Let $g$ be a metric on a 4-dimensional manifold with signature  $\left[1\,-1\,-1\,-1\right]$. A list of 4 vectors $\left[L\,N\,M\,\bar{M}\right]$ defines a null tetrad if $L$ and $N$ are real, $\bar{M}$ is the complex conjugate of $M$,

A Lorentz transformation is a (linear) change of frame which transforms a null tetrad $\left[L\,N\,M\,\bar{M}\right]$ into another null tetrad $\left[L\'\,N\'\,M\'\,\bar{M}\'\right]$. Every Lorentz transformation can be expressed as the composition of the following 4 basic Lorentz transformations.

1.  A null rotation about the $L$ axis ($\mathrm{\θ}$ complex):

2.  A null rotation about the N axis (θ complex)

3.  A spatial rotation in the $M-\bar{M}$ plane ($\mathrm{\θ}$ real):

4.  A boost ($\mathrm{\θ}$ real and non-zero):

The command NullTetradTransformation(NullTetrad, TransType, $\mathbf{\θ}$, axis) returns the new null tetrad $\[L\'\, N\'\, M\'\, \bar{M}\'$$\]$ obtained from NullTetrad = $\[L\, N\, M\, \bar{M}$$\]$ through the application of one of the above Lorentz transformations.

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

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

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

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

$g≔\mathrm{evalDG}\left(2\mathrm{du}\&s\mathrm{dv}-\frac{1}{2}\left(\mathrm{dx}\&t\mathrm{dx}+\mathrm{dy}\&t\mathrm{dy}\right)\right)$

$g:=\mathrm{du}\mathrm{dv}\+\mathrm{dv}\mathrm{du}-\frac{1}{2}\mathrm{dx}\mathrm{dx}-\frac{1}{2}\mathrm{dy}\mathrm{dy}$

$L,N,M,\mathrm{barM}≔\mathrm{D\_u},\mathrm{D\_v},\mathrm{evalDG}\left(\mathrm{D\_x}+\mathrm{I}\mathrm{D\_y}\right),\mathrm{evalDG}\left(\mathrm{D\_x}-\mathrm{I}\mathrm{D\_y}\right)$

$L\,N\,M\,\mathrm{barM}:=\mathrm{D\_u}\,\mathrm{D\_v}\,\mathrm{D\_x}\+\mathrm{I}\mathrm{D\_y}\,\mathrm{D\_x}-\mathrm{I}\mathrm{D\_y}$

$T≔\left[L\,N\,M\,\mathrm{barM}\right]$

$T:=\left[\mathrm{D\_u}\,\mathrm{D\_v}\,\mathrm{D\_x}\+\mathrm{I}\mathrm{D\_y}\,\mathrm{D\_x}-\mathrm{I}\mathrm{D\_y}\right]$

$\mathrm{GRQuery}\left(T\,g\,''NullTetrad''\right)$

$\mathrm{true}$

$\mathrm{T1a}≔\mathrm{NullTetradTransformation}\left(T\,''null\; rotation''\,a\,''l''\right)\mathbf{assuming}a::\mathrm{real}$

$\mathrm{T1a}:=\left[\mathrm{D\_u}\,a^2\mathrm{D\_u}\+\mathrm{D\_v}\+2a\mathrm{D\_x}\,a\mathrm{D\_u}\+\mathrm{D\_x}\+\mathrm{I}\mathrm{D\_y}\,a\mathrm{D\_u}\+\mathrm{D\_x}-\mathrm{I}\mathrm{D\_y}\right]$

$\mathrm{GRQuery}\left(\mathrm{T1a}\,g\,''NullTetrad''\right)$

$\mathrm{true}$

$\mathrm{T1b}≔\mathrm{NullTetradTransformation}\left(T\,''null\; rotation''\,\mathrm{I}b\,''l''\right)\mathbf{assuming}b::\mathrm{real}$

$\mathrm{T1b}:=\left[\mathrm{D\_u}\,b^2\mathrm{D\_u}\+\mathrm{D\_v}-2b\mathrm{D\_y}\,-\mathrm{I}b\mathrm{D\_u}\+\mathrm{D\_x}\+\mathrm{I}\mathrm{D\_y}\,\mathrm{I}b\mathrm{D\_u}\+\mathrm{D\_x}-\mathrm{I}\mathrm{D\_y}\right]$

$\mathrm{GRQuery}\left(\mathrm{T1b}\,g\,''NullTetrad''\right)$

$\mathrm{true}$

$\mathrm{T2a}≔\mathrm{NullTetradTransformation}\left(T\,''null\; rotation''\,a\,''n''\right)\mathbf{assuming}a::\mathrm{real}$

$\mathrm{T2a}:=\left[\mathrm{D\_u}\+a^2\mathrm{D\_v}\+2a\mathrm{D\_x}\,\mathrm{D\_v}\,a\mathrm{D\_v}\+\mathrm{D\_x}\+\mathrm{I}\mathrm{D\_y}\,a\mathrm{D\_v}\+\mathrm{D\_x}-\mathrm{I}\mathrm{D\_y}\right]$

$\mathrm{GRQuery}\left(\mathrm{T2a}\,g\,''NullTetrad''\right)$

$\mathrm{true}$

$\mathrm{T2b}≔\mathrm{NullTetradTransformation}\left(T\,''null\; rotation''\,\mathrm{I}b\,''n''\right)\mathbf{assuming}b::\mathrm{real}$

$\mathrm{T2b}:=\left[\mathrm{D\_u}\+b^2\mathrm{D\_v}-2b\mathrm{D\_y}\,\mathrm{D\_v}\,-\mathrm{I}b\mathrm{D\_v}\+\mathrm{D\_x}\+\mathrm{I}\mathrm{D\_y}\,\mathrm{I}b\mathrm{D\_v}\+\mathrm{D\_x}-\mathrm{I}\mathrm{D\_y}\right]$

$\mathrm{GRQuery}\left(\mathrm{T2b}\,g\,''NullTetrad''\right)$

$\mathrm{true}$

$\mathrm{T3}≔\mathrm{NullTetradTransformation}\left(T\,''spatial\; rotation''\,\mathrm{\theta }\,''n''\right)\mathbf{assuming}\mathrm{\theta }::\mathrm{real}$

$\mathrm{T3}:=\left[\mathrm{D\_u}\,\mathrm{D\_v}\,\left(\cos \left(\mathrm{\θ}\right)\+\mathrm{I}\sin \left(\mathrm{\θ}\right)\right)\mathrm{D\_x}\+\left(\mathrm{I}\cos \left(\mathrm{\θ}\right)-\sin \left(\mathrm{\θ}\right)\right)\mathrm{D\_y}\,\left(\cos \left(\mathrm{\θ}\right)-\mathrm{I}\sin \left(\mathrm{\θ}\right)\right)\mathrm{D\_x}-\left(\mathrm{I}\cos \left(\mathrm{\θ}\right)\+\sin \left(\mathrm{\θ}\right)\right)\mathrm{D\_y}\right]$

$\mathrm{GRQuery}\left(\mathrm{T3}\,g\,''NullTetrad''\right)$

$\mathrm{true}$

$\mathrm{T4}≔\mathrm{NullTetradTransformation}\left(T\,''spatial\; rotation''\,\mathrm{\theta }\,''n''\right)\mathbf{assuming}\mathrm{\theta }::\mathrm{real}$

$\mathrm{T4}:=\left[\mathrm{D\_u}\,\mathrm{D\_v}\,\left(\cos \left(\mathrm{\θ}\right)\+\mathrm{I}\sin \left(\mathrm{\θ}\right)\right)\mathrm{D\_x}\+\left(\mathrm{I}\cos \left(\mathrm{\θ}\right)-\sin \left(\mathrm{\θ}\right)\right)\mathrm{D\_y}\,\left(\cos \left(\mathrm{\θ}\right)-\mathrm{I}\sin \left(\mathrm{\θ}\right)\right)\mathrm{D\_x}-\left(\mathrm{I}\cos \left(\mathrm{\θ}\right)\+\sin \left(\mathrm{\θ}\right)\right)\mathrm{D\_y}\right]$

$\mathrm{GRQuery}\left(\mathrm{T4}\,g\,''NullTetrad''\right)$

$\mathrm{true}$

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

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

$\mathrm{T1}≔\mathrm{evalDG}\left(\left[\frac{1}{2}{2}^{\frac{1}{2}}\mathrm{D\_t}+\frac{1}{2}{2}^{\frac{1}{2}}\mathrm{D\_z}\,\frac{1}{2}{2}^{\frac{1}{2}}\mathrm{D\_t}-\frac{1}{2}{2}^{\frac{1}{2}}\mathrm{D\_z}\,\frac{1}{2}{2}^{\frac{1}{2}}{z}^2\mathrm{D\_x}+\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}{x}^2\mathrm{D\_y}\,\frac{1}{2}{2}^{\frac{1}{2}}{z}^2\mathrm{D\_x}-\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}{x}^2\mathrm{D\_y}\right]\right)$

$\mathrm{T1}:=\left[\frac{1}{2}\sqrt{2}\mathrm{D\_t}\+\frac{1}{2}\sqrt{2}\mathrm{D\_z}\,\frac{1}{2}\sqrt{2}\mathrm{D\_t}-\frac{1}{2}\sqrt{2}\mathrm{D\_z}\,\frac{1}{2}\sqrt{2}{z}^2\mathrm{D\_x}\+\frac{1}{2}\mathrm{I}\sqrt{2}{x}^2\mathrm{D\_y}\,\frac{1}{2}\sqrt{2}{z}^2\mathrm{D\_x}-\frac{1}{2}\mathrm{I}\sqrt{2}{x}^2\mathrm{D\_y}\right]$

$\mathrm{T2}≔\mathrm{NullTetradTransformation}\left(\mathrm{T1}\,''null\; rotation''\,a\,''l''\right)\mathbf{assuming}a::\mathrm{real}$

$\mathrm{T2}:=\left[\frac{1}{2}\sqrt{2}\mathrm{D\_t}\+\frac{1}{2}\sqrt{2}\mathrm{D\_z}\,\left(\frac{1}{2}{a}^2\sqrt{2}\+\frac{1}{2}\sqrt{2}\right)\mathrm{D\_t}\+a\sqrt{2}{z}^2\mathrm{D\_x}\+\left(\frac{1}{2}{a}^2\sqrt{2}-\frac{1}{2}\sqrt{2}\right)\mathrm{D\_z}\,\frac{1}{2}a\sqrt{2}\mathrm{D\_t}\+\frac{1}{2}\sqrt{2}{z}^2\mathrm{D\_x}\+\frac{1}{2}\mathrm{I}\sqrt{2}{x}^2\mathrm{D\_y}\+\frac{1}{2}a\sqrt{2}\mathrm{D\_z}\,\frac{1}{2}a\sqrt{2}\mathrm{D\_t}\+\frac{1}{2}\sqrt{2}{z}^2\mathrm{D\_x}-\frac{1}{2}\mathrm{I}\sqrt{2}{x}^2\mathrm{D\_y}\+\frac{1}{2}a\sqrt{2}\mathrm{D\_z}\right]$

$\mathrm{NPCurvatureScalars}\left(\mathrm{T2}\,\mathrm{output}=\left[''WeylScalars''\right]\right)$

$\mathrm{table}\left(\left[''Psi3''\=-\frac{1}{2}\frac{6z^3{a}^{2}x-2z^{3}x-6z^{6}a\+3a^3{x}^2\+3a{x}^2}{z^2{x}^2}\,''Psi1''\=-\frac{1}{2}\frac{2z^3\+3ax}{z^{2}x}\,''Psi2''\=-\frac{1}{2}\frac{-2z^6\+x^2\+3a^2{x}^2\+4xa{z}^3}{z^2{x}^2}\,''Psi0''\=-\frac{3}{2z^2}\,''Psi4''\=-\frac{1}{2}\frac{8x{a}^3{z}^3-8xa{z}^3\+3a^4{x}^2\+6a^2{x}^2-12a^2{z}^6\+3x^2}{z^2{x}^2}\right]\right)$

$\mathrm{T3}≔\mathrm{eval}\left(\mathrm{T2}\,a=-\frac{\frac{2}{3}{z}^3}{x}\right)$

$\mathrm{T3}:=\left[\frac{1}{2}\sqrt{2}\mathrm{D\_t}\+\frac{1}{2}\sqrt{2}\mathrm{D\_z}\,\left(\frac{2}{9}\frac{z^6\sqrt{2}}{x^2}\+\frac{1}{2}\sqrt{2}\right)\mathrm{D\_t}-\frac{2}{3}\frac{z^5\sqrt{2}\mathrm{D\_x}}{x}\+\left(\frac{2}{9}\frac{z^6\sqrt{2}}{x^2}-\frac{1}{2}\sqrt{2}\right)\mathrm{D\_z}\,-\frac{1}{3}\frac{z^3\sqrt{2}\mathrm{D\_t}}{x}\+\frac{1}{2}\sqrt{2}{z}^2\mathrm{D\_x}\+\frac{1}{2}\mathrm{I}\sqrt{2}{x}^2\mathrm{D\_y}-\frac{1}{3}\frac{z^3\sqrt{2}\mathrm{D\_z}}{x}\,-\frac{1}{3}\frac{z^3\sqrt{2}\mathrm{D\_t}}{x}\+\frac{1}{2}\sqrt{2}{z}^2\mathrm{D\_x}-\frac{1}{2}\mathrm{I}\sqrt{2}{x}^2\mathrm{D\_y}-\frac{1}{3}\frac{z^3\sqrt{2}\mathrm{D\_z}}{x}\right]$

$\mathrm{NPCurvatureScalars}\left(\mathrm{T3}\,\mathrm{output}=\left[''WeylScalars''\right]\right)$

$\mathrm{table}\left(\left[''Psi3''\=\frac{2}{9}\frac{z\left(-13z^6\+9x^2\right)}{x^3}\,''Psi1''\=0\,''Psi2''\=-\frac{1}{6}\frac{-10z^6\+3x^2}{z^2{x}^2}\,''Psi0''\=-\frac{3}{2z^2}\,''Psi4''\=-\frac{1}{18}\frac{-64z^{12}\+72z^6{x}^2\+27x^4}{z^2{x}^4}\right]\right)$