# Question on Einstein Field Equations: charged black holes and fluids

## Introduction

Suppose then you're asked to find the metric of a charged black hole. Given a generic, static and spherically symmetric metric tensor,

$$\text{d}s^{2} = A(r)\text{d}t^{2} + B(r)\text{d}r^{2} +r^{2}(\text{d}\theta ^{2} + \sin^{2}\theta \text{d}\phi^{2}) \tag{1},$$

the one should use the stress-energy-momentum tensor:

$$T^{\mathrm{(EM)}}_{\mu\nu} =\frac{1}{4\pi}\big(F_{\mu \lambda} F_{\nu}\hspace{0.1mm}^{\lambda} - g_{\mu\nu}F_{\alpha \beta}F^{\alpha \beta}\big)\tag{2},$$

to solve the Einstein Field Equations (EFE), $$G_{\mu\nu} = 8\pi T^{\mathrm{(EM)}}_{\mu\nu}$$. The final result yields the Reissner-Nordström geometry:

$$\text{d}s^{2} = -\Big(1-\frac{2m}{r}+\frac{q^2}{r^2}\Big)\text{d}t^{2} + \frac{\text{d}r^{2}}{\Big(1-\frac{2m}{r}+\frac{q^2}{r^2}\Big)} +r^{2}(\text{d}\theta ^{2} + \sin^{2}\theta \text{d}\phi^{2}) \tag{3}.$$

## My Question

Suppose now you are starting with metric tensor $$(3)$$ in your hands. Then you calculate the Einstein tensors $$G^{(c)}_{\mu\nu}$$(*); with the calculated Einstein tensors, you give, by hand, a matter tensor (in a suitable tetrad basis):

$$T^{(F)}_{\mu\nu} = \mathrm{Diag}[\rho(r),p_{r},p_{\theta},p_{\phi}].\tag{4}$$

Now, my question is: the EFE will be $$G^{(c)}_{\mu\nu} = 8\pi\big[T^{(F)}_{\mu\nu} + T^{\mathrm{(EM)}}_{\mu\nu} \big]$$ or just $$G^{(c)}_{\mu\nu} = 8\pi T^{(F)}_{\mu\nu}$$? Since the information about the charge $$q$$ is already in the spacetime (in the metric).

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(*) the $$^{(c)}$$ is for "the components were calculated with the given metric $$(3)$$"

Once you have the metric, you have already set the stress-energy tensor automatically. You'll have $$G_{\mu\nu}^{(\text{c})} = 8 \pi T_{\mu\nu}^{(\text{EM})}$$. What you might then attempt to do is to set $$T_{\mu\nu}^{(\text{EM})} = T_{\mu\nu}^{(\text{F})}$$ and try to solve for the densities and pressures, with the possibility of failure. Otherwise, you'd need to solve the full EFE's with $$G_{\mu\nu}^{(\text{c})} = 8\pi\left(T_{\mu\nu}^{(\text{EM})} + T_{\mu\nu}^{(\text{F})}\right)$$, but then they wouldn't need to give rise to the Reissner–Nordström solution. For example, if the fluid you are adding behaves as a cosmological constant, you might end with the charged version of the Schwarzschild–De Sitter solution.