 DifferentialGeometry/Tensor/NPSpinCoefficients - Maple Help

Tensor[NPSpinCoefficients] - find the Newman-Penrose spin coefficients

Calling Sequences

NPSpinCoefficients(Fr, output)

Parameters

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 $\left(1,-1,-1,-1\right)$ and ) 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  be the basis of 1-forms dual to the given null tetrad ). With respect to this basis, the metric $g$ becomes

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

 k = $\left({\nabla }_{L}{\mathrm{Θ}}_{N}\right)\left(M\right)$ $\mathrm{σ}=\left({\nabla }_{M}{\mathrm{\Theta }}_{N}\right)\left(M\right)$ $\mathrm{τ}=-\left({\nabla }_{N}{\mathrm{\Theta }}_{N}\right)\left(M\right)$ $\mathrm{π}=-\left({\nabla }_{N}{\mathrm{\Theta }}_{L}\right)\left(\stackrel{‾}{M}\right)$ $\mathrm{λ}=-\left({\nabla }_{N}{\mathrm{\Theta }}_{L}\right)$$\left(\stackrel{‾}{M}\right)$ $\mathrm{μ}=-\left({\nabla }_{N}{\mathrm{Θ}}_{M}\right)\left(\stackrel{‾}{M}\right)$ $\mathrm{ν}=\left({\nabla }_{N}{\mathrm{Θ}}_{N}\right)\left(\stackrel{‾}{M}\right)$ $\mathrm{α}=\frac{1}{2}\left({\nabla }_{\stackrel{‾}{M}}{\mathrm{Θ}}_{N}\right)\left(N\right)+\frac{1}{2}\left({\nabla }_{\stackrel{‾}{M}}{\mathrm{\Theta }}_{\stackrel{‾}{M}}\right)\left(\stackrel{‾}{M}\right)$ $\mathrm{β}=\frac{1}{2}\left({\nabla }_{M}{\mathrm{Θ}}_{N}\right)\left(N\right)+\frac{1}{2}\left({\nabla }_{M}{\mathrm{Θ}}_{\stackrel{‾}{M}}\right)\left(\stackrel{‾}{M}\right)$ $\mathrm{γ}=\frac{1}{2}\left({\nabla }_{N}{\mathrm{\Theta }}_{N}\right)\left(N\right)+\frac{1}{2}\left({\nabla }_{N}{\mathrm{\Theta }}_{\stackrel{‾}{M}}\right)\left(\stackrel{‾}{M}\right)$ $\mathrm{ε}=\frac{1}{2}\left({\nabla }_{L}{\mathrm{\Theta }}_{N}\right)\left(N\right)+\frac{1}{2}\left({\nabla }_{L}{\mathrm{\Theta }}_{\stackrel{‾}{M}}\right)\left(\stackrel{‾}{M}\right)$

 • 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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$

Example 1.

Define a manifold $S$ with coordinates $\left(t,x,y,z\right)$.

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],S\right)$
 ${\mathrm{frame name: S}}$ (2.1)

Define a metric $g$.

 S > $g≔\mathrm{evalDG}\left({x}^{2}\mathrm{dt}&t\mathrm{dt}-{y}^{2}\mathrm{dx}&t\mathrm{dx}-{z}^{2}\mathrm{dy}&t\mathrm{dy}-{t}^{2}\mathrm{dz}&t\mathrm{dz}\right)$
 ${g}{≔}{{x}}^{{2}}{}{\mathrm{dt}}{}{\mathrm{dt}}{-}{{y}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}{{z}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}{-}{{t}}^{{2}}{}{\mathrm{dz}}{}{\mathrm{dz}}$ (2.2)

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

 S > $\mathrm{OTetrad}≔\left[\frac{1\mathrm{D_t}}{x},\frac{1\mathrm{D_x}}{y},\frac{1\mathrm{D_y}}{z},\frac{1\mathrm{D_z}}{t}\right]$
 ${\mathrm{OTetrad}}{≔}\left[\frac{{\mathrm{D_t}}}{{x}}{,}\frac{{\mathrm{D_x}}}{{y}}{,}\frac{{\mathrm{D_y}}}{{z}}{,}\frac{{\mathrm{D_z}}}{{t}}\right]$ (2.3)
 S > $\mathrm{GRQuery}\left(\mathrm{OTetrad},g,"OrthonormalTetrad"\right)$
 ${\mathrm{true}}$ (2.4)

 S > $\mathrm{NTetrad}≔\mathrm{NullTetrad}\left(\mathrm{OTetrad}\right)$
 ${\mathrm{NTetrad}}{≔}\left[\frac{\sqrt{{2}}}{{2}{}{x}}{}{\mathrm{D_t}}{+}\frac{\sqrt{{2}}}{{2}{}{t}}{}{\mathrm{D_z}}{,}\frac{\sqrt{{2}}}{{2}{}{x}}{}{\mathrm{D_t}}{-}\frac{\sqrt{{2}}}{{2}{}{t}}{}{\mathrm{D_z}}{,}\frac{\sqrt{{2}}}{{2}{}{y}}{}{\mathrm{D_x}}{+}\frac{\frac{{I}}{{2}}{}\sqrt{{2}}}{{z}}{}{\mathrm{D_y}}{,}\frac{\sqrt{{2}}}{{2}{}{y}}{}{\mathrm{D_x}}{-}\frac{\frac{{I}}{{2}}{}\sqrt{{2}}}{{z}}{}{\mathrm{D_y}}\right]$ (2.5)

 S > $\mathrm{SpinCoeff}≔\mathrm{NPSpinCoefficients}\left(\mathrm{NTetrad}\right)$
 ${\mathrm{SpinCoeff}}{≔}{table}{}\left(\left[{"epsilon"}{=}\frac{\sqrt{{2}}}{{4}{}{t}{}{x}}{,}{"tau"}{=}{-}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{,}{"kappa"}{=}{-}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{,}{"alpha"}{=}\frac{\frac{{I}}{{4}}{}\sqrt{{2}}}{{z}{}{y}}{,}{"beta"}{=}\frac{\frac{{I}}{{4}}{}\sqrt{{2}}}{{z}{}{y}}{,}{"mu"}{=}{-}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{,}{"rho"}{=}{-}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{,}{"lambda"}{=}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{,}{"gamma"}{=}{-}\frac{\sqrt{{2}}}{{4}{}{t}{}{x}}{,}{"sigma"}{=}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{,}{"nu"}{=}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{,}{"pi"}{=}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}\right]\right)$ (2.6)

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

 S > ${\mathrm{SpinCoeff}}_{"tau"}$
 ${-}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}$ (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 > $\mathrm{κ},\mathrm{ρ},\mathrm{σ},\mathrm{τ},\mathrm{pi},\mathrm{λ},\mathrm{μ},\mathrm{ν},\mathrm{α},\mathrm{β},\mathrm{gam},\mathrm{ε}≔\mathrm{NPSpinCoefficients}\left(\mathrm{NTetrad},\mathrm{output}="Sequence"\right)$
 ${\mathrm{\kappa }}{,}{\mathrm{\rho }}{,}{\mathrm{\sigma }}{,}{\mathrm{\tau }}{,}{\mathrm{π}}{,}{\mathrm{\lambda }}{,}{\mathrm{\mu }}{,}{\mathrm{\nu }}{,}{\mathrm{\alpha }}{,}{\mathrm{\beta }}{,}{\mathrm{gam}}{,}{\mathrm{ϵ}}{≔}{-}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{,}{-}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{,}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{,}{-}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{,}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{,}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{,}{-}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{,}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{,}\frac{\frac{{I}}{{4}}{}\sqrt{{2}}}{{z}{}{y}}{,}\frac{\frac{{I}}{{4}}{}\sqrt{{2}}}{{z}{}{y}}{,}{-}\frac{\sqrt{{2}}}{{4}{}{t}{}{x}}{,}\frac{\sqrt{{2}}}{{4}{}{t}{}{x}}$ (2.8)

Example 3.

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

 S > $L,N,M,\mathrm{barM}≔\mathrm{op}\left(\mathrm{NTetrad}\right)$
 ${L}{,}{N}{,}{M}{,}{\mathrm{barM}}{≔}\frac{\sqrt{{2}}}{{2}{}{x}}{}{\mathrm{D_t}}{+}\frac{\sqrt{{2}}}{{2}{}{t}}{}{\mathrm{D_z}}{,}\frac{\sqrt{{2}}}{{2}{}{x}}{}{\mathrm{D_t}}{-}\frac{\sqrt{{2}}}{{2}{}{t}}{}{\mathrm{D_z}}{,}\frac{\sqrt{{2}}}{{2}{}{y}}{}{\mathrm{D_x}}{+}\frac{\frac{{I}}{{2}}{}\sqrt{{2}}}{{z}}{}{\mathrm{D_y}}{,}\frac{\sqrt{{2}}}{{2}{}{y}}{}{\mathrm{D_x}}{-}\frac{\frac{{I}}{{2}}{}\sqrt{{2}}}{{z}}{}{\mathrm{D_y}}$ (2.9)

Define the dual basis.

 S > $\mathrm{Theta_L},\mathrm{Theta_N},\mathrm{Theta_M},\mathrm{Theta_barM}≔\mathrm{op}\left(\mathrm{DualBasis}\left(\mathrm{NTetrad}\right)\right)$
 ${\mathrm{Theta_L}}{,}{\mathrm{Theta_N}}{,}{\mathrm{Theta_M}}{,}{\mathrm{Theta_barM}}{≔}\frac{\sqrt{{2}}{}{x}}{{2}}{}{\mathrm{dt}}{+}\frac{\sqrt{{2}}{}{t}}{{2}}{}{\mathrm{dz}}{,}\frac{\sqrt{{2}}{}{x}}{{2}}{}{\mathrm{dt}}{-}\frac{\sqrt{{2}}{}{t}}{{2}}{}{\mathrm{dz}}{,}\frac{\sqrt{{2}}{}{y}}{{2}}{}{\mathrm{dx}}{-}\frac{{I}}{{2}}{}\sqrt{{2}}{}{z}{}{\mathrm{dy}}{,}\frac{\sqrt{{2}}{}{y}}{{2}}{}{\mathrm{dx}}{+}\frac{{I}}{{2}}{}\sqrt{{2}}{}{z}{}{\mathrm{dy}}$ (2.10)

Calculate the Christoffel connection.

 S > $C≔\mathrm{Christoffel}\left(g\right)$
 ${C}{≔}\frac{{1}}{{x}}{}{\mathrm{D_t}}{}{\mathrm{dt}}{}{\mathrm{dx}}{+}\frac{{1}}{{x}}{}{\mathrm{D_t}}{}{\mathrm{dx}}{}{\mathrm{dt}}{+}\frac{{t}}{{{x}}^{{2}}}{}{\mathrm{D_t}}{}{\mathrm{dz}}{}{\mathrm{dz}}{+}\frac{{x}}{{{y}}^{{2}}}{}{\mathrm{D_x}}{}{\mathrm{dt}}{}{\mathrm{dt}}{+}\frac{{1}}{{y}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{1}}{{y}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dx}}{-}\frac{{y}}{{{z}}^{{2}}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}\frac{{1}}{{z}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}\frac{{1}}{{z}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{}{\mathrm{dy}}{+}\frac{{1}}{{t}}{}{\mathrm{D_z}}{}{\mathrm{dt}}{}{\mathrm{dz}}{-}\frac{{z}}{{{t}}^{{2}}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}\frac{{1}}{{t}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{}{\mathrm{dt}}$ (2.11)

1. k = $\left({\nabla }_{L}{\mathrm{Θ}}_{N}\right)\left(M\right)$

 S > $\mathrm{κ}=\mathrm{Hook}\left(\left[M\right],\mathrm{DirectionalCovariantDerivative}\left(L,\mathrm{Theta_N},C\right)\right)$
 ${-}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{=}{-}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}$ (2.12)

2.

 S > $\mathrm{ρ}=\mathrm{Hook}\left(\left[M\right],\mathrm{DirectionalCovariantDerivative}\left(\mathrm{barM},\mathrm{Theta_N},C\right)\right)$
 ${-}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{=}{-}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}$ (2.13)

3. $\mathrm{σ}=\left({\nabla }_{M}{\mathrm{\Theta }}_{N}\right)\left(M\right)$

 S > $\mathrm{σ}=\mathrm{Hook}\left(\left[M\right],\mathrm{DirectionalCovariantDerivative}\left(M,\mathrm{Theta_N},C\right)\right)$
 $\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{=}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}$ (2.14)

4. $\mathrm{τ}=-\left({\nabla }_{N}{\mathrm{\Theta }}_{N}\right)\left(M\right)$

 S > $\mathrm{τ}=\mathrm{Hook}\left(\left[M\right],\mathrm{DirectionalCovariantDerivative}\left(N,\mathrm{Theta_N},C\right)\right)$
 ${-}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{=}{-}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}$ (2.15)

5. $\mathrm{π}=-\left({\nabla }_{N}{\mathrm{\Theta }}_{L}\right)\left(\stackrel{‾}{M}\right)$

 S > $\mathrm{pi}=-\mathrm{Hook}\left(\left[\mathrm{barM}\right],\mathrm{DirectionalCovariantDerivative}\left(L,\mathrm{Theta_L},C\right)\right)$
 $\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{=}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}$ (2.16)

6. $\mathrm{λ}=-\left({\nabla }_{N}{\mathrm{\Theta }}_{L}\right)$$\left(\stackrel{‾}{M}\right)$

 S > $\mathrm{λ}=-\mathrm{Hook}\left(\left[\mathrm{barM}\right],\mathrm{DirectionalCovariantDerivative}\left(\mathrm{barM},\mathrm{Theta_L},C\right)\right)$
 $\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{=}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}$ (2.17)

7. $\mathrm{μ}=-\left({\nabla }_{N}{\mathrm{Θ}}_{M}\right)\left(\stackrel{‾}{M}\right)$

 S > $\mathrm{μ}=-\mathrm{Hook}\left(\left[\mathrm{barM}\right],\mathrm{DirectionalCovariantDerivative}\left(M,\mathrm{Theta_L},C\right)\right)$
 ${-}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{=}{-}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}$ (2.18)

8. $\mathrm{ν}=\left({\nabla }_{N}{\mathrm{Θ}}_{N}\right)\left(\stackrel{‾}{M}\right)$

 S > $\mathrm{ν}=-\mathrm{Hook}\left(\left[\mathrm{barM}\right],\mathrm{DirectionalCovariantDerivative}\left(N,\mathrm{Theta_L},C\right)\right)$
 $\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{=}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}$ (2.19)

9. $\mathrm{α}=\frac{1}{2}\left({\nabla }_{\stackrel{‾}{M}}{\mathrm{Θ}}_{N}\right)\left(N\right)+\frac{1}{2}\left({\nabla }_{\stackrel{‾}{M}}{\mathrm{\Theta }}_{\stackrel{‾}{M}}\right)\left(\stackrel{‾}{M}\right)$

 S > $\mathrm{α}=\frac{1\mathrm{Hook}\left(\left[N\right],\mathrm{DirectionalCovariantDerivative}\left(\mathrm{barM},\mathrm{Theta_N},C\right)\right)}{2}+\frac{1\mathrm{Hook}\left(\left[\mathrm{barM}\right],\mathrm{DirectionalCovariantDerivative}\left(\mathrm{barM},\mathrm{Theta_barM},C\right)\right)}{2}$
 $\frac{\frac{{I}}{{4}}{}\sqrt{{2}}}{{z}{}{y}}{=}\frac{\frac{{I}}{{4}}{}\sqrt{{2}}}{{z}{}{y}}$ (2.20)

10. $\mathrm{β}=\frac{1}{2}\left({\nabla }_{M}{\mathrm{Θ}}_{N}\right)\left(N\right)+\frac{1}{2}\left({\nabla }_{M}{\mathrm{Θ}}_{\stackrel{‾}{M}}\right)\left(\stackrel{‾}{M}\right)$

 S > $\mathrm{β}=\frac{1\mathrm{Hook}\left(\left[N\right],\mathrm{DirectionalCovariantDerivative}\left(M,\mathrm{Theta_N},C\right)\right)}{2}+\frac{1\mathrm{Hook}\left(\left[\mathrm{barM}\right],\mathrm{DirectionalCovariantDerivative}\left(\mathrm{barM},\mathrm{Theta_barM},C\right)\right)}{2}$
 $\frac{\frac{{I}}{{4}}{}\sqrt{{2}}}{{z}{}{y}}{=}\frac{\frac{{I}}{{4}}{}\sqrt{{2}}}{{z}{}{y}}$ (2.21)

11. $\mathrm{γ}=\frac{1}{2}\left({\nabla }_{N}{\mathrm{\Theta }}_{N}\right)\left(N\right)+\frac{1}{2}\left({\nabla }_{N}{\mathrm{\Theta }}_{\stackrel{‾}{M}}\right)\left(\stackrel{‾}{M}\right)$

 S > $\mathrm{gam}=\frac{1\mathrm{Hook}\left(\left[N\right],\mathrm{DirectionalCovariantDerivative}\left(N,\mathrm{Theta_N},C\right)\right)}{2}+\frac{1\mathrm{Hook}\left(\left[\mathrm{barM}\right],\mathrm{DirectionalCovariantDerivative}\left(N,\mathrm{Theta_barM},C\right)\right)}{2}$
 ${-}\frac{\sqrt{{2}}}{{4}{}{t}{}{x}}{=}{-}\frac{\sqrt{{2}}}{{4}{}{t}{}{x}}$ (2.22)

12. $\mathrm{ε}=\frac{1}{2}\left({\nabla }_{L}{\mathrm{\Theta }}_{N}\right)\left(N\right)+\frac{1}{2}\left({\nabla }_{L}{\mathrm{\Theta }}_{\stackrel{‾}{M}}\right)\left(\stackrel{‾}{M}\right)$

 S > $\mathrm{ε}=\frac{1\mathrm{Hook}\left(\left[N\right],\mathrm{DirectionalCovariantDerivative}\left(L,\mathrm{Theta_N},C\right)\right)}{2}+\frac{1\mathrm{Hook}\left(\left[\mathrm{barM}\right],\mathrm{DirectionalCovariantDerivative}\left(L,\mathrm{Theta_barM},C\right)\right)}{2}$
 $\frac{\sqrt{{2}}}{{4}{}{t}{}{x}}{=}\frac{\sqrt{{2}}}{{4}{}{t}{}{x}}$ (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 > $\mathrm{FD}≔\mathrm{FrameData}\left(\mathrm{NTetrad},\mathrm{NP}\right)$
 ${\mathrm{FD}}{≔}\left[\left[{\mathrm{E1}}{,}{\mathrm{E2}}\right]{=}\frac{\sqrt{{2}}{}{\mathrm{E1}}}{{2}{}{t}{}{x}}{-}\frac{\sqrt{{2}}{}{\mathrm{E2}}}{{2}{}{t}{}{x}}{,}\left[{\mathrm{E1}}{,}{\mathrm{E3}}\right]{=}\frac{\sqrt{{2}}{}{\mathrm{E1}}}{{4}{}{x}{}{y}}{+}\frac{\sqrt{{2}}{}{\mathrm{E2}}}{{4}{}{x}{}{y}}{-}\frac{\sqrt{{2}}{}{\mathrm{E3}}}{{4}{}{t}{}{z}}{+}\frac{\sqrt{{2}}{}{\mathrm{E4}}}{{4}{}{t}{}{z}}{,}\left[{\mathrm{E1}}{,}{\mathrm{E4}}\right]{=}\frac{\sqrt{{2}}{}{\mathrm{E1}}}{{4}{}{x}{}{y}}{+}\frac{\sqrt{{2}}{}{\mathrm{E2}}}{{4}{}{x}{}{y}}{+}\frac{\sqrt{{2}}{}{\mathrm{E3}}}{{4}{}{t}{}{z}}{-}\frac{\sqrt{{2}}{}{\mathrm{E4}}}{{4}{}{t}{}{z}}{,}\left[{\mathrm{E2}}{,}{\mathrm{E3}}\right]{=}\frac{\sqrt{{2}}{}{\mathrm{E1}}}{{4}{}{x}{}{y}}{+}\frac{\sqrt{{2}}{}{\mathrm{E2}}}{{4}{}{x}{}{y}}{+}\frac{\sqrt{{2}}{}{\mathrm{E3}}}{{4}{}{t}{}{z}}{-}\frac{\sqrt{{2}}{}{\mathrm{E4}}}{{4}{}{t}{}{z}}{,}\left[{\mathrm{E2}}{,}{\mathrm{E4}}\right]{=}\frac{\sqrt{{2}}{}{\mathrm{E1}}}{{4}{}{x}{}{y}}{+}\frac{\sqrt{{2}}{}{\mathrm{E2}}}{{4}{}{x}{}{y}}{-}\frac{\sqrt{{2}}{}{\mathrm{E3}}}{{4}{}{t}{}{z}}{+}\frac{\sqrt{{2}}{}{\mathrm{E4}}}{{4}{}{t}{}{z}}{,}\left[{\mathrm{E3}}{,}{\mathrm{E4}}\right]{=}{-}\frac{{I}{}\sqrt{{2}}{}{\mathrm{E3}}}{{2}{}{y}{}{z}}{-}\frac{{I}{}\sqrt{{2}}{}{\mathrm{E4}}}{{2}{}{y}{}{z}}\right]$ (2.24)
 S > $\mathrm{DGsetup}\left(\mathrm{FD}\right)$
 ${\mathrm{frame name: NP}}$ (2.25)

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

 NP > $\mathrm{NPSpinCoefficients}\left(\mathrm{NP}\right)$
 ${table}{}\left(\left[{"epsilon"}{=}\frac{\sqrt{{2}}}{{4}{}{t}{}{x}}{,}{"tau"}{=}{-}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{,}{"kappa"}{=}{-}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{,}{"alpha"}{=}\frac{\frac{{I}}{{4}}{}\sqrt{{2}}}{{z}{}{y}}{,}{"beta"}{=}\frac{\frac{{I}}{{4}}{}\sqrt{{2}}}{{z}{}{y}}{,}{"mu"}{=}{-}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{,}{"rho"}{=}{-}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{,}{"lambda"}{=}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{,}{"gamma"}{=}{-}\frac{\sqrt{{2}}}{{4}{}{t}{}{x}}{,}{"sigma"}{=}\frac{\sqrt{{2}}}{{4}{}{t}{}{z}}{,}{"nu"}{=}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}{,}{"pi"}{=}\frac{\sqrt{{2}}}{{4}{}{x}{}{y}}\right]\right)$ (2.26) See Also