The Newman-Penrose Ricci identities are a set of 24 equations which encode the usual formulas for the curvature tensor in terms of the Christoffel connection in terms of the NP spin coefficients and the NP curvature scalars.  The relative simplicity of the Newman-Penrose Ricci identities underscores the importance of this formalism.

Given the tetrad, the spin-coefficients and the curvature scalars, the command NPRicciIdentities will calculate a specified list of the Ricci 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", ..., "r"} or {"all"}.

If the current frame is an anholonomic frame, then the 5th argument NTetrad is not required.

See Details for Ricci and Bianchi Identities for a complete list of the Newman-Penrose Ricci Identities.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form NPRicciIdentities(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-NPRicciIdentities.

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{Eqa}≔\mathrm{NPRicciIdentities}\left(\mathrm{SpinCoeff}\,\mathrm{RS}\,\mathrm{WS}\,\left[''a''\right]\,\mathrm{NTetrad}\right)$

$\mathrm{Eqa}≔\left[\frac{1}{4x{t}^{2}z}+\frac{1}{4t^2{z}^2}-\frac{1}{4y^2{x}^2}+\frac{\mathrm{I}}{4zx{y}^2}=\frac{1}{4t^2{z}^2}+\frac{1}{4x{t}^{2}z}-\frac{1}{8y^2{x}^2}+\frac{\sqrt{2}\left(\frac{\mathrm{I}\sqrt{2}}{2zy}-\frac{\sqrt{2}}{4xy}\right)}{4xy}\right]$

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

$0$