>
|
|
Example 1.
First create a vector bundle over with base coordinates and fiber coordinates.
>
|
|
Define a metric on . For this example we use a pure radiation metric of Petrov type N. (See (22.70) in Stephani, Kramer et al.) Note that we have changed the sign of the metric to conform to the signature convention [1, -1, -1, -1] used by the spinor formalism in DifferentialGeometry.
M >
|
|
| (2.2) |
Use DGGramSchmidt to calculate an orthonormal frame F for the metric .
M >
|
|
| (2.3) |
Use SolderForm to compute the solder form sigma from the frame F.
| (2.4) |
Calculate the Weyl spinor from the solder form sigma.
| (2.5) |
Example 2.
We check that the same answer for the Weyl spinor Psi1 is obtained if we first calculate the Weyl tensor of the metric defined by .
| (2.6) |
M >
|
|
| (2.7) |
Example 3.
We see that the rank 4 spinor Psi1 calculated by WeylSpinor factors as the 4-th tensor product of a rank 1 spinor psi. This affirms the fact that the metric g has Petrov type N.
M >
|
|
| (2.9) |
| (2.10) |
M >
|
|
| (2.11) |
Example 4.
We check that the Weyl spinor Psi1 satisfies the decomposition (1) given above. From the Weyl tensor W calculated in Example 2, we find the left-hand side LHS of (1). (The intermediate expressions, even in this simple example, are too long to display.)
M >
|
|
We re-arrange the indices to place all the unbarred (unprimed) indices first and all the barred (primed) indices last.
M >
|
|
We calculate the first terms on the right-hand side of (1) as RHS1.
M >
|
|
| (2.13) |
M >
|
|
We use the command ConjugateSpinor to find the complex conjugate of Psi1. Then we calculate the second terms on the right-hand side of (1) as RHS2.
M >
|
|
M >
|
|
| (2.14) |
M >
|
|
We check that the left-hand side and right-hand side of (1) are the same.
| (2.15) |
Example 5.
We use the second calling sequence to calculate a Weyl spinor from a spinor dyad and a set of Newman-Penrose coefficients.
M >
|
|
| (2.17) |
>
|
|
| (2.18) |
>
|
|
| (2.19) |
Example 6.
We use the third calling sequence to calculate a Weyl spinor in adapted normal form.
| (2.20) |
M >
|
|
M >
|
|
| (2.22) |