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 >

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

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

Maple

Maple 附加模块

MapleSim

MapleSim 附加模块

系统工程

项目服务

Maple T.A. and Möbius

教育

行业

汽车与航空航天

机器人

机器设计和工业自动化

其他

应用领域

产品价格

购买

院系/全校正版授权

Maplesoft精英维护升级计划

支持

产品培训

产品在线帮助

研讨会与活动

出版

资源中心

示例和应用

社区

关于 Maplesoft

媒体中心

用户社区

联系我们

Online Help

All Products Maple MapleSim

Maple

数学软件

Maple 附加模块

MapleSim

多学科系统级建模仿真

MapleSim 附加模块

系统工程

项目服务

Maple T.A. and Möbius

教育

行业

汽车与航空航天

机器人

机器设计和工业自动化

其他

应用领域

产品价格

购买

院系/全校正版授权

Maplesoft精英维护升级计划

支持

产品培训

产品在线帮助

研讨会与活动

出版

资源中心

示例和应用

社区

关于 Maplesoft

媒体中心

用户社区

联系我们

Online Help

All Products Maple MapleSim