# Conservation of Electromagnetic Energy-Momentum Tensor in GR

• In the follow Planck units have been used.

• The Electromagnetic Energy-Momentum Tensor is $$T^{\mu\nu} = \frac{1}{4 \pi} \left[ F^{\mu \alpha}F^\nu{}_{\alpha} - \frac{1}{4} g^{\mu\nu}F_{\alpha\beta} F^{\alpha\beta}\right]$$ where $$F^{\mu \nu}$$ denotes the Faraday Tensor.

• From the second Bianchi identity we know that the 4-divergence of the Einstein Tensor $$G^{\mu \nu}$$ is null, i.e. $$\nabla_{\mu} G^{\mu \nu}=0$$.

• Putting this in the Einstein Equations $$G^{\mu \nu}=8\pi T^{\mu \nu}$$ we also have the conservation of the energy-momentum tensor: $$\nabla_{\mu} T^{\mu \nu}=0$$.

• Now, the problem is that for the Electromagnetic Energy-Momentum Tensor the 4-divergence is not null, but we have instead $$\nabla_{\mu} T^{\mu \nu}=F^{\mu \nu}j_{\nu}$$, with $$j^{\nu}$$ the 4-current.

• So is it not compatible with the Einstein Equations? Am I missing something?

• Have you read Dirac's 75 page booklet on GR and electrodynamics? Commented Aug 22, 2020 at 18:02
• Commented Aug 22, 2020 at 18:12
• @DanielC nope, what's the specific name of that book? I can't find it Commented Aug 22, 2020 at 20:54
• The great answer by professore Moretti can be supplemented by reading the relevant paragraphs of Dirac's book. I won't give you the amazon link (Jeff Bezos has enough money already), but he LoC one. lccn.loc.gov/75008690 Commented Aug 23, 2020 at 0:34

As soon as you have a non-zero current $$J$$, the EM field is not isolated and it is not the complete source of the gravitational field. On the right-hand side of the Einstein equations you should insert the whole stress-energy tensor and not only the EM one. With the added part you find a consistent system of equations.
• $T^{ab}=T_{EM}^{ab}+ \mu V^aV^b$ and where $J^a=\rho V^a$ with $\rho = g \mu$ for some constant $g$. Commented Aug 22, 2020 at 21:59