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.

Define an orthonormal frame on M with respect to the metric g.

Calculate the solder form sigma from the frame F.

Calculate the spin-connection for the solder form sigma.

Example 2.

Define a rank 1 spinor phi.  Calculate the covariant derivative of phi.  Calculate the directional derivatives of phi.

Example 3.

Check that the covariant derivative of sigma vanishes.  Because sigma is a spin-tensor, both connections are required.  Calculate the Christoffel connection for the metric g.

Define an epsilon spinor and check that its covariant derivative vanishes.

Example 4.

Calculate the curvature spin-tensor for the spin-connection Gamma2.

The curvature tensor R for the Christoffel connection can be expressed in terms of the curvature spin-tensor SpinR, its complex conjugate barSpinR and the bivector solder forms S and barS by the identity

2*R^i_{jhk} = S^i_j_A^B*R^A_{Bhk} + S^i_j_A'^B'*R^A'_{B'hk}    (*)

Let's check this formula for the Christoffel connection Gamma1 and the spin-connection Gamma2.  First calculate the curvature tensor for Gamma1.

Calculate the complex conjugate of the spinor curvature SpinR.

Calculate the bivector soldering forms S and barS.

The first term on the right-hand side of (*) is

The second term on the right-hand side of (*) is

DifferentialGeometry, Tensor, BivectorSolderForm, Connection, Physics[Christoffel], CovariantDerivative, Physics[D_], DirectionalCovariantDerivative, CurvatureTensor, Physics[Riemann], EnergyMomentumTensor, EpsilonSpinor, MatterFieldEquations, SpacetimeConventions