The principal null directions of a spacetime are a fundamental set of invariant directions which play an important role in studying the geometry of the spacetime. Let $g$be the metric of the spacetime and let $W$ be the Weyl tensor for $g$. A principal null direction is a null vector $k$ which satisfies the system of non-linear algebraic equations

The command PrincipalNullDirections returns a list of distinct principal null directions for a spacetime as determined by a given null tetrad.

The complicated algebraic equations that must be solved to determine the principal null directions can often be simplified a priori by using an adapted null tetrad. See AdaptedNullTetrad.

The principal null directions are calculated from the factorization of the Weyl spinor as a product of rank 1 spinors -- each distinct spinor in this factorization determines a principal null direction. See WeylSpinor, FactorWeylSpinor.

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

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

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

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

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

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

$\mathrm{g1}:=-t^2\mathrm{dx}\mathrm{dx}-x^2\mathrm{dy}\mathrm{dy}\+\mathrm{dt}\mathrm{dt}-\mathrm{dz}\mathrm{dz}$

$\mathrm{OT1}≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D\_t}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]\&comma;\mathrm{g1}\&comma;\mathrm{signature}=\left[1\&comma;-1\&comma;-1\&comma;-1\right]\right)\mathbf{assuming}0<t,0<x$

$\mathrm{OT1}:=\left[\mathrm{D\_t}\&comma;\frac{\mathrm{D\_x}}{t}\&comma;\frac{\mathrm{D\_y}}{x}\&comma;\mathrm{D\_z}\right]$

$\mathrm{NT1}≔\mathrm{NullTetrad}\left(\mathrm{OT1}\right)$

$\mathrm{NT1}:=\left[\frac{1}{2}\sqrt{2}\mathrm{D\_t}\&plus;\frac{1}{2}\sqrt{2}\mathrm{D\_z}\&comma;\frac{1}{2}\sqrt{2}\mathrm{D\_t}-\frac{1}{2}\sqrt{2}\mathrm{D\_z}\&comma;\frac{1}{2}\frac{\sqrt{2}\mathrm{D\_x}}{t}\&plus;\frac{\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{D\_y}}{x}\&comma;\frac{1}{2}\frac{\sqrt{2}\mathrm{D\_x}}{t}-\frac{\frac{1}{2}\mathrm{I}\sqrt{2}\mathrm{D\_y}}{x}\right]$

$\mathrm{PetrovType}\left(\mathrm{NT1}\right)$

$''I''$

$\mathrm{PND1}≔\mathrm{PrincipalNullDirections}\left(\mathrm{NT1}\&comma;''I''\right)$

$\mathrm{PND1}:=\left[-\sqrt{2}\left(\sqrt{2}-2\right)\mathrm{D\_t}\&plus;\sqrt{2}\left(\sqrt{2}-2\right)\mathrm{D\_z}\&comma;-\sqrt{2}\left(\sqrt{2}-2\right)\mathrm{D\_t}\&plus;\frac{\sqrt{2}\left(\sqrt{2}-2\right)\mathrm{D\_y}}{x}\&comma;-\sqrt{2}\left(\sqrt{2}-2\right){\left(1\&plus;\sqrt{2}\right)}^2\mathrm{D\_t}-\frac{\sqrt{2}\left(\sqrt{2}-2\right){\left(1\&plus;\sqrt{2}\right)}^2\mathrm{D\_y}}{x}\&comma;-\frac{1}{2}\sqrt{2}\left(\sqrt{2}-2\right){\left(2\&plus;\sqrt{2}\right)}^2\mathrm{D\_t}-\frac{1}{2}\sqrt{2}\left(\sqrt{2}-2\right){\left(2\&plus;\sqrt{2}\right)}^2\mathrm{D\_z}\right]$

$\mathrm{W1}≔\mathrm{WeylTensor}\left(\mathrm{g1}\right)\&colon;$

$\mathrm{map}\left(\mathrm{GRQuery}\&comma;\mathrm{PND1}\&comma;\mathrm{g1}\&comma;\mathrm{W1}\&comma;''PrincipalNullDirection''\right)$

$\left[\mathrm{true}\&comma;\mathrm{true}\&comma;\mathrm{true}\&comma;\mathrm{true}\right]$

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

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

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

$\mathrm{g2}:=-\mathrm{dr}\mathrm{du}-\mathrm{du}\mathrm{dr}\&plus;\frac{2\left(3rx\&plus;m\right)\mathrm{du}\mathrm{du}}{r}\&plus;\frac{1}{4}\frac{r^2\mathrm{dx}\mathrm{dx}}{x^3}\&plus;\frac{1}{4}\frac{r^2\mathrm{dy}\mathrm{dy}}{x^3}$

$\mathrm{OT2}≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D\_r}\&comma;\mathrm{D\_u}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\right]\&comma;\mathrm{g2}\&comma;\mathrm{signature}=\left[\left[-1\&comma;1\right]\&comma;1\&comma;1\right]\right)\mathbf{assuming}0<r,0<x,0<m$

$\mathrm{OT2}:=\left[\frac{1}{2}\frac{\sqrt{2}\left(3rx\&plus;m\&plus;r\right)\mathrm{D\_r}}{r}\&plus;\frac{1}{2}\sqrt{2}\mathrm{D\_u}\&comma;-\frac{1}{2}\frac{\sqrt{2}\left(3rx\&plus;m-r\right)\mathrm{D\_r}}{r}-\frac{1}{2}\sqrt{2}\mathrm{D\_u}\&comma;\frac{2x^{3\&sol;2}\mathrm{D\_x}}{r}\&comma;\frac{2x^{3\&sol;2}\mathrm{D\_y}}{r}\right]$

$\mathrm{NT2}≔\mathrm{NullTetrad}\left(\mathrm{OT2}\right)$

$\mathrm{NT2}:=\left[\frac{1}{2}\frac{\left(3rx\&plus;m\&plus;r\right)\mathrm{D\_r}}{r}\&plus;\frac{1}{2}\mathrm{D\_u}\&plus;\frac{\sqrt{2}{x}^{3\&sol;2}\mathrm{D\_y}}{r}\&comma;\frac{1}{2}\frac{\left(3rx\&plus;m\&plus;r\right)\mathrm{D\_r}}{r}\&plus;\frac{1}{2}\mathrm{D\_u}-\frac{\sqrt{2}{x}^{3\&sol;2}\mathrm{D\_y}}{r}\&comma;-\frac{1}{2}\frac{\left(3rx\&plus;m-r\right)\mathrm{D\_r}}{r}-\frac{1}{2}\mathrm{D\_u}\&plus;\frac{\mathrm{I}\sqrt{2}{x}^{3\&sol;2}\mathrm{D\_x}}{r}\&comma;-\frac{1}{2}\frac{\left(3rx\&plus;m-r\right)\mathrm{D\_r}}{r}-\frac{1}{2}\mathrm{D\_u}-\frac{\mathrm{I}\sqrt{2}{x}^{3\&sol;2}\mathrm{D\_x}}{r}\right]$

$\mathrm{NT2}≔\mathrm{NullTetrad}\left(\left[\mathrm{OT2}\left[1\right]\&comma;\mathrm{OT2}\left[4\right]\&comma;\mathrm{OT2}\left[3\right]\&comma;\mathrm{OT2}\left[2\right]\right]\right)$

$\mathrm{NT2}:=\left[\mathrm{D\_r}\&comma;\frac{\left(3rx\&plus;m\right)\mathrm{D\_r}}{r}\&plus;\mathrm{D\_u}\&comma;\frac{\mathrm{I}\sqrt{2}{x}^{3\&sol;2}\mathrm{D\_x}}{r}\&plus;\frac{\sqrt{2}{x}^{3\&sol;2}\mathrm{D\_y}}{r}\&comma;-\frac{\mathrm{I}\sqrt{2}{x}^{3\&sol;2}\mathrm{D\_x}}{r}\&plus;\frac{\sqrt{2}{x}^{3\&sol;2}\mathrm{D\_y}}{r}\right]$

$\mathrm{PetrovType}\left(\mathrm{NT2}\right)$

$''II''$

$\mathrm{W2}≔\mathrm{WeylTensor}\left(\mathrm{g2}\right)\&colon;$

$\mathrm{PND2}≔\mathrm{PrincipalNullDirections}\left(\mathrm{NT2}\&comma;''II''\&comma;\mathrm{W2}\right)\mathbf{assuming}0<x,0<y,0<r,3m-2xr<0,m<0$

$\mathrm{PND2}:=\left[-\frac{\sqrt{r}\sqrt{2rx-3m}x\mathrm{D\_r}}{m}\&comma;-\frac{\left(2x^{5\&sol;2}\sqrt{2}{r}^{5\&sol;2}\sqrt{2rx-3m}\&plus;4r^3{x}^3-3m{r}^2{x}^2\&plus;3m^{2}rx\&plus;m^3\right)\mathrm{D\_r}}{r^{3\&sol;2}\sqrt{2rx-3m}xm}-\frac{m\mathrm{D\_u}}{\sqrt{r}\sqrt{2rx-3m}x}-\frac{2\sqrt{2}x\left(\sqrt{2}\sqrt{r}x\&plus;\sqrt{x}\sqrt{2rx-3m}\right)\mathrm{D\_x}}{r\sqrt{2rx-3m}}\&comma;\frac{\left(2x^{5\&sol;2}\sqrt{2}{r}^{5\&sol;2}\sqrt{2rx-3m}-4r^3{x}^3\&plus;3m{r}^2{x}^2-3m^{2}rx-m^3\right)\mathrm{D\_r}}{r^{3\&sol;2}\sqrt{2rx-3m}xm}-\frac{m\mathrm{D\_u}}{\sqrt{r}\sqrt{2rx-3m}x}\&plus;\frac{2\sqrt{2}x\left(-\sqrt{2}\sqrt{r}x\&plus;\sqrt{x}\sqrt{2rx-3m}\right)\mathrm{D\_x}}{r\sqrt{2rx-3m}}\right]$

$\mathrm{GRQuery}\left(\mathrm{PND2}\left[1\right]\&comma;\mathrm{g2}\&comma;\mathrm{W2}\&comma;''PrincipalNullDirection''\right)$

$\mathrm{true}$

$\mathrm{GRQuery}\left(\mathrm{PND2}\left[2\right]\&comma;\mathrm{g2}\&comma;\mathrm{W2}\&comma;''PrincipalNullDirection''\right)$

$\mathrm{true}$

$\mathrm{GRQuery}\left(\mathrm{PND2}\left[3\right]\&comma;\mathrm{g2}\&comma;\mathrm{W2}\&comma;''PrincipalNullDirection''\right)$

$\mathrm{true}$

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

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

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

$\mathrm{g3}:=-\mathrm{dr}\mathrm{du}-\mathrm{du}\mathrm{dr}\&plus;\frac{3}{2}x\mathrm{du}\mathrm{du}\&plus;\frac{r^2\mathrm{dx}\mathrm{dx}}{x^3}\&plus;\frac{r^2\mathrm{dy}\mathrm{dy}}{x^3}$

$\mathrm{OT3}≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D\_r}\&comma;\mathrm{D\_u}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\right]\&comma;\mathrm{g3}\&comma;\mathrm{signature}=\left[\left[-1\&comma;1\right]\&comma;1\&comma;1\right]\right)\mathbf{assuming}0<r,0<x$

$\mathrm{OT3}:=\left[\frac{1}{8}\sqrt{2}\left(3x\&plus;4\right)\mathrm{D\_r}\&plus;\frac{1}{2}\sqrt{2}\mathrm{D\_u}\&comma;-\frac{1}{8}\sqrt{2}\left(3x-4\right)\mathrm{D\_r}-\frac{1}{2}\sqrt{2}\mathrm{D\_u}\&comma;\frac{x^{3\&sol;2}\mathrm{D\_x}}{r}\&comma;\frac{x^{3\&sol;2}\mathrm{D\_y}}{r}\right]$

$\mathrm{NT3}≔\mathrm{NullTetrad}\left(\left[\mathrm{OT3}\left[1\right]\&comma;\mathrm{OT3}\left[4\right]\&comma;\mathrm{OT3}\left[3\right]\&comma;\mathrm{OT3}\left[2\right]\right]\right)$

$\mathrm{NT3}:=\left[\mathrm{D\_r}\&comma;\frac{3}{4}x\mathrm{D\_r}\&plus;\mathrm{D\_u}\&comma;\frac{\frac{1}{2}\mathrm{I}\sqrt{2}{x}^{3\&sol;2}\mathrm{D\_x}}{r}\&plus;\frac{1}{2}\frac{\sqrt{2}{x}^{3\&sol;2}\mathrm{D\_y}}{r}\&comma;-\frac{\frac{1}{2}\mathrm{I}\sqrt{2}{x}^{3\&sol;2}\mathrm{D\_x}}{r}\&plus;\frac{1}{2}\frac{\sqrt{2}{x}^{3\&sol;2}\mathrm{D\_y}}{r}\right]$

$\mathrm{PetrovType}\left(\mathrm{NT3}\right)$

$''III''$

$\mathrm{PND3a}≔\mathrm{PrincipalNullDirections}\left(\mathrm{NT3}\&comma;''III''\right)$

$\mathrm{PND3a}:=\left[\frac{32}{9}\frac{r^4\mathrm{D\_r}}{{\left|x\right|}^3}\&comma;\frac{27}{1024}\frac{x\left(8{\left|x\right|}^3\&plus;3x^3\right)\mathrm{D\_r}}{r^4}\&plus;\frac{9}{32}\frac{{\left|x\right|}^3\mathrm{D\_u}}{r^4}\&plus;\frac{27}{256}\frac{x^2\left({\left|x\right|}^3\&plus;x^3\right)\mathrm{D\_x}}{r^5}-\frac{\frac{27}{256}\mathrm{I}{x}^2\left({\left|x\right|}^3-x^3\right)\mathrm{D\_y}}{r^5}\right]$

$\mathrm{PND3b}≔\mathrm{simplify}\left(\mathrm{PND3a}\right)\mathbf{assuming}0<x$

$\mathrm{PND3b}:=\left[\frac{32}{9}\frac{r^4\mathrm{D\_r}}{x^3}\&comma;\frac{297}{1024}\frac{x^4\mathrm{D\_r}}{r^4}\&plus;\frac{9}{32}\frac{x^3\mathrm{D\_u}}{r^4}\&plus;\frac{27}{128}\frac{x^5\mathrm{D\_x}}{r^5}\&plus;0\mathrm{D\_y}\right]$

$\mathrm{PND3}≔\mathrm{evalDG}\left(\left[\mathrm{D\_r}\&comma;\frac{\frac{32}{3}{r}^4}{x^3}\mathrm{PND3b}\left[2\right]\right]\right)$

$\mathrm{PND3}:=\left[\mathrm{D\_r}\&comma;\frac{99}{32}x\mathrm{D\_r}\&plus;3\mathrm{D\_u}\&plus;\frac{9}{4}\frac{x^2\mathrm{D\_x}}{r}\right]$

$\mathrm{W3}≔\mathrm{WeylTensor}\left(\mathrm{g3}\right)\&colon;$

$\mathrm{GRQuery}\left(\mathrm{PND3}\left[1\right]\&comma;\mathrm{g3}\&comma;\mathrm{W3}\&comma;''PrincipalNullDirection''\right)$

$\mathrm{true}$

$\mathrm{GRQuery}\left(\mathrm{PND3}\left[2\right]\&comma;\mathrm{g3}\&comma;\mathrm{W3}\&comma;''PrincipalNullDirection''\right)$

$\mathrm{true}$

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

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

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

$\mathrm{g4}:=-\mathrm{dt}\mathrm{dt}-{\&ExponentialE;}^x\mathrm{dt}\mathrm{dz}\&plus;\mathrm{dx}\mathrm{dx}\&plus;\mathrm{dy}\mathrm{dy}-{\&ExponentialE;}^x\mathrm{dz}\mathrm{dt}-\frac{1}{2}{\&ExponentialE;}^{2x}\mathrm{dz}\mathrm{dz}$

$\mathrm{OT4}≔\mathrm{DGGramSchmidt}\left(\left[\mathrm{D\_t}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;\mathrm{D\_z}\right]\&comma;\mathrm{g4}\&comma;\mathrm{signature}=\left[-1\&comma;1\&comma;1\&comma;1\right]\right)$

$\mathrm{OT4}:=\left[\mathrm{D\_t}\&comma;\mathrm{D\_x}\&comma;\mathrm{D\_y}\&comma;-\frac{\sqrt{2}{\&ExponentialE;}^x\mathrm{D\_t}}{\sqrt{{\&ExponentialE;}^{2x}}}\&plus;\frac{\sqrt{2}\mathrm{D\_z}}{\sqrt{{\&ExponentialE;}^{2x}}}\right]$

$\mathrm{NT4}≔\mathrm{NullTetrad}\left(\mathrm{OT4}\right)\mathbf{assuming}0<x$

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

$\mathrm{PetrovType}\left(\mathrm{NT4}\right)$

$''D''$

$\mathrm{PND4}≔\mathrm{PrincipalNullDirections}\left(\mathrm{NT4}\&comma;''D''\right)$

$\mathrm{PND4}:=\left[\sqrt{2}\mathrm{D\_t}-\sqrt{2}\mathrm{D\_y}\&comma;\frac{1}{4}\sqrt{2}\mathrm{D\_t}\&plus;\frac{1}{4}\sqrt{2}\mathrm{D\_y}\right]$

$\mathrm{W4}≔\mathrm{WeylTensor}\left(\mathrm{g4}\right)\&colon;$

$\mathrm{GRQuery}\left(\mathrm{PND4}\left[1\right]\&comma;\mathrm{g4}\&comma;\mathrm{W4}\&comma;''PrincipalNullDirection''\right)$

$\mathrm{true}$

$\mathrm{GRQuery}\left(\mathrm{PND4}\left[2\right]\&comma;\mathrm{g4}\&comma;\mathrm{W4}\&comma;''PrincipalNullDirection''\right)$

$\mathrm{true}$

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

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

$\mathrm{g5},\mathrm{NT5}≔\mathrm{op}\left(\mathrm{Library}:-\mathrm{Retrieve}\left(''Stephani''\&comma;1\&comma;\left[12\&comma;34\&comma;1\right]\&comma;\mathrm{manifoldname}=M\&comma;\mathrm{output}=\left[''Fields''\&comma;''NullTetrad''\right]\right)\right)\&colon;$

$\mathrm{g5}$

${\&ExponentialE;}^{-2z}\mathrm{du}\mathrm{dx}\&plus;{\&ExponentialE;}^{-2z}\mathrm{dx}\mathrm{du}\&plus;\mathrm{\_\&varepsilon;}{\&ExponentialE;}^z\mathrm{dx}\mathrm{dx}\&plus;{\&ExponentialE;}^{-2z}\mathrm{dy}\mathrm{dy}\&plus;\frac{3\mathrm{dz}\mathrm{dz}}{\left|\mathrm{\_\&Lambda;}\right|}$

$\mathrm{NT5}$

$\left[2{\&ExponentialE;}^z\mathrm{D\_u}\&comma;\left(\frac{1}{4}{\&ExponentialE;}^{4z}\mathrm{\_\&varepsilon;}\&plus;\frac{1}{2}{\&ExponentialE;}^z\right)\mathrm{D\_u}-\frac{1}{2}{\&ExponentialE;}^z\mathrm{D\_x}\&plus;\frac{1}{6}\sqrt{2}\sqrt{3}\sqrt{-\mathrm{\_\&Lambda;}}\mathrm{D\_z}\&comma;{\&ExponentialE;}^z\mathrm{D\_u}\&plus;\frac{1}{2}\mathrm{I}\sqrt{2}{\&ExponentialE;}^z\mathrm{D\_y}\&plus;\frac{1}{6}\sqrt{2}\sqrt{3}\sqrt{-\mathrm{\_\&Lambda;}}\mathrm{D\_z}\&comma;{\&ExponentialE;}^z\mathrm{D\_u}-\frac{1}{2}\mathrm{I}\sqrt{2}{\&ExponentialE;}^z\mathrm{D\_y}\&plus;\frac{1}{6}\sqrt{2}\sqrt{3}\sqrt{-\mathrm{\_\&Lambda;}}\mathrm{D\_z}\right]$

$\mathrm{PetrovType}\left(\mathrm{NT5}\right)$

$''N''$

$\mathrm{PND5}≔\mathrm{PrincipalNullDirections}\left(\mathrm{NT5}\&comma;''N''\right)$

$\mathrm{PND5}:=\left[2{\&ExponentialE;}^z\mathrm{D\_u}\right]$

$\mathrm{W5}≔\mathrm{WeylTensor}\left(\mathrm{g5}\right)\&colon;$

$\mathrm{GRQuery}\left(\mathrm{PND5}\left[1\right]\&comma;\mathrm{g5}\&comma;\mathrm{W5}\&comma;''PrincipalNullDirection''\right)$

$\mathrm{true}$