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

Calling Sequences

WeylSpinor( $σ$ , W)

WeylSpinor(dyad, NP)

WeylSpinor(dyad, PT, $η$ , $χ$ )

Parameters

$σ$ - 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 coefficients for the Weyl spinor to be constructed

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

$η$ , $χ$ - the complex numbers used to construct the Penrose normal form of the Weyl spinor

Description

Examples

See Also

Description

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

$W_{AA' BB' CC' DD} = Ψ_{ABCD} {ϵ_{A' B'}}_{} ϵ_{C' D'} + {\overline{Ψ}}_{A' B' C' D'} ϵ_{AB} ϵ_{CD} &period; (1)$

The symmetric rank 4 spinor $Ψ_{ABCD}$ is called the Weyl spinor. If $\{ι_{A}, ο_{A}\}$ is a spinor dyad (a pair of rank-2 spinors with $ι_{A} ο^{A} = 1$ ) then the spinor $Ψ_{ABCD}$

can be expressed as

$Ψ_{ABCD} = Ψ_{0} ι_{A} ι_{B} ι_{C} ι_{D} - 4 Ψ_{1} ι_{(A} ι_{B} ι_{C} ο_{D)} + 6 Ψ_{2} ι_{(A} ι_{B} ο_{C} ο_{D)} - 4 Ψ_{3} ι_{(A} ο_{B} ο_{C} ο_{D)} + Ψ_{4} ο_{A} ο_{B} ο_{C} ο_{D} &period; (2)$

The complex scalars $Ψ_{0}, Ψ_{1}, Ψ_{2}, Ψ_{3}, Ψ_{4}$ 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 $g$ has been previously computed, then the Weyl spinor will be computed more quickly using the calling sequence WeylSpinor( $σ$ , W).

•	In the second calling sequence the Weyl spinor is calculated directly from the a spinor dyad $\{ι_{A}, ο_{A}\}$ and a set of Newman-Penrose coefficients using equation (2).

•	The third calling sequence also uses equation (2), but the Newman-Penrose coefficients are calculated from the Petrov type according to the following normal forms rules:

Type I. $Ψ_{0} = \frac{3}{2} η χ, Ψ_{1} = 0, Ψ_{2} = \frac{1}{2} η (2 - χ), Ψ_{3} = 0, Ψ_{4} = \frac{3}{2} η χ &period;$

Type II. $Ψ_{0}$ $= 0, Ψ_{1} = 0, Ψ_{2} = η, Ψ_{3} = 0, Ψ_{4} = 6 η &period;$

Type III. $Ψ_{0} = 0, Ψ_{1} = 0, Ψ_{2} = 0, Ψ_{3} = 1, Ψ_{4} = 0.$

Type D. $Ψ_{0} = 0, Ψ_{1} = 0, Ψ_{2} = η, Ψ_{3} = 0, Ψ_{4} = 0.$

Type N. $Ψ_{0} = 0, Ψ_{1} = 0, Ψ_{2} = 0, Ψ_{3} = 0, Ψ_{4} = 1.$

Type O. $Ψ_{0} = 0, Ψ_{1} = 0, Ψ_{2} = 0, Ψ_{3} = 0, Ψ_{4} = 0.$

•	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

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

Example 1.

First create a vector bundle over $M$ with base coordinates $(t, ρ, φ, z)$ and fiber coordinates $(z1, z2, w1, w2)$ .

>	$DGsetup ([t, ρ, φ, z], [z1, z2, w1, w2], M)$

$frame name: M$

(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.

M >	$g ≔ evalDG (\exp (2 k (t - ρ)) (dt &t dt - drho &t drho) - ρ^{2} dphi &t dphi - dz &t dz)$

$g := {&ExponentialE;}^{- 2 k (- t + ρ)} dt dt - {&ExponentialE;}^{- 2 k (- t + ρ)} drho drho - ρ^{2} dphi dphi - dz dz$

(2.2)

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

M >	$F ≔ DGGramSchmidt ([D_t, D_rho, D_phi, D_z], g, signature = [1, - 1, - 1, - 1]) assuming k :: real, t :: real, 0 < ρ$

$F := [{&ExponentialE;}^{k (- t + ρ)} D_t, {&ExponentialE;}^{k (- t + ρ)} D_rho, \frac{D_phi}{ρ}, D_z]$

(2.3)

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

M >	$σ ≔ SolderForm (F)$

$σ := \frac{1}{2} {&ExponentialE;}^{- k (- t + ρ)} \sqrt{2} dt D_z1 D_w1 + \frac{1}{2} {&ExponentialE;}^{- k (- t + ρ)} \sqrt{2} dt D_z2 D_w2 + \frac{1}{2} {&ExponentialE;}^{- k (- t + ρ)} \sqrt{2} drho D_z1 D_w2 + \frac{1}{2} {&ExponentialE;}^{- k (- t + ρ)} \sqrt{2} drho D_z2 D_w1 - \frac{1}{2} I ρ \sqrt{2} dphi D_z1 D_w2 + \frac{1}{2} I ρ \sqrt{2} dphi D_z2 D_w1 + \frac{1}{2} \sqrt{2} dz D_z1 D_w1 - \frac{1}{2} \sqrt{2} dz D_z2 D_w2$

(2.4)

Calculate the Weyl spinor from the solder form sigma.

M >	$Ψ1 ≔ WeylSpinor (σ)$

$Ψ1 := - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz1 dz1 dz1}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz1 dz1 dz2}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz1 dz2 dz1}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz1 dz2 dz2}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz2 dz1 dz1}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz2 dz1 dz2}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz2 dz2 dz1}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz2 dz2 dz2}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz1 dz1 dz1}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz1 dz1 dz2}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz1 dz2 dz1}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz1 dz2 dz2}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz2 dz1 dz1}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz2 dz1 dz2}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz2 dz2 dz1}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz2 dz2 dz2}{ρ}$

(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 $σ$ .

M >	$W ≔ WeylTensor (g)$

$W := \frac{1}{2} k ρ dt dphi dt dphi - \frac{1}{2} k ρ dt dphi drho dphi - \frac{1}{2} k ρ dt dphi dphi dt + \frac{1}{2} k ρ dt dphi dphi drho - \frac{1}{2} \frac{k dt dz dt dz}{ρ} + \frac{1}{2} \frac{k dt dz drho dz}{ρ} + \frac{1}{2} \frac{k dt dz dz dt}{ρ} - \frac{1}{2} \frac{k dt dz dz drho}{ρ} - \frac{1}{2} k ρ drho dphi dt dphi + \frac{1}{2} k ρ drho dphi drho dphi + \frac{1}{2} k ρ drho dphi dphi dt - \frac{1}{2} k ρ drho dphi dphi drho + \frac{1}{2} \frac{k drho dz dt dz}{ρ} - \frac{1}{2} \frac{k drho dz drho dz}{ρ} - \frac{1}{2} \frac{k drho dz dz dt}{ρ} + \frac{1}{2} \frac{k drho dz dz drho}{ρ} - \frac{1}{2} k ρ dphi dt dt dphi + \frac{1}{2} k ρ dphi dt drho dphi + \frac{1}{2} k ρ dphi dt dphi dt - \frac{1}{2} k ρ dphi dt dphi drho + \frac{1}{2} k ρ dphi drho dt dphi - \frac{1}{2} k ρ dphi drho drho dphi - \frac{1}{2} k ρ dphi drho dphi dt + \frac{1}{2} k ρ dphi drho dphi drho + \frac{1}{2} \frac{k dz dt dt dz}{ρ} - \frac{1}{2} \frac{k dz dt drho dz}{ρ} - \frac{1}{2} \frac{k dz dt dz dt}{ρ} + \frac{1}{2} \frac{k dz dt dz drho}{ρ} - \frac{1}{2} \frac{k dz drho dt dz}{ρ} + \frac{1}{2} \frac{k dz drho drho dz}{ρ} + \frac{1}{2} \frac{k dz drho dz dt}{ρ} - \frac{1}{2} \frac{k dz drho dz drho}{ρ}$

(2.6)

M >	$Ψ2 ≔ WeylSpinor (σ, W)$

$Ψ2 := - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz1 dz1 dz1}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz1 dz1 dz2}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz1 dz2 dz1}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz1 dz2 dz2}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz2 dz1 dz1}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz2 dz1 dz2}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz2 dz2 dz1}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz2 dz2 dz2}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz1 dz1 dz1}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz1 dz1 dz2}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz1 dz2 dz1}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz1 dz2 dz2}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz2 dz1 dz1}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz2 dz1 dz2}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz2 dz2 dz1}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz2 dz2 dz2}{ρ}$

(2.7)

M >	$Ψ1 &minus Ψ2$

$0 dz1 dz1 dz1 dz1$

(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.

M >	$α ≔ - \frac{\frac{1}{4} \cdot 1}{ρ} \exp (2 k (- t + ρ)) k$

$α := - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k}{ρ}$

(2.9)

M >	$ψ ≔ α^{\frac{1}{4}} evalDG (dz1 - dz2)$

$ψ := {(- \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k}{ρ})}^{1 / 4} (dz1 - dz2)$

(2.10)

M >	$ψ4 ≔ evalDG (((ψ &t ψ) &t ψ) &t ψ)$

$ψ4 := - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz1 dz1 dz1}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz1 dz1 dz2}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz1 dz2 dz1}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz1 dz2 dz2}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz2 dz1 dz1}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz2 dz1 dz2}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz2 dz2 dz1}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz1 dz2 dz2 dz2}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz1 dz1 dz1}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz1 dz1 dz2}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz1 dz2 dz1}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz1 dz2 dz2}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz2 dz1 dz1}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz2 dz1 dz2}{ρ} + \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz2 dz2 dz1}{ρ} - \frac{1}{4} \frac{{&ExponentialE;}^{2 k (- t + ρ)} k dz2 dz2 dz2 dz2}{ρ}$

(2.11)

M >	$Ψ1 &minus ψ4$

$0 dz1 dz1 dz1 dz1$

(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.)

M >	$WS ≔ convert (W, DGspinor, σ, [1, 2, 3, 4]) &colon;$

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

M >	$LHS ≔ RearrangeIndices (WS, [1, 5, 2, 6, 3, 7, 4, 8]) &colon;$

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

M >	$barE ≔ EpsilonSpinor (cov, barspinor)$

$barE := dw1 dw2 - dw2 dw1$

(2.13)

M >	$RHS1 ≔ (Ψ1 &tensor barE) &tensor barE &colon;$

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.

M >	$barPsi1 ≔ ConjugateSpinor (Ψ1) assuming k :: real &colon;$

M >	$E ≔ EpsilonSpinor (cov, spinor)$

$E := dz1 dz2 - dz2 dz1$

(2.14)

M >	$RHS2 ≔ (E &tensor E) &tensor barPsi1 &colon;$

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

M >	$evalDG (LHS - (RHS1 + RHS2))$

$0 dz1 dz1 dz1 dz1 dz1 dz1 dz1 dz1$

(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.

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

$frame name: M$

(2.16)

>	$dyad ≔ [dz1, dz2]$

$dyad := [dz1, dz2]$

(2.17)

>	$NP ≔ table ([Psi0 = 0, Psi1 = 0, Psi2 = 0, Psi3 = z, Psi4 = t])$

$NP := table ([Psi1 = 0, Psi2 = 0, Psi0 = 0, Psi4 = t, Psi3 = z])$

(2.18)

>	$WeylSpinor (dyad, NP)$

$t dz1 dz1 dz1 dz1 - z dz1 dz1 dz1 dz2 - z dz1 dz1 dz2 dz1 - z dz1 dz2 dz1 dz1 - z dz2 dz1 dz1 dz1$

(2.19)

Example 6.

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

M >	$dyad ≔ [dz1, dz2]$

$dyad := [dz1, dz2]$

(2.20)

M >	$WeylSpinor (dyad, N)$

$dz1 dz1 dz1 dz1$

(2.21)

M >	$WeylSpinor (dyad, II, A)$

$6 A dz1 dz1 dz1 dz1 + A dz1 dz1 dz2 dz2 + A dz1 dz2 dz1 dz2 + A dz1 dz2 dz2 dz1 + A dz2 dz1 dz1 dz2 + A dz2 dz1 dz2 dz1 + A dz2 dz2 dz1 dz1$

(2.22)

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

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