DifferentialGeometry/Tensor/NPSpinCoefficients

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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

•

Let g be a metric with signature [1, -1, -1, -1] and [L, N, M, barM] 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 [Theta_L, Theta_N, Theta_M, Theta_barM] be the basis of 1-forms dual to the given null tetrad [L, N, M, barM]. With respect to this basis, the metric g becomes g = Theta_L &s Theta_N - Theta_M &s Theta_barM, where &s is the symmetric tensor product. Let nabla_X be the directional covariant derivative operator (in the direction of a vector X) defined by the Christoffel connection for the metric g. If omega is a 1-form, then nabla_X(omega) is a 1-form which can be evaluated on a vector Y to give the scalar nabla_X(omega)(Y). In terms of this notation, the spin coefficients are:

–	kappa = nabla_L(Theta_N)(M)

–	rho = nabla_barM(Theta_N)(M)

–	sigma = nabla_M(Theta_N)(M)

–	tau = -nabla_N(Theta_N)(M)

–	pi = -nabla_N(Theta_L)(barM)

–	lambda = -nabla_N(Theta_L)(barM)

–	mu = -nabla_N(Theta_M)(barM)

–	nu = nabla_N(Theta_N(barM)

–	alpha = 1/2(nabla_barM Theta_N)(N) + 1/2(nabla_barM (Theta_barM)(barM)

–	beta = 1/2(nabla_M Theta_N)(N) + 1/2(nabla_M (Theta_barM)(barM)

–	gamma = 1/2(nabla_N Theta_N)(N) + 1/2(nabla_N (Theta_barM)(barM)

–	epsilon = 1/2(nabla_L Theta_N)(N) + 1/2(nabla_L(Theta_barM)(barM)

•

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

Example 1.

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

(2.1)

Define a metric g.

S >

(2.2)

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

S >

(2.3)

S >

(2.4)

Construct a null tetrad NTetrad from the orthonormal tetrad OTetrad.

S >

[_DG([["vector", S, []], [[[1], (1/2)*2^(1/2)/x], [[4], (1/2)*2^(1/2)/t]]]), _DG([["vector", S, []], [[[1], (1/2)*2^(1/2)/x], [[4], -(1/2)*2^(1/2)/t]]]), _DG([["vector", S, []], [[[2], (1/2)*2^(1/2)/y], [[3], ((1/2)*I)*2^(1/2)/z]]]), _DG([["vector", S, []], [[[2], (1/2)*2^(1/2)/y], [[3], -((1/2)*I)*2^(1/2)/z]]])]

(2.5)

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

S >

(2.6)

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

S >

(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 >

-(1/4)*2^(1/2)/(x*y), -(1/4)*2^(1/2)/(t*z), (1/4)*2^(1/2)/(t*z), -(1/4)*2^(1/2)/(x*y), (1/4)*2^(1/2)/(x*y), (1/4)*2^(1/2)/(t*z), -(1/4)*2^(1/2)/(t*z), (1/4)*2^(1/2)/(x*y), ((1/4)*I)*2^(1/2)/(z*y), ((1/4)*I)*2^(1/2)/(z*y), -(1/4)*2^(1/2)/(t*x), (1/4)*2^(1/2)/(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 >

_DG([["vector", S, []], [[[1], (1/2)*2^(1/2)/x], [[4], (1/2)*2^(1/2)/t]]]), _DG([["vector", S, []], [[[1], (1/2)*2^(1/2)/x], [[4], -(1/2)*2^(1/2)/t]]]), _DG([["vector", S, []], [[[2], (1/2)*2^(1/2)/y], [[3], ((1/2)*I)*2^(1/2)/z]]]), _DG([["vector", S, []], [[[2], (1/2)*2^(1/2)/y], [[3], -((1/2)*I)*2^(1/2)/z]]])

(2.9)

Define the dual basis.

S >

_DG([["form", S, 1], [[[1], (1/2)*2^(1/2)*x], [[4], (1/2)*2^(1/2)*t]]]), _DG([["form", S, 1], [[[1], (1/2)*2^(1/2)*x], [[4], -(1/2)*2^(1/2)*t]]]), _DG([["form", S, 1], [[[2], (1/2)*2^(1/2)*y], [[3], -((1/2)*I)*2^(1/2)*z]]]), _DG([["form", S, 1], [[[2], (1/2)*2^(1/2)*y], [[3], ((1/2)*I)*2^(1/2)*z]]])

(2.10)

Calculate the Christoffel connection.

S >

_DG([["connection", S, [["cov_bas", "con_bas", "cov_bas"], []]], [[[1, 1, 2], 1/x], [[1, 2, 1], x/y^2], [[1, 4, 4], 1/t], [[2, 1, 1], 1/x], [[2, 2, 3], 1/y], [[2, 3, 2], -y/z^2], [[3, 2, 2], 1/y], [[3, 3, 4], 1/z], [[3, 4, 3], -z/t^2], [[4, 1, 4], t/x^2], [[4, 3, 3], 1/z], [[4, 4, 1], 1/t]]])

(2.11)

1. kappa = nabla_L(Theta_N)(M)

S >

(2.12)

2. rho = nabla_barM(Theta_N)(M)

S >

(2.13)

3. sigma = nabla_M(Theta_N)(M)

S >

(2.14)

4. tau = nabla_N(Theta_N)(M)

S >

(2.15)

5. pi = - nabla_N(Theta_L)(barM)

S >

(2.16)

6. lambda = -nabla_barM(Theta_L)(barM)

S >

(2.17)

7. mu = nabla_M(Theta_L)(barM)

S >

(2.18)

8. nu = nabla_N(Theta_L)(barM)

S >

(2.19)

9. alpha = 1/2*nabla_barM(Theta_N)(N) + nabla_barM(Theta_barM)(barM)

S >

alpha = (1/2)*Hook([N], DirectionalCovariantDerivative(barM, Theta_N, C))+(1/2)*Hook([barM], DirectionalCovariantDerivative(barM, Theta_barM, C))

((1/4)*I)*2^(1/2)/(z*y) = ((1/4)*I)*2^(1/2)/(z*y)

(2.20)

10. beta = 1/2*nabla_M(Theta_N)(N) + 1/2*nabla_M(Theta_barM)(barM)

S >

beta = (1/2)*Hook([N], DirectionalCovariantDerivative(M, Theta_N, C))+(1/2)*Hook([barM], DirectionalCovariantDerivative(barM, Theta_barM, C))

((1/4)*I)*2^(1/2)/(z*y) = ((1/4)*I)*2^(1/2)/(z*y)

(2.21)

11. gamma =1/2*nabla_N(Theta_N)(N) + 1/2*nabla_N(Theta_barM)(barM)

S >

gam = (1/2)*Hook([N], DirectionalCovariantDerivative(N, Theta_N, C))+(1/2)*Hook([barM], DirectionalCovariantDerivative(N, Theta_barM, C))

(2.22)

12. epsilon = 1/2*nabla_L(Theta_N)(N) + 1/2*nabla_L(Theta_barM)(barM)

S >

epsilon = (1/2)*Hook([N], DirectionalCovariantDerivative(L, Theta_N, C))+(1/2)*Hook([barM], DirectionalCovariantDerivative(L, Theta_barM, C))

(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 >