The Newman-Penrose Weyl scalars are a set of 5 complex scalars, labeled ${\mathrm{\Ψ}}_0$, ${\mathrm{\Ψ}}_1\,$${\mathrm{\Ψ}}_2\,$${\mathrm{\Ψ}}_3\,$${\mathrm{\Ψ}}_4$, and defined by certain components of the Weyl tensor with respect to a given null tetrad in a four dimensional spacetime of signature [1, -1, -1, -1]. Under local Lorentz transformations, the Newman-Penrose Weyl scalars transform among themselves in a natural way. Depending upon the Petrov type of the spacetime it is possible to transform the Newman-Penrose Weyl scalars to one of following normal forms. Below, $\mathrm{\η}$and $\mathrm{\χ}$are complex scalars.See NPCurvatureScalars, NullTetradTransformation.

Null tetrads for which the Newman-Penrose Weyl scalars are in the above normal form are called adapted null tetrads. Calculations are often simplified by using an adapted null tetrad.

The command AdaptedNullTetrad returns a null tetrad which will put the Newman-Penrose Weyl scalars in the above normal form.

The procedure AdaptedNullTetrad first calculates the Weyl spinor and calls the procedure AdaptedSpinorDyad to find a spinor dyad which transforms the Weyl spinor to normal form. The adapted null tetrad is then constructed from the spinor dyad.

The command AdaptedNullTetrad is part of the DifferentialGeometry:-Tensor package. It can be used in the form AdaptedNullTetrad(...) only after executing the commands with(DifferentialGeometry) and with(Tensor), but can always be used by executing DifferentialGeometry:-Tensor:-AdaptedNullTetrad(...).

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

$\mathrm{\_EnvExplicit}≔\mathrm{true}\:$

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

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

$\mathrm{g1}≔\mathrm{evalDG}\left(\mathrm{dt}\&t\mathrm{dt}-t^2\mathrm{dx}\&t\mathrm{dx}-x^2\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g1}:=\mathrm{dt}\mathrm{dt}-\left(\left(t^2\mathrm{dx}\right)\mathrm{dx}\right)-\left(\left(x^2\mathrm{dy}\right)\mathrm{dy}\right)-\left(\mathrm{dz}\mathrm{dz}\right)$

$\mathrm{NT1}≔\mathrm{evalDG}\left(\left[\mathrm{D\_t}+\mathrm{D\_z}\,\frac{1}{2}\left(\mathrm{D\_t}-\mathrm{D\_z}\right)\,\frac{\frac{1}{2}\mathrm{sqrt}\left(2\right)\mathrm{D\_x}}{t}+\frac{\frac{1}{2}\mathrm{I}\mathrm{sqrt}\left(2\right)\mathrm{D\_y}}{x}\,\frac{\frac{1}{2}\mathrm{sqrt}\left(2\right)\mathrm{D\_x}}{t}-\frac{\frac{1}{2}\mathrm{I}\mathrm{sqrt}\left(2\right)\mathrm{D\_y}}{x}\right]\right)$

$\mathrm{NT1}:=\left[\mathrm{D\_t}+\mathrm{D\_z}\,\frac{1}{2}\mathrm{D\_t}-\left(\frac{1}{2}\mathrm{D\_z}\right)\,\frac{\sqrt{2}}{2t}\mathrm{D\_x}+\frac{\frac{\mathrm{I}}{2}\sqrt{2}}{x}\mathrm{D\_y}\,\frac{\sqrt{2}}{2t}\mathrm{D\_x}-\left(\frac{\frac{\mathrm{I}}{2}\sqrt{2}}{x}\mathrm{D\_y}\right)\right]$

$\mathrm{GRQuery}\left(\mathrm{NT1}\,\mathrm{g1}\,''NullTetrad''\right)$

$\mathrm{true}$

$\mathrm{NP1}≔\mathrm{NPCurvatureScalars}\left(\mathrm{NT1}\,\mathrm{output}=\left[''WeylScalars''\right]\right)$

$\mathrm{NP1}:=\mathrm{table}\left(\left[''Psi1''\=-\frac{1}{4}\frac{\sqrt{2}}{x{t}^2}\,''Psi0''\=0\,''Psi2''\=0\,''Psi4''\=0\,''Psi3''\=-\frac{1}{8}\frac{\sqrt{2}}{x{t}^2}\right]\right)$

$\mathrm{PetrovType}\left(\mathrm{NP1}\right)$

$''I''$

$\mathrm{newNT1}≔\mathrm{combine}\left(\mathrm{AdaptedNullTetrad}\left(\mathrm{NT1}\,''I''\right)\,\mathrm{symbolic}\right)$

$\mathrm{newNT1}:=\left[\frac{\sqrt{2}}{2}\mathrm{D\_t}-\left(\frac{\sqrt{2}}{2t}\mathrm{D\_x}\right)\,\frac{\sqrt{2}}{2}\mathrm{D\_t}+\frac{\sqrt{2}}{2t}\mathrm{D\_x}\,\frac{\frac{1}{2}+\frac{\mathrm{I}}{2}}{x}\mathrm{D\_y}+\left(\frac{1}{2}-\frac{\mathrm{I}}{2}\right)\mathrm{D\_z}\,\frac{\frac{1}{2}-\frac{\mathrm{I}}{2}}{x}\mathrm{D\_y}+\left(\frac{1}{2}+\frac{\mathrm{I}}{2}\right)\mathrm{D\_z}\right]$

$\mathrm{newNP1}≔\mathrm{NPCurvatureScalars}\left(\mathrm{newNT1}\,\mathrm{output}=\left[''WeylScalars''\right]\right)$

$\mathrm{newNP1}:=\mathrm{table}\left(\left[''Psi1''\=0\,''Psi0''\=-\frac{\frac{1}{2}\mathrm{I}}{t^{2}x}\,''Psi2''\=0\,''Psi4''\=-\frac{\frac{1}{2}\mathrm{I}}{t^{2}x}\,''Psi3''\=0\right]\right)$

$\mathrm{DGsetup}\left(\left[r\,u\,x\,y\right]\,M\right)$

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

$\mathrm{g2}≔\mathrm{evalDG}\left(-\frac{2r^2}{{\left(2x\right)}^3}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}\right)+2\mathrm{du}\&s\mathrm{dr}-\left(32x+\frac{2m}{r}\right)\mathrm{du}\&t\mathrm{du}\right)$

$\mathrm{g2}:=\mathrm{dr}\mathrm{du}+\mathrm{du}\mathrm{dr}-\left(\left(\frac{2\left(3xr+m\right)}{r}\mathrm{du}\right)\mathrm{du}\right)-\left(\left(\frac{r^2}{4x^3}\mathrm{dx}\right)\mathrm{dx}\right)-\left(\left(\frac{r^2}{4x^3}\mathrm{dy}\right)\mathrm{dy}\right)$

$\mathrm{NT2}≔\mathrm{evalDG}\left(\left[\mathrm{D\_r}\,\frac{\left(3xr+m\right)\mathrm{D\_r}}{r}+\mathrm{D\_u}\,\frac{\mathrm{I}\mathrm{sqrt}\left(2\right)x^{\frac{3}{2}}\mathrm{D\_x}}{r}+\frac{\mathrm{sqrt}\left(2\right)x^{\frac{3}{2}}\mathrm{D\_y}}{r}\,-\frac{\mathrm{I}\mathrm{sqrt}\left(2\right)x^{\frac{3}{2}}\mathrm{D\_x}}{r}+\frac{\mathrm{sqrt}\left(2\right)x^{\frac{3}{2}}\mathrm{D\_y}}{r}\right]\right)$

$\mathrm{NT2}:=\left[\mathrm{D\_r}\,\frac{3xr+m}{r}\mathrm{D\_r}+\mathrm{D\_u}\,\frac{\mathrm{I}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\mathrm{D\_x}+\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\mathrm{D\_y}\,-\left(\frac{\mathrm{I}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\mathrm{D\_x}\right)+\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\mathrm{D\_y}\right]$

$\mathrm{GRQuery}\left(\mathrm{NT2}\,\mathrm{g2}\,''NullTetrad''\right)$

$\mathrm{true}$

$\mathrm{NP2}≔\mathrm{NPCurvatureScalars}\left(\mathrm{NT2}\,\mathrm{output}=\left[''WeylScalars''\right]\right)$

$\mathrm{NP2}:=\mathrm{table}\left(\left[''Psi1''\=0\,''Psi0''\=0\,''Psi2''\=-\frac{m}{r^3}\,''Psi4''\=\frac{18x^2}{r^2}\,''Psi3''\=-\frac{3\mathrm{I}\sqrt{2}{x}^{3\/2}}{r^2}\right]\right)$

$\mathrm{PetrovType}\left(\mathrm{NP2}\right)$

$''II''$

$\mathrm{newNT2}≔\mathrm{AdaptedNullTetrad}\left(\mathrm{NT2}\&comma;''II''\&comma;\mathrm{NP2}\right)\mathbf{assuming}0<x,0<y,0<r,3m-2xr<0,m<0$

$\mathrm{newNT2}:=\left[-\left(\frac{\sqrt{r}\sqrt{2xr-3m}x}{m}\mathrm{D\_r}\right)\&comma;-\left(\frac{2r^3{x}^3+3m^{2}rx+m^3}{r^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{2xr-3m}xm}\mathrm{D\_r}\right)-\left(\frac{m}{\sqrt{r}\sqrt{2xr-3m}x}\mathrm{D\_u}\right)-\left(\frac{4x^2}{\sqrt{r}\sqrt{2xr-3m}}\mathrm{D\_x}\right)\&comma;-\left(\frac{\mathrm{I}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}r}{m}\mathrm{D\_r}\right)-\left(\frac{\mathrm{I}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\mathrm{D\_x}\right)-\left(\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\mathrm{D\_y}\right)\&comma;\frac{\mathrm{I}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}r}{m}\mathrm{D\_r}+\frac{\mathrm{I}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\mathrm{D\_x}-\left(\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\mathrm{D\_y}\right)\right]$

$\mathrm{newNP2}≔\mathrm{NPCurvatureScalars}\left(\mathrm{newNT2}\&comma;\mathrm{output}=\left[''WeylScalars''\right]\right)$

$\mathrm{newNP2}:=\mathrm{table}\left(\left[''Psi1''\&equals;0\&comma;''Psi0''\&equals;0\&comma;''Psi2''\&equals;-\frac{m}{r^3}\&comma;''Psi4''\&equals;-\frac{6m}{r^3}\&comma;''Psi3''\&equals;0\right]\right)$

$\mathrm{DGsetup}\left(\left[r\&comma;u\&comma;x\&comma;y\right]\&comma;M\right)$

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

$\mathrm{g3}≔\mathrm{evalDG}\left(-\frac{r^2}{x^3}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}\right)+2\mathrm{du}\&s\mathrm{dr}-\frac{3}{2}x\mathrm{du}\&t\mathrm{du}\right)$

$\mathrm{g3}:=\mathrm{dr}\mathrm{du}+\mathrm{du}\mathrm{dr}-\left(\left(\frac{3x}{2}\mathrm{du}\right)\mathrm{du}\right)-\left(\left(\frac{r^2}{x^3}\mathrm{dx}\right)\mathrm{dx}\right)-\left(\left(\frac{r^2}{x^3}\mathrm{dy}\right)\mathrm{dy}\right)$

$\mathrm{NT3}≔\mathrm{evalDG}\left(\left[\left(\frac{3}{8}x+\frac{1}{2}\right)\mathrm{D\_r}+\frac{1}{2}\mathrm{D\_u}+\frac{\frac{1}{2}\mathrm{sqrt}\left(2\right)x^{\frac{3}{2}}\mathrm{D\_y}}{r}\&comma;\left(\frac{3}{8}x+\frac{1}{2}\right)\mathrm{D\_r}+\frac{1}{2}\mathrm{D\_u}-\frac{\frac{1}{2}\mathrm{sqrt}\left(2\right)x^{\frac{3}{2}}\mathrm{D\_y}}{r}\&comma;\left(-\frac{3}{8}x+\frac{1}{2}\right)\mathrm{D\_r}-\frac{1}{2}\mathrm{D\_u}+\frac{\frac{1}{2}\mathrm{I}\mathrm{sqrt}\left(2\right)x^{\frac{3}{2}}\mathrm{D\_x}}{r}\&comma;\left(-\frac{3}{8}x+\frac{1}{2}\right)\mathrm{D\_r}-\frac{1}{2}\mathrm{D\_u}-\frac{\frac{1}{2}\mathrm{I}\mathrm{sqrt}\left(2\right)x^{\frac{3}{2}}\mathrm{D\_x}}{r}\right]\right)$

$\mathrm{NT3}:=\left[\left(\frac{3x}{8}+\frac{1}{2}\right)\mathrm{D\_r}+\frac{1}{2}\mathrm{D\_u}+\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2r}\mathrm{D\_y}\&comma;\left(\frac{3x}{8}+\frac{1}{2}\right)\mathrm{D\_r}+\frac{1}{2}\mathrm{D\_u}-\left(\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2r}\mathrm{D\_y}\right)\&comma;-\left(\left(\frac{3x}{8}-\frac{1}{2}\right)\mathrm{D\_r}\right)-\left(\frac{1}{2}\mathrm{D\_u}\right)+\frac{\frac{\mathrm{I}}{2}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\mathrm{D\_x}\&comma;-\left(\left(\frac{3x}{8}-\frac{1}{2}\right)\mathrm{D\_r}\right)-\left(\frac{1}{2}\mathrm{D\_u}\right)-\left(\frac{\frac{\mathrm{I}}{2}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\mathrm{D\_x}\right)\right]$

$\mathrm{GRQuery}\left(\mathrm{NT3}\&comma;\mathrm{g3}\&comma;''NullTetrad''\right)$

$\mathrm{true}$

$\mathrm{NP3}≔\mathrm{NPCurvatureScalars}\left(\mathrm{NT3}\&comma;\mathrm{output}=\left[''WeylScalars''\right]\right)$

$\mathrm{NP3}:=\mathrm{table}\left(\left[''Psi1''\&equals;-\frac{3}{32}\frac{\left(2\mathrm{I}\sqrt{2}\&plus;3\sqrt{x}\right)x^{3\&sol;2}}{r^2}\&comma;''Psi0''\&equals;\frac{3}{32}\frac{x\left(4\mathrm{I}\sqrt{2}\sqrt{x}\&plus;3x\right)}{r^2}\&comma;''Psi2''\&equals;\frac{9}{32}\frac{x^2}{r^2}\&comma;''Psi4''\&equals;-\frac{3}{32}\frac{x\left(4\mathrm{I}\sqrt{2}\sqrt{x}-3x\right)}{r^2}\&comma;''Psi3''\&equals;\frac{3}{32}\frac{\left(2\mathrm{I}\sqrt{2}-3\sqrt{x}\right)x^{3\&sol;2}}{r^2}\right]\right)$

$\mathrm{PetrovType}\left(\mathrm{NP3}\right)$

$''III''$

$\mathrm{newNT3}≔\mathrm{AdaptedNullTetrad}\left(\mathrm{NT3}\&comma;''III''\&comma;\mathrm{NP3}\right)\mathbf{assuming}0<x$

$\mathrm{newNT3}:=\left[\frac{3\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{8r^2}\mathrm{D\_r}\&comma;\frac{11\sqrt{2}{r}^2}{8\sqrt{x}}\mathrm{D\_r}+\frac{4\sqrt{2}{r}^2}{3x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\mathrm{D\_u}+r\sqrt{x}\sqrt{2}\mathrm{D\_x}\&comma;\frac{3\sqrt{2}\sqrt{x}}{8}\mathrm{D\_r}+\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2r}\mathrm{D\_x}+\frac{\frac{\mathrm{I}}{2}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\mathrm{D\_y}\&comma;\frac{3\sqrt{2}\sqrt{x}}{8}\mathrm{D\_r}+\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2r}\mathrm{D\_x}-\left(\frac{\frac{\mathrm{I}}{2}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\mathrm{D\_y}\right)\right]$

$\mathrm{NPCurvatureScalars}\left(\mathrm{newNT3}\&comma;\mathrm{output}=\left[''WeylScalars''\right]\right)$

$\mathrm{table}\left(\left[''Psi1''\&equals;0\&comma;''Psi0''\&equals;0\&comma;''Psi2''\&equals;0\&comma;''Psi4''\&equals;0\&comma;''Psi3''\&equals;1\right]\right)$

$\mathrm{DGsetup}\left(\left[t\&comma;x\&comma;y\&comma;z\right]\&comma;M\right)$

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

$\mathrm{g4}≔\mathrm{evalDG}\left(-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\frac{1}{2}\exp \left(2x\right)\mathrm{dz}\&t\mathrm{dz}+\left(\mathrm{dt}+\exp \left(x\right)\mathrm{dz}\right)\&s\left(\mathrm{dt}+\exp \left(x\right)\mathrm{dz}\right)\right)$

$\mathrm{g4}:=\mathrm{dt}\mathrm{dt}+\left({\&ExponentialE;}^x\mathrm{dt}\right)\mathrm{dz}-\left(\mathrm{dx}\mathrm{dx}\right)-\left(\mathrm{dy}\mathrm{dy}\right)+\left({\&ExponentialE;}^x\mathrm{dz}\right)\mathrm{dt}+\left(\frac{{\&ExponentialE;}^{2x}}{2}\mathrm{dz}\right)\mathrm{dz}$

$\mathrm{NT4}≔\mathrm{evalDG}\left(\left[-\frac{1}{2}\mathrm{sqrt}\left(2\right)\left(\mathrm{sqrt}\left(2\right)-1\right)\mathrm{D\_t}+\exp \left(-x\right)\mathrm{D\_z}\&comma;\frac{1}{2}\mathrm{sqrt}\left(2\right)\left(1+\mathrm{sqrt}\left(2\right)\right)\mathrm{D\_t}-\exp \left(-x\right)\mathrm{D\_z}\&comma;\frac{1}{2}\mathrm{sqrt}\left(2\right)\mathrm{D\_x}+\frac{1}{2}\mathrm{I}\mathrm{sqrt}\left(2\right)\mathrm{D\_y}\&comma;\frac{1}{2}\mathrm{sqrt}\left(2\right)\mathrm{D\_x}-\frac{1}{2}\mathrm{I}\mathrm{sqrt}\left(2\right)\mathrm{D\_y}\right]\right)$

$\mathrm{NT4}:=\left[-\left(\frac{\sqrt{2}\left(\sqrt{2}-1\right)}{2}\mathrm{D\_t}\right)+{\&ExponentialE;}^{-x}\mathrm{D\_z}\&comma;\frac{\sqrt{2}\left(\sqrt{2}+1\right)}{2}\mathrm{D\_t}-\left({\&ExponentialE;}^{-x}\mathrm{D\_z}\right)\&comma;\frac{\sqrt{2}}{2}\mathrm{D\_x}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\&comma;\frac{\sqrt{2}}{2}\mathrm{D\_x}-\left(\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\right)\right]$

$\mathrm{GRQuery}\left(\mathrm{NT4}\&comma;\mathrm{g4}\&comma;''NullTetrad''\right)$

$\mathrm{true}$

$\mathrm{NP4}≔\mathrm{NPCurvatureScalars}\left(\mathrm{NT4}\&comma;\mathrm{output}=\left[''WeylScalars''\right]\right)$

$\mathrm{NP4}:=\mathrm{table}\left(\left[''Psi1''\&equals;0\&comma;''Psi0''\&equals;\frac{1}{4}\&comma;''Psi2''\&equals;\frac{1}{12}\&comma;''Psi4''\&equals;\frac{1}{4}\&comma;''Psi3''\&equals;0\right]\right)$

$\mathrm{PetrovType}\left(\mathrm{NP4}\right)$

$''D''$

$\mathrm{newNT4}≔\mathrm{AdaptedNullTetrad}\left(\mathrm{NT4}\&comma;''D''\right)$

$\mathrm{newNT4}:=\left[\sqrt{2}\mathrm{D\_t}-\left(\sqrt{2}\mathrm{D\_y}\right)\&comma;\frac{\sqrt{2}}{4}\mathrm{D\_t}+\frac{\sqrt{2}}{4}\mathrm{D\_y}\&comma;-\left(\mathrm{I}\mathrm{D\_t}\right)+\frac{\sqrt{2}}{2}\mathrm{D\_x}+\mathrm{I}{\&ExponentialE;}^{-x}\mathrm{D\_z}\&comma;\mathrm{I}\mathrm{D\_t}+\frac{\sqrt{2}}{2}\mathrm{D\_x}-\left(\mathrm{I}{\&ExponentialE;}^{-x}\mathrm{D\_z}\right)\right]$

$\mathrm{newNP}≔\mathrm{NPCurvatureScalars}\left(\mathrm{newNT4}\&comma;\mathrm{output}=\left[''WeylScalars''\right]\right)$

$\mathrm{newNP}:=\mathrm{table}\left(\left[''Psi1''\&equals;0\&comma;''Psi0''\&equals;0\&comma;''Psi2''\&equals;-\frac{1}{6}\&comma;''Psi4''\&equals;0\&comma;''Psi3''\&equals;0\right]\right)$

$\mathrm{DGsetup}\left(\left[u\&comma;x\&comma;y\&comma;z\right]\&comma;M\right)$

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

$\mathrm{g5}≔\mathrm{evalDG}\left(\exp \left(-2z\right)\mathrm{du}\&t\mathrm{dx}+\exp \left(-2z\right)\mathrm{dx}\&t\mathrm{du}+\exp \left(z\right)\mathrm{dx}\&t\mathrm{dx}-\exp \left(-2z\right)\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g5}:=\left({\&ExponentialE;}^{-2z}\mathrm{du}\right)\mathrm{dx}+\left({\&ExponentialE;}^{-2z}\mathrm{dx}\right)\mathrm{du}+\left({\&ExponentialE;}^z\mathrm{dx}\right)\mathrm{dx}-\left(\left({\&ExponentialE;}^{-2z}\mathrm{dy}\right)\mathrm{dy}\right)-\left(\mathrm{dz}\mathrm{dz}\right)$

$\mathrm{NT5}≔\mathrm{evalDG}\left(\left[-\frac{1}{4}\left(\exp \left(3z\right)-2\right)\exp \left(z\right)\mathrm{D\_u}+\frac{1}{2}\exp \left(z\right)\mathrm{D\_x}+\frac{1}{2}\mathrm{sqrt}\left(2\right)\mathrm{D\_z}\&comma;-\frac{1}{4}\left(\exp \left(3z\right)-2\right)\exp \left(z\right)\mathrm{D\_u}+\frac{1}{2}\exp \left(z\right)\mathrm{D\_x}-\frac{1}{2}\mathrm{sqrt}\left(2\right)\mathrm{D\_z}\&comma;\frac{1}{4}\left(\exp \left(3z\right)+2\right)\exp \left(z\right)\mathrm{D\_u}-\frac{1}{2}\exp \left(z\right)\mathrm{D\_x}+\frac{1}{2}\mathrm{I}\mathrm{sqrt}\left(2\right)\exp \left(z\right)\mathrm{D\_y}\&comma;\frac{1}{4}\left(\exp \left(3z\right)+2\right)\exp \left(z\right)\mathrm{D\_u}-\frac{1}{2}\exp \left(z\right)\mathrm{D\_x}-\frac{1}{2}\mathrm{I}\mathrm{sqrt}\left(2\right)\exp \left(z\right)\mathrm{D\_y}\right]\right)$

$\mathrm{NT5}:=\left[-\left(\frac{\left({\&ExponentialE;}^{3z}-2\right){\&ExponentialE;}^z}{4}\mathrm{D\_u}\right)+\frac{{\&ExponentialE;}^z}{2}\mathrm{D\_x}+\frac{\sqrt{2}}{2}\mathrm{D\_z}\&comma;-\left(\frac{\left({\&ExponentialE;}^{3z}-2\right){\&ExponentialE;}^z}{4}\mathrm{D\_u}\right)+\frac{{\&ExponentialE;}^z}{2}\mathrm{D\_x}-\left(\frac{\sqrt{2}}{2}\mathrm{D\_z}\right)\&comma;\frac{\left({\&ExponentialE;}^{3z}+2\right){\&ExponentialE;}^z}{4}\mathrm{D\_u}-\left(\frac{{\&ExponentialE;}^z}{2}\mathrm{D\_x}\right)+\frac{\mathrm{I}}{2}\sqrt{2}{\&ExponentialE;}^z\mathrm{D\_y}\&comma;\frac{\left({\&ExponentialE;}^{3z}+2\right){\&ExponentialE;}^z}{4}\mathrm{D\_u}-\left(\frac{{\&ExponentialE;}^z}{2}\mathrm{D\_x}\right)-\left(\frac{\mathrm{I}}{2}\sqrt{2}{\&ExponentialE;}^z\mathrm{D\_y}\right)\right]$

$\mathrm{GRQuery}\left(\mathrm{NT5}\&comma;\mathrm{g5}\&comma;''NullTetrad''\right)$

$\mathrm{true}$

$\mathrm{NP5}≔\mathrm{NPCurvatureScalars}\left(\mathrm{NT5}\&comma;\mathrm{output}=\left[''WeylScalars''\right]\right)$

$\mathrm{NP5}:=\mathrm{table}\left(\left[''Psi1''\&equals;-\frac{3}{8}{\&ExponentialE;}^{3z}\&comma;''Psi0''\&equals;\frac{3}{8}{\&ExponentialE;}^{3z}\&comma;''Psi2''\&equals;\frac{3}{8}{\&ExponentialE;}^{3z}\&comma;''Psi4''\&equals;\frac{3}{8}{\&ExponentialE;}^{3z}\&comma;''Psi3''\&equals;-\frac{3}{8}{\&ExponentialE;}^{3z}\right]\right)$

$\mathrm{PetrovType}\left(\mathrm{NP5}\right)$

$''N''$

$\mathrm{newNT5}≔\mathrm{AdaptedNullTetrad}\left(\mathrm{NT5}\&comma;''N''\&comma;\mathrm{NP5}\right)\mathbf{assuming}0<z$

$\mathrm{newNT5}:=\left[\frac{{\&ExponentialE;}^{\frac{5z}{2}}\sqrt{6}}{2}\mathrm{D\_u}\&comma;-\left(\frac{\sqrt{6}\left({\&ExponentialE;}^{3z}-2\right){\&ExponentialE;}^{-\frac{z}{2}}}{6}\mathrm{D\_u}\right)+\frac{{\&ExponentialE;}^{-\frac{z}{2}}\sqrt{6}}{3}\mathrm{D\_x}+\frac{2\sqrt{3}{\&ExponentialE;}^{-\frac{3z}{2}}}{3}\mathrm{D\_z}\&comma;-\left({\&ExponentialE;}^z\mathrm{D\_u}\right)+\frac{\mathrm{I}}{2}\sqrt{2}{\&ExponentialE;}^z\mathrm{D\_y}-\left(\frac{\sqrt{2}}{2}\mathrm{D\_z}\right)\&comma;-\left({\&ExponentialE;}^z\mathrm{D\_u}\right)-\left(\frac{\mathrm{I}}{2}\sqrt{2}{\&ExponentialE;}^z\mathrm{D\_y}\right)-\left(\frac{\sqrt{2}}{2}\mathrm{D\_z}\right)\right]$

$\mathrm{newNP5}≔\mathrm{NPCurvatureScalars}\left(\mathrm{newNT5}\&comma;\mathrm{output}=\left[''WeylScalars''\right]\right)$

$\mathrm{newNP5}:=\mathrm{table}\left(\left[''Psi1''\&equals;0\&comma;''Psi0''\&equals;0\&comma;''Psi2''\&equals;0\&comma;''Psi4''\&equals;1\&comma;''Psi3''\&equals;0\right]\right)$