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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 $W$

Description

Examples

Description

•	A rank 4 symmetric spinor $W_{ABCD}$ can always be factorized as the symmetric product of rank 1 spinors,

$W_{ABCD} = α_{(A} β_{B} γ_{C} δ_{D)}$ .

The (non-unique) spinors $α_{A}, β_{B}, γ_{C}, δ_{D}$ are called the principal spinors of $W$ and the corresponding null vectors are called the principal null directions. The Petrov type of the spinor $W_{ABCD}$ (see AdaptedSpinorDyad or PetrovType) determines the multiplicities of the principal spinors.

TYPE I. The principal spinors are all distinct, that is, non -proportional and $W_{ABCD} = α_{(A} β_{B} γ_{C} δ_{D)}$ .

TYPE II. Two of the principal spinors are proportional, $W_{ABCD} = α_{(A} α_{B} γ_{C} δ_{D)},$ where $α_{A}, γ_{C}, δ_{D}$ are non-proportional.

TYPE III. Three of principal spinors are proportional, $W_{ABCD} = α_{(A} α_{B} α_{C} δ_{D)},$ where $α_{A}, δ_{D}$ are non-proportional.

TYPE D. Two pairs of the principal spinors are proportional, $W_{ABCD} = α_{(A} α_{B} β_{C} β_{D)},$ where $α_{A}, β_{C}$ are non-proportional. Equivalently, there is a spinor dyad ( $ι_{A}, ο_{C}$ ) and a complex number $η$ such that $W_{ABCD} = η \cdot ι_{(A} ι_{B} ο_{C} ο_{D)}$ .

TYPE N. The principal spinors are all proportional, $W_{ABCD} = α_{(A} α_{B} α_{C} α_{D)}$ .

•	The command FactorWeylSpinor returns a list of 4 spinors $[α_{A}, β_{B}, γ_{C}, δ_{D}$ ] and a scaling factor $η$ such that $W_{ABCD} = η \cdot α_{(A} β_{B} γ_{C} δ_{D)}$ .

•	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

>	$with (DifferentialGeometry) &colon;$ $with (Tensor) &colon;$

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.

>	$DGsetup ([t, x, y, z], [z1, z2, w1, w2], Spin)$

$frame name: Spin$

(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 >

$S ≔ GenerateSymmetricTensors ([dz1, dz2], 4)$

$S := [dz1 dz1 dz1 dz1, \frac{1}{4} dz1 dz1 dz1 dz2 + \frac{1}{4} dz1 dz1 dz2 dz1 + \frac{1}{4} dz1 dz2 dz1 dz1 + \frac{1}{4} dz2 dz1 dz1 dz1, \frac{1}{6} dz1 dz1 dz2 dz2 + \frac{1}{6} dz1 dz2 dz1 dz2 + \frac{1}{6} dz1 dz2 dz2 dz1 + \frac{1}{6} dz2 dz1 dz1 dz2 + \frac{1}{6} dz2 dz1 dz2 dz1 + \frac{1}{6} dz2 dz2 dz1 dz1, \frac{1}{4} dz1 dz2 dz2 dz2 + \frac{1}{4} dz2 dz1 dz2 dz2 + \frac{1}{4} dz2 dz2 dz1 dz2 + \frac{1}{4} dz2 dz2 dz2 dz1, dz2 dz2 dz2 dz2]$

(2.2)

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

Spin >

$_EnvExplicit ≔ true &colon;$

Example 1. Type I

Define a rank 4 spinor $W_{1} &period;$

Spin >

$W1 ≔ DGzip ([6, 12, 30, 24, 6], S, plus)$

$W1 := 6 dz1 dz1 dz1 dz1 + 3 dz1 dz1 dz1 dz2 + 3 dz1 dz1 dz2 dz1 + 5 dz1 dz1 dz2 dz2 + 3 dz1 dz2 dz1 dz1 + 5 dz1 dz2 dz1 dz2 + 5 dz1 dz2 dz2 dz1 + 6 dz1 dz2 dz2 dz2 + 3 dz2 dz1 dz1 dz1 + 5 dz2 dz1 dz1 dz2 + 5 dz2 dz1 dz2 dz1 + 6 dz2 dz1 dz2 dz2 + 5 dz2 dz2 dz1 dz1 + 6 dz2 dz2 dz1 dz2 + 6 dz2 dz2 dz2 dz1 + 6 dz2 dz2 dz2 dz2$

(2.3)

Calculate the Newman-Penrose coefficients for $W_{1}$ with respect to the given dyad basis $[dz1, dz2]$ .

Spin >

$NP1 ≔ NPCurvatureScalars (W1, [dz1, dz2])$

$NP1 := table ([Psi4 = 6, Psi1 = - 6, Psi2 = 5, Psi3 = - 3, Psi0 = 6])$

(2.4)

Use the Newman-Penrose coefficients to find the Petrov type of $W_{1} &period;$

Spin >

$PetrovType (NP1)$

$I$

(2.5)

Factor $W_{1} &period;$

Spin >

$PS1, η1 ≔ FactorWeylSpinor (W1, I)$

$PS1, η1 := [(\frac{1}{2} + I - \frac{1}{2} I \sqrt{3}) dz1 + I dz2, (\frac{1}{2} - I - \frac{1}{2} I \sqrt{3}) dz1 - I dz2, (\frac{1}{2} + I + \frac{1}{2} I \sqrt{3}) dz1 + I dz2, (\frac{1}{2} - I + \frac{1}{2} I \sqrt{3}) dz1 - I dz2], 6$

(2.6)

We check that this answer is correct by computing the symmetric tensor product of the 4 spinors ${PS}_{1}$ .

Spin >

$W1Check ≔ η1 &mult SymmetrizeIndices (((PS1 [1] &t PS1 [2]) &t PS1 [3]) &t PS1 [4], [1, 2, 3, 4], Symmetric)$

$W1Check := 6 dz1 dz1 dz1 dz1 + 3 dz1 dz1 dz1 dz2 + 3 dz1 dz1 dz2 dz1 + 5 dz1 dz1 dz2 dz2 + 3 dz1 dz2 dz1 dz1 + 5 dz1 dz2 dz1 dz2 + 5 dz1 dz2 dz2 dz1 + 6 dz1 dz2 dz2 dz2 + 3 dz2 dz1 dz1 dz1 + 5 dz2 dz1 dz1 dz2 + 5 dz2 dz1 dz2 dz1 + 6 dz2 dz1 dz2 dz2 + 5 dz2 dz2 dz1 dz1 + 6 dz2 dz2 dz1 dz2 + 6 dz2 dz2 dz2 dz1 + 6 dz2 dz2 dz2 dz2$

(2.7)

Spin >

$W1 &minus W1Check$

$0 dz1 dz1 dz1 dz1$

(2.8)

Example 2. Type II

Define a rank 4 spinor $W_{2} &period;$

Spin >

$W2 ≔ DGzip ([4, 4, 6, 16, 10], S, plus)$

$W2 := 4 dz1 dz1 dz1 dz1 + dz1 dz1 dz1 dz2 + dz1 dz1 dz2 dz1 + dz1 dz1 dz2 dz2 + dz1 dz2 dz1 dz1 + dz1 dz2 dz1 dz2 + dz1 dz2 dz2 dz1 + 4 dz1 dz2 dz2 dz2 + dz2 dz1 dz1 dz1 + dz2 dz1 dz1 dz2 + dz2 dz1 dz2 dz1 + 4 dz2 dz1 dz2 dz2 + dz2 dz2 dz1 dz1 + 4 dz2 dz2 dz1 dz2 + 4 dz2 dz2 dz2 dz1 + 10 dz2 dz2 dz2 dz2$

(2.9)

Calculate the Newman-Penrose coefficients for $W_{2}$ with respect to the given dyad basis $[dz1, dz2]$ .

Spin >

$NP2 ≔ NPCurvatureScalars (W2, [dz1, dz2])$

$NP2 := table ([Psi4 = 4, Psi1 = - 4, Psi2 = 1, Psi3 = - 1, Psi0 = 10])$

(2.10)

Find the Petrov type of $W_{2} &period;$

Spin >

$PetrovType (NP2)$

$II$

(2.11)

Factor $W_{2} &period;$

Spin >

$PS2, η2 ≔ FactorWeylSpinor (W2, II)$

$PS2, η2 := [\frac{1}{3} \sqrt{3} dz1 + \frac{1}{3} \sqrt{3} dz2, \frac{1}{3} \sqrt{3} dz1 + \frac{1}{3} \sqrt{3} dz2, (\frac{1}{3} I \sqrt{3} + \frac{1}{3} \sqrt{3}) dz1 + (- \frac{2}{3} I \sqrt{3} + \frac{1}{3} \sqrt{3}) dz2, (- \frac{1}{3} I \sqrt{3} + \frac{1}{3} \sqrt{3}) dz1 + (\frac{2}{3} I \sqrt{3} + \frac{1}{3} \sqrt{3}) dz2], 18$

(2.12)

Note that the first two factors are identical.

Spin >

$PS2 [1] &minus PS2 [2]$

$0 dt$

(2.13)

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors ${PS}_{2} &period;$

Spin >

$W2Check ≔ η2 &mult SymmetrizeIndices (((PS2 [1] &t PS2 [2]) &t PS2 [3]) &t PS2 [4], [1, 2, 3, 4], Symmetric)$

$W2Check := 4 dz1 dz1 dz1 dz1 + dz1 dz1 dz1 dz2 + dz1 dz1 dz2 dz1 + dz1 dz1 dz2 dz2 + dz1 dz2 dz1 dz1 + dz1 dz2 dz1 dz2 + dz1 dz2 dz2 dz1 + 4 dz1 dz2 dz2 dz2 + dz2 dz1 dz1 dz1 + dz2 dz1 dz1 dz2 + dz2 dz1 dz2 dz1 + 4 dz2 dz1 dz2 dz2 + dz2 dz2 dz1 dz1 + 4 dz2 dz2 dz1 dz2 + 4 dz2 dz2 dz2 dz1 + 10 dz2 dz2 dz2 dz2$

(2.14)

Spin >

$W2 &minus W2Check$

$0 dz1 dz1 dz1 dz1$

(2.15)

Example 3. Type III

Define a rank 4 spinor $W_{3} &period;$

Spin >

$W3 ≔ DGzip ([- 8, - 20 I, 12, - 4 I, 4], S, plus)$

$W3 := - 8 dz1 dz1 dz1 dz1 - I dz2 dz2 dz1 dz2 - 5 I dz1 dz1 dz2 dz1 + 2 dz1 dz1 dz2 dz2 - I dz2 dz1 dz2 dz2 + 2 dz1 dz2 dz1 dz2 + 2 dz1 dz2 dz2 dz1 - I dz2 dz2 dz2 dz1 - 5 I dz2 dz1 dz1 dz1 + 2 dz2 dz1 dz1 dz2 + 2 dz2 dz1 dz2 dz1 - 5 I dz1 dz1 dz1 dz2 + 2 dz2 dz2 dz1 dz1 - 5 I dz1 dz2 dz1 dz1 - I dz1 dz2 dz2 dz2 + 4 dz2 dz2 dz2 dz2$

(2.16)

Calculate the Newman-Penrose coefficients for $W_{3}$ with respect to the given dyad basis $[dz1, dz2]$ .

Spin >

$NP3 ≔ NPCurvatureScalars (W3, [dz1, dz2])$

$NP3 := table ([Psi4 = - 8, Psi1 = I, Psi2 = 2, Psi3 = 5 I, Psi0 = 4])$

(2.17)

Find the Petrov type of $W_{3} &period;$

Spin >

$PetrovType (NP3)$

$III$

(2.18)

Factor $W_{3.}$

Spin >

$PS3, η3 ≔ FactorWeylSpinor (W3, III)$

$PS3, η3 := [(- \frac{1}{2} \sqrt{6} + \frac{1}{2} I \sqrt{6}) dz1 + (- \frac{1}{2} - \frac{1}{2} I) \sqrt{6} dz2, (- \frac{1}{2} \sqrt{6} + \frac{1}{2} I \sqrt{6}) dz1 + (- \frac{1}{2} - \frac{1}{2} I) \sqrt{6} dz2, (- \frac{1}{2} \sqrt{6} + \frac{1}{2} I \sqrt{6}) dz1 + (- \frac{1}{2} - \frac{1}{2} I) \sqrt{6} dz2, (\frac{1}{9} - \frac{1}{9} I) \sqrt{6} dz1 + (- \frac{1}{18} - \frac{1}{18} I) \sqrt{6} dz2], - 4$

(2.19)

Note that the first three factors are identical.

Spin >

$PS3 [1] &minus PS3 [2], PS3 [1] &minus PS3 [3]$

$0 dt, 0 dt$

(2.20)

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors ${PS}_{3} &period;$

Spin >

$W3Check ≔ η3 &mult SymmetrizeIndices (((PS3 [1] &t PS3 [2]) &t PS3 [3]) &t PS3 [4], [1, 2, 3, 4], Symmetric)$

$W3Check := - 8 dz1 dz1 dz1 dz1 - 5 I dz1 dz2 dz1 dz1 - I dz1 dz2 dz2 dz2 + 2 dz1 dz1 dz2 dz2 - 5 I dz2 dz1 dz1 dz1 + 2 dz1 dz2 dz1 dz2 + 2 dz1 dz2 dz2 dz1 - 5 I dz1 dz1 dz1 dz2 - I dz2 dz1 dz2 dz2 + 2 dz2 dz1 dz1 dz2 + 2 dz2 dz1 dz2 dz1 - 5 I dz1 dz1 dz2 dz1 + 2 dz2 dz2 dz1 dz1 - I dz2 dz2 dz2 dz1 - I dz2 dz2 dz1 dz2 + 4 dz2 dz2 dz2 dz2$

(2.21)

Spin >

$W3 &minus W3Check$

$0 dz1 dz1 dz1 dz1$

(2.22)

Example 4. Type D

Define a rank 4 spinor $W_{4} &period;$

Spin >

$W4 ≔ DGzip ([3, - 18, 3, 72, 48], S, plus)$

$W4 := 3 dz1 dz1 dz1 dz1 - \frac{9}{2} dz1 dz1 dz1 dz2 - \frac{9}{2} dz1 dz1 dz2 dz1 + \frac{1}{2} dz1 dz1 dz2 dz2 - \frac{9}{2} dz1 dz2 dz1 dz1 + \frac{1}{2} dz1 dz2 dz1 dz2 + \frac{1}{2} dz1 dz2 dz2 dz1 + 18 dz1 dz2 dz2 dz2 - \frac{9}{2} dz2 dz1 dz1 dz1 + \frac{1}{2} dz2 dz1 dz1 dz2 + \frac{1}{2} dz2 dz1 dz2 dz1 + 18 dz2 dz1 dz2 dz2 + \frac{1}{2} dz2 dz2 dz1 dz1 + 18 dz2 dz2 dz1 dz2 + 18 dz2 dz2 dz2 dz1 + 48 dz2 dz2 dz2 dz2$

(2.23)

Calculate the Newman-Penrose coefficients for $W_{4}$ with respect to the given dyad basis $[dz1, dz2]$ .

Spin >

$NP4 ≔ NPCurvatureScalars (W4, [dz1, dz2])$

$NP4 := table ([Psi4 = 3, Psi1 = - 18, Psi2 = \frac{1}{2}, Psi3 = \frac{9}{2}, Psi0 = 48])$

(2.24)

Find the Petrov type of $W_{4} &period;$

Spin >

$PetrovType (NP4)$

$D$

(2.25)

Factor $W_{4.}$

Spin >

$PS4, η4 ≔ FactorWeylSpinor (W4, D)$

$PS4, η4 := [dz1 + dz2, dz1 + dz2, - \frac{1}{5} dz1 + \frac{4}{5} dz2, - \frac{1}{5} dz1 + \frac{4}{5} dz2], 75$

(2.26)

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

Spin >

$PS4 [1] &minus PS4 [2], PS4 [3] &minus PS4 [4]$

$0 dt, 0 dt$

(2.27)

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors ${PS}_{4} &period;$

Spin >

$W4Check ≔ η4 &mult SymmetrizeIndices (((PS4 [1] &t PS4 [2]) &t PS4 [3]) &t PS4 [4], [1, 2, 3, 4], Symmetric)$

$W4Check := 3 dz1 dz1 dz1 dz1 - \frac{9}{2} dz1 dz1 dz1 dz2 - \frac{9}{2} dz1 dz1 dz2 dz1 + \frac{1}{2} dz1 dz1 dz2 dz2 - \frac{9}{2} dz1 dz2 dz1 dz1 + \frac{1}{2} dz1 dz2 dz1 dz2 + \frac{1}{2} dz1 dz2 dz2 dz1 + 18 dz1 dz2 dz2 dz2 - \frac{9}{2} dz2 dz1 dz1 dz1 + \frac{1}{2} dz2 dz1 dz1 dz2 + \frac{1}{2} dz2 dz1 dz2 dz1 + 18 dz2 dz1 dz2 dz2 + \frac{1}{2} dz2 dz2 dz1 dz1 + 18 dz2 dz2 dz1 dz2 + 18 dz2 dz2 dz2 dz1 + 48 dz2 dz2 dz2 dz2$

(2.28)

Spin >

$W4 &minus W4Check$

$0 dz1 dz1 dz1 dz1$

(2.29)

Example 5. Type N

Define a rank 4 spinor $W_{5} &period;$

Spin >

$W5 ≔ DGzip ([1, 12, 54, 108, 81], S, plus)$

$W5 := dz1 dz1 dz1 dz1 + 3 dz1 dz1 dz1 dz2 + 3 dz1 dz1 dz2 dz1 + 9 dz1 dz1 dz2 dz2 + 3 dz1 dz2 dz1 dz1 + 9 dz1 dz2 dz1 dz2 + 9 dz1 dz2 dz2 dz1 + 27 dz1 dz2 dz2 dz2 + 3 dz2 dz1 dz1 dz1 + 9 dz2 dz1 dz1 dz2 + 9 dz2 dz1 dz2 dz1 + 27 dz2 dz1 dz2 dz2 + 9 dz2 dz2 dz1 dz1 + 27 dz2 dz2 dz1 dz2 + 27 dz2 dz2 dz2 dz1 + 81 dz2 dz2 dz2 dz2$

(2.30)

Calculate the Newman-Penrose coefficients for $W_{5}$ with respect to the given dyad basis $[dz1, dz2]$ .

Spin >

$NP5 ≔ NPCurvatureScalars (W5, [dz1, dz2])$

$NP5 := table ([Psi4 = 1, Psi1 = - 27, Psi2 = 9, Psi3 = - 3, Psi0 = 81])$

(2.31)

Find the Petrov type of $W_{5} &period;$

Spin >

$PetrovType (NP5)$

$N$

(2.32)

Factor $W_{5.}$

Spin >

$PS5, η5 ≔ FactorWeylSpinor (W5, N)$

$PS5, η5 := [dz1 + 3 dz2, dz1 + 3 dz2, dz1 + 3 dz2, dz1 + 3 dz2], 1$

(2.33)

Note that all four factors are identical.

Spin >

$PS5 [1] &minus PS5 [2], PS5 [1] &minus PS5 [3], PS5 [1] &minus PS5 [4]$

$0 dt, 0 dt, 0 dt$

(2.34)

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors ${PS}_{5} &period;$

Spin >

$W5Check ≔ η5 &mult SymmetrizeIndices (((PS5 [1] &t PS5 [2]) &t PS5 [3]) &t PS5 [4], [1, 2, 3, 4], Symmetric)$

$W5Check := dz1 dz1 dz1 dz1 + 3 dz1 dz1 dz1 dz2 + 3 dz1 dz1 dz2 dz1 + 9 dz1 dz1 dz2 dz2 + 3 dz1 dz2 dz1 dz1 + 9 dz1 dz2 dz1 dz2 + 9 dz1 dz2 dz2 dz1 + 27 dz1 dz2 dz2 dz2 + 3 dz2 dz1 dz1 dz1 + 9 dz2 dz1 dz1 dz2 + 9 dz2 dz1 dz2 dz1 + 27 dz2 dz1 dz2 dz2 + 9 dz2 dz2 dz1 dz1 + 27 dz2 dz2 dz1 dz2 + 27 dz2 dz2 dz2 dz1 + 81 dz2 dz2 dz2 dz2$

(2.35)

Spin >

$W5 &minus W5Check$

$0 dz1 dz1 dz1 dz1$

(2.36)

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