DifferentialGeometry/Tensor/NPBianchiIdentities - Maple Help

Tensor[NPBianchiIdentities] - calculate the Newman-Penrose version of the Bianchi identities

Calling Sequences

NPBianchiIdentities(SpinCoeff, RicciCoeff, WeylCoeff, Idlist, NTetrad, ConjCoord)

Parameters

SpinCoeff  - a table, the Newman Penrose spin coefficients

RicciCoeff - a table, the Newman Penrose Ricci coefficients

WeylCoeff  - a table, the Newman Penrose Weyl coefficients

Idlist     - a list of strings, the Bianchi identities to be computed

NTetrad    - (optional) a list of 4 vectors defining a null tetrad

ConjCoord  - (optional) keyword argument conjugatecoordinates = CC, where CC is a list of lists specifying conjugate coordinates

Description

 • The Newman-Penrose Bianchi identities are a set of 11 equations which encode the usual differential Bianchi identities for the curvature tensor in terms of the NP spin coefficients and the NP curvature scalars. The relative simplicity of the Newman-Penrose Bianchi identities underscores the importance of this formalism.
 • Given the tetrad, the spin-coefficients and the curvature scalars, the command NPBianchiIdentities will calculate a specified list of the Bianchi identities.
 • The index set for the table SpinCoeff must be {"mu", "nu", "pi", "rho", "tau", "alpha", "beta", "epsilon", "gamma", "kappa", "lambda", "sigma"}.
 • The index set for the table RicciCoeff must be {"Lambda", "Phi00", "Phi01", "Phi02", "Phi11", "Phi12", "Phi22"}.
 • The index set for the table WeylCoeff must be {"Psi0", "Psi1", "Psi2", "Psi3", "Psi4"}.
 • The equation list Idlist is a list of letters, chosen from {"a", "b", ..., "k"} or {"all"}.
 • If the current frame is defined by a null tetrad, then the 5th argument NTetrad is not required.
 • See Details for Ricci and Bianchi Identities for a complete list of the Newman-Penrose Bianchi Identities.
 • This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form NPBianchiIdentities(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-NPBianchiIdentities.

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 > $g≔\mathrm{evalDG}\left({x}^{2}\mathrm{dt}&t\mathrm{dt}-{y}^{2}\mathrm{dx}&t\mathrm{dx}-{z}^{2}\mathrm{dy}&t\mathrm{dy}-{t}^{2}\mathrm{dz}&t\mathrm{dz}\right)$
 ${g}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{S}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{x}}^{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}{{y}}^{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-}{{z}}^{{2}}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-}{{t}}^{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{S}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{x}}^{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}{{y}}^{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-}{{z}}^{{2}}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-}{{t}}^{{2}}\right]\right]\right]\right)$ (2.2)

Define a null tetrad NTetrad for the metric $g$. Verify that this is a null tetrad using GRQuery.

 S > $\mathrm{NTetrad}≔\mathrm{evalDG}\left(\left[\frac{1{2}^{\frac{1}{2}}\mathrm{D_t}}{2x}+\frac{1{2}^{\frac{1}{2}}\mathrm{D_z}}{2t},\frac{1{2}^{\frac{1}{2}}\mathrm{D_t}}{2x}-\frac{1{2}^{\frac{1}{2}}\mathrm{D_z}}{2t},\frac{1{2}^{\frac{1}{2}}\mathrm{D_x}}{2y}+\frac{1I{2}^{\frac{1}{2}}\mathrm{D_y}}{2z},\frac{1{2}^{\frac{1}{2}}\mathrm{D_x}}{2y}-\frac{1I{2}^{\frac{1}{2}}\mathrm{D_y}}{2z}\right]\right)$
 ${\mathrm{NTetrad}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{\sqrt{{2}}}{{2}{}{x}}\right]{,}\left[\left[{4}\right]{,}\frac{\sqrt{{2}}}{{2}{}{t}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{\sqrt{{2}}}{{2}{}{x}}\right]{,}\left[\left[{4}\right]{,}\frac{\sqrt{{2}}}{{2}{}{t}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{\sqrt{{2}}}{{2}{}{x}}\right]{,}\left[\left[{4}\right]{,}{-}\frac{\sqrt{{2}}}{{2}{}{t}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{\sqrt{{2}}}{{2}{}{x}}\right]{,}\left[\left[{4}\right]{,}{-}\frac{\sqrt{{2}}}{{2}{}{t}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{\sqrt{{2}}}{{2}{}{y}}\right]{,}\left[\left[{3}\right]{,}\frac{\frac{{I}}{{2}}{}\sqrt{{2}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{\sqrt{{2}}}{{2}{}{y}}\right]{,}\left[\left[{3}\right]{,}\frac{\frac{{I}}{{2}}{}\sqrt{{2}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{\sqrt{{2}}}{{2}{}{y}}\right]{,}\left[\left[{3}\right]{,}\frac{{-}\frac{{I}}{{2}}{}\sqrt{{2}}}{{z}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{S}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{\sqrt{{2}}}{{2}{}{y}}\right]{,}\left[\left[{3}\right]{,}\frac{{-}\frac{{I}}{{2}}{}\sqrt{{2}}}{{z}}\right]\right]\right]\right)\right]$ (2.3)
 S > $\mathrm{GRQuery}\left(\mathrm{NTetrad},g,"NullTetrad"\right)$
 ${\mathrm{true}}$ (2.4)

 S > $\mathrm{SpinCoeff}≔\mathrm{NPSpinCoefficients}\left(\mathrm{NTetrad}\right)$
 ${\mathrm{SpinCoeff}}{:=}{\mathrm{table}}\left(\left[{"epsilon"}{=}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{t}{}{x}}{,}{"mu"}{=}{-}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{t}{}{z}}{,}{"nu"}{=}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{x}{}{y}}{,}{"rho"}{=}{-}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{t}{}{z}}{,}{"lambda"}{=}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{t}{}{z}}{,}{"tau"}{=}{-}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{x}{}{y}}{,}{"beta"}{=}\frac{\frac{{1}}{{4}}{}{I}{}\sqrt{{2}}}{{z}{}{y}}{,}{"gamma"}{=}{-}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{t}{}{x}}{,}{"alpha"}{=}\frac{\frac{{1}}{{4}}{}{I}{}\sqrt{{2}}}{{z}{}{y}}{,}{"pi"}{=}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{x}{}{y}}{,}{"kappa"}{=}{-}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{x}{}{y}}{,}{"sigma"}{=}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{t}{}{z}}\right]\right)$ (2.5)

Calculate the Newman-Penrose curvature scalars.

 S > $\mathrm{RS},\mathrm{WS}≔\mathrm{NPCurvatureScalars}\left(\mathrm{SpinCoeff},\mathrm{NTetrad}\right)$
 ${\mathrm{RS}}{,}{\mathrm{WS}}{:=}{\mathrm{table}}\left(\left[{"Phi12"}{=}{-}\frac{{1}}{{4}}{}\frac{{I}{}{{x}}^{{2}}{-}{{z}}^{{2}}}{{y}{}{t}{}{{x}}^{{2}}{}{{z}}^{{2}}}{,}{"Phi01"}{=}\frac{{1}}{{4}}{}\frac{{I}{}{{x}}^{{2}}{+}{{z}}^{{2}}}{{y}{}{t}{}{{x}}^{{2}}{}{{z}}^{{2}}}{,}{"Phi22"}{=}{-}\frac{{1}}{{2}{}{x}{}{{t}}^{{2}}{}{z}}{,}{"Phi00"}{=}\frac{{1}}{{2}{}{x}{}{{t}}^{{2}}{}{z}}{,}{"Phi11"}{=}{0}{,}{"Lambda"}{=}{0}{,}{"Phi02"}{=}\frac{\frac{{1}}{{2}}{}{I}}{{z}{}{x}{}{{y}}^{{2}}}\right]\right){,}{\mathrm{table}}\left(\left[{"Psi4"}{=}\frac{{1}}{{2}}{}\frac{{I}{}{{t}}^{{2}}{+}{{y}}^{{2}}}{{x}{}{{t}}^{{2}}{}{z}{}{{y}}^{{2}}}{,}{"Psi1"}{=}{-}\frac{{1}}{{4}}{}\frac{{I}{}{{x}}^{{2}}{-}{{z}}^{{2}}}{{y}{}{t}{}{{x}}^{{2}}{}{{z}}^{{2}}}{,}{"Psi0"}{=}{-}\frac{{1}}{{2}}{}\frac{{I}{}{{t}}^{{2}}{+}{{y}}^{{2}}}{{x}{}{{t}}^{{2}}{}{z}{}{{y}}^{{2}}}{,}{"Psi2"}{=}{0}{,}{"Psi3"}{=}{-}\frac{{1}}{{4}}{}\frac{{I}{}{{x}}^{{2}}{-}{{z}}^{{2}}}{{y}{}{t}{}{{x}}^{{2}}{}{{z}}^{{2}}}\right]\right)$ (2.6)

Compute Bianchi identity "c" and verify that it is satisfied.

 S > $\mathrm{Eqc}≔\mathrm{NPBianchiIdentities}\left(\mathrm{SpinCoeff},\mathrm{RS},\mathrm{WS},\left["c"\right],\mathrm{NTetrad}\right)$
 ${\mathrm{Eqc}}{:=}\left[{-}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}\left({-}\frac{\frac{{1}}{{2}}{}{I}}{{x}{}{y}{}{t}{}{{z}}^{{2}}}{-}\frac{{2}{}\left({-}\frac{{1}}{{4}}{}{I}{}{{x}}^{{2}}{+}\frac{{1}}{{4}}{}{{z}}^{{2}}\right)}{{y}{}{t}{}{{x}}^{{3}}{}{{z}}^{{2}}}\right)}{{y}}{-}\frac{\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}\left({-}\frac{{1}}{{4}}{}{I}{}{{x}}^{{2}}{+}\frac{{1}}{{4}}{}{{z}}^{{2}}\right)}{{{z}}^{{3}}{}{{y}}^{{2}}{}{t}{}{{x}}^{{2}}}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}}{{{x}}^{{2}}{}{{t}}^{{3}}{}{z}}{+}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{{t}}^{{3}}{}{{z}}^{{2}}{}{x}}{-}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}\left(\frac{\frac{{1}}{{2}}{}{I}}{{x}{}{y}{}{t}{}{{z}}^{{2}}}{-}\frac{{2}{}\left(\frac{{1}}{{4}}{}{I}{}{{x}}^{{2}}{+}\frac{{1}}{{4}}{}{{z}}^{{2}}\right)}{{y}{}{t}{}{{x}}^{{3}}{}{{z}}^{{2}}}\right)}{{y}}{-}\frac{\frac{{1}}{{2}}{}{I}{}\sqrt{{2}}{}\left(\frac{{1}}{{4}}{}{I}{}{{x}}^{{2}}{+}\frac{{1}}{{4}}{}{{z}}^{{2}}\right)}{{{z}}^{{3}}{}{{y}}^{{2}}{}{t}{}{{x}}^{{2}}}{=}\frac{{1}}{{8}}{}\frac{\sqrt{{2}}{}\left({I}{}{{t}}^{{2}}{+}{{y}}^{{2}}\right)}{{{t}}^{{3}}{}{{z}}^{{2}}{}{x}{}{{y}}^{{2}}}{-}\frac{{1}}{{2}}{}\frac{\left(\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{x}{}{y}}{-}\frac{\frac{{1}}{{4}}{}{I}{}\sqrt{{2}}}{{z}{}{y}}\right){}\left({I}{}{{x}}^{{2}}{-}{{z}}^{{2}}\right)}{{y}{}{t}{}{{x}}^{{2}}{}{{z}}^{{2}}}{-}\frac{{1}}{{8}}{}\frac{\sqrt{{2}}{}\left({I}{}{{x}}^{{2}}{-}{{z}}^{{2}}\right)}{{{x}}^{{3}}{}{{y}}^{{2}}{}{t}{}{{z}}^{{2}}}{+}\frac{{1}}{{2}}{}\frac{{-}\frac{\sqrt{{2}}}{{t}{}{x}}{+}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{t}{}{z}}}{{x}{}{{t}}^{{2}}{}{z}}{-}\frac{{1}}{{2}}{}\frac{\left(\frac{\frac{{1}}{{4}}{}{I}{}\sqrt{{2}}}{{z}{}{y}}{-}\frac{{1}}{{4}}{}\frac{\sqrt{{2}}}{{x}{}{y}}\right){}\left({I}{}{{x}}^{{2}}{+}{{z}}^{{2}}\right)}{{y}{}{t}{}{{x}}^{{2}}{}{{z}}^{{2}}}{+}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}\left({-}\frac{{1}}{{4}}{}{I}{}{{x}}^{{2}}{+}\frac{{1}}{{4}}{}{{z}}^{{2}}\right)}{{{x}}^{{3}}{}{{y}}^{{2}}{}{t}{}{{z}}^{{2}}}{+}\frac{\frac{{1}}{{8}}{}{I}{}\sqrt{{2}}}{{t}{}{{z}}^{{2}}{}{x}{}{{y}}^{{2}}}\right]$ (2.7)
 S > $\mathrm{simplify}\left(\mathrm{lhs}\left({\mathrm{Eqc}}_{1}\right)-\mathrm{rhs}\left({\mathrm{Eqc}}_{1}\right)\right)$
 ${0}$ (2.8)