=     

enjoys all the symmetries 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 defined by the solder form .  

.    (*)

The coefficients   coincide with the Newman-Penrose Weyl scalars for the Weyl tensor  constructed from the null tetrad defined by the spinor dyad

Type I.  

Type II.  

Type III.  

Type  D.  

Type N.  

Type  O.  

See Penrose and Rindle  Vol. 2, Section 8.3.

Thus, for example, if the Petrov type of the Weyl spinor is , then there is a spinor dyad such that the Weyl spinor takes the form . Such a spinor dyad is said to be an adapted spinor dyad.

Set the global environment variable _EnvExplicit to true to insure that our results are free of expressions.

We give examples of Weyl spinors of each Petrov type and calculate an adapted spinor dyad.  We check that the spinor dyad has the desired properties.

First create the spinor bundle over a 4 dimensional spacetime.

In order to construct the Weyl spinors for our examples, we need a basis for the vector space of symmetric rank 4 spinors. This we obtain from the GenerateSymmetricTensors command.

Example 1. Type I

Define a rank 4 spinor

Calculate the Newman-Penrose coefficients for  with respect to the initial dyad basis .

Use these coefficients to find the Petrov type of

Compute an adapted contravariant spinor dyad for

Here is the covariant form of the spinor dyad.

We check that this answer is correct in two ways.  First we can re-calculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.

This is the correct normal form since and

Second, we can calculate a type I Weyl spinor from this spinor dyad using the command WeylSpinor and check that the result coincides with the original spinor .

Alternative dyads will be computed with the keyword argument  output

Example 2. Type II

Define a rank 4 spinor

Calculate the Newman-Penrose coefficients for  with respect to the standard dyad basis .

Find the Petrov type of

Compute an adapted contravariant spinor dyad for .

Here is the covariant form of the adapted spinor dyad.

We check that this answer is correct in two ways.  First we can re-caculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.

This is the correct normal form since and

Second, we can calculate a type II Weyl spinor from this spinor dyad and check that the result coincides with the original spinor .

Example 3. Type III

Define a rank 4 spinor

Calculate the Newman-Penrose coefficients for  with respect to the initial dyad basis .

Find the Petrov type of

Compute an adapted contravariant spinor dyad for

Here is the covariant form of the spinor dyad.

We check that this answer is correct in two ways.  First we can re-caculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.

This is the correct normal form since

Second, we can calculate a type III Weyl spinor from this spinor dyad and check that the result coincides with the original spinor .

Example 4. Type D

Define a rank 4 spinor

Calculate the Newman-Penrose coefficients for  with respect to the initial dyad basis .

Find the Petrov type of

Compute an adapted contravariant spinor dyad for

Here is the covariant form of the spinor dyad.

We check that this answer is correct in two ways.  First we can re-calculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.

This is the correct normal form since

Second, we can calculate a type D Weyl spinor from this spinor dyad and check that the result coincides with the original spinor .

Example 5.  Type N

Define a rank 4 spinor

Calculate the Newman-Penrose coefficients for  with respect to the initial dyad basis .

Find the Petrov type of

Compute an adapted contravariant spinor dyad for

Here is the covariant form of the spinor dyad.

We check that this answer is correct in two ways.  First we can re-caculate the Newman-Penrose coefficients and confirm that they are in the correct normal form.

This is the correct normal form since

Second, we can calculate a type N Weyl spinor from this spinor dyad and check that the result coincides with the original spinor .

DifferentialGeometry, Tensor,  AdaptedSpinorDyad, AdaptedNullTetrad,  NPCurvatureScalars, NullVector, PetrovType,  PrincipalNullDirections, WeylSpinor