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Details for EnergyMomentumTensor, MatterFieldEquations, DivergenceIdentities

 

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Description

• 

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 .

 

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

 

See Also

EnergyMomentumTensor


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