FactorWeylSpinor - Maple Help

All Products Maple MapleSim

Home : Support : Online Help : Mathematics : DifferentialGeometry : Tensor : FactorWeylSpinor

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)

Maple

Add-ons Maple

Student Success Platform

Améliorer les taux de retention

MapleSim

Add-ons MapleSim

Ingénierie systèmes

Services-conseils

Produits d’enseignement en ligne

Enseignement

Industrie

Automobile et aéronautique

Robotique

Conception de machine & automatismes industriels

Autres

Domaines d’application

Prix et d'achats

Achats

Licences institutionnelles Etudiants

Vue d’ensemble du service EMP

Support

Formation produit

Aide en ligne Produit

Webinaires et évènements

Publications

Hubs de contenu

Exemples et applications

Communauté

A propos de Maplesoft

Ressources presse

Communauté

Contacts

Online Help

All Products Maple MapleSim

Maple

Logiciel de mathématiques puissant facile à utiliser

Add-ons Maple

Student Success Platform

Améliorer les taux de retention

MapleSim

Modélisation avancée au niveau système

Add-ons MapleSim

Ingénierie systèmes

Services-conseils

Produits d’enseignement en ligne

Enseignement

Industrie

Automobile et aéronautique

Robotique

Conception de machine & automatismes industriels

Autres

Domaines d’application

Prix et d'achats

Achats

Licences institutionnelles Etudiants

Vue d’ensemble du service EMP

Support

Formation produit

Aide en ligne Produit

Webinaires et évènements

Publications

Hubs de contenu

Exemples et applications

Communauté

A propos de Maplesoft

Ressources presse

Communauté

Contacts

Online Help

All Products Maple MapleSim