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I have read these questions:

Does charge bend spacetime like mass?

Why is spacetime curved by mass but not charge?

Where John Rennie says:

"Charge does curve spacetime."

And where Frederic Thomas says:

"On the other hand there no compulsory relationship between the charge (or spin) and the inertial mass, better said, there is no relation at all. Therefore charge or spin have a priori no effect on space-time, at least not a direct one. "

So one says yes, charges curve spacetime, the other says no.

Question:

  1. Which one is right? Do charges curve spacetime like stress-energy or not?
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    $\begingroup$ Seems to me that all of the answers agree; Frederic Thomas is saying that the electromagnetic fields created by charged particles curve spacetime, which is what the other answers also say. The only difference is that he believes that the charge's contribution to the EM field is an indirect result of the charge existing, whereas the other answers do not make such a philosophical distinction. It really depends on what your philosophy is on whether you can talk about electric charge without the electromagnetic field. $\endgroup$ – probably_someone Sep 6 '18 at 0:48
  • $\begingroup$ Experimentally. this has not been confirmed. Theoretically, electromagnetism is quantum, but gravity is not. Until the theories are made compatible, no one knows. $\endgroup$ – safesphere Sep 6 '18 at 5:29
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First, stress-energy tensor (of matter fields) $T_{\mu \nu}$ is something that you have to put in by hand in Einstein's equations:

$$R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = \frac{8\pi G}{c^4} T_{\mu \nu}$$

to see how it determines the curvature $(g_{\mu \nu})$. Charge is not to be treated exclusively from $T_{\mu \nu}$, as you seem to think. Everything that can contribute to $T_{\mu \nu}$ must be included in it. If the stress-energy tensor is zero, it implies that the geometry is Ricci flat: $R_{\mu \nu}=0$. (Note that spacetime can still be curved for $R_{\mu \nu}=0$ because $R_{\mu \nu \rho \sigma} \neq 0$, in general).

Now, a charge creates an electric field around it. The electromagnetic (electric only, for our case) field is described by the Lagrangian for classical electromagnetism: $\mathcal{L} = -\frac{1}{4} F^{\mu \nu} F_{\mu \nu}$. To find the $T_{\mu \nu}$ for this electromagnetic field, we need to vary the action for $\mathcal{L}$ with respect to the metric tensor: $T_{\mu \nu} \sim \frac{\delta S}{\delta g^{\mu \nu}}$. This electromagnetic stress-energy tensor is not zero. (It is traceless, however, so the Ricci scalar $R=0$). So $R_{\mu \nu} \neq 0$ and spacetime is curved.

The typical example given for such spacetimes is the Reissner–Nordström metric, which can be derived from above calculations, and some other assumptions.

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I wouldn't state it like Frederic Thomas did. Charges produce an electromagnetic field, which in turn carries energy-momentum, which in turn produce a gravitational field.

So it is true that they do not have a direct effect on spacetime curvature, but they do (always!) produce an EM field which affects the curvature. Hence, as John Rennie said, charge does curve spacetime, albeit indirectly. If all the charges in your system where set to zero, the spacetime curvature would be different than its charged analogue (compare e.g. the charged Reissner-Nordström black hole to the neutral Schwarzschild black hole).

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