calculate the spinor form of the Weyl tensor

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

Calling Sequences

WeylSpinor(sigma, W)

WeylSpinor(dyad, NP)

WeylSpinor(dyad, PT, eta, chi)

Parameters

sigma - 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 coeffficients for the Weyl spinor to be constructed

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

eta, chi - the complex numbers used to contruct the Penrose normal form of the Weyl spinor

Description

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

The symmetric rank 4 spinor is called the Weyl spinor. Ifis a spinor dyad (a pair of rank-2 spinors with ) then the spinor

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(sigma, W).

•	In the second calling sequence the Weyl spinor is calculated directly from the a spinor dyad and a set of Newman-Penrose coefficients using equation (2).

•	The third calling seqeunce 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.

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

Example 1.

First create a vector bundle over M with base coordinates [t, rho, phi, z] and fiber coordinates [z1, z2, w1, w2].

(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. See SpacetimeConventions.)

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(2.2)

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

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(2.3)

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

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_DG([["tensor", M, [["cov_bas", "con_vrt", "con_vrt"], []]], [[[1, 5, 7], (1/2)*exp(-k*(-t+rho))*2^(1/2)], [[1, 6, 8], (1/2)*exp(-k*(-t+rho))*2^(1/2)], [[2, 5, 8], (1/2)*exp(-k*(-t+rho))*2^(1/2)], [[2, 6, 7], (1/2)*exp(-k*(-t+rho))*2^(1/2)], [[3, 5, 8], -((1/2)*I)*rho*2^(1/2)], [[3, 6, 7], ((1/2)*I)*rho*2^(1/2)], [[4, 5, 7], (1/2)*2^(1/2)], [[4, 6, 8], -(1/2)*2^(1/2)]]])

(2.4)

Calculate the Weyl spinor from the solder form sigma.

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(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.

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(2.6)

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(2.7)

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(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.

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alpha := -(1*1)*exp(2*k*(-t+rho))*k/(4*rho)

-(1/4)*exp(2*k*(-t+rho))*k/rho

(2.9)

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(-(1/4)*exp(2*k*(-t+rho))*k/rho)^(1/4)*_DG([["form", M, 1], [[[5], 1], [[6], -1]]])

(2.10)

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(2.11)

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(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.)

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

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

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(2.13)

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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.

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(2.14)

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

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(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.

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(2.16)

(2.17)

(2.18)

(2.19)

Example 6.

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

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(2.20)

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(2.21)

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(2.22)

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