The tensor has all the algebraic properties of the Weyl tensor.  It is skew-symmetric in the indices  and satisfies the cyclic identity on  and is trace-free with respect to the metric .

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.)

Define a metric tensor .

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

Compute the Plebanski tensor of

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

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

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

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  .

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  and using this tetrad and the command SolderForm to construct a solder form for the metric .

The command RicciSpinor gives the spinor form of

We calculate the Plebanski tensor in its spinor form.

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

Example 3.

In this example we will calculate the Newman-Penrose coefficients of the tensor  from the Newman-Penrose coefficients of the Plebanski tensor  For these calculations we need the NullTetrad determined by the orthonormal tetrad .

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

DifferentialGeometry, Tensor, convert, NPCurvatureScalars, NullTetrad, OrthonormalTetrad, RicciSpinor, SolderForm, WeylSpinor