Let $g$ be a metric on a 4-dimensional manifold with signature $\left[1\,-1\,-1\,-1\right]$. A list of 4 vectors $\left[E_t\,E_x\,E_y\,E_z\right]$ defines an orthonormal tetrad if

Given an orthonormal tetrad OrthTetrad = $\left[E_t\,E_x\,E_y\,E_z\right]$, the command NullTetrad(OrthTetrad) constructs the null tetrad given by

Let sigma be a solder form (index type ["con", " cov", "cov"]), with components ${\mathrm{\σ}}_{\mathrm{AA}\'}^i \,$for the metric $g$. Let ${\mathrm{o}}^A$and ${\mathrm{\iota }}^B$ be rank 1, unprimed spinors with ${\mathrm{\ε}}_{\mathrm{AB}}{\mathrm{\ο}}^A{\mathrm{\ι}}^B\=1$. Let $\overline{\mathrm{\ο}}$ and $\overline{\mathrm{\ι}}$ be their conjugates (see ConjugateSpinor).  Then the following vectors

Given a null tetrad NullTetrad =$$$\[L\,N\,M\,{\overline{M\]}}^{}$, the command OrthonormalTetrad(NullTetrad) constructs the orthonormal tetrad defined by

The command DGGramSchmidt can also be used to construct an orthonormal tetrad.

The command GRQuery can be used to check that a given tetrad is a null tetrad or an orthonormal tetrad.

These commands are part of the DifferentialGeometry:-Tensor package, and so can be used in the form NullTetrad(...) or OrthonormalTetrad(...) only after executing the commands with(DifferentialGeometry); with(Tensor) in that order. They can always be used in the long form DifferentialGeometry:-Tensor:-NullTetrad or DifferentialGeometry:-Tensor:-OrthonormalTetrad.

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

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

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

$g≔\mathrm{evalDG}\left(\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$g:=\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$F≔\left[\mathrm{D\_t}\,\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

$F:=\left[\mathrm{D\_t}\,\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

$\mathrm{GRQuery}\left(F\,g\,''OrthonormalTetrad''\right)$

$\mathrm{true}$

$\mathrm{NT}≔\mathrm{NullTetrad}\left(F\right)$

$\mathrm{NT}:=\left[\frac{\sqrt{2}}{2}\mathrm{D\_t}+\frac{\sqrt{2}}{2}\mathrm{D\_z}\,\frac{\sqrt{2}}{2}\mathrm{D\_t}-\frac{\sqrt{2}}{2}\mathrm{D\_z}\,\frac{\sqrt{2}}{2}\mathrm{D\_x}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\,\frac{\sqrt{2}}{2}\mathrm{D\_x}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\right]$

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

$\mathrm{true}$

$\mathrm{TensorInnerProduct}\left(g\,\mathrm{NT}\,\mathrm{NT}\right)$

$\mathrm{DGsetup}\left(\left[t\,x\,y\,z\right]\,\left[\mathrm{z1}\,\mathrm{z2}\,\mathrm{w1}\,\mathrm{w2}\right]\,E\right)$

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

$\mathrm{g2}≔\mathrm{evalDG}\left(\mathrm{dt}\&t\mathrm{dt}-\mathrm{dx}\&t\mathrm{dx}-\mathrm{dy}\&t\mathrm{dy}-\mathrm{dz}\&t\mathrm{dz}\right)$

$\mathrm{g2}:=\mathrm{dt}\mathrm{dt}-\mathrm{dx}\mathrm{dx}-\mathrm{dy}\mathrm{dy}-\mathrm{dz}\mathrm{dz}$

$\mathrm{F2}≔\left[\mathrm{D\_t}\,\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

$\mathrm{F2}:=\left[\mathrm{D\_t}\,\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

$\mathrm{\sigma }≔\mathrm{SolderForm}\left(\mathrm{F2}\,\mathrm{indextype}=\left[''con''\,''cov''\,''cov''\right]\right)$

$\mathrm{\σ}:=\frac{\sqrt{2}}{2}\mathrm{D\_t}\mathrm{dz1}\mathrm{dw1}+\frac{\sqrt{2}}{2}\mathrm{D\_t}\mathrm{dz2}\mathrm{dw2}+\frac{\sqrt{2}}{2}\mathrm{D\_x}\mathrm{dz1}\mathrm{dw2}+\frac{\sqrt{2}}{2}\mathrm{D\_x}\mathrm{dz2}\mathrm{dw1}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\mathrm{dz1}\mathrm{dw2}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}\mathrm{dz2}\mathrm{dw1}+\frac{\sqrt{2}}{2}\mathrm{D\_z}\mathrm{dz1}\mathrm{dw1}-\frac{\sqrt{2}}{2}\mathrm{D\_z}\mathrm{dz2}\mathrm{dw2}$

$\mathrm{o}≔\mathrm{evalDG}\left(\mathrm{D\_z1}+2\mathrm{D\_z2}\right)$

$\mathrm{\ο}:=\mathrm{D\_z1}+2\mathrm{D\_z2}$

$\mathrm{\iota }≔\mathrm{evalDG}\left(2\mathrm{D\_z1}+5\mathrm{D\_z2}\right)$

$\mathrm{\ι}:=2\mathrm{D\_z1}+5\mathrm{D\_z2}$

$\mathrm{SpinorInnerProduct}\left(\mathrm{o}\,\mathrm{\iota }\right)$

$1$

$\mathrm{N2}≔\mathrm{NullTetrad}\left(\mathrm{\sigma }\,\left[\mathrm{o}\,\mathrm{\iota }\right]\right)$

$\mathrm{N2}:=\left[\frac{5\sqrt{2}}{2}\mathrm{D\_t}+2\sqrt{2}\mathrm{D\_x}-\frac{3\sqrt{2}}{2}\mathrm{D\_z}\,\frac{29\sqrt{2}}{2}\mathrm{D\_t}+10\sqrt{2}\mathrm{D\_x}-\frac{21\sqrt{2}}{2}\mathrm{D\_z}\,6\sqrt{2}\mathrm{D\_t}+\frac{9\sqrt{2}}{2}\mathrm{D\_x}+\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}-4\sqrt{2}\mathrm{D\_z}\,6\sqrt{2}\mathrm{D\_t}+\frac{9\sqrt{2}}{2}\mathrm{D\_x}-\frac{\mathrm{I}}{2}\sqrt{2}\mathrm{D\_y}-4\sqrt{2}\mathrm{D\_z}\right]$

$\mathrm{TensorInnerProduct}\left(\mathrm{g2}\,\mathrm{N2}\,\mathrm{N2}\right)$

$T≔\mathrm{OrthonormalTetrad}\left(\mathrm{N2}\right)$

$T:=\left[17\mathrm{D\_t}+12\mathrm{D\_x}-12\mathrm{D\_z}\,12\mathrm{D\_t}+9\mathrm{D\_x}-8\mathrm{D\_z}\,\mathrm{D\_y}\,-12\mathrm{D\_t}-8\mathrm{D\_x}+9\mathrm{D\_z}\right]$

$\mathrm{TensorInnerProduct}\left(\mathrm{g2}\,T\,T\right)$