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

Examples

See Also

Description

•

Let $g$ be a metric with signature $(1, - 1, - 1, - 1)$ and $(L, N, M, \overline{M}$ ) 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 $(Θ_{L}, Θ_{N}, Θ_{M}, Θ_{\overline{M}})$ be the basis of 1-forms dual to the given null tetrad $(L, N, M, \overline{M}$ ). With respect to this basis, the metric $g$ becomes

$g = 2 Θ_{L} ⊙ Θ_{N} - 2 Θ_{M} ⊙ Θ_{\overline{M}} = Θ_{L} \otimes Θ_{N} + Θ_{N} \otimes Θ_{L} - Θ_{M} \otimes Θ_{\overline{M}} - Θ_{\overline{M}} \otimes Θ_{M},$

where $⊙$ 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 $ω$ is a 1-form, then $\nabla_{X} ω$ is a 1-form which can be evaluated on a vector $Y$ to give the scalar $\nabla_{X} ω (Y)$ . In terms of this notation, the spin coefficients are:

k = $(\nabla_{L} Θ_{N}) (M)$	$ρ = (\nabla_{\overline{M}} Θ_{N}) (M)$	$σ = (\nabla_{M} Θ_{N}) (M)$	$τ = - (\nabla_{N} Θ_{N}) (M)$
$π = - (\nabla_{N} Θ_{L}) (\overline{M})$	$λ = - (\nabla_{N} Θ_{L})$ $(\overline{M})$	$μ = - (\nabla_{N} Θ_{M}) (\overline{M})$	$ν = (\nabla_{N} Θ_{N}) (\overline{M})$
$α = \frac{1}{2} (\nabla_{\overline{M}} Θ_{N}) (N) + \frac{1}{2} (\nabla_{\overline{M}} Θ_{\overline{M}}) (\overline{M})$	$β = \frac{1}{2} (\nabla_{M} Θ_{N}) (N) + \frac{1}{2} (\nabla_{M} Θ_{\overline{M}}) (\overline{M})$	$γ = \frac{1}{2} (\nabla_{N} Θ_{N}) (N) + \frac{1}{2} (\nabla_{N} Θ_{\overline{M}}) (\overline{M})$	$ε = \frac{1}{2} (\nabla_{L} Θ_{N}) (N) + \frac{1}{2} (\nabla_{L} Θ_{\overline{M}}) (\overline{M})$

•

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

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

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 >	$g ≔ evalDG (x^{2} dt &t dt - y^{2} dx &t dx - z^{2} dy &t dy - t^{2} dz &t dz)$

$g ≔ x^{2} dt dt - y^{2} dx dx - z^{2} dy dy - t^{2} dz dz$

(2.2)

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

S >	$OTetrad ≔ [\frac{1}{x} D_t, \frac{1}{y} D_x, \frac{1}{z} D_y, \frac{1}{t} D_z]$

$OTetrad ≔ [\frac{D_t}{x}, \frac{D_x}{y}, \frac{D_y}{z}, \frac{D_z}{t}]$

(2.3)

S >	$GRQuery (OTetrad, g, OrthonormalTetrad)$

$true$

(2.4)

Construct a null tetrad NTetrad from the orthonormal tetrad OTetrad.

S >	$NTetrad ≔ NullTetrad (OTetrad)$

$NTetrad ≔ [\frac{\sqrt{2}}{2 x} D_t + \frac{\sqrt{2}}{2 t} D_z, \frac{\sqrt{2}}{2 x} D_t - \frac{\sqrt{2}}{2 t} D_z, \frac{\sqrt{2}}{2 y} D_x + \frac{\frac{I}{2} \sqrt{2}}{z} D_y, \frac{\sqrt{2}}{2 y} D_x - \frac{\frac{I}{2} \sqrt{2}}{z} D_y]$

(2.5)

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

S >	$SpinCoeff ≔ NPSpinCoefficients (NTetrad)$

$SpinCoeff ≔ table ([epsilon = \frac{\sqrt{2}}{4 t x}, tau = - \frac{\sqrt{2}}{4 x y}, kappa = - \frac{\sqrt{2}}{4 x y}, alpha = \frac{\frac{I}{4} \sqrt{2}}{z y}, beta = \frac{\frac{I}{4} \sqrt{2}}{z y}, mu = - \frac{\sqrt{2}}{4 t z}, rho = - \frac{\sqrt{2}}{4 t z}, lambda = \frac{\sqrt{2}}{4 t z}, gamma = - \frac{\sqrt{2}}{4 t x}, sigma = \frac{\sqrt{2}}{4 t z}, nu = \frac{\sqrt{2}}{4 x y}, pi = \frac{\sqrt{2}}{4 x y}])$

(2.6)

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

S >	$SpinCoeff [tau]$

$- \frac{\sqrt{2}}{4 x y}$

(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 >	$κ, ρ, σ, τ, pi, λ, μ, ν, α, β, gam, ε ≔ NPSpinCoefficients (NTetrad, output = Sequence)$

$κ, ρ, σ, τ, π, λ, μ, ν, α, β, gam, ϵ ≔ - \frac{\sqrt{2}}{4 x y}, - \frac{\sqrt{2}}{4 t z}, \frac{\sqrt{2}}{4 t z}, - \frac{\sqrt{2}}{4 x y}, \frac{\sqrt{2}}{4 x y}, \frac{\sqrt{2}}{4 t z}, - \frac{\sqrt{2}}{4 t z}, \frac{\sqrt{2}}{4 x y}, \frac{\frac{I}{4} \sqrt{2}}{z y}, \frac{\frac{I}{4} \sqrt{2}}{z y}, - \frac{\sqrt{2}}{4 t x}, \frac{\sqrt{2}}{4 t x}$

(2.8)

Example 3.

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

S >	$L, N, M, barM ≔ op (NTetrad)$

$L, N, M, barM ≔ \frac{\sqrt{2}}{2 x} D_t + \frac{\sqrt{2}}{2 t} D_z, \frac{\sqrt{2}}{2 x} D_t - \frac{\sqrt{2}}{2 t} D_z, \frac{\sqrt{2}}{2 y} D_x + \frac{\frac{I}{2} \sqrt{2}}{z} D_y, \frac{\sqrt{2}}{2 y} D_x - \frac{\frac{I}{2} \sqrt{2}}{z} D_y$

(2.9)

Define the dual basis.

S >	$Theta_L, Theta_N, Theta_M, Theta_barM ≔ op (DualBasis (NTetrad))$

$Theta_L, Theta_N, Theta_M, Theta_barM ≔ \frac{\sqrt{2} x}{2} dt + \frac{\sqrt{2} t}{2} dz, \frac{\sqrt{2} x}{2} dt - \frac{\sqrt{2} t}{2} dz, \frac{\sqrt{2} y}{2} dx - \frac{I}{2} \sqrt{2} z dy, \frac{\sqrt{2} y}{2} dx + \frac{I}{2} \sqrt{2} z dy$

(2.10)

Calculate the Christoffel connection.

S >	$C ≔ Christoffel (g)$

$C ≔ \frac{1}{x} D_t dt dx + \frac{1}{x} D_t dx dt + \frac{t}{x^{2}} D_t dz dz + \frac{x}{y^{2}} D_x dt dt + \frac{1}{y} D_x dx dy + \frac{1}{y} D_x dy dx - \frac{y}{z^{2}} D_y dx dx + \frac{1}{z} D_y dy dz + \frac{1}{z} D_y dz dy + \frac{1}{t} D_z dt dz - \frac{z}{t^{2}} D_z dy dy + \frac{1}{t} D_z dz dt$

(2.11)

1. k = $(\nabla_{L} Θ_{N}) (M)$

S >	$κ = Hook ([M], DirectionalCovariantDerivative (L, Theta_N, C))$

$- \frac{\sqrt{2}}{4 x y} = - \frac{\sqrt{2}}{4 x y}$

(2.12)

2. $ρ = (\nabla_{\overline{M}} Θ_{N}) (M)$

S >	$ρ = Hook ([M], DirectionalCovariantDerivative (barM, Theta_N, C))$

$- \frac{\sqrt{2}}{4 t z} = - \frac{\sqrt{2}}{4 t z}$

(2.13)

3. $σ = (\nabla_{M} Θ_{N}) (M)$

S >	$σ = Hook ([M], DirectionalCovariantDerivative (M, Theta_N, C))$

$\frac{\sqrt{2}}{4 t z} = \frac{\sqrt{2}}{4 t z}$

(2.14)

4. $τ = - (\nabla_{N} Θ_{N}) (M)$

S >	$τ = Hook ([M], DirectionalCovariantDerivative (N, Theta_N, C))$

$- \frac{\sqrt{2}}{4 x y} = - \frac{\sqrt{2}}{4 x y}$

(2.15)

5. $π = - (\nabla_{N} Θ_{L}) (\overline{M})$

S >	$pi = - Hook ([barM], DirectionalCovariantDerivative (L, Theta_L, C))$

$\frac{\sqrt{2}}{4 x y} = \frac{\sqrt{2}}{4 x y}$

(2.16)

6. $λ = - (\nabla_{N} Θ_{L})$ $(\overline{M})$

S >	$λ = - Hook ([barM], DirectionalCovariantDerivative (barM, Theta_L, C))$

$\frac{\sqrt{2}}{4 t z} = \frac{\sqrt{2}}{4 t z}$

(2.17)

7. $μ = - (\nabla_{N} Θ_{M}) (\overline{M})$

S >	$μ = - Hook ([barM], DirectionalCovariantDerivative (M, Theta_L, C))$

$- \frac{\sqrt{2}}{4 t z} = - \frac{\sqrt{2}}{4 t z}$

(2.18)

8. $ν = (\nabla_{N} Θ_{N}) (\overline{M})$

S >	$ν = - Hook ([barM], DirectionalCovariantDerivative (N, Theta_L, C))$

$\frac{\sqrt{2}}{4 x y} = \frac{\sqrt{2}}{4 x y}$

(2.19)

9. $α = \frac{1}{2} (\nabla_{\overline{M}} Θ_{N}) (N) + \frac{1}{2} (\nabla_{\overline{M}} Θ_{\overline{M}}) (\overline{M})$

S >	$α = \frac{1}{2} Hook ([N], DirectionalCovariantDerivative (barM, Theta_N, C)) + \frac{1}{2} Hook ([barM], DirectionalCovariantDerivative (barM, Theta_barM, C))$

$\frac{\frac{I}{4} \sqrt{2}}{z y} = \frac{\frac{I}{4} \sqrt{2}}{z y}$

(2.20)

10. $β = \frac{1}{2} (\nabla_{M} Θ_{N}) (N) + \frac{1}{2} (\nabla_{M} Θ_{\overline{M}}) (\overline{M})$

S >	$β = \frac{1}{2} Hook ([N], DirectionalCovariantDerivative (M, Theta_N, C)) + \frac{1}{2} Hook ([barM], DirectionalCovariantDerivative (barM, Theta_barM, C))$

$\frac{\frac{I}{4} \sqrt{2}}{z y} = \frac{\frac{I}{4} \sqrt{2}}{z y}$

(2.21)

11. $γ = \frac{1}{2} (\nabla_{N} Θ_{N}) (N) + \frac{1}{2} (\nabla_{N} Θ_{\overline{M}}) (\overline{M})$

S >	$gam = \frac{1}{2} Hook ([N], DirectionalCovariantDerivative (N, Theta_N, C)) + \frac{1}{2} Hook ([barM], DirectionalCovariantDerivative (N, Theta_barM, C))$

$- \frac{\sqrt{2}}{4 t x} = - \frac{\sqrt{2}}{4 t x}$

(2.22)

12. $ε = \frac{1}{2} (\nabla_{L} Θ_{N}) (N) + \frac{1}{2} (\nabla_{L} Θ_{\overline{M}}) (\overline{M})$

S >	$ε = \frac{1}{2} Hook ([N], DirectionalCovariantDerivative (L, Theta_N, C)) + \frac{1}{2} Hook ([barM], DirectionalCovariantDerivative (L, Theta_barM, C))$

$\frac{\sqrt{2}}{4 t x} = \frac{\sqrt{2}}{4 t x}$

(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 >	$FD ≔ FrameData (NTetrad, NP)$

$FD ≔ [[E1, E2] = \frac{\sqrt{2} E1}{2 t x} - \frac{\sqrt{2} E2}{2 t x}, [E1, E3] = \frac{\sqrt{2} E1}{4 x y} + \frac{\sqrt{2} E2}{4 x y} - \frac{\sqrt{2} E3}{4 t z} + \frac{\sqrt{2} E4}{4 t z}, [E1, E4] = \frac{\sqrt{2} E1}{4 x y} + \frac{\sqrt{2} E2}{4 x y} + \frac{\sqrt{2} E3}{4 t z} - \frac{\sqrt{2} E4}{4 t z}, [E2, E3] = \frac{\sqrt{2} E1}{4 x y} + \frac{\sqrt{2} E2}{4 x y} + \frac{\sqrt{2} E3}{4 t z} - \frac{\sqrt{2} E4}{4 t z}, [E2, E4] = \frac{\sqrt{2} E1}{4 x y} + \frac{\sqrt{2} E2}{4 x y} - \frac{\sqrt{2} E3}{4 t z} + \frac{\sqrt{2} E4}{4 t z}, [E3, E4] = - \frac{I \sqrt{2} E3}{2 y z} - \frac{I \sqrt{2} E4}{2 y z}]$

(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 >	$NPSpinCoefficients (NP)$

$table ([epsilon = \frac{\sqrt{2}}{4 t x}, tau = - \frac{\sqrt{2}}{4 x y}, kappa = - \frac{\sqrt{2}}{4 x y}, alpha = \frac{\frac{I}{4} \sqrt{2}}{z y}, beta = \frac{\frac{I}{4} \sqrt{2}}{z y}, mu = - \frac{\sqrt{2}}{4 t z}, rho = - \frac{\sqrt{2}}{4 t z}, lambda = \frac{\sqrt{2}}{4 t z}, gamma = - \frac{\sqrt{2}}{4 t x}, sigma = \frac{\sqrt{2}}{4 t z}, nu = \frac{\sqrt{2}}{4 x y}, pi = \frac{\sqrt{2}}{4 x y}])$

(2.26)

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

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