Details for EnergyMomentumTensor, MatterFieldEquations, DivergenceIdentities
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Description
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Here we give the precise formulas for the energy-momentum tensors, matter field equations and divergence identities, as computed by these commands. In the formulas below, the indices are raised and lowered using the metric and denotes the covariant derivative compatible with .
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1. "DiracWeyl". The fields are a solder form , a rank 1 covariant spinor and the complex conjugate spinor The energy-momentum tensor is the contravariant, symmetric rank 2 tensor:
and the matter field equations are the rank 1 contravariant spinors with components
The divergence of the energy-momentum tensor is given in terms of the matter field equations by
Here is the bivector solder form and denotes the complex conjugate of the previous terms.
2. "Dust". The fields are a four-vector , with and a scalar μ (energy density). The energy-momentum tensor is the contravariant, symmetric rank 2 tensor
and the matter field equations consist of the scalar and vector
and
The divergence of the energy momentum tensor is given in terms of the matter field equations by
3. "Electromagnetic". The field is a 1-form or a 2-form The energy-momentum tensor is the contravariant, symmetric rank 2 tensor
and the matter field equations are given by
The divergence of the energy-momentum tensor is given in terms of the matter field equations by
4. "PerfectFluid". The fields are a four-vector , with and scalars and (energy density and pressure). The energy-momentum tensor is the contravariant, symmetric rank 2 tensor
The matter field equations are defined by the divergence of the energy-momentum tensor:
5. "Scalar". The field is a scalar The energy-momentum tensor is the contravariant, symmetric rank 2 tensor
where is a constant. The matter field equations are defined by the scalar
The divergence of the energy momentum tensor is given in terms of the matter field equations by
6. "NMCScalar". The field is a scalar . The energy-momentum tensor is the contravariant, symmetric rank 2 tensor
where is the Einstein tensor and and are constants. The matter field equations are defined by the scalar
where is the Ricci scalar. The divergence of the energy momentum tensor is given in terms of the matter field equations by
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