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)\:$

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

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

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

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

$\mathrm{g1}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,1\right]\,\left[\left[2\,2\right]\,-t^2\right]\,\left[\left[3\,3\right]\,-x^2\right]\,\left[\left[4\,4\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,1\right]\,\left[\left[2\,2\right]\,-t^2\right]\,\left[\left[3\,3\right]\,-x^2\right]\,\left[\left[4\,4\right]\,\mathrm{-1}\right]\right]\right]\right)$

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

$\mathrm{NT1}:=\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,\frac{1}{2}\right]\,\left[\left[4\right]\,-\frac{1}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,\frac{1}{2}\right]\,\left[\left[4\right]\,-\frac{1}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,\frac{\sqrt{2}}{2t}\right]\,\left[\left[3\right]\,\frac{\frac{\mathrm{I}}{2}\sqrt{2}}{x}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,\frac{\sqrt{2}}{2t}\right]\,\left[\left[3\right]\,\frac{\frac{\mathrm{I}}{2}\sqrt{2}}{x}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,\frac{\sqrt{2}}{2t}\right]\,\left[\left[3\right]\,\frac{-\frac{\mathrm{I}}{2}\sqrt{2}}{x}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,\frac{\sqrt{2}}{2t}\right]\,\left[\left[3\right]\,\frac{-\frac{\mathrm{I}}{2}\sqrt{2}}{x}\right]\right]\right]\right)\right]$

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

$\mathrm{true}$

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

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

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

$''I''$

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

$\mathrm{newNT1}:=\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[2\right]\,-\frac{\sqrt{2}}{2t}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[2\right]\,-\frac{\sqrt{2}}{2t}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[2\right]\,\frac{\sqrt{2}}{2t}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,\frac{\sqrt{2}}{2}\right]\,\left[\left[2\right]\,\frac{\sqrt{2}}{2t}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,\frac{\frac{1}{2}+\frac{\mathrm{I}}{2}}{x}\right]\,\left[\left[4\right]\,\frac{1}{2}-\frac{\mathrm{I}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,\frac{\frac{1}{2}+\frac{\mathrm{I}}{2}}{x}\right]\,\left[\left[4\right]\,\frac{1}{2}-\frac{\mathrm{I}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,\frac{\frac{1}{2}-\frac{\mathrm{I}}{2}}{x}\right]\,\left[\left[4\right]\,\frac{1}{2}+\frac{\mathrm{I}}{2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,\frac{\frac{1}{2}-\frac{\mathrm{I}}{2}}{x}\right]\,\left[\left[4\right]\,\frac{1}{2}+\frac{\mathrm{I}}{2}\right]\right]\right]\right)\right]$

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

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

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

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

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

$\mathrm{g2}:=\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,1\right]\,\left[\left[2\,1\right]\,1\right]\,\left[\left[2\,2\right]\,-\frac{2\left(3xr+m\right)}{r}\right]\,\left[\left[3\,3\right]\,-\frac{r^2}{4x^3}\right]\,\left[\left[4\,4\right]\,-\frac{r^2}{4x^3}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,1\right]\,\left[\left[2\,1\right]\,1\right]\,\left[\left[2\,2\right]\,-\frac{2\left(3xr+m\right)}{r}\right]\,\left[\left[3\,3\right]\,-\frac{r^2}{4x^3}\right]\,\left[\left[4\,4\right]\,-\frac{r^2}{4x^3}\right]\right]\right]\right)$

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

$\mathrm{NT2}:=\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,\frac{3xr+m}{r}\right]\,\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,\frac{3xr+m}{r}\right]\,\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,\frac{\mathrm{I}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\,\left[\left[4\right]\,\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,\frac{\mathrm{I}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\,\left[\left[4\right]\,\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,\frac{\mathrm{-I}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\,\left[\left[4\right]\,\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[3\right]\,\frac{\mathrm{-I}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\,\left[\left[4\right]\,\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\right]\right]\right)\right]$

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

$\mathrm{true}$

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

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

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

$''II''$

$\mathrm{newNT2}≔\mathrm{AdaptedNullTetrad}\left(\mathrm{NT2}\&comma;''II''\&comma;\mathrm{NP2}\right)assuming0<x\&comma;0<y\&comma;0<r\&comma;3m-2xr<0\&comma;m<0$

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

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

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

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

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

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

$\mathrm{g3}:=\mathrm{\_DG}\left(\left[\left[''tensor''\&comma;M\&comma;\left[\left[''cov\_bas''\&comma;''cov\_bas''\right]\&comma;\left[\right]\right]\right]\&comma;\left[\left[\left[1\&comma;2\right]\&comma;1\right]\&comma;\left[\left[2\&comma;1\right]\&comma;1\right]\&comma;\left[\left[2\&comma;2\right]\&comma;-\frac{3x}{2}\right]\&comma;\left[\left[3\&comma;3\right]\&comma;-\frac{r^2}{x^3}\right]\&comma;\left[\left[4\&comma;4\right]\&comma;-\frac{r^2}{x^3}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''tensor''\&comma;M\&comma;\left[\left[''cov\_bas''\&comma;''cov\_bas''\right]\&comma;\left[\right]\right]\right]\&comma;\left[\left[\left[1\&comma;2\right]\&comma;1\right]\&comma;\left[\left[2\&comma;1\right]\&comma;1\right]\&comma;\left[\left[2\&comma;2\right]\&comma;-\frac{3x}{2}\right]\&comma;\left[\left[3\&comma;3\right]\&comma;-\frac{r^2}{x^3}\right]\&comma;\left[\left[4\&comma;4\right]\&comma;-\frac{r^2}{x^3}\right]\right]\right]\right)$

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

$\mathrm{NT3}:=\left[\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{3x}{8}+\frac{1}{2}\right]\&comma;\left[\left[2\right]\&comma;\frac{1}{2}\right]\&comma;\left[\left[4\right]\&comma;\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2r}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{3x}{8}+\frac{1}{2}\right]\&comma;\left[\left[2\right]\&comma;\frac{1}{2}\right]\&comma;\left[\left[4\right]\&comma;\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2r}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{3x}{8}+\frac{1}{2}\right]\&comma;\left[\left[2\right]\&comma;\frac{1}{2}\right]\&comma;\left[\left[4\right]\&comma;-\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2r}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{3x}{8}+\frac{1}{2}\right]\&comma;\left[\left[2\right]\&comma;\frac{1}{2}\right]\&comma;\left[\left[4\right]\&comma;-\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2r}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;-\frac{3x}{8}+\frac{1}{2}\right]\&comma;\left[\left[2\right]\&comma;-\frac{1}{2}\right]\&comma;\left[\left[3\right]\&comma;\frac{\frac{\mathrm{I}}{2}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;-\frac{3x}{8}+\frac{1}{2}\right]\&comma;\left[\left[2\right]\&comma;-\frac{1}{2}\right]\&comma;\left[\left[3\right]\&comma;\frac{\frac{\mathrm{I}}{2}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;-\frac{3x}{8}+\frac{1}{2}\right]\&comma;\left[\left[2\right]\&comma;-\frac{1}{2}\right]\&comma;\left[\left[3\right]\&comma;\frac{-\frac{\mathrm{I}}{2}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;-\frac{3x}{8}+\frac{1}{2}\right]\&comma;\left[\left[2\right]\&comma;-\frac{1}{2}\right]\&comma;\left[\left[3\right]\&comma;\frac{-\frac{\mathrm{I}}{2}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\right]\right]\right)\right]$

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

$\mathrm{true}$

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

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

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

$''III''$

$\mathrm{newNT3}≔\mathrm{AdaptedNullTetrad}\left(\mathrm{NT3}\&comma;''III''\&comma;\mathrm{NP3}\right)assuming0<x$

$\mathrm{newNT3}:=\left[\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{3\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{8r^2}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{3\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{8r^2}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{11\sqrt{2}{r}^2}{8\sqrt{x}}\right]\&comma;\left[\left[2\right]\&comma;\frac{4\sqrt{2}{r}^2}{3x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right]\&comma;\left[\left[3\right]\&comma;r\sqrt{x}\sqrt{2}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{11\sqrt{2}{r}^2}{8\sqrt{x}}\right]\&comma;\left[\left[2\right]\&comma;\frac{4\sqrt{2}{r}^2}{3x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right]\&comma;\left[\left[3\right]\&comma;r\sqrt{x}\sqrt{2}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{3\sqrt{2}\sqrt{x}}{8}\right]\&comma;\left[\left[3\right]\&comma;\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2r}\right]\&comma;\left[\left[4\right]\&comma;\frac{\frac{\mathrm{I}}{2}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{3\sqrt{2}\sqrt{x}}{8}\right]\&comma;\left[\left[3\right]\&comma;\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2r}\right]\&comma;\left[\left[4\right]\&comma;\frac{\frac{\mathrm{I}}{2}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{3\sqrt{2}\sqrt{x}}{8}\right]\&comma;\left[\left[3\right]\&comma;\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2r}\right]\&comma;\left[\left[4\right]\&comma;\frac{-\frac{\mathrm{I}}{2}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{3\sqrt{2}\sqrt{x}}{8}\right]\&comma;\left[\left[3\right]\&comma;\frac{\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2r}\right]\&comma;\left[\left[4\right]\&comma;\frac{-\frac{\mathrm{I}}{2}\sqrt{2}{x}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{r}\right]\right]\right]\right)\right]$

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

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

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

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

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

$\mathrm{g4}:=\mathrm{\_DG}\left(\left[\left[''tensor''\&comma;M\&comma;\left[\left[''cov\_bas''\&comma;''cov\_bas''\right]\&comma;\left[\right]\right]\right]\&comma;\left[\left[\left[1\&comma;1\right]\&comma;1\right]\&comma;\left[\left[1\&comma;4\right]\&comma;{\&ExponentialE;}^x\right]\&comma;\left[\left[2\&comma;2\right]\&comma;\mathrm{-1}\right]\&comma;\left[\left[3\&comma;3\right]\&comma;\mathrm{-1}\right]\&comma;\left[\left[4\&comma;1\right]\&comma;{\&ExponentialE;}^x\right]\&comma;\left[\left[4\&comma;4\right]\&comma;\frac{{\&ExponentialE;}^{2x}}{2}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''tensor''\&comma;M\&comma;\left[\left[''cov\_bas''\&comma;''cov\_bas''\right]\&comma;\left[\right]\right]\right]\&comma;\left[\left[\left[1\&comma;1\right]\&comma;1\right]\&comma;\left[\left[1\&comma;4\right]\&comma;{\&ExponentialE;}^x\right]\&comma;\left[\left[2\&comma;2\right]\&comma;\mathrm{-1}\right]\&comma;\left[\left[3\&comma;3\right]\&comma;\mathrm{-1}\right]\&comma;\left[\left[4\&comma;1\right]\&comma;{\&ExponentialE;}^x\right]\&comma;\left[\left[4\&comma;4\right]\&comma;\frac{{\&ExponentialE;}^{2x}}{2}\right]\right]\right]\right)$

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

$\mathrm{NT4}:=\left[\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;-\frac{\sqrt{2}\left(\sqrt{2}-1\right)}{2}\right]\&comma;\left[\left[4\right]\&comma;{\&ExponentialE;}^{-x}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;-\frac{\sqrt{2}\left(\sqrt{2}-1\right)}{2}\right]\&comma;\left[\left[4\right]\&comma;{\&ExponentialE;}^{-x}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{\sqrt{2}\left(\sqrt{2}+1\right)}{2}\right]\&comma;\left[\left[4\right]\&comma;-{\&ExponentialE;}^{-x}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{\sqrt{2}\left(\sqrt{2}+1\right)}{2}\right]\&comma;\left[\left[4\right]\&comma;-{\&ExponentialE;}^{-x}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[2\right]\&comma;\frac{\sqrt{2}}{2}\right]\&comma;\left[\left[3\right]\&comma;\frac{\mathrm{I}}{2}\sqrt{2}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[2\right]\&comma;\frac{\sqrt{2}}{2}\right]\&comma;\left[\left[3\right]\&comma;\frac{\mathrm{I}}{2}\sqrt{2}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[2\right]\&comma;\frac{\sqrt{2}}{2}\right]\&comma;\left[\left[3\right]\&comma;-\frac{\mathrm{I}}{2}\sqrt{2}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[2\right]\&comma;\frac{\sqrt{2}}{2}\right]\&comma;\left[\left[3\right]\&comma;-\frac{\mathrm{I}}{2}\sqrt{2}\right]\right]\right]\right)\right]$

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

$\mathrm{true}$

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

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

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

$''D''$

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

$\mathrm{newNT4}:=\left[\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\sqrt{2}\right]\&comma;\left[\left[3\right]\&comma;-\sqrt{2}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\sqrt{2}\right]\&comma;\left[\left[3\right]\&comma;-\sqrt{2}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{\sqrt{2}}{4}\right]\&comma;\left[\left[3\right]\&comma;\frac{\sqrt{2}}{4}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\frac{\sqrt{2}}{4}\right]\&comma;\left[\left[3\right]\&comma;\frac{\sqrt{2}}{4}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\mathrm{-I}\right]\&comma;\left[\left[2\right]\&comma;\frac{\sqrt{2}}{2}\right]\&comma;\left[\left[4\right]\&comma;\mathrm{I}{\&ExponentialE;}^{-x}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\mathrm{-I}\right]\&comma;\left[\left[2\right]\&comma;\frac{\sqrt{2}}{2}\right]\&comma;\left[\left[4\right]\&comma;\mathrm{I}{\&ExponentialE;}^{-x}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\mathrm{I}\right]\&comma;\left[\left[2\right]\&comma;\frac{\sqrt{2}}{2}\right]\&comma;\left[\left[4\right]\&comma;\mathrm{-I}{\&ExponentialE;}^{-x}\right]\right]\right]\right)\&comma;\mathrm{\_DG}\left(\left[\left[''vector''\&comma;M\&comma;\left[\right]\right]\&comma;\left[\left[\left[1\right]\&comma;\mathrm{I}\right]\&comma;\left[\left[2\right]\&comma;\frac{\sqrt{2}}{2}\right]\&comma;\left[\left[4\right]\&comma;\mathrm{-I}{\&ExponentialE;}^{-x}\right]\right]\right]\right)\right]$