sigma = sqrt(2)/2*omega^a*P_a   (sum on a).

Example 1.

First create a vector bundle over M with base coordinates [t, x, y, z] and fiber coordinates [z1, z2, w1, w2].

Define a spacetime metric g on M with signature [1, -1, -1, -1].

Define an orthonormal frame on M with respect to the metric g.  Verify using the command GRQuery.

Calculate the solder form sigma from the frame F.

Let us obtain this result directly from the the definition. First we define the 4 Pauli matrices.

Define the corresponding rank 2 Hermitian spinors.

Define the dual coframe to F.

This coincides with sigma.

Example 2.

The solder form satisfies two important identities.  The first identity involves contracting a pair of solder forms over their spinor indices:

sigma_i^{AA'}*sigma_j_{AA'} = g_{ij}.

The second identity involves contracting a pair of solder forms over their tensor indices:

sigma_j^{AA'}*sigma^j^{BB'} = epsilon^{AB}epsilon^{A'B'}.

Let us check the first identity using the the solder form from Example 1.  First calculate the covariant form of the solder form, using the orthonormal frame of the previous example.

Note that this coincides with the result of using RaiseLowerSpinorIndices to lower the spinor indices of sigma using the epsilon spinor.

The contraction of sigma and sigmaCov over their spinor indices gives the metric g.

The same result can be obtained using SpinorInnerProduct.

To check the second identity calculate the contravariant form of sigma.

Note that this coincides with the result of using RaiseLowerIndices to raise the tensor index of sigma using the inverse of the metric g.

The contraction of sigma and sigmaCon over their tensor indices gives a product of epsilon spinors (EpsilonSpinor).

Rearrange the indices so that the spinor indices are first, the barred spinor indices second.

Example 3.

Here we compute a solder form for the Godel spacetime.  (See (12.26) in Stephani Kramer et al.) First create a vector bundle over M with base coordinates [t, x, y, z] and fiber coordinates [z1, z2, w1, w2].

Define the Godel metric g on M.  (Note that we have adjusted the metric to conform to the signature conventions [1, -1, -1, -1] used by the spinor formalism in DifferentialGeometry.See SpacetimeConventions.)

Use DGGramSchmidt to SolderForm calculate an orthonormal frame F for the metric g.

Use SolderForm to compute the solder form sigma from the frame F.

Example 4.

For any metric of Lorentz signature [1, -1, -1, -1], a compatible solder form can be constructed.

Define a spacetime metric g.

Use the command DGGramSchmidt to find a orthonormal frame.

Calculate the solder form from F3.

Use SpinorInnerProduct to check that sigma3 is compatible with the metric g3.

DifferentialGeometry, Tensor, BivectorSolderForm, convert/DGspinor, convert/DGtensor, DGGramSchmidt, DualBasis, EpsilonSpinor, GRQuery, NullTetrad, OrthonormalTetrad, RaiseLowerIndices, RaiseLowerSpinorIndices, RicciSpinor, SpacetimeConventions, SpinConnection, SpinorInnerProduct, WeylSpinor