Under General Relativity, nothing can escape a black hole's event horizon, not even light, which is of course made of photons.
However, it is said that black holes can have charge. This presents a problem when the Standard Model is included in describing a black hole.
The Standard Model requires the exchange of force carrier particles, which exchange momentum and thus impart a force between particles. The force carrier particle for the electromagnetic force is the photon, which we have just stated is incapable of escaping a black hole's event horizon.
How then, is it possible for a black hole to have an electric charge, if for that charge to exert a force, it requires the exchange of photons?
It seems like if a black hole could have a charge, then you could send information across its event horizon, for example by sending a charge across the event horizon and triangulating its position inside the horizon by looking at its effects on the EM field outside of the horizon. You could then move this charge inside the horizon and look at the "wobbling" as a means of sending bits.
How is this scenario prevented while still allowing black holes to have charge?
Edit: Most of the suggested answers treat the problem classically and sweep it under the rug, usually by treating the observer at infinity, or by saying the charge never crosses the horizon from an external observer's view, which doesn't provide an answer for how matter inside the horizon can affect the EM field outside it. Some of the ones that do attempt to provide an answer within QM even imply the virtual photons exchanging the forces of the charges travel at infinite speed, or that information can even pass the event horizon under QM, which doesn't erase the information passing problem at all nor provide an answer of how it is that a black hole can have a charge.
I understand that QM and GR are far from being unified into a coherent framework, but we seem pretty convinced that black holes can have a charge under classical GR treatment, but QM seems to imply that either can't be possible (the photons that exchange the force cannot cross the horizon, and thus black holes should not be able to have a charge classically), or that information can be exchanged across the horizon under QM, which clearly breaks GR.