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I learned about the Einstein Field Equations and also I know they depend on energy-momentum tensors. The solution of the field equations is a space-time metric.

$$G_{\mu \nu}=8\pi GT_{\mu \nu}$$

Now, I want to know what is the solution of Einstein field equations in an electromagnetic field? Clearly, I tend to know its metric. Could you answer me, please?

I could not explain my question truly. I know about some metrics such as Reissner–Nordström metric, which is a static solution to the Einstein-Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass $M$:

$$ds^2=\left(1-\frac{r_S}{r}+\frac{r_Q^2}{r^2}\right)c^2dt^2-\left(1-\frac{r_S}{r}+\frac{r_Q^2}{r^2}\right)^{-1}dr^2-r^2d\Omega_{(2)}^2$$

and the Kerr–Newman metric, which is a solution of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding a charged, rotating mass:

$$c^2d\tau^2=-\left(\frac{dr^2}{\Delta}+d\theta^2\right)\rho^2+\left(cdt-\alpha\sin^2d\phi\right)^2\frac{\Delta}{\rho^2}-\left(\left(r^2+\alpha^2\right)d\phi-\alpha cdt\right)^2\frac{\sin^2\theta}{\rho^2}$$

But I don't know; is the Reissner-Nordstrom metric really a solution of the field equations where an electromagnetic field EM is present?

Could you please let me know how the form of the solution heavily depends on the exact configuration of the EM field? Could you give me some references or a few examples?

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marked as duplicate by Kyle Kanos, John Rennie general-relativity Oct 6 '15 at 14:19

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ This might be related somewhat: physics.stackexchange.com/q/55660 $\endgroup$ – user81619 Oct 5 '15 at 21:04
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    $\begingroup$ Essentially a duplicate of physics.stackexchange.com/q/55660 $\endgroup$ – Kyle Kanos Oct 5 '15 at 21:05
  • $\begingroup$ The form of the solution heavily depends on the exact configuration of the EM field. There are charged black hole solutions (see Reissner-Nordstrom metric), metrics in the Universe filled with uniform EM radiation and etc. $\endgroup$ – Prof. Legolasov Oct 6 '15 at 1:25
  • $\begingroup$ So is your question now about how the Reissner-Nordstrom metric is a solution to the EFE? $\endgroup$ – Kyle Kanos Oct 6 '15 at 21:13
  • $\begingroup$ is the Reissner-Nordstrom metric really a solution of the field equations where an electromagnetic field EM is present? - yes. $\endgroup$ – John Rennie Oct 7 '15 at 7:12
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The stress energy tensor of the EM field is

\begin{equation} T^{\mu \nu} = \frac{1}{\mu_0} \left( F^{\mu \alpha} g_{\alpha \beta} F^{\nu \beta} - \frac{1}{4} g^{\mu \nu} F_{\delta \gamma} F^{\delta \gamma} \right) \end{equation}

Which stems from the usual definition of the stress energy tensor as a variation of the Lagrangian. And Maxwell's equation in curved spacetime is

\begin{equation} {F^{ab}}_{;a} = \mu_0 j^b \end{equation}

Be careful with the gauge because due to the definition of the derivative, it will be this in the Lorentz gauge :

\begin{equation} - {A^{\alpha ; \beta}}_{ \beta} + {R^{\alpha}}_{\beta} A^{\beta} = \mu_0 J^{\alpha} \end{equation}

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