Electromagnetic interaction in the stress-energy momentum tensor

Question

In Phys. Rev. D 4, 2185, the author uses the following stress-energy tensor for a charged spherically-symmetric fluid:

$$ T^{\mu \nu} := (\delta + P) u^{\mu} u^{\nu} + P g^{\mu \nu} + \frac{\pi}{4} [ F^{\mu \alpha} F ^{\nu}_{ \alpha} - \frac{1}{4} g^{\mu \nu} F^{\alpha \beta} F_{\alpha \beta} ] $$

for energy density $\delta$, hydrostatic pressure $P$ and electromagnetic fields $F$.

unexplainably absent, is any accounting for the matter-field interaction, which I would expect to be:

$$ T^{\mu \nu}_{\text{interaction}} = g^{\mu \nu} J_{\alpha} A^{\alpha} - ( J^{\mu} A^{\nu} + J^{\nu} A^{\mu} )$$

The paper does not offer any justification for dropping the interaction term, and although this seems to be one of the oldest papers that I could find that write a charged fluid stress-energy tensor making this omission, I've seen this same omission made essentially automatically in lots of other papers on the same subject since then, and I have yet to find a reasonable argument for dropping it.

Question: Is there a reasonable justification for dropping this term on general grounds?

Answer 1

I've no access to the paper, but I'd expect current to appear in the forcing term of the divergence of the stress-energy tensor, in an momentum-energy equation that can be written as

$$\partial_{\mu} T^{\mu \nu} = - F^{\nu}_{\mu} J^{\mu} \ .$$

Update from the comment below. The total momentum-energy equation of an isolated system (no external force, no mass flux) reads

$$ \partial_{\mu} T^{tot,\mu \nu} = 0 \ ,$$

translating the conservation of energy and momentum.

Total momentum-energy is the sum of the "mass", $T^{m, \mu\nu}$, contribution and the electromagnetic contribution, $T^{EM, \mu\nu}$.

If you write the equation for the mass part and the electromagnetic part separately, the divergences of the two stress-energy tensors are forced by opposite forcing terms that cancel out after sum, namely

$$\partial_{\mu} T^{m, \mu\nu} = F^{\mu}_{\nu} J^{\nu}$$ $$\partial_{\mu} T^{EM, \mu\nu} = - F^{\mu}_{\nu} J^{\nu} \ ,$$

with the current that is related to the motion of matter, something like $J^{\mu} = q U^{\mu}$ for the "current" of a point charge (as you can write the dynamical equation for a point charge in an electromagnetic field, and thus subject to Lorentz force, $m d_{\tau} U^{\mu} = q F^{\mu}_{\nu} U^{\nu} = F^{\mu}_{\nu} J^{\nu}$).

If you want to get the two equations for every single part of the system, you need to start from the Lagrangian of the system. Momentum-energy equation put all the contributions together and you can't recover single contributions.

For reference, Landau&Lifshitz, Classical Theory of Fields, par.33

Remark. In your question, I guess there's something wrong in the index notation, since indices are not balanced, in the term $\{\}^{\mu \nu} = \dots - \frac{1}{4} g^{\mu \nu} F^{\alpha \beta} F^{\alpha \beta}$: maybe it's $- \frac{1}{4} g^{\mu \nu} F^{\alpha \beta} F_{\alpha \beta}$.

Answer 2

I am not sure of the actual reason but I can try a guess. From the title of the paper, it is clear that we are talking about some fluid solution in (or near) an hydrostatic equilibrium. In such a limit, we can guess that the effective electrostatic repulsive force experienced by a particle due to the surrounding particles is almost balanced by their mutual gravitational interactions. Under such a scenario we may treat the fluid particles to be effectively non-interacting. A non-interactive fluid also means that it has a zero viscosity, while the local conservation of energy-momentum (which is required as a consequence of the Einstein's field equations) means that there is no effective heat or momentum transfer to the surrounding, hence an adiabatic system. A non-viscous and adiabatic fluids are governed by Euler's equation and hence it seems more reasonable to start with an ansatz for ideal fluid (Note that the conservation equations for ideal fluid SET gives the Euler equations).