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The conservation of momentum and energy in electromagnetism is usually written as $$ \partial_\nu T^{\mu \nu} + \eta^{\mu \rho} f_\rho = 0 $$ where $T^{\mu\nu}$ is the electromagnetic stress-energy tensor and $f_\rho$ is the Lorentz force.

However the conservation of momentum and energy in general relativity is stated as the divergence of the total stress-energy tensor being zero. So is there a symmetric tensor $T_\text{Lorentz}^{\mu\nu}$ which divergence is $\eta^{\mu \rho} f_\rho$, so that $\partial_\nu(T^{\mu\nu}+T_\text{Lorentz}^{\mu\nu})=0$ ?

$T_\text{Lorentz}^{\mu\nu}$ would measure the energy and momentum of charged particules.

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  • $\begingroup$ Doesn't this answer your question? en.wikipedia.org/wiki/… $\endgroup$
    – Kosm
    Jun 1, 2017 at 23:19
  • $\begingroup$ I think you basically answered your question in the last sentence. $\endgroup$
    – Javier
    Jun 2, 2017 at 0:07
  • $\begingroup$ @Javier Then I expect a formula of $T^{\mu\nu}_\text{Lorentz}$ as a function of the 4-current $J^\mu$, but I can't find it. $\endgroup$
    – V. Semeria
    Jun 2, 2017 at 7:38

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