PlebanskiTensor - Maple Help

Tensor[PlebanskiTensor] - calculate the Plebanski tensor from a trace-free rank 2 symmetric tensor, the Plebanski spinor from a symmetric (2, 2) spinor, the Plebanski Newman-Penrose coefficients from a table of Newman-Penrose Ricci coefficients

Calling Sequences

PlebanskiTensor(g, A)

PlebanskiTensor(S)

PlebanskiTensor(T$)$

Parameters

g     - the metric tensor on a 4 dimensional manifold $M$

S     -  a symmetric, trace-free, covariant rank 2 tensor  or a rank 4 spinor on $M$

T     - a table with indices

Description

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

The tensor has all the algebraic properties of the Weyl tensor. It is skew-symmetric in the indices $\mathrm{ab}$ and satisfies the cyclic identity onand is trace-free with respect to the metric $g$.

 • The 2 component spinors and corresponding to the tensors and are related by

 • The Newman Penrose coefficientsfor are given in terms of the Newman-Penrose coefficients  for by

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

Examples

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

Example 1.

First create a spinor bundle with space-time coordinates (and spinor coordinates (Spinors are not needed for this first example but will be used in Example 2.)

 > $\mathrm{DGsetup}\left(\left[t,x,y,z\right],\left[\mathrm{z1},\mathrm{z2},\mathrm{w1},\mathrm{w2}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)

Define a metric tensor $g$.

 M > $g≔\mathrm{evalDG}\left({x}^{2}\mathrm{dt}&t\mathrm{dt}-{t}^{2}\mathrm{dx}&t\mathrm{dx}-\mathrm{dy}&t\mathrm{dy}-\mathrm{dz}&t\mathrm{dz}\right)$
 ${g}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{x}}^{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}{{t}}^{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{{x}}^{{2}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}{{t}}^{{2}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-1}\right]\right]\right]\right)$ (2.2)

Define a symmetric, trace-free, rank 2 tensor.

 M > $S≔\mathrm{evalDG}\left(\mathrm{dx}&s\mathrm{dy}\right)$
 ${S}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{2}{,}{3}\right]{,}\frac{{1}}{{2}}\right]{,}\left[\left[{3}{,}{2}\right]{,}\frac{{1}}{{2}}\right]\right]\right]\right)$ (2.3)

Compute the Plebanski tensor of $S.$

 M > $P≔\mathrm{PlebanskiTensor}\left(g,S\right)$
 ${P}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{1}{,}{2}\right]{,}{-}\frac{{{x}}^{{2}}}{{12}}\right]{,}\left[\left[{1}{,}{2}{,}{2}{,}{1}\right]{,}\frac{{{x}}^{{2}}}{{12}}\right]{,}\left[\left[{1}{,}{3}{,}{1}{,}{3}\right]{,}{-}\frac{{{x}}^{{2}}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{1}{,}{3}{,}{3}{,}{1}\right]{,}\frac{{{x}}^{{2}}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{1}{,}{4}{,}{1}{,}{4}\right]{,}\frac{{{x}}^{{2}}}{{6}{}{{t}}^{{2}}}\right]{,}\left[\left[{1}{,}{4}{,}{4}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}}{{6}{}{{t}}^{{2}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}{,}{2}\right]{,}\frac{{{x}}^{{2}}}{{12}}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}}{{12}}\right]{,}\left[\left[{2}{,}{3}{,}{2}{,}{3}\right]{,}{-}\frac{{1}}{{6}}\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{2}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{2}{,}{4}{,}{2}{,}{4}\right]{,}\frac{{1}}{{12}}\right]{,}\left[\left[{2}{,}{4}{,}{4}{,}{2}\right]{,}{-}\frac{{1}}{{12}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}\frac{{{x}}^{{2}}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{3}{,}{2}{,}{2}{,}{3}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{3}{,}{2}{,}{3}{,}{2}\right]{,}{-}\frac{{1}}{{6}}\right]{,}\left[\left[{3}{,}{4}{,}{3}{,}{4}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{3}{,}{4}{,}{4}{,}{3}\right]{,}{-}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{4}{,}{1}{,}{1}{,}{4}\right]{,}{-}\frac{{{x}}^{{2}}}{{6}{}{{t}}^{{2}}}\right]{,}\left[\left[{4}{,}{1}{,}{4}{,}{1}\right]{,}\frac{{{x}}^{{2}}}{{6}{}{{t}}^{{2}}}\right]{,}\left[\left[{4}{,}{2}{,}{2}{,}{4}\right]{,}{-}\frac{{1}}{{12}}\right]{,}\left[\left[{4}{,}{2}{,}{4}{,}{2}\right]{,}\frac{{1}}{{12}}\right]{,}\left[\left[{4}{,}{3}{,}{3}{,}{4}\right]{,}{-}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{4}{,}{3}{,}{4}{,}{3}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{1}{,}{2}\right]{,}{-}\frac{{{x}}^{{2}}}{{12}}\right]{,}\left[\left[{1}{,}{2}{,}{2}{,}{1}\right]{,}\frac{{{x}}^{{2}}}{{12}}\right]{,}\left[\left[{1}{,}{3}{,}{1}{,}{3}\right]{,}{-}\frac{{{x}}^{{2}}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{1}{,}{3}{,}{3}{,}{1}\right]{,}\frac{{{x}}^{{2}}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{1}{,}{4}{,}{1}{,}{4}\right]{,}\frac{{{x}}^{{2}}}{{6}{}{{t}}^{{2}}}\right]{,}\left[\left[{1}{,}{4}{,}{4}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}}{{6}{}{{t}}^{{2}}}\right]{,}\left[\left[{2}{,}{1}{,}{1}{,}{2}\right]{,}\frac{{{x}}^{{2}}}{{12}}\right]{,}\left[\left[{2}{,}{1}{,}{2}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}}{{12}}\right]{,}\left[\left[{2}{,}{3}{,}{2}{,}{3}\right]{,}{-}\frac{{1}}{{6}}\right]{,}\left[\left[{2}{,}{3}{,}{3}{,}{2}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{2}{,}{4}{,}{2}{,}{4}\right]{,}\frac{{1}}{{12}}\right]{,}\left[\left[{2}{,}{4}{,}{4}{,}{2}\right]{,}{-}\frac{{1}}{{12}}\right]{,}\left[\left[{3}{,}{1}{,}{1}{,}{3}\right]{,}\frac{{{x}}^{{2}}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{3}{,}{1}{,}{3}{,}{1}\right]{,}{-}\frac{{{x}}^{{2}}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{3}{,}{2}{,}{2}{,}{3}\right]{,}\frac{{1}}{{6}}\right]{,}\left[\left[{3}{,}{2}{,}{3}{,}{2}\right]{,}{-}\frac{{1}}{{6}}\right]{,}\left[\left[{3}{,}{4}{,}{3}{,}{4}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{3}{,}{4}{,}{4}{,}{3}\right]{,}{-}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{4}{,}{1}{,}{1}{,}{4}\right]{,}{-}\frac{{{x}}^{{2}}}{{6}{}{{t}}^{{2}}}\right]{,}\left[\left[{4}{,}{1}{,}{4}{,}{1}\right]{,}\frac{{{x}}^{{2}}}{{6}{}{{t}}^{{2}}}\right]{,}\left[\left[{4}{,}{2}{,}{2}{,}{4}\right]{,}{-}\frac{{1}}{{12}}\right]{,}\left[\left[{4}{,}{2}{,}{4}{,}{2}\right]{,}\frac{{1}}{{12}}\right]{,}\left[\left[{4}{,}{3}{,}{3}{,}{4}\right]{,}{-}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{4}{,}{3}{,}{4}{,}{3}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]\right]\right]\right)$ (2.4)

We check that the tensor $P$ has the same algebraic properties as the Weyl tensor.  We use the command SymmetrizeIndices to show that  $P$ is skew-symmetric on its 1st and 2nd indices

 M > $\mathrm{SymmetrizeIndices}\left(P,\left[1,2\right],"Symmetric"\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)$ (2.5)

The Plebanski tensor is skew-symmetric on its 3rd and 4th indices

 M > $\mathrm{SymmetrizeIndices}\left(P,\left[3,4\right],"Symmetric"\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)$ (2.6)

The Plebanski tensor satisfies the cyclic identity on its first 3 indices.

 M > $\mathrm{SymmetrizeIndices}\left(P,\left[1,2,3\right],"SkewSymmetric"\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}{,}{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)$ (2.7)

The Plebanski tensor is also trace-free on its 1st and 3rd indices.  To check this we use the commands InverseMetric and ContractIndices  to evaluate  ${g}^{\mathrm{ac}}{P}_{\mathrm{abcd}}$.

 M > $h≔\mathrm{InverseMetric}\left(g\right)$
 ${h}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{x}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}\frac{{1}}{{{t}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"con_bas"}{,}{"con_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}\frac{{1}}{{{x}}^{{2}}}\right]{,}\left[\left[{2}{,}{2}\right]{,}{-}\frac{{1}}{{{t}}^{{2}}}\right]{,}\left[\left[{3}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{4}{,}{4}\right]{,}{-1}\right]\right]\right]\right)$ (2.8)
 M > $\mathrm{ContractIndices}\left(h,P,\left[\left[1,1\right],\left[2,3\right]\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"cov_bas"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{1}\right]{,}{0}\right]\right]\right]\right)$ (2.9)

Example 2.

In this example we will convert the tensor S to a spinor and compute the spinor form of the Plebanski tensor. We start by defining an orthonormal tetrad for the metric $g$ and using this tetrad and the command SolderForm to construct a solder form for the metric $g$.

 M > $\mathrm{ot}≔\mathrm{evalDG}\left(\left[\frac{1\mathrm{D_t}}{x},\frac{1\mathrm{D_x}}{t},\mathrm{D_y},\mathrm{D_z}\right]\right)$
 ${\mathrm{ot}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{{x}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{1}}{{x}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{{1}}{{t}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{{1}}{{t}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.10)
 M > $\mathrm{σ}≔\mathrm{SolderForm}\left(\mathrm{ot}\right)$
 ${\mathrm{σ}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{5}{,}{7}\right]{,}\frac{{x}{}\sqrt{{2}}}{{2}}\right]{,}\left[\left[{1}{,}{6}{,}{8}\right]{,}\frac{{x}{}\sqrt{{2}}}{{2}}\right]{,}\left[\left[{2}{,}{5}{,}{8}\right]{,}\frac{{t}{}\sqrt{{2}}}{{2}}\right]{,}\left[\left[{2}{,}{6}{,}{7}\right]{,}\frac{{t}{}\sqrt{{2}}}{{2}}\right]{,}\left[\left[{3}{,}{5}{,}{8}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{4}{,}{6}{,}{8}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_bas"}{,}{"con_vrt"}{,}{"con_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{1}{,}{5}{,}{7}\right]{,}\frac{{x}{}\sqrt{{2}}}{{2}}\right]{,}\left[\left[{1}{,}{6}{,}{8}\right]{,}\frac{{x}{}\sqrt{{2}}}{{2}}\right]{,}\left[\left[{2}{,}{5}{,}{8}\right]{,}\frac{{t}{}\sqrt{{2}}}{{2}}\right]{,}\left[\left[{2}{,}{6}{,}{7}\right]{,}\frac{{t}{}\sqrt{{2}}}{{2}}\right]{,}\left[\left[{3}{,}{5}{,}{8}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{3}{,}{6}{,}{7}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]{,}\left[\left[{4}{,}{5}{,}{7}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]{,}\left[\left[{4}{,}{6}{,}{8}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}\right]\right]\right]\right)$ (2.11)

The command RicciSpinor gives the spinor form of $S.$

 M > $\mathrm{φ}≔\mathrm{RicciSpinor}\left(\mathrm{σ},S\right)$
 ${\mathrm{φ}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{8}{,}{8}\right]{,}\frac{{-}\frac{{I}}{{4}}}{{t}}\right]{,}\left[\left[{6}{,}{6}{,}{7}{,}{7}\right]{,}\frac{\frac{{I}}{{4}}}{{t}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{8}{,}{8}\right]{,}\frac{{-}\frac{{I}}{{4}}}{{t}}\right]{,}\left[\left[{6}{,}{6}{,}{7}{,}{7}\right]{,}\frac{\frac{{I}}{{4}}}{{t}}\right]\right]\right]\right)$ (2.12)

We calculate the Plebanski tensor in its spinor form.

 M > $\mathrm{ψ}≔\mathrm{PlebanskiTensor}\left(\mathrm{φ}\right)$
 ${\mathrm{ψ}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]\right]\right]\right)$ (2.13)

We can check the consistency of this result using the command WeylSpinor to calculate the spinor form of $P.$

 M > $\mathrm{ψ1}≔\mathrm{WeylSpinor}\left(\mathrm{σ},P\right)$
 ${\mathrm{ψ1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{6}{,}{6}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{5}{,}{6}{,}{5}{,}{6}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{5}{,}{6}{,}{6}{,}{5}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{6}{,}{5}{,}{5}{,}{6}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{6}{,}{5}{,}{6}{,}{5}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]{,}\left[\left[{6}{,}{6}{,}{5}{,}{5}\right]{,}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]\right]\right]\right)$ (2.14)
 M > $\mathrm{ψ}&minus\mathrm{ψ}$
 ${\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"tensor"}{,}{M}{,}\left[\left[{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}{,}{"cov_vrt"}\right]{,}\left[\right]\right]\right]{,}\left[\left[\left[{5}{,}{5}{,}{5}{,}{5}\right]{,}{0}\right]\right]\right]\right)$ (2.15)

Example 3.

In this example we will calculate the Newman-Penrose coefficients of the tensor  from the Newman-Penrose coefficients of the Plebanski tensor $P.$ For these calculations we need the NullTetrad determined by the orthonormal tetrad $\mathrm{ot}$.

 M > $\mathrm{nt}≔\mathrm{NullTetrad}\left(\mathrm{ot}\right)$
 ${\mathrm{nt}}{:=}\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{\sqrt{{2}}}{{2}{}{x}}\right]{,}\left[\left[{4}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{\sqrt{{2}}}{{2}{}{x}}\right]{,}\left[\left[{4}\right]{,}\frac{\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{\sqrt{{2}}}{{2}{}{x}}\right]{,}\left[\left[{4}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{\sqrt{{2}}}{{2}{}{x}}\right]{,}\left[\left[{4}\right]{,}{-}\frac{\sqrt{{2}}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{\sqrt{{2}}}{{2}{}{t}}\right]{,}\left[\left[{3}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{\sqrt{{2}}}{{2}{}{t}}\right]{,}\left[\left[{3}\right]{,}\frac{{I}}{{2}}{}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{\sqrt{{2}}}{{2}{}{t}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{M}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}\frac{\sqrt{{2}}}{{2}{}{t}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{I}}{{2}}{}\sqrt{{2}}\right]\right]\right]\right)\right]$ (2.16)
 M > $\mathrm{NPRicciForS}≔\mathrm{NPCurvatureScalars}\left(\mathrm{nt},g,S,\mathrm{output}=\left["AllRicciScalars"\right]\right)$
 ${\mathrm{NPRicciForS}}{:=}{\mathrm{table}}\left(\left[{"Phi22"}{=}{0}{,}{"Phi12"}{=}{0}{,}{"Phi20"}{=}{-}\frac{\frac{{1}}{{4}}{}{I}}{{t}}{,}{"Phi21"}{=}{0}{,}{"Phi11"}{=}{0}{,}{"Phi10"}{=}{0}{,}{"Phi01"}{=}{0}{,}{"Phi02"}{=}\frac{\frac{{1}}{{4}}{}{I}}{{t}}{,}{"Phi00"}{=}{0}{,}{"Lambda"}{=}{0}\right]\right)$ (2.17)
 M > $\mathrm{NPWeylForP}≔\mathrm{PlebanskiTensor}\left(\mathrm{NPRicciForS}\right)$
 ${\mathrm{NPWeylForP}}{:=}{\mathrm{table}}\left(\left[{"Psi4"}{=}{0}{,}{"Psi1"}{=}{0}{,}{"Psi0"}{=}{0}{,}{"Psi3"}{=}{0}{,}{"Psi2"}{=}{-}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]\right)$ (2.18)

We can check the consistency of this result using the command NPCurvatureScalars  to calculate the Newman-Penrose coefficients of  $P.$ 

 M > $\mathrm{NPCurvatureScalars}\left(\mathrm{nt},g,P,\mathrm{output}=\left["WeylScalars"\right]\right)$
 ${\mathrm{table}}\left(\left[{"Psi4"}{=}{0}{,}{"Psi1"}{=}{0}{,}{"Psi0"}{=}{0}{,}{"Psi3"}{=}{0}{,}{"Psi2"}{=}{-}\frac{{1}}{{12}{}{{t}}^{{2}}}\right]\right)$ (2.19)