DifferentialGeometry/Tensor/NPSpinCoefficients - Maple Help
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Tensor[NPSpinCoefficients] - find the Newman-Penrose spin coefficients

Calling Sequences

     NPSpinCoefficients(NTetrad, output)

     NPSpinCoefficients(Fr, output)

Parameters

   NTetrad - a list of 4 vectors defining a null tetrad

   Fr      - the name of an initialized anholonomic frame, created from a null tetrad

   output  - (optional) keyword argument output = "sequence"

 

Description

Examples

See Also

Description

• 

Let g be a metric with signature 1,1,1,1 and (L, N, M, M) a null tetrad for g. The Newman-Penrose spin coefficients are the connection coefficients defined by the null tetrad. They are thus certain complex linear combinations of the Christoffel connection coefficients. The NP spin coefficients provide for a very compact and efficient formalism for connection and curvature computations in general relativity. See Newman and Penrose, Stewart.

• 

The NPSpinCoefficients command returns a table with 12 entries "kappa", "rho", "sigma", "tau", "pi", "lambda", "mu", "nu", "alpha ", "beta ", " gamma ", "epsilon". These are the customary labels assigned to the spin coefficients. With the optional keyword argument output = "sequence", the spin coefficients are returned as a sequence of 12 Maple expressions.

• 

Here are the formulas that are used to compute the NP spin coefficients. Let ΘL, ΘN,ΘM,ΘM be the basis of 1-forms dual to the given null tetrad (L, N, M, M). With respect to this basis, the metric g becomes

 g= 2 ΘLΘN2 ΘMΘM =ΘLΘN+ΘNΘLΘMΘMΘMΘM , 

whereis the symmetric tensor product. Let X be the directional covariant derivative operator (in the direction of a vector X) defined by the Christoffel connection for the metric g. If ω is a 1-form, then Xω is a 1-form which can be evaluated on a vector Y to give the scalar XωY.  In terms of this notation, the spin coefficients are:

 k = LΘNM

 ρ=M ΘNM

σ=MΘNM

τ=NΘNM

π=NΘLM

λ=NΘLM

μ=NΘMM

ν=NΘNM

α=12MΘNN+12MΘMM

β=12MΘNN+12MΘMM

γ=12NΘNN+12NΘMM

ε=12LΘNN+12LΘMM

    

• 

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form NPSpinCoefficients(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-NPSpinCoefficients.

Examples

withDifferentialGeometry:withTensor:

 

Example 1.

Define a manifold S with coordinates t,x,y,z.

DGsetupt,x,y,z,S

frame name: S

(2.1)

 

Define a metric g.

S > 

gevalDGx2dt&tdty2dx&tdxz2dy&tdyt2dz&tdz

gx2dtdty2dxdxz2dydyt2dzdz

(2.2)

 

Define an orthonormal tetrad OTetrad for the metric g.  Use GRQuery to check that OTetrad is indeed an orthonormal tetrad.

S > 

OTetrad1xD_t,1yD_x,1zD_y,1tD_z

OTetradD_tx,D_xy,D_yz,D_zt

(2.3)
S > 

GRQueryOTetrad,g,OrthonormalTetrad

true

(2.4)

 

Construct a null tetrad NTetrad from the orthonormal tetrad OTetrad.

S > 

NTetradNullTetradOTetrad

NTetrad22xD_t+22tD_z,22xD_t22tD_z,22yD_x+I22zD_y,22yD_xI22zD_y

(2.5)

 

Calculate the NP spin coefficients defined by the null tetrad NTetrad.

S > 

SpinCoeffNPSpinCoefficientsNTetrad

SpinCoefftableepsilon=24tx,tau=24xy,kappa=24xy,alpha=I42zy,beta=I42zy,mu=24tz,rho=24tz,lambda=24tz,gamma=24tx,sigma=24tz,nu=24xy,pi=24xy

(2.6)

 

The individual spin coefficients can be extracted from the table SpinCoeff.

S > 

SpinCoefftau

24xy

(2.7)

 

Example 2.

With the keyword argument output = "sequence", the command NPSpinCoefficients will return the spin coefficients as a sequence.  (Note that gamma is protected by Maple.)

S > 

κ,ρ,σ,τ,pi,λ,μ,ν,α,β,gam,εNPSpinCoefficientsNTetrad,output=Sequence

κ,ρ,σ,τ,π,λ,μ,ν,α,β,gam,ϵ24xy,24tz,24tz,24xy,24xy,24tz,24tz,24xy,I42zy,I42zy,24tx,24tx

(2.8)

 

Example 3.

We check the results from Example 2 against the definitions of the spin-coefficients.  First define the null tetrad.

S > 

L,N,M,barMopNTetrad

L,N,M,barM22xD_t+22tD_z,22xD_t22tD_z,22yD_x+I22zD_y,22yD_xI22zD_y

(2.9)

 

Define the dual basis.

S > 

Theta_L,Theta_N,Theta_M,Theta_barMopDualBasisNTetrad

Theta_L,Theta_N,Theta_M,Theta_barM2x2dt+2t2dz,2x2dt2t2dz,2y2dxI22zdy,2y2dx+I22zdy

(2.10)

 

Calculate the Christoffel connection.

S > 

CChristoffelg

C1xD_tdtdx+1xD_tdxdt+tx2D_tdzdz+xy2D_xdtdt+1yD_xdxdy+1yD_xdydxyz2D_ydxdx+1zD_ydydz+1zD_ydzdy+1tD_zdtdzzt2D_zdydy+1tD_zdzdt

(2.11)

 

1. k = LΘNM

S > 

κ=HookM,DirectionalCovariantDerivativeL,Theta_N,C

24xy=24xy

(2.12)

 

2. ρ=M ΘNM

S > 

ρ=HookM,DirectionalCovariantDerivativebarM,Theta_N,C

24tz=24tz

(2.13)

 

3. σ=MΘNM

S > 

σ=HookM,DirectionalCovariantDerivativeM,Theta_N,C

24tz=24tz

(2.14)

 

4. τ=NΘNM

S > 

τ=HookM,DirectionalCovariantDerivativeN,Theta_N,C

24xy=24xy

(2.15)

 

5. π=NΘLM

S > 

pi=HookbarM,DirectionalCovariantDerivativeL,Theta_L,C

24xy=24xy

(2.16)

 

6. λ=NΘLM

S > 

λ=HookbarM,DirectionalCovariantDerivativebarM,Theta_L,C

24tz=24tz

(2.17)

 

7. μ=NΘMM

S > 

μ=HookbarM,DirectionalCovariantDerivativeM,Theta_L,C

24tz=24tz

(2.18)

 

8. ν=NΘNM

S > 

ν=HookbarM,DirectionalCovariantDerivativeN,Theta_L,C

24xy=24xy

(2.19)

 

9. α=12MΘNN+12MΘMM

S > 

α=12HookN,DirectionalCovariantDerivativebarM,Theta_N,C+12HookbarM,DirectionalCovariantDerivativebarM,Theta_barM,C

I42zy=I42zy

(2.20)

 

10. β=12MΘNN+12MΘMM

S > 

β=12HookN,DirectionalCovariantDerivativeM,Theta_N,C+12HookbarM,DirectionalCovariantDerivativebarM,Theta_barM,C

I42zy=I42zy

(2.21)

 

11. γ=12NΘNN+12NΘMM

S > 

gam=12HookN,DirectionalCovariantDerivativeN,Theta_N,C+12HookbarM,DirectionalCovariantDerivativeN,Theta_barM,C

24tx=24tx

(2.22)

 

12. ε=12LΘNN+12LΘMM

S > 

ε=12HookN,DirectionalCovariantDerivativeL,Theta_N,C+12HookbarM,DirectionalCovariantDerivativeL,Theta_barM,C

24tx=24tx

(2.23)

 

Example 4

When working with the NP formalism, it is usually advantageous to work with the anholonomic frame defined by the null tetrad.  To create anholonomic frames in DifferentialGeometry, see FrameData.

S > 

FDFrameDataNTetrad,NP

FDE1,E2=2E12tx2E22tx,E1,E3=2E14xy+2E24xy2E34tz+2E44tz,E1,E4=2E14xy+2E24xy+2E34tz2E44tz,E2,E3=2E14xy+2E24xy+2E34tz2E44tz,E2,E4=2E14xy+2E24xy2E34tz+2E44tz,E3,E4=I2E32yzI2E42yz

(2.24)
S > 

DGsetupFD

frame name: NP

(2.25)

 

We can now calculate the spin coefficients for the null tetrad with the second calling sequence.

NP > 

NPSpinCoefficientsNP

tableepsilon=24tx,tau=24xy,kappa=24xy,alpha=I42zy,beta=I42zy,mu=24tz,rho=24tz,lambda=24tz,gamma=24tx,sigma=24tz,nu=24xy,pi=24xy

(2.26)

See Also

DifferentialGeometry, Tensor, Christoffel, CovariantDerivative, DirectionalCovariantDerivative, DualBasis, FrameData, GRQuery, NullTetrad