In a charged black hole, how does the electromagnetic field of the charge get out from inside?
It doesn’t so far as OUTSIDE observers are concerned. For real black holes, the charge, mass and angular momentum of a black hole are properties we associate and measure at the horizon itself, if I have read my copy of ‘Gravitation’ by Misner, Thorne and Wheeler correctly. The general relativistic solutions for such black holes also predict what the interior geometry ‘inside’ the outer event horizon ought to look like, but these features are not visible to external observers. However, if you were falling into the black hole, your reference frame would show you a very different view of the interior of a black hole and this information would not be hidden from you. For distant observers, the interior electromagnetic field of the black hole does not escape to ‘infinity’ to be observed, BUT paradoxically, it is observable as a continuous phenomenon by an infalling observer who passes across the horizon in a finite amount of time as measured by clocks in their rest frame. To distant observe
