DifferentialGeometry/Tensor/EnergyMomentumTensorDetails - Maple Help

Details for EnergyMomentumTensor, MatterFieldEquations, DivergenceIdentities

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 $g,$and denotes the covariant derivative compatible with $g$.

1. "DiracWeyl". The fields are a solder form $\mathrm{σ}$, a rank 1 covariant spinor $\mathrm{ψ}$ and the complex conjugate spinor $\stackrel{&conjugate0;}{\mathrm{\psi }}.$ 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 $S$ 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 $g\left(u,u\right)=±1,$ 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 $A$ 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 $u$, with $g\left(u,u\right)=±1,$and scalars $\mathrm{μ}$ and $p$ (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 $\mathrm{ϕ}.$ The energy-momentum tensor is the contravariant, symmetric rank 2 tensor

where $m$ 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 $\mathrm{ϕ}$. The energy-momentum tensor is the contravariant, symmetric rank 2 tensor

where is the Einstein tensor and $m$ and $\mathrm{ξ}$ are constants. The matter field equations are defined by the scalar

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