Example 1.  "DiracWeyl"

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

Define a metric of signature  [+1, -1, -1, -1] and an orthonormal tetrad.

Calculate the solder form.

Define a rank 1-spinor field psi1 and its complex conjugate.

Calculate the Dirac-Weyl energy momentum tensor T.

Evaluate the Dirac-Weyl field equations E for the given spinor field psi.

Check the divergence identity for the dust energy momentum tensor T. The LHS is the covariant divergence of the energy momentum tensor and the RHS is a combination of the field equations.

We note that f(x) = h(x) = 1/x is a solution of the Dirac-Weyl field equations:

 The covariant divergence of the energy momentum tensor vanishes on this solution:

Example 2.  "Dust"

First create a manifold M with base coordinates [t, x, y, z]:

Define a metric.

Define the normalized 4-vector representing the 4-velocity of the dust.

Define the energy density.

Calculate the dust energy- momentum tensor T2.

Evaluate the dust field equations E2 for the given u and mu.

Check that the follow values for f(t) and h(t) solve the dust field equations.

Check the divergence identity for the dust energy-momentum tensor T2. The LHS is the covariant divergence of the energy-momentum tensor and the RHS is a combination of the field equations.

Example 3.  "Electromagnetic"

First create a manifold M with base coordinates [t, x, y, z, t].

Define a metric.

Define an electromagnetic 4-potential A3.

Calculate the electromagnetic energy-momentum tensor T3.

Note that the energy-momentum tensor can also be computed from the field strength tensor F = dA.

Evaluate the electromagnetic field equations E3 for the given 4-potential A.

Note that the electromagnetic field equations E3 can also be computed from the field strength tensor F = dA.

Check the divergence identity for the electromagnetic energy-momentum tensor T3. The LHS is the covariant divergence of  the energy momentum tensor and the RHS is a combination of the matter field equations.

We note that f1(x) = x^2, f2(x) = ln(x) is a solution of the electromagnetic field equations:

The covariant divergence of the energy-momentum tensor vanishes on this solution:

Example 4.  "PerfectFluid"

First create a manifold M with base coordinates [t, x, y, z]:

Define a metric.

Define the normalized 4-velocity.

Define the energy density.

Define the pressure.

Calculate the perfect fluid energy-momentum tensor T4.

Evaluate the fluid field equations E4 for the given fluid.

We can use the dsolve command to find the energy density k(t) and the pressure h(t) which satisfy the matter field equations.

Example 5.  "Scalar"

First create a manifold M with base coordinates [t, x, y, z].

Define a metric.

Define a scalar field.

Calculate the energy- momentum tensor T5 for the scalar field phi5.

Evaluate the matter field equations E for the given scalar field phi.

Check the divergence identity for the scalar energy-momentum tensor T.  The LHS is the covariant divergence of the energy-momentum tensor and the RHS is a combination of the matter field equations.

Example 6.  "NMCScalar"

First create a manifold M with base coordinates [t, x, y, z].

Define a metric.

Define a scalar field

Calculate the energy-momentum tensor T for the non-minimally coupled scalar field phi6.

Evaluate the matter field equations E for the given scalar field phi6.

Check the divergence identity for the scalar energy-momentum tensor T6. The LHS is the covariant divergence of the energy-momentum tensor and the RHS is a combination of the matter field equations.

DifferentialGeometry, Tensor