinor
E:=dz1dz2−dz2dz1
RHS2≔E&tensorE&tensorbarPsi1:
We check that the left-hand side and right-hand side of (1) are the same.
evalDGLHS−RHS1+RHS2
0dz1dz1dz1dz1dz1dz1dz1dz1
Example 5.
We use the second calling sequence to calculate a Weyl spinor from a spinor dyad and a set of Newman-Penrose coefficients.
DGsetupt,x,y,z,z1,z2,w1,w2,M
dyad≔dz1,dz2
dyad:=dz1,dz2
NP≔tablePsi0=0,Psi1=0,Psi2=0,Psi3=z,Psi4=t
NP:=tablePsi1=0,Psi2=0,Psi0=0,Psi4=t,Psi3=z
WeylSpinordyad,NP
tdz1dz1dz1dz1−zdz1dz1dz1dz2−zdz1dz1dz2dz1−zdz1dz2dz1dz1−zdz2dz1dz1dz1
Example 6.
We use the third calling sequence to calculate a Weyl spinor in adapted normal form.
WeylSpinordyad,N
dz1dz1dz1dz1
WeylSpinordyad,II,A
6Adz1dz1dz1dz1+Adz1dz1dz2dz2+Adz1dz2dz1dz2+Adz1dz2dz2dz1+Adz2dz1dz1dz2+Adz2dz1dz2dz1+Adz2dz2dz1dz1
DifferentialGeometry, Tensor, AdaptedSpinorDyad, AdaptedNullTetrad, ConjugateSpinor, DGGramSchmidt, NPCurvatureScalars, Physics[Riemann], PetrovType, RicciSpinor, Physics[Ricci], SolderForm, WeylTensor, Physics[Weyl]
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