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 metric g on M.  For this example we use the Godel metric.  (See (12.26) in Exact Solutions to Einstein's Field Equations.)  Note that we have adjusted the metric to conform to the signature conventions [1, -1, -1, -1] used by the spinor formalism in the DifferentialGeometry  package.  See SpacetimeConventions.

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

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

Calculate the Ricci spinor from the solder form sigma.

Example 2.

In this example we first calculate the Ricci tensor of the metric g and then use the second calling sequence for RicciSpinor.

Example 3.

We can check the result of Example 1 by direct computation, starting from the solder form sigma.  First use the command SpinorInnerProduct to calculate the metric g3 from sigma.  (Note that g3 coincides with the original metric g.)

Second, calculate the curvature tensor C, the Ricci tensor R, and the Ricci scalar S.

Calculate the trace-free Ricci tensor T.

Convert T to a spinor U.

Rearrange the indices of U and scale by (-1/2) to arrive at the Ricci spinor Phi1 (or Phi2).

DifferentialGeometry, Tensor, Convert, CurvatureTensor, Physics[Riemann],  RicciTensor, Physics[Ricci], SpacetimeConventions, SolderForm, SpinorInnerProduct, WeylSpinor, Physics[Weyl]