Let $g$be a metric on a 4-dimensional manifold with signature$\left[1\, -1\, -1\, -1\right]\.$ A null vector $X$satisfies $g\left(X\, X\right) \= 0.$

Let $\mathrm{\σ}$ be a solder form for the metric$g\,$that is, $\mathrm{\σ}$ is a rank 3 spin-tensor such that $g_{\mathrm{ij}} \= {\mathrm{\σ}}_i^{ \mathrm{AA}\'}{\mathrm{\σ}}_{\mathrm{jAA}\' }^{}\.$ The NullVector command accepts, as its first argument, a solder form with either covariant or contravariant tensor and spinor indices.

With two arguments, the NullVector command returns the real vector with components $X^{i }\= {\mathrm{\σ}}^{\mathrm{iAA}\'}{\mathrm{\φ}}_A {\overline{\mathrm{\φ}}}_{A\'}$

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

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

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

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

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

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

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

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

$\mathrm{\sigma }≔\mathrm{SolderForm}\left(F\right)$

$\mathrm{\σ}:=\frac{1}{2}\sqrt{2}\mathrm{dt}\mathrm{D\_z1}\mathrm{D\_w1}\+\frac{1}{2}\sqrt{2}\mathrm{dt}\mathrm{D\_z2}\mathrm{D\_w2}\+\frac{1}{2}\sqrt{2}\mathrm{dx}\mathrm{D\_z1}\mathrm{D\_w2}\+\frac{1}{2}\sqrt{2}\mathrm{dx}\mathrm{D\_z2}\mathrm{D\_w1}-\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{dy}\mathrm{D\_z1}\mathrm{D\_w2}\+\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{dy}\mathrm{D\_z2}\mathrm{D\_w1}\+\frac{1}{2}\sqrt{2}\mathrm{dz}\mathrm{D\_z1}\mathrm{D\_w1}-\frac{1}{2}\sqrt{2}\mathrm{dz}\mathrm{D\_z2}\mathrm{D\_w2}$

$\mathrm{\φ1}≔\mathrm{D\_z1}$

$\mathrm{\φ1}:=\mathrm{D\_z1}$

$\mathrm{\φ2}≔\mathrm{evalDG}\left(a\mathrm{D\_z1}+b\mathrm{D\_z2}\right)$

$\mathrm{\φ2}:=a\mathrm{D\_z1}\+b\mathrm{D\_z2}$

$\mathrm{\φ3}≔\mathrm{D\_w2}$

$\mathrm{\φ3}:=\mathrm{D\_w2}$

$X≔\mathrm{NullVector}\left(\mathrm{\sigma }\,\mathrm{\φ1}\right)$

$X:=\frac{1}{2}\sqrt{2}\mathrm{D\_t}\+\frac{1}{2}\sqrt{2}\mathrm{D\_z}$

$Y≔\mathrm{NullVector}\left(\mathrm{\sigma }\,\mathrm{\φ2}\right)\mathbf{assuming}a::\mathrm{real},b::\mathrm{real}$

$Y:=\left(\frac{1}{2}\sqrt{2}{b}^2\+\frac{1}{2}\sqrt{2}{a}^2\right)\mathrm{D\_t}\+\sqrt{2}ab\mathrm{D\_x}\+\left(-\frac{1}{2}\sqrt{2}{b}^2\+\frac{1}{2}\sqrt{2}{a}^2\right)\mathrm{D\_z}$

$Z≔\mathrm{NullVector}\left(\mathrm{\sigma }\,\mathrm{\φ1}\,\mathrm{\φ3}\right)$

$Z:=\frac{1}{2}\sqrt{2}\mathrm{D\_x}\+\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{D\_y}$

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

$0$

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

$0$

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

$0$