The Plebanski tensor $P_{ }$is the rank 4 covariant tensor constructed the metric tensor $g$and a symmetric covariant trace-free rank two tensor $S$ by the formula

 The 2 component spinors $P_{\mathrm{ABCD}}$and $S_{\mathrm{ABA}\'B\' }$corresponding to the tensors $P_{\mathrm{abcd} }$and $S_{\mathrm{ab}}$are related by

The Newman Penrose coefficients$\{{\mathrm{\Ψ}}_0 \,$${\mathrm{\Psi }}_1 \,$${\mathrm{\Psi }}_2 \, {\mathrm{\Psi }}_3 \, {\mathrm{\Psi }}_4 \}$for $P_{\mathrm{abcd}}$are given in terms of the Newman-Penrose coefficients $\left\{{\mathrm{\Φ}}_{00}\, {\mathrm{\Phi }}_{01}\,{\mathrm{\Phi }}_{02}\,{\mathrm{\Phi }}_{10}\,{\mathrm{\Phi }}_{11}\,{\mathrm{\Phi }}_{12}\,{\mathrm{\Phi }}_{20}\,{\mathrm{\Φ}}_{21}\,{\mathrm{\Φ}}_{22} \right\}$ for $S_{\mathrm{ab} }$by

The command PlebanskiTensor calculates the Plebanski tensor $P$for a given tensor $S$. If a tensorial form of $S$is given, the tensorial form of $P$ is returned (first calling sequence); if the spinor components of $S$are given, the spinor components of $P$ are returned (second calling sequence); and if the Newman-Penrose components of $S$are given, then the Newman-Penrose components of $P$are returned (third calling sequence).

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

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

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

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

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

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

$S≔\mathrm{evalDG}\left(\mathrm{dx}\&s\mathrm{dy}\right)$

$S:=\frac{1}{2}\mathrm{dx}\mathrm{dy}+\frac{1}{2}\mathrm{dy}\mathrm{dx}$

$P≔\mathrm{PlebanskiTensor}\left(g\,S\right)$

$P:=-\frac{x^2}{12}\mathrm{dt}\mathrm{dx}\mathrm{dt}\mathrm{dx}+\frac{x^2}{12}\mathrm{dt}\mathrm{dx}\mathrm{dx}\mathrm{dt}-\frac{x^2}{12t^2}\mathrm{dt}\mathrm{dy}\mathrm{dt}\mathrm{dy}+\frac{x^2}{12t^2}\mathrm{dt}\mathrm{dy}\mathrm{dy}\mathrm{dt}+\frac{x^2}{6t^2}\mathrm{dt}\mathrm{dz}\mathrm{dt}\mathrm{dz}-\frac{x^2}{6t^2}\mathrm{dt}\mathrm{dz}\mathrm{dz}\mathrm{dt}+\frac{x^2}{12}\mathrm{dx}\mathrm{dt}\mathrm{dt}\mathrm{dx}-\frac{x^2}{12}\mathrm{dx}\mathrm{dt}\mathrm{dx}\mathrm{dt}-\frac{1}{6}\mathrm{dx}\mathrm{dy}\mathrm{dx}\mathrm{dy}+\frac{1}{6}\mathrm{dx}\mathrm{dy}\mathrm{dy}\mathrm{dx}+\frac{1}{12}\mathrm{dx}\mathrm{dz}\mathrm{dx}\mathrm{dz}-\frac{1}{12}\mathrm{dx}\mathrm{dz}\mathrm{dz}\mathrm{dx}+\frac{x^2}{12t^2}\mathrm{dy}\mathrm{dt}\mathrm{dt}\mathrm{dy}-\frac{x^2}{12t^2}\mathrm{dy}\mathrm{dt}\mathrm{dy}\mathrm{dt}+\frac{1}{6}\mathrm{dy}\mathrm{dx}\mathrm{dx}\mathrm{dy}-\frac{1}{6}\mathrm{dy}\mathrm{dx}\mathrm{dy}\mathrm{dx}+\frac{1}{12t^2}\mathrm{dy}\mathrm{dz}\mathrm{dy}\mathrm{dz}-\frac{1}{12t^2}\mathrm{dy}\mathrm{dz}\mathrm{dz}\mathrm{dy}-\frac{x^2}{6t^2}\mathrm{dz}\mathrm{dt}\mathrm{dt}\mathrm{dz}+\frac{x^2}{6t^2}\mathrm{dz}\mathrm{dt}\mathrm{dz}\mathrm{dt}-\frac{1}{12}\mathrm{dz}\mathrm{dx}\mathrm{dx}\mathrm{dz}+\frac{1}{12}\mathrm{dz}\mathrm{dx}\mathrm{dz}\mathrm{dx}-\frac{1}{12t^2}\mathrm{dz}\mathrm{dy}\mathrm{dy}\mathrm{dz}+\frac{1}{12t^2}\mathrm{dz}\mathrm{dy}\mathrm{dz}\mathrm{dy}$

$\mathrm{SymmetrizeIndices}\left(P\,\left[1\,2\right]\,''Symmetric''\right)$

$0\mathrm{dt}\mathrm{dt}\mathrm{dt}\mathrm{dt}$

$\mathrm{SymmetrizeIndices}\left(P\,\left[3\,4\right]\,''Symmetric''\right)$

$0\mathrm{dt}\mathrm{dt}\mathrm{dt}\mathrm{dt}$

$\mathrm{SymmetrizeIndices}\left(P\,\left[1\,2\,3\right]\,''SkewSymmetric''\right)$

$0\mathrm{dt}\mathrm{dt}\mathrm{dt}\mathrm{dt}$

$h≔\mathrm{InverseMetric}\left(g\right)$

$h:=\frac{1}{x^2}\mathrm{D\_t}\mathrm{D\_t}-\frac{1}{t^2}\mathrm{D\_x}\mathrm{D\_x}-\mathrm{D\_y}\mathrm{D\_y}-\mathrm{D\_z}\mathrm{D\_z}$

$\mathrm{ContractIndices}\left(h\,P\,\left[\left[1\,1\right]\,\left[2\,3\right]\right]\right)$

$0\mathrm{dt}\mathrm{dt}$

$\mathrm{ot}≔\mathrm{evalDG}\left(\left[\frac{1}{x}\mathrm{D\_t}\,\frac{1}{t}\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]\right)$

$\mathrm{ot}:=\left[\frac{1}{x}\mathrm{D\_t}\,\frac{1}{t}\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]$

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

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

$\mathrm{\phi }≔\mathrm{RicciSpinor}\left(\mathrm{\sigma }\,S\right)$

$\mathrm{\φ}:=-\frac{\frac{\mathrm{I}}{4}}{t}\mathrm{dz1}\mathrm{dz1}\mathrm{dw2}\mathrm{dw2}+\frac{\frac{\mathrm{I}}{4}}{t}\mathrm{dz2}\mathrm{dz2}\mathrm{dw1}\mathrm{dw1}$

$\mathrm{\psi }≔\mathrm{PlebanskiTensor}\left(\mathrm{\phi }\right)$

$\mathrm{\ψ}:=\frac{1}{12t^2}\mathrm{dz1}\mathrm{dz1}\mathrm{dz2}\mathrm{dz2}+\frac{1}{12t^2}\mathrm{dz1}\mathrm{dz2}\mathrm{dz1}\mathrm{dz2}+\frac{1}{12t^2}\mathrm{dz1}\mathrm{dz2}\mathrm{dz2}\mathrm{dz1}+\frac{1}{12t^2}\mathrm{dz2}\mathrm{dz1}\mathrm{dz1}\mathrm{dz2}+\frac{1}{12t^2}\mathrm{dz2}\mathrm{dz1}\mathrm{dz2}\mathrm{dz1}+\frac{1}{12t^2}\mathrm{dz2}\mathrm{dz2}\mathrm{dz1}\mathrm{dz1}$

$\mathrm{\ψ1}≔\mathrm{WeylSpinor}\left(\mathrm{\sigma }\,P\right)$

$\mathrm{\ψ1}:=\frac{1}{12t^2}\mathrm{dz1}\mathrm{dz1}\mathrm{dz2}\mathrm{dz2}+\frac{1}{12t^2}\mathrm{dz1}\mathrm{dz2}\mathrm{dz1}\mathrm{dz2}+\frac{1}{12t^2}\mathrm{dz1}\mathrm{dz2}\mathrm{dz2}\mathrm{dz1}+\frac{1}{12t^2}\mathrm{dz2}\mathrm{dz1}\mathrm{dz1}\mathrm{dz2}+\frac{1}{12t^2}\mathrm{dz2}\mathrm{dz1}\mathrm{dz2}\mathrm{dz1}+\frac{1}{12t^2}\mathrm{dz2}\mathrm{dz2}\mathrm{dz1}\mathrm{dz1}$

$\mathrm{\psi }\&minus\mathrm{\psi }$

$0\mathrm{dz1}\mathrm{dz1}\mathrm{dz1}\mathrm{dz1}$

$\mathrm{nt}≔\mathrm{NullTetrad}\left(\mathrm{ot}\right)$

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

$\mathrm{NPRicciForS}≔\mathrm{NPCurvatureScalars}\left(\mathrm{nt}\,g\,S\,\mathrm{output}=\left[''AllRicciScalars''\right]\right)$

$\mathrm{NPRicciForS}:=\mathrm{table}\left(\left[''Phi22''\=0\,''Phi12''\=0\,''Phi20''\=-\frac{\frac{1}{4}\mathrm{I}}{t}\,''Phi21''\=0\,''Phi11''\=0\,''Phi10''\=0\,''Phi01''\=0\,''Phi02''\=\frac{\frac{1}{4}\mathrm{I}}{t}\,''Phi00''\=0\,''Lambda''\=0\right]\right)$

$\mathrm{NPWeylForP}≔\mathrm{PlebanskiTensor}\left(\mathrm{NPRicciForS}\right)$

$\mathrm{NPWeylForP}:=\mathrm{table}\left(\left[''Psi4''\=0\,''Psi1''\=0\,''Psi0''\=0\,''Psi3''\=0\,''Psi2''\=-\frac{1}{12t^2}\right]\right)$

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

$\mathrm{table}\left(\left[''Psi4''\=0\,''Psi1''\=0\,''Psi0''\=0\,''Psi3''\=0\,''Psi2''\=-\frac{1}{12t^2}\right]\right)$