# Is the Lorentz force the divergence of a symmetric tensor?

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.

• @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. Jun 2 '17 at 7:38