DifferentialGeometry/Tensor/SpinConnection

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Tensor[SpinConnection] - compute the spin connection defined by a solder form

Calling Sequences

SpinConnection(sigma)

Parameters

sigma - a solder form

Description

•

The DifferentialGeometry Tensor package supports general computations with connections on vector bundles (Connection, Example 3; CovariantDerivative, Example 3; DirectionalCovariantDerivative, Example 3; and CurvatureTensor, Example 3). This functionality naturally provides for covariant differentiation of spinors.

•

Given a solder form sigma, let g be the associated metric. There is a unique spin connection nabla such that nabla(sigma) = 0 and nabla(epsilon) = 0, where epsilon denotes either of the covariant epsilon spinors (EpsilonSpinor). In the definition of nabla(sigma) the tensorial argument or index is covariantly differentiated with respect to the Christoffel connection for g. It is this connection nabla which is computed by the command SpinConnection(sigma).

•	Note that a generic connection for the differentiation of spinors can be constructed using the Connection command.

•	This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form SpinConnection(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-SpinConnection.

Examples

Example 1.

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

(2.1)

Define a spacetime metric g on M.

M >

(2.2)

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

M >

(2.3)

Calculate the solder form sigma from the frame F.

M >

sigma := x^2*2^(1/2)*dt*D_z1*D_w1/2+x^2*2^(1/2)*dt*D_z2*D_w2/2+2^(1/2)*dx*D_z1*D_w2/2+2^(1/2)*dx*D_z2*D_w1/2-((1/2)*I)*2^(1/2)*dy*D_z1*D_w2+((1/2)*I)*2^(1/2)*dy*D_z2*D_w1+2^(1/2)*dz*D_z1*D_w1/2-2^(1/2)*dz*D_z2*D_w2/2

(2.4)

Calculate the spin-connection for the solder form sigma.

M >

(2.5)

Example 2.

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

M >

(2.6)

M >

(2.7)

M >

(2.8)

M >

(2.9)

M >

(2.10)

M >

(2.11)

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.

M >

(2.12)

M >

(2.13)

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

M >

(2.14)

M >

(2.15)

Example 4.

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

M >

(2.16)

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.

M >

(2.17)

Calculate the complex conjugate of the spinor curvature SpinR.

M >

(2.18)

Calculate the bivector soldering forms S and barS.

M >

(2.19)

M >

(2.20)

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

M >

R1 := -2*D_t*dx*dt*dx/x^2+2*D_t*dx*dx*dt/x^2-2*x^2*D_x*dt*dt*dx+2*x^2*D_x*dt*dx*dt+(2*I)*D_y*dz*dt*dx-(2*I)*D_y*dz*dx*dt-(2*I)*D_z*dy*dt*dx+(2*I)*D_z*dy*dx*dt

(2.21)

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

M >

R2 := -2*D_t*dx*dt*dx/x^2+2*D_t*dx*dx*dt/x^2-2*x^2*D_x*dt*dt*dx+2*x^2*D_x*dt*dx*dt-(2*I)*D_y*dz*dt*dx+(2*I)*D_y*dz*dx*dt+(2*I)*D_z*dy*dt*dx-(2*I)*D_z*dy*dx*dt

(2.22)

M >

(2.23)

M >

(2.24)

M >

(2.25)

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