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 ΘL⊙ΘN−2 ΘM⊙ΘM‾ =ΘL⊗ΘN+ΘN⊗ΘL−ΘM⊗ΘM‾−ΘM‾⊗ΘM , 

where⊙is 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=12∇M‾ΘNN+12∇M‾ΘM‾M‾

β=12∇MΘNN+12∇MΘM‾M‾

γ=12∇NΘNN+12∇NΘM‾M‾

ε=12∇LΘNN+12∇LΘM‾M‾

    

• 

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

> 

with⁡DifferentialGeometry:with⁡Tensor:

 

Example 1.

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

> 

DGsetup⁡t,x,y,z,S

frame name: S

(2.1)

 

Define a metric g.

S > 

g≔evalDG⁡x2⁢dt&tdt−y2⁢dx&tdx−z2⁢dy&tdy−t2⁢dz&tdz

g≔x2⁢dt⁢dt−y2⁢dx⁢dx−z2⁢dy⁢dy−t2⁢dz⁢dz

(2.2)

 

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

S > 

OTetrad≔1x⁢D_t,1y⁢D_x,1z⁢D_y,1t⁢D_z

OTetrad≔D_tx,D_xy,D_yz,D_zt

(2.3)
S > 

GRQuery⁡OTetrad,g,OrthonormalTetrad

true

(2.4)

 

Construct a null tetrad NTetrad from the orthonormal tetrad OTetrad.

S > 

NTetrad≔NullTetrad⁡OTetrad

NTetrad≔22⁢x⁢D_t+22⁢t⁢D_z,22⁢x⁢D_t−22⁢t⁢D_z,22⁢y⁢D_x+I2⁢2z⁢D_y,22⁢y⁢D_x−I2⁢2z⁢D_y

(2.5)

 

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

S > 

SpinCoeff≔NPSpinCoefficients⁡NTetrad

SpinCoeff≔table⁡epsilon=24⁢t⁢x,tau=−24⁢x⁢y,kappa=−24⁢x⁢y,alpha=I4⁢2z⁢y,beta=I4⁢2z⁢y,mu=−24⁢t⁢z,rho=−24⁢t⁢z,lambda=24⁢t⁢z,gamma=−24⁢t⁢x,sigma=24⁢t⁢z,nu=24⁢x⁢y,pi=24⁢x⁢y

(2.6)

 

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

S > 

SpinCoefftau

−24⁢x⁢y

(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,ε≔NPSpinCoefficients⁡NTetrad,output=Sequence

κ,ρ,σ,τ,π,λ,μ,ν,α,β,gam,ϵ≔−24⁢x⁢y,−24⁢t⁢z,24⁢t⁢z,−24⁢x⁢y,24⁢x⁢y,24⁢t⁢z,−24⁢t⁢z,24⁢x⁢y,I4⁢2z⁢y,I4⁢2z⁢y,−24⁢t⁢x,24⁢t⁢x

(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,barM≔op⁡NTetrad

L,N,M,barM≔22⁢x⁢D_t+22⁢t⁢D_z,22⁢x⁢D_t−22⁢t⁢D_z,22⁢y⁢D_x+I2⁢2z⁢D_y,22⁢y⁢D_x−I2⁢2z⁢D_y

(2.9)

 

Define the dual basis.

S > 

Theta_L,Theta_N,Theta_M,Theta_barM≔op⁡DualBasis⁡NTetrad

Theta_L,Theta_N,Theta_M,Theta_barM≔2⁢x2⁢dt+2⁢t2⁢dz,2⁢x2⁢dt−2⁢t2⁢dz,2⁢y2⁢dx−I2⁢2⁢z⁢dy,2⁢y2⁢dx+I2⁢2⁢z⁢dy

(2.10)

 

Calculate the Christoffel connection.

S > 

C≔Christoffel⁡g

C≔1x⁢D_t⁢dt⁢dx+1x⁢D_t⁢dx⁢dt+tx2⁢D_t⁢dz⁢dz+xy2⁢D_x⁢dt⁢dt+1y⁢D_x⁢dx⁢dy+1y⁢D_x⁢dy⁢dx−yz2⁢D_y⁢dx⁢dx+1z⁢D_y⁢dy⁢dz+1z⁢D_y⁢dz⁢dy+1t⁢D_z⁢dt⁢dz−zt2⁢D_z⁢dy⁢dy+1t⁢D_z⁢dz⁢dt

(2.11)

 

1. k = ∇LΘNM

S > 

κ=Hook⁡M,DirectionalCovariantDerivative⁡L,Theta_N,C

−24⁢x⁢y=−24⁢x⁢y

(2.12)

 

2. ρ=∇M‾ ΘNM

S > 

ρ=Hook⁡M,DirectionalCovariantDerivative⁡barM,Theta_N,C

−24⁢t⁢z=−24⁢t⁢z

(2.13)

 

3. σ=∇MΘNM

S > 

σ=Hook⁡M,DirectionalCovariantDerivative⁡M,Theta_N,C

24⁢t⁢z=24⁢t⁢z

(2.14)

 

4. τ=−∇NΘNM

S > 

τ=Hook⁡M,DirectionalCovariantDerivative⁡N,Theta_N,C

−24⁢x⁢y=−24⁢x⁢y

(2.15)

 

5. π=−∇NΘLM‾

S > 

pi=−Hook⁡barM,DirectionalCovariantDerivative⁡L,Theta_L,C

24⁢x⁢y=24⁢x⁢y

(2.16)

 

6. λ=−∇NΘLM‾

S > 

λ=−Hook⁡barM,DirectionalCovariantDerivative⁡barM,Theta_L,C

24⁢t⁢z=24⁢t⁢z

(2.17)

 

7. μ=−∇NΘMM‾

S > 

μ=−Hook⁡barM,DirectionalCovariantDerivative⁡M,Theta_L,C

−24⁢t⁢z=−24⁢t⁢z

(2.18)

 

8. ν=∇NΘNM‾

S > 

ν=−Hook⁡barM,DirectionalCovariantDerivative⁡N,Theta_L,C

24⁢x⁢y=24⁢x⁢y

(2.19)

 

9. α=12∇M‾ΘNN+12∇M‾ΘM‾M‾

S > 

α=12⁢Hook⁡N,DirectionalCovariantDerivative⁡barM,Theta_N,C+12⁢Hook⁡barM,DirectionalCovariantDerivative⁡barM,Theta_barM,C

I4⁢2z⁢y=I4⁢2z⁢y

(2.20)

 

10. β=12∇MΘNN+12∇MΘM‾M‾

S > 

β=12⁢Hook⁡N,DirectionalCovariantDerivative⁡M,Theta_N,C+12⁢Hook⁡barM,DirectionalCovariantDerivative⁡barM,Theta_barM,C

I4⁢2z⁢y=I4⁢2z⁢y

(2.21)

 

11. γ=12∇NΘNN+12∇NΘM‾M‾

S > 

gam=12⁢Hook⁡N,DirectionalCovariantDerivative⁡N,Theta_N,C+12⁢Hook⁡barM,DirectionalCovariantDerivative⁡N,Theta_barM,C

−24⁢t⁢x=−24⁢t⁢x

(2.22)

 

12. ε=12∇LΘNN+12∇LΘM‾M‾

S > 

ε=12⁢Hook⁡N,DirectionalCovariantDerivative⁡L,Theta_N,C+12⁢Hook⁡barM,DirectionalCovariantDerivative⁡L,Theta_barM,C

24⁢t⁢x=24⁢t⁢x

(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 > 

FD≔FrameData⁡NTetrad,NP

FD≔E1,E2=2⁢E12⁢t⁢x−2⁢E22⁢t⁢x,E1,E3=2⁢E14⁢x⁢y+2⁢E24⁢x⁢y−2⁢E34⁢t⁢z+2⁢E44⁢t⁢z,E1,E4=2⁢E14⁢x⁢y+2⁢E24⁢x⁢y+2⁢E34⁢t⁢z−2⁢E44⁢t⁢z,E2,E3=2⁢E14⁢x⁢y+2⁢E24⁢x⁢y+2⁢E34⁢t⁢z−2⁢E44⁢t⁢z,E2,E4=2⁢E14⁢x⁢y+2⁢E24⁢x⁢y−2⁢E34⁢t⁢z+2⁢E44⁢t⁢z,E3,E4=−I⁢2⁢E32⁢y⁢z−I⁢2⁢E42⁢y⁢z

(2.24)
S > 

DGsetup⁡FD

frame name: NP

(2.25)

 

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

NP > 

NPSpinCoefficients⁡NP

table⁡epsilon=24⁢t⁢x,tau=−24⁢x⁢y,kappa=−24⁢x⁢y,alpha=I4⁢2z⁢y,beta=I4⁢2z⁢y,mu=−24⁢t⁢z,rho=−24⁢t⁢z,lambda=24⁢t⁢z,gamma=−24⁢t⁢x,sigma=24⁢t⁢z,nu=24⁢x⁢y,pi=24⁢x⁢y

(2.26)

See Also

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