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.

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

$\mathrm{DGsetup}\left(\left[t\,x\,y\,z\right]\,S\right)$

$\mathrm{frame\; name:\; S}$

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

$g:=x^2\mathrm{dt}\mathrm{dt}-y^2\mathrm{dx}\mathrm{dx}-z^2\mathrm{dy}\mathrm{dy}-t^2\mathrm{dz}\mathrm{dz}$

$\mathrm{NTetrad}≔\mathrm{evalDG}\left(\left[\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{x}\mathrm{D\_t}+\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{t}\mathrm{D\_z}\,\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{x}\mathrm{D\_t}-\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{t}\mathrm{D\_z}\,\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{y}\mathrm{D\_x}+\frac{\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}}{z}\mathrm{D\_y}\,\frac{\frac{1}{2}{2}^{\frac{1}{2}}}{y}\mathrm{D\_x}-\frac{\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}}{z}\mathrm{D\_y}\right]\right)$

$\mathrm{NTetrad}:=\left[\frac{\sqrt{2}}{2x}\mathrm{D\_t}+\frac{\sqrt{2}}{2t}\mathrm{D\_z}\,\frac{\sqrt{2}}{2x}\mathrm{D\_t}-\frac{\sqrt{2}}{2t}\mathrm{D\_z}\,\frac{\sqrt{2}}{2y}\mathrm{D\_x}+\frac{\frac{\mathrm{I}}{2}\sqrt{2}}{z}\mathrm{D\_y}\,\frac{\sqrt{2}}{2y}\mathrm{D\_x}-\frac{\frac{\mathrm{I}}{2}\sqrt{2}}{z}\mathrm{D\_y}\right]$

$\mathrm{GRQuery}\left(\mathrm{NTetrad}\,g\,''NullTetrad''\right)$

$\mathrm{true}$

$\mathrm{SpinCoeff}≔\mathrm{NPSpinCoefficients}\left(\mathrm{NTetrad}\right)$

$\mathrm{SpinCoeff}:=\mathrm{table}\left(\left[''epsilon''\=\frac{1}{4}\frac{\sqrt{2}}{tx}\,''mu''\=-\frac{1}{4}\frac{\sqrt{2}}{tz}\,''nu''\=\frac{1}{4}\frac{\sqrt{2}}{xy}\,''rho''\=-\frac{1}{4}\frac{\sqrt{2}}{tz}\,''lambda''\=\frac{1}{4}\frac{\sqrt{2}}{tz}\,''tau''\=-\frac{1}{4}\frac{\sqrt{2}}{xy}\,''beta''\=\frac{\frac{1}{4}\mathrm{I}\sqrt{2}}{zy}\,''gamma''\=-\frac{1}{4}\frac{\sqrt{2}}{tx}\,''alpha''\=\frac{\frac{1}{4}\mathrm{I}\sqrt{2}}{zy}\,''pi''\=\frac{1}{4}\frac{\sqrt{2}}{xy}\,''kappa''\=-\frac{1}{4}\frac{\sqrt{2}}{xy}\,''sigma''\=\frac{1}{4}\frac{\sqrt{2}}{tz}\right]\right)$

$\mathrm{RS},\mathrm{WS}≔\mathrm{NPCurvatureScalars}\left(\mathrm{SpinCoeff}\,\mathrm{NTetrad}\right)$

$\mathrm{RS}\,\mathrm{WS}:=\mathrm{table}\left(\left[''Phi12''\=-\frac{1}{4}\frac{\mathrm{I}{x}^2-z^2}{yt{x}^2{z}^2}\,''Phi01''\=\frac{1}{4}\frac{\mathrm{I}{x}^2\+z^2}{yt{x}^2{z}^2}\,''Phi22''\=-\frac{1}{2x{t}^{2}z}\,''Phi00''\=\frac{1}{2x{t}^{2}z}\,''Phi11''\=0\,''Lambda''\=0\,''Phi02''\=\frac{\frac{1}{2}\mathrm{I}}{zx{y}^2}\right]\right)\,\mathrm{table}\left(\left[''Psi4''\=\frac{1}{2}\frac{\mathrm{I}{t}^2\+y^2}{x{t}^{2}z{y}^2}\,''Psi1''\=-\frac{1}{4}\frac{\mathrm{I}{x}^2-z^2}{yt{x}^2{z}^2}\,''Psi0''\=-\frac{1}{2}\frac{\mathrm{I}{t}^2\+y^2}{x{t}^{2}z{y}^2}\,''Psi2''\=0\,''Psi3''\=-\frac{1}{4}\frac{\mathrm{I}{x}^2-z^2}{yt{x}^2{z}^2}\right]\right)$

$\mathrm{Eqc}≔\mathrm{NPBianchiIdentities}\left(\mathrm{SpinCoeff}\,\mathrm{RS}\,\mathrm{WS}\,\left[''c''\right]\,\mathrm{NTetrad}\right)$

$\mathrm{Eqc}:=\left[-\frac{1}{2}\frac{\sqrt{2}\left(-\frac{\frac{1}{2}\mathrm{I}}{xyt{z}^2}-\frac{2\left(-\frac{1}{4}\mathrm{I}{x}^2\+\frac{1}{4}{z}^2\right)}{yt{x}^3{z}^2}\right)}{y}-\frac{\frac{1}{2}\mathrm{I}\sqrt{2}\left(-\frac{1}{4}\mathrm{I}{x}^2\+\frac{1}{4}{z}^2\right)}{z^3{y}^{2}t{x}^2}-\frac{1}{2}\frac{\sqrt{2}}{x^2{t}^{3}z}\+\frac{1}{4}\frac{\sqrt{2}}{t^3{z}^{2}x}-\frac{1}{2}\frac{\sqrt{2}\left(\frac{\frac{1}{2}\mathrm{I}}{xyt{z}^2}-\frac{2\left(\frac{1}{4}\mathrm{I}{x}^2\+\frac{1}{4}{z}^2\right)}{yt{x}^3{z}^2}\right)}{y}-\frac{\frac{1}{2}\mathrm{I}\sqrt{2}\left(\frac{1}{4}\mathrm{I}{x}^2\+\frac{1}{4}{z}^2\right)}{z^3{y}^{2}t{x}^2}\=\frac{1}{8}\frac{\sqrt{2}\left(\mathrm{I}{t}^2\+y^2\right)}{t^3{z}^{2}x{y}^2}-\frac{1}{2}\frac{\left(\frac{1}{4}\frac{\sqrt{2}}{xy}-\frac{\frac{1}{4}\mathrm{I}\sqrt{2}}{zy}\right)\left(\mathrm{I}{x}^2-z^2\right)}{yt{x}^2{z}^2}-\frac{1}{8}\frac{\sqrt{2}\left(\mathrm{I}{x}^2-z^2\right)}{x^3{y}^{2}t{z}^2}\+\frac{1}{2}\frac{-\frac{\sqrt{2}}{tx}\+\frac{1}{4}\frac{\sqrt{2}}{tz}}{x{t}^{2}z}-\frac{1}{2}\frac{\left(\frac{\frac{1}{4}\mathrm{I}\sqrt{2}}{zy}-\frac{1}{4}\frac{\sqrt{2}}{xy}\right)\left(\mathrm{I}{x}^2\+z^2\right)}{yt{x}^2{z}^2}\+\frac{1}{2}\frac{\sqrt{2}\left(-\frac{1}{4}\mathrm{I}{x}^2\+\frac{1}{4}{z}^2\right)}{x^3{y}^{2}t{z}^2}\+\frac{\frac{1}{8}\mathrm{I}\sqrt{2}}{t{z}^{2}x{y}^2}\right]$

$\mathrm{simplify}\left(\mathrm{lhs}\left(\mathrm{Eqc}\left[1\right]\right)-\mathrm{rhs}\left(\mathrm{Eqc}\left[1\right]\right)\right)$

$0$