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Details for EnergyMomentumTensor, MatterFieldEquations, DivergenceIdentities

 

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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 g,and  denotes the covariant derivative compatible with g.

 

1. "DiracWeyl". The fields are a solder form σ, a rank 1 covariant spinor ψ and the complex conjugate spinor ψ&conjugate0;. The energy-momentum tensor is the contravariant, symmetric rank 2 tensor: 

 Tij = i2σiAB'ψA j ψ&conjugate0;B'  ψ&conjugate0;B'j ψA  + σjAB'ψA i ψ&conjugate0;B'  ψ&conjugate0;B'i ψA 

 

and the matter field equations are the rank 1 contravariant spinors with components

 

EA = 1i σkAB' kψ&conjugate0;B'   and   EB' = i σkAB' kψA .

 

The divergence of the energy-momentum tensor is given in terms of the matter field equations by

 

j Tij = 2 i ψA EA + S  B ijAψAj EB + c.c.

 

Here S is the bivector solder form and c.c. denotes the complex conjugate of the previous terms.

 

2. "Dust". The fields are a four-vector u, with gu,u=±1, and a scalar μ (energy density). The energy-momentum tensor is the contravariant, symmetric rank 2 tensor Tij= μ uiuj,

and the matter field equations consist of the scalar and vector

E = iμ ui and Vi=uj jui .

 

The divergence of the energy momentum tensor is given in terms of the matter field equations by

 

jTij=Eui + μVi.

 

3. "Electromagnetic". The field is a 1-form A or a 2-form F = dA. The energy-momentum tensor is the contravariant, symmetric rank 2 tensor

 

T ij = FihF  hj   14 gijFhkFhk ,

and the matter field equations are given by

Ei=j Fij .

 

The divergence of the energy-momentum tensor is given in terms of the matter field equations by

 

jTij = F ji Ej + Ai j Ej .

 

4. "PerfectFluid". The fields are a four-vector u, with gu,u=±1,and scalars μ and p (energy density and pressure). The energy-momentum tensor is the contravariant, symmetric rank 2 tensor

Tij = μ+puiuj+pgij .

 

The matter field equations are defined by the divergence of the energy-momentum tensor:

Ei=j T ij .

 

5. "Scalar". The field is a scalar ϕ. The energy-momentum tensor is the contravariant, symmetric rank 2 tensor

 

T ij=iϕjϕ  gij12kϕkϕ +m2ϕ2

 

where m is a constant. The matter field equations are defined by the scalar

 

E=i iϕm2ϕ.

 

The divergence of the energy momentum tensor is given in terms of the matter field equations by

 

jT ij=iϕ E.

 

6. "NMCScalar". The field is a scalar ϕ. The energy-momentum tensor is the contravariant, symmetric rank 2 tensor

 

Tij = 12 ξiϕjϕ+2 ξ12kϕkϕ gij2 ξi jϕ+2 ξk kϕ gij+ξϕ2Gij12m2ϕ2gij,

 

where Gij is the Einstein tensor and m and ξ are constants. The matter field equations are defined by the scalar

 

E=i iϕξR+m2ϕ,

 

where R is the Ricci scalar. The divergence of the energy momentum tensor is given in terms of the matter field equations by

 

jT ij=iϕ E.

See Also

EnergyMomentumTensor