The Newman-Penrose curvature scalars are components of the trace-free Ricci tensor and Weyl tensor with respect to a null tetrad [L, N, M, barM] for a Lorentz metric g on a 4-dimensional manifold.

If S is the trace-free Ricci tensor, then the Ricci scalars are

Phi00 = -S(L, L)

Phi01 = -S(L, M)

Phi02 = -S(M, M)

Phi11 = -S(L, N)

Phi12 = -S(N, M)

Phi22 = -S(N, N)

Lambda = 1/24*RicciScalar

Phi[1,0] = conj(Phi[0,1]) = -S(L, barM)

Phi[2,0] = conj(Phi[0,2]) = -S(barM, barM)

Phi[2,1] = conj(Phi[1,2]) = -S(N, barM)

If W is the Weyl tensor, then the Weyl scalars are

Psi[0] = -W(L, M, L, M)

Psi[1] = -W(L, N, L, M)

Psi[2] = -W(L, M, barM, N)

Psi[3] = -W(L, N, barM, N)

Psi[4] = -W(N, barM, N, barM)

barPsi[0] = -W(L, barM, L, barM)

barPsi[1] = -W(L, N, L, barM)

barPsi[2] = -W(L, barM, M, N)

barPsi[3] = -W(L, N, M, N)

barPsi[4] = -W(N, M, N, M)

The minus sign appears in these formulas because there is a sign difference in the definition of the curvature between what is used in the Newman-Penrose formalism and what is used in the DifferentialGeometry CurvatureTensor command.

The command NPCurvatureScalars returns a pair of tables NPRicci, NPWeyl.  The table NPRicci contains the values of the NP Ricci scalars and has indices "Phi00", "Phi01", "Phi02", "Phi11","Phi12", "Phi22".  The table NPWeyl contains the values of the NP Weyl scalars and has indices "Psi0", "Psi1", "Psi2", "Psi3", "Psi4".

With output = ["RicciScalars"] only the table of Ricci scalars is computed.  With output = ["AllRicciScalars"] the values of the conjugate Ricci scalars are included.  With output = ["WeylScalars"] only the table of Weyl scalars is computed.  With output = ["AllWeylScalars"] the values of the conjugate Weyl scalars are included.

If the curvature tensor R of the metric g has been previously calculated, then passing the value of the curvature to the command NPCurvatureScalars will avoid the need to recalculate it.  For this calling sequence, any rank 4 tensor R with indices ["cov_bas", "cov_bas", "cov_bas", "cov_bas"] can be used.

If the Ricci tensor R of the metric g has been previously calculated and only the Ricci scalars are to be computed, then passing the value of the Ricci tensor to the command NPCurvatureScalars will avoid the need to recalculate it.  For this calling sequence, any symmetric, covariant rank 2 tensor R can be used.

With either of the first two calling sequences, the Ricci and Weyl scalars are computed directly from the above formulas.

With the third calling sequence the Ricci and Weyl scalars are compute from the NP Ricci identities using the Ricci and Weyl scalars.  These identities require knowledge of the null tetrad.  If the current frame is a coordinate frame, then the null tetrad must be specified as the second argument.  If the current frame is an anholonomic frame, then that frame is taken to be the null tetrad.

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