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 $\nabla$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: 

 $T^{\mathrm{ij}} \= \frac{i}{2}\left[{\mathrm{\sigma }}^{\mathrm{iAB}\'}\left({\mathrm{\psi }}_A {\nabla }^{j }{\stackrel{\&conjugate0;}{\mathrm{\psi }}}_{B\'} - {\stackrel{\&conjugate0;}{\mathrm{\psi }}}_{B\'}{\nabla }^{j }{\mathrm{\psi }}_A \right) \+ {\mathrm{\sigma }}^{\mathrm{jAB}\'}\left({\mathrm{\psi }}_A {\nabla }^{i }{\stackrel{\&conjugate0;}{\mathrm{\psi }}}_{B\'} - {\stackrel{\&conjugate0;}{\mathrm{\psi }}}_{B\'}{\nabla }^{i }{\mathrm{\psi }}_A \right)\right]$

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

$E^A \= \frac{1}{i} {\mathrm{\σ}}^{\mathrm{kAB}\'} {\nabla }_k{\stackrel{\&conjugate0;}{\mathrm{\psi }}}_{B\'} and E^{B\'} \= i {\mathrm{\σ}}^{\mathrm{kAB}\'} {\nabla }_k{\mathrm{\ψ}}_A \.$

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

${\nabla }_j T^{\mathrm{ij}} \= -2 {\nabla }^{i }{\mathrm{\ψ}}_A E^A \+ S_{ B}^{ \mathrm{ijA}}{\mathrm{\ψ}}_A{\nabla }_{j }{E}^B \+ c\.c\.$

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 $g\left(u\,u\right)\=\pm 1\,$ and a scalar μ (energy density). The energy-momentum tensor is the contravariant, symmetric rank 2 tensor$T^{\mathrm{ij}}\= \mathrm{\μ} u^i{u}^j\,$

and the matter field equations consist of the scalar and vector

$E \= {\nabla }_i\left(\mathrm{\μ} u^i\right)$and $V^i\=u^{j }{\nabla }_j{u}^i \.$

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

${\nabla }_j{T}^{\mathrm{ij}}\={\mathrm{Eu}}^i \+ {\mathrm{\μV}}^i\.$

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

$T^{ \mathrm{ij}} \= F^{\mathrm{ih}}{F}_{ h}^{j } - \frac{1}{4}$ $g^{\mathrm{ij}}{F}^{\mathrm{hk}}{F}_{\mathrm{hk}} \,$

and the matter field equations are given by

$E^i\={\nabla }_{j }{F}^{\mathrm{ij}} \.$

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

${\nabla }_j{T}^{\mathrm{ij}} \= F_{ j}^i E^j \+ A^{i }{\nabla }_{j }{E}^j \.$

4. "PerfectFluid". The fields are a four-vector $u$, with $g\left(u\,u\right)\=\pm 1\,$and scalars $\mathrm{\μ}$ and $p$ (energy density and pressure). The energy-momentum tensor is the contravariant, symmetric rank 2 tensor

$T^{\mathrm{ij}} \= \left(\mathrm{\μ}\+p\right)u^i{u}^j\+{\mathrm{pg}}^{\mathrm{ij}} \.$

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

$E^i\={\nabla }_{j }{T}^{ \mathrm{ij}} \.$

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

$T^{ \mathrm{ij}}\={\nabla }^i\mathrm{\ϕ}{\nabla }^j\mathrm{\ϕ} - g^{\mathrm{ij}}\left(\frac{1}{2}{\nabla }^k\mathrm{\ϕ}{\nabla }_k\mathrm{\ϕ} \+m^2{\mathrm{\ϕ}}^2\right)$

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

$E\={\nabla }^{i }{\nabla }_i\mathrm{\ϕ}-m^2\mathrm{\ϕ}\.$

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

${\nabla }_j{T}^{ \mathrm{ij}}\={\nabla }^i\mathrm{\ϕ} E\.$

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

$T^{\mathrm{ij}} \= \left(1-2 \mathrm{\ξ}\right){\nabla }^i\mathrm{\ϕ}{\nabla }^j\mathrm{\ϕ}\+\left(2 \mathrm{\ξ}-\frac{1}{2}\right){\nabla }^k\mathrm{\varphi }{\nabla }_k\mathrm{\ϕ} g^{\mathrm{ij}}-2 \mathrm{\ξ}{\nabla }^{i }{\nabla }^j\mathrm{\ϕ}\+2 \mathrm{\ξ}{\nabla }^{k }{\nabla }_k\mathrm{\ϕ} g^{\mathrm{ij}}\+{\mathrm{\ξ\ϕ}}^2{G}^{\mathrm{ij}}-\frac{1}{2}{m}^2{\mathrm{\ϕ}}^2{g}^{\mathrm{ij}}\,$

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

$E\={\nabla }^{i }{\nabla }_i\mathrm{\ϕ}-\left(\mathrm{\ξR}\+m^2\right)\mathrm{\ϕ}\,$

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

${\nabla }_j{T}^{ \mathrm{ij}}\={\nabla }^i\mathrm{\varphi } E\.$

	See Also
	EnergyMomentumTensor