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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

• 

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

WABCD = α(A βB γC δD).

The (non-unique) spinors  αA , βB, γC , δD are called the principal spinors of W and the corresponding null vectors are called the principal null directions. The Petrov type of the spinor WABCD (see AdaptedSpinorDyad or PetrovType) determines the multiplicities of the principal spinors.

TYPE I. The principal spinors are all distinct, that is, non -proportional and WABCD = α(A βB γC δD).

TYPE II. Two of the principal spinors are proportional, WABCD = α(A αB γC δD), where αA , γC, δD  are non-proportional.

TYPE III. Three of principal spinors are proportional, WABCD = α(A αB αC δD), where αA , δD  are non-proportional.

TYPE D. Two pairs of the principal spinors are proportional, WABCD= α(A αB βC βD), where  αA , βC  are non-proportional.  Equivalently, there is a spinor dyad (ιA , οC) and a complex number η such that WABCD= η ι(A ιB οC οD).

TYPE N. The principal spinors are all proportional, WABCD= α(A αB αC αD).

• 

The command FactorWeylSpinor returns a list of 4 spinors [αA , βB, γC , δD ] and a scaling factor η such that WABCD=  ηα(A βB γC δD).

• 

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

withDifferentialGeometry:withTensor:

 

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.

DGsetupt,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:=GenerateSymmetricTensorsdz1,dz2,4

S:=dz1dz1dz1dz1,14dz1dz1dz1dz2+14dz1dz1dz2dz1+14dz1dz2dz1dz1+14dz2dz1dz1dz1,16dz1dz1dz2dz2+16dz1dz2dz1dz2+16dz1dz2dz2dz1+16dz2dz1dz1dz2+16dz2dz1dz2dz1+16dz2dz2dz1dz1,14dz1dz2dz2dz2+14dz2dz1dz2dz2+14dz2dz2dz1dz2+14dz2dz2dz2dz1,dz2dz2dz2dz2

(2.2)

 

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

Spin > 

_EnvExplicit:=true:

 

Example 1. Type I

Define a rank 4 spinor W1.

Spin > 

W1:=DGzip6,12,30,24,6,S,plus

W1:=6dz1dz1dz1dz1+3dz1dz1dz1dz2+3dz1dz1dz2dz1+5dz1dz1dz2dz2+3dz1dz2dz1dz1+5dz1dz2dz1dz2+5dz1dz2dz2dz1+6dz1dz2dz2dz2+3dz2dz1dz1dz1+5dz2dz1dz1dz2+5dz2dz1dz2dz1+6dz2dz1dz2dz2+5dz2dz2dz1dz1+6dz2dz2dz1dz2+6dz2dz2dz2dz1+6dz2dz2dz2dz2

(2.3)

 

Calculate the Newman-Penrose coefficients for W1 with respect to the given dyad basis dz1, dz2.

Spin > 

NP1:=NPCurvatureScalarsW1,dz1,dz2

NP1:=tablePsi4=6,Psi1=6,Psi2=5,Psi3=3,Psi0=6

(2.4)

 

Use the Newman-Penrose coefficients to find the Petrov type of W1.

Spin > 

PetrovTypeNP1

I

(2.5)

 

Factor W1.

Spin > 

PS1,η1:=FactorWeylSpinorW1,I

PS1,η1:=12+I12I3dz1+Idz2,12I12I3dz1Idz2,12+I+12I3dz1+Idz2,12I+12I3dz1Idz2,6

(2.6)

 

We check that this answer is correct by computing the symmetric tensor product of the 4 spinors PS1.

Spin > 

W1Check:=η1 &mult SymmetrizeIndicesPS11 &t PS12 &t PS13 &t PS14,1,2,3,4,Symmetric

W1Check:=6dz1dz1dz1dz1+3dz1dz1dz1dz2+3dz1dz1dz2dz1+5dz1dz1dz2dz2+3dz1dz2dz1dz1+5dz1dz2dz1dz2+5dz1dz2dz2dz1+6dz1dz2dz2dz2+3dz2dz1dz1dz1+5dz2dz1dz1dz2+5dz2dz1dz2dz1+6dz2dz1dz2dz2+5dz2dz2dz1dz1+6dz2dz2dz1dz2+6dz2dz2dz2dz1+6dz2dz2dz2dz2

(2.7)
Spin > 

W1 &minus W1Check

0dz1dz1dz1dz1

(2.8)

 

Example 2. Type II

Define a rank 4 spinor W2.

Spin > 

W2:=DGzip4,4,6,16,10,S,plus

W2:=4dz1dz1dz1dz1+dz1dz1dz1dz2+dz1dz1dz2dz1+dz1dz1dz2dz2+dz1dz2dz1dz1+dz1dz2dz1dz2+dz1dz2dz2dz1+4dz1dz2dz2dz2+dz2dz1dz1dz1+dz2dz1dz1dz2+dz2dz1dz2dz1+4dz2dz1dz2dz2+dz2dz2dz1dz1+4dz2dz2dz1dz2+4dz2dz2dz2dz1+10dz2dz2dz2dz2

(2.9)

 

Calculate the Newman-Penrose coefficients for W2 with respect to the given dyad basis dz1, dz2.

Spin > 

NP2:=NPCurvatureScalarsW2,dz1,dz2

NP2:=tablePsi4=4,Psi1=4,Psi2=1,Psi3=1,Psi0=10

(2.10)

 

Find the Petrov type of W2.

Spin > 

PetrovTypeNP2

II

(2.11)

 

Factor W2.

Spin > 

PS2,η2:=FactorWeylSpinorW2,II

PS2,η2:=133dz1+133dz2,133dz1+133dz2,13I3+133dz1+23I3+133dz2,13I3+133dz1+23I3+133dz2,18

(2.12)

 

Note that the first two factors are identical.

Spin > 

PS21 &minus PS22

0dt

(2.13)

 

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors PS2.

Spin > 

W2Check:=η2 &mult SymmetrizeIndicesPS21 &t PS22 &t PS23 &t PS24,1,2,3,4,Symmetric

W2Check:=4dz1dz1dz1dz1+dz1dz1dz1dz2+dz1dz1dz2dz1+dz1dz1dz2dz2+dz1dz2dz1dz1+dz1dz2dz1dz2+dz1dz2dz2dz1+4dz1dz2dz2dz2+dz2dz1dz1dz1+dz2dz1dz1dz2+dz2dz1dz2dz1+4dz2dz1dz2dz2+dz2dz2dz1dz1+4dz2dz2dz1dz2+4dz2dz2dz2dz1+10dz2dz2dz2dz2

(2.14)
Spin > 

W2 &minus W2Check

0dz1dz1dz1dz1

(2.15)

 

Example 3. Type III

Define a rank 4 spinor W3.

Spin > 

W3:=DGzip8,20I,12,4I,4,S,plus

W3:=8dz1dz1dz1dz1Idz2dz2dz1dz25Idz1dz1dz2dz1+2dz1dz1dz2dz2Idz2dz1dz2dz2+2dz1dz2dz1dz2+2dz1dz2dz2dz1Idz2dz2dz2dz15Idz2dz1dz1dz1+2dz2dz1dz1dz2+2dz2dz1dz2dz15Idz1dz1dz1dz2+2dz2dz2dz1dz15Idz1dz2dz1dz1Idz1dz2dz2dz2+4dz2dz2dz2dz2

(2.16)

 

Calculate the Newman-Penrose coefficients for W3 with respect to the given dyad basis dz1, dz2.

Spin > 

NP3:=NPCurvatureScalarsW3,dz1,dz2

NP3:=tablePsi4=8,Psi1=I,Psi2=2,Psi3=5I,Psi0=4

(2.17)

 

Find the Petrov type of W3.

Spin > 

PetrovTypeNP3

III

(2.18)

 

Factor W3.

Spin > 

PS3,η3:=FactorWeylSpinorW3,III

PS3,η3:=126+12I6dz1+1212I6dz2,126+12I6dz1+1212I6dz2,126+12I6dz1+1212I6dz2,1919I6dz1+118118I6dz2,4

(2.19)

 

Note that the first three factors are identical.

Spin > 

PS31 &minus PS32,PS31 &minus PS33

0dt,0dt

(2.20)

 

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors PS3.

Spin > 

W3Check:=η3 &mult SymmetrizeIndicesPS31 &t PS32 &t PS33 &t PS34,1,2,3,4,Symmetric

W3Check:=8dz1dz1dz1dz15Idz1dz2dz1dz1Idz1dz2dz2dz2+2dz1dz1dz2dz25Idz2dz1dz1dz1+2dz1dz2dz1dz2+2dz1dz2dz2dz15Idz1dz1dz1dz2Idz2dz1dz2dz2+2dz2dz1dz1dz2+2dz2dz1dz2dz15Idz1dz1dz2dz1+2dz2dz2dz1dz1Idz2dz2dz2dz1Idz2dz2dz1dz2+4dz2dz2dz2dz2

(2.21)
Spin > 

W3 &minus W3Check

0dz1dz1dz1dz1

(2.22)

 

Example 4. Type D

Define a rank 4 spinor W4.

Spin > 

W4:=DGzip3,18,3,72,48,S,plus

W4:=3dz1dz1dz1dz192dz1dz1dz1dz292dz1dz1dz2dz1+12dz1dz1dz2dz292dz1dz2dz1dz1+12dz1dz2dz1dz2+12dz1dz2dz2dz1+18dz1dz2dz2dz292dz2dz1dz1dz1+12dz2dz1dz1dz2+12dz2dz1dz2dz1+18dz2dz1dz2dz2+12dz2dz2dz1dz1+18dz2dz2dz1dz2+18dz2dz2dz2dz1+48dz2dz2dz2dz2

(2.23)

 

Calculate the Newman-Penrose coefficients for W4 with respect to the given dyad basis dz1, dz2.

Spin > 

NP4:=NPCurvatureScalarsW4,dz1,dz2

NP4:=tablePsi4=3,Psi1=18,Psi2=12,Psi3=92,Psi0=48

(2.24)

 

Find the Petrov type of W4.

Spin > 

PetrovTypeNP4

D

(2.25)

 

Factor W4.

Spin > 

PS4,η4:=FactorWeylSpinorW4,D

PS4,η4:=dz1+dz2,dz1+dz2,15dz1+45dz2,15dz1+45dz2,75

(2.26)

 

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

Spin > 

PS41 &minus PS42,PS43 &minus PS44

0dt,0dt

(2.27)

 

We check that this factorization is correct by computing the symmetric tensor product of the 4 spinors PS4.

Spin > 

W4Check:=η4 &mult SymmetrizeIndicesPS41 &t PS42 &t PS43 &t PS44,1,2,3,4,Symmetric

W4Check:=3dz1dz1dz1dz192dz1dz1dz1dz292dz1dz1dz2dz1+12dz1dz1dz2dz292dz1dz2dz1dz1+12dz1dz2dz1dz2+12dz1dz2dz2dz1+18dz1dz2dz2dz292dz2dz1dz1dz1+12dz2dz1dz1dz2+12dz2dz1dz2dz1+18dz2dz1dz2dz2+12dz2dz2dz1dz1+18dz2dz2dz1dz2+18dz2dz2dz2dz1+48dz2dz2dz2dz2

(2.28)
Spin > 

W4 &minus W4Check

0dz1dz1dz1dz1

(2.29)

 

Example 5. Type N

Define a rank 4 spinor W5.

Spin > 

W5:=DGzip1,12,54,108,81,S,plus

W5:=dz1dz1dz1dz1+3dz1dz1dz1dz2+3dz1dz1dz2dz1+9dz1dz1dz2dz2+3dz1dz2dz1dz1+9dz1dz2dz1dz2+9dz1dz2dz2dz1+27dz1dz2dz2dz2+3dz2dz1dz1dz1+9dz2dz1dz1dz2+9dz2dz1dz2dz1+27dz2dz1dz2dz2+9dz2dz2dz1dz1+27dz2dz2dz1dz2+27dz2dz2dz2dz1+81dz2dz2dz2dz2

(2.30)

 

Calculate the Newman-Penrose coefficients for W5 with respect to the given dyad basis dz1, dz2.

Spin > 

NP5:=NPCurvatureScalarsW5,dz1,dz2

NP5:=table