factorize a rank 4 symmetric spinor

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Tensor[FactorWeylSpinor] - factorize a rank 4 symmetric spinor

Calling Sequences

FactorWeylSpinor( W, PT)

Parameters

W- a symmetric rank 4 covariant spinor

PT - the Petrov type of the spinor

Description

•	A rank 4 symmetric spinor can always be factorized as the symmetric product of rank 1 spinors,

The (non-unique) spinors are called the principal spinors of and the corresponding null vectors are called the principal null directions. The Petrov type of the spinor (see AdaptedSpinorDyad or PetrovType) determines the multiplicities of the principal spinors.

TYPE I. The principal spinors are all distinct, that is, non -proportional and .

TYPE II. Two of the principal spinors are proportional, where are non-proportional.

TYPE III. Three of principal spinors are proportional, where are non-proportional.

TYPE D. Two pairs of the principal spinors are proportional, where are non-proportional. Equivalently, there is a spinor dyad () and a complex number such that

TYPE N. The principal spinors are all proportional, .

•	The command FactorWeylSpinor returns a list of 4 spinors] and a scaling factor such that .

•	The factorization of the Weyl spinor is computed from an adapted spinor dyad. See AdaptedSpinorDyad.

•	The command FactorWeylSpinor is part of the DifferentialGeometry:-Tensor package. It can be used in the form FactorWeylSpinor(...) only after executing the commands with(DifferentialGeometry) and with(Tensor), but can always be used by executing DifferentialGeometry:-Tensor:-FactorWeylSpinor(...).

Examples

We calculate a factorization of Weyl spinors of each Petrov type and we use the command SymmetrizeIndices to verify that the factorization is correct.

We first create a spinor bundle over a 4-dimensional spacetime.

(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 >

(2.2)

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

Spin >

Example 1. Type I

Define a rank 4 spinor

Spin >

_DG([["tensor", Spin, [["cov_vrt", "cov_vrt", "cov_vrt", "cov_vrt"], []]], [[[5, 5, 5, 5], 6], [[5, 5, 5, 6], 3], [[5, 5, 6, 5], 3], [[5, 5, 6, 6], 5], [[5, 6, 5, 5], 3], [[5, 6, 5, 6], 5], [[5, 6, 6, 5], 5], [[5, 6, 6, 6], 6], [[6, 5, 5, 5], 3], [[6, 5, 5, 6], 5], [[6, 5, 6, 5], 5], [[6, 5, 6, 6], 6], [[6, 6, 5, 5], 5], [[6, 6, 5, 6], 6], [[6, 6, 6, 5], 6], [[6, 6, 6, 6], 6]]])

(2.3)

Calculate the Newman-Penrose coefficients for with respect to the given dyad basis .

Spin >

(2.4)

Use the Newman-Penrose coefficients to find the Petrov type of

Spin >

(2.5)

Factor

Spin >

[_DG([["form", Spin, 1], [[[5], 1/2+I-((1/2)*I)*3^(1/2)], [[6], I]]]), _DG([["form", Spin, 1], [[[5], 1/2-I-((1/2)*I)*3^(1/2)], [[6], -I]]]), _DG([["form", Spin, 1], [[[5], 1/2+I+((1/2)*I)*3^(1/2)], [[6], I]]]), _DG([["form", Spin, 1], [[[5], 1/2-I+((1/2)*I)*3^(1/2)], [[6], -I]]])], 6

(2.6)

We check that this answer is correct by computing the symmetric tensor product of the 4 spinors .

Spin >

(2.7)

Spin >

(2.8)

Example 2. Type II

Define a rank 4 spinor

Spin >

_DG([["tensor", Spin, [["cov_vrt", "cov_vrt", "cov_vrt", "cov_vrt"], []]], [[[5, 5, 5, 5], 4], [[5, 5, 5, 6], 1], [[5, 5, 6, 5], 1], [[5, 5, 6, 6], 1], [[5, 6, 5, 5], 1], [[5, 6, 5, 6], 1], [[5, 6, 6, 5], 1], [[5, 6, 6, 6], 4], [[6, 5, 5, 5], 1], [[6, 5, 5, 6], 1], [[6, 5, 6, 5], 1], [[6, 5, 6, 6], 4], [[6, 6, 5, 5], 1], [[6, 6, 5, 6], 4], [[6, 6, 6, 5], 4], [[6, 6, 6, 6], 10]]])

(2.9)

Calculate the Newman-Penrose coefficients for with respect to the given dyad basis .

Spin >

(2.10)

Find the Petrov type of

Spin >

(2.11)

Factor

Spin >

[_DG([["form", Spin, 1], [[[5], (1/3)*3^(1/2)], [[6], (1/3)*3^(1/2)]]]), _DG([["form", Spin, 1], [[[5], (1/3)*3^(1/2)], [[6], (1/3)*3^(1/2)]]]), _DG([["form", Spin, 1], [[[5], ((1/3)*I)*3^(1/2)+(1/3)*3^(1/2)], [[6], -((2/3)*I)*3^(1/2)+(1/3)*3^(1/2)]]]), _DG([["form", Spin, 1], [[[5], -((1/3)*I)*3^(1/2)+(1/3)*3^(1/2)], [[6], ((2/3)*I)*3^(1/2)+(1/3)*3^(1/2)]]])], 18

(2.12)

Note that the first two factors are identical.

Spin >

(2.13)

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors

Spin >

(2.14)

Spin >

(2.15)

Example 3. Type III

Define a rank 4 spinor

Spin >

_DG([["tensor", Spin, [["cov_vrt", "cov_vrt", "cov_vrt", "cov_vrt"], []]], [[[5, 5, 5, 5], -8], [[5, 5, 5, 6], -5*I], [[5, 5, 6, 5], -5*I], [[5, 5, 6, 6], 2], [[5, 6, 5, 5], -5*I], [[5, 6, 5, 6], 2], [[5, 6, 6, 5], 2], [[5, 6, 6, 6], -I], [[6, 5, 5, 5], -5*I], [[6, 5, 5, 6], 2], [[6, 5, 6, 5], 2], [[6, 5, 6, 6], -I], [[6, 6, 5, 5], 2], [[6, 6, 5, 6], -I], [[6, 6, 6, 5], -I], [[6, 6, 6, 6], 4]]])

(2.16)

Calculate the Newman-Penrose coefficients for with respect to the given dyad basis .

Spin >

(2.17)

Find the Petrov type of

Spin >

(2.18)

Factor

Spin >

[_DG([["form", Spin, 1], [[[5], -(1/2)*6^(1/2)+((1/2)*I)*6^(1/2)], [[6], (-1/2-(1/2)*I)*6^(1/2)]]]), _DG([["form", Spin, 1], [[[5], -(1/2)*6^(1/2)+((1/2)*I)*6^(1/2)], [[6], (-1/2-(1/2)*I)*6^(1/2)]]]), _DG([["form", Spin, 1], [[[5], -(1/2)*6^(1/2)+((1/2)*I)*6^(1/2)], [[6], (-1/2-(1/2)*I)*6^(1/2)]]]), _DG([["form", Spin, 1], [[[5], (1/9-(1/9)*I)*6^(1/2)], [[6], (-1/18-(1/18)*I)*6^(1/2)]]])], -4

(2.19)

Note that the first three factors are identical.

Spin >

(2.20)

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors

Spin >

(2.21)

Spin >

(2.22)

Example 4. Type D

Define a rank 4 spinor

Spin >

(2.23)

Calculate the Newman-Penrose coefficients for with respect to the given dyad basis .

Spin >

(2.24)

Find the Petrov type of

Spin >

(2.25)

Factor

Spin >

(2.26)

Note that the first two factors and last two factors are identical.

Spin >

(2.27)

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors

Spin >

(2.28)

Spin >

(2.29)

Example 5. Type N

Define a rank 4 spinor

Spin >

_DG([["tensor", Spin, [["cov_vrt", "cov_vrt", "cov_vrt", "cov_vrt"], []]], [[[5, 5, 5, 5], 1], [[5, 5, 5, 6], 3], [[5, 5, 6, 5], 3], [[5, 5, 6, 6], 9], [[5, 6, 5, 5], 3], [[5, 6, 5, 6], 9], [[5, 6, 6, 5], 9], [[5, 6, 6, 6], 27], [[6, 5, 5, 5], 3], [[6, 5, 5, 6], 9], [[6, 5, 6, 5], 9], [[6, 5, 6, 6], 27], [[6, 6, 5, 5], 9], [[6, 6, 5, 6], 27], [[6, 6, 6, 5], 27], [[6, 6, 6, 6], 81]]])

(2.30)

Calculate the Newman-Penrose coefficients for with respect to the given dyad basis .

Spin >

(2.31)

Find the Petrov type of

Spin >

(2.32)

Factor

Spin >

(2.33)

Note that all four factors are identical.

Spin >

(2.34)

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors

Spin >

(2.35)

Spin >

(2.36)

	See Also
	DifferentialGeometry, Tensor, AdaptedSpinorDyad, AdaptedNullTetrad, NPCurvatureScalars, NullVector, PetrovType, PrincipalNullDirections, WeylSpinor , Physics[Weyl]