Form of energy momentum tensor of matter in EM fields

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• Sunny Singh
Sunny Singh
TL;DR Summary
I'm trying to understand if there is a scientific consensus on the form of the energy momentum tensor for a polarizable matter in electromagnetic fields
I've been reading about the form of the energy momentum tensor for a polarizable medium in electromagnetic fields and i'm not sure if there is a scientific consensus on its form. Starting from the series of papers by Kluitenberg in the 1950s to works by Israel and Dixon in the 70s... various sources give various expressions for the form of the energy momentum tensor. In "Classical Field Theory" by Davison Soper, there are 7 different contributions to the energy momentum tensor while in "Relativistic fluids and magnetofluids" by A.M.Anile, there are four such terms which don't seem to agree with others' works. More recent papers on magnetohydrodynamics use the following form of the matter + field ##T^{\mu\nu}##: $$T^{\mu\nu} = \varepsilon u^\mu u^\nu-P \Delta^{\mu \nu}-\frac{1}{2}\left(M^{\mu \lambda} F_\lambda{ }^\nu+M^{\nu \lambda} F_\lambda{ }^\mu\right) -F^{\mu \lambda} F_{\;\lambda}^\nu+\frac{g^{\mu \nu}}{4} F^{\rho \sigma} F_{\rho \sigma}$$
And some papers like https://arxiv.org/pdf/0812.0801 uses $$T^{\mu\nu} = \varepsilon u^\mu u^\nu-P \Delta^{\mu \nu}-M^{\mu \lambda} F_\lambda{ }^\nu -F^{\mu \lambda} F_{\;\lambda}^\nu+\frac{g^{\mu \nu}}{4} F^{\rho \sigma} F_{\rho \sigma}$$
Which doesn't even look symmetric. And according to "Dynamics of Polarisation" -- General Relativity and Gravitation, 9, 5 (1978) by Israel, the last term gets replaced by ##\frac{g^{\mu \nu}}{4} F^{\rho \sigma} H_{\rho \sigma}## when minkowski tensor is considered. These modern expressions look way more simplified than some of the older sources' forms.
My question is if there is a scientific consensus on what the energy-momentum tensor of such a polarizable fluid in electromagnetic fields look like. Any help or direction will be greatly appreciated since i'm thoroughly confused now.

The metric i used is the westcoast metric and ##\Delta^{\mu\nu}=g^{\mu\nu}-u^\mu u^\nu## is the projection operator orthogonal to fluid flow.

Frabjous said:
I was also going to suggest that exact same paper.

The bottom line is that there is no agreed upon form for either the matter tensor or the field tensor separately. But the matter+field tensor is well defined

Sunny Singh and Frabjous

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