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Example 1. "DiracWeyl"
First create a vector bundle with base coordinates and fiber coordinates .
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Define a metric of signature and an orthonormal tetrad.
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| (2.2) |
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| (2.3) |
Calculate the solder form.
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| (2.4) |
Define a rank 1-spinor field and its complex conjugate.
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| (2.5) |
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| (2.6) |
Calculate the Dirac-Weyl energy momentum tensor .
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| (2.7) |
Evaluate the Dirac-Weyl field equations for the given spinor field .
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| (2.8) |
Check the divergence identity for the dust energy momentum tensor . The LHS is the covariant divergence of the energy momentum tensor and the RHS is a combination of the field equations.
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| (2.9) |
We note that is a solution of the Dirac-Weyl field equations:
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| (2.11) |
The covariant divergence of the energy momentum tensor vanishes on this solution:
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Example 2. "Dust"
First create a manifold with base coordinates :
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Define a metric.
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| (2.14) |
Define the normalized 4-vector representing the 4-velocity of the dust.
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| (2.15) |
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Define the energy density.
Calculate the dust energy- momentum tensor .
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| (2.18) |
Evaluate the dust field equations for the given and .
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| (2.19) |
Check that the following values for and solve the dust field equations.
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| (2.20) |
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| (2.21) |
Check the divergence identity for the dust energy-momentum tensor . The LHS is the covariant divergence of the energy-momentum tensor and the RHS is a combination of the field equations.
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| (2.22) |
Example 3. "Electromagnetic"
First create a manifold with base coordinates .
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Define a metric.
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| (2.25) |
Define an electromagnetic 4-potential .
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| (2.26) |
Calculate the electromagnetic energy-momentum tensor .
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| (2.27) |
Note that the energy-momentum tensor can also be computed from the field strength tensor .
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| (2.28) |
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| (2.29) |
Evaluate the electromagnetic field equations for the given 4-potential .
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| (2.30) |
Note that the electromagnetic field equations can also be computed from the field strength tensor .
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| (2.31) |
Check the divergence identity for the electromagnetic energy-momentum tensor . The LHS is the covariant divergence of the energy momentum tensor and the RHS is a combination of the matter field equations.
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| (2.32) |
We note that is a solution of the electromagnetic field equations:
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The covariant divergence of the energy-momentum tensor vanishes on this solution:
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Example 4. "PerfectFluid"
First create a manifold with base coordinates :
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Define a metric.
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| (2.37) |
Define the normalized 4-velocity.
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Define the energy density.
Define the pressure.
Calculate the perfect fluid energy-momentum tensor .
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| (2.42) |
Evaluate the fluid field equations for the given fluid.
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| (2.43) |
We can use the dsolve command to find the energy density and the pressure which satisfy the field equations.
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| (2.44) |
| (2.45) |
Example 5. "Scalar"
First create a manifold with base coordinates .
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Define a metric.
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| (2.47) |
Define a scalar field.
Calculate the energy- momentum tensor for the scalar field .
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| (2.49) |
Evaluate the matter field equations for the given scalar field .
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| (2.50) |
Check the divergence identity for the scalar energy-momentum tensor . The LHS is the covariant divergence of the energy-momentum tensor and the RHS is a combination of the matter field equations.
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| (2.51) |
Example 6. "NMCScalar"
First create a manifold with base coordinates .
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Define a metric.
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| (2.54) |
Define a scalar field
Calculate the energy-momentum tensor for the non-minimally coupled scalar field .
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| (2.56) |
Evaluate the matter field equations for the given scalar field .
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| (2.57) |
Check the divergence identity for the scalar energy-momentum tensor . The LHS is the covariant divergence of the energy-momentum tensor and the RHS is a combination of the matter field equations.
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| (2.58) |