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Here in this forum (and elsewhre) I read about "charged" black holes. What is the background for this questions? Is this more a question of theroretical interest?

In "practice" :=) black holes should be neutral, because matter is neutral globally. A star is charged a little bit, depending on surface temperature and plasma physics in corona. But this charge will be blast away (with surface material) when the interior of the star collapses to form a black hole. Again, when a black hole "eats" matter from around, charge separation might occur, but only to a limited extent, until the voltage developed would cancel the separation mechanism. (As in case of a star)

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up vote 8 down vote accepted

"Realistic" black holes, by which I mean black holes of the sort for whose existence we have significant evidence, are almost certainly very close to electrically neutral, pretty much for the reasons you suggest. The two main types of black holes that seem to exist in nature are those of stellar-mass and supermassive ones at galactic centers. Both of these almost certainly formed out of large amounts of matter that was, to an excellent approximation, electrically neutral. You could get small amounts of charge separation, but I don't think anyone expects that to occur at any measurable level.

Of course it's always possible that this is wrong: either the black holes we know about could be charged for some reason, or there could be whole other populations of charged black holes. So we'll have to do observations and experiments to check. People are always trying to improve our observations of black holes, not specifically to check for charge but to measure other interesting things such as angular momentum. I'd guess that, if these black holes had a large charge, such observations would reveal it, but I've never looked into details.

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I believe the other effect that people will cite about this is that if a black hole WERE to acquire a net charge such that $Q$ were comparable to $M$, it would polarize the nearby matter and neutralize itself pretty quickly. I've never seen a detailed proof of this, but the heuristic sounds right. –  Jerry Schirmer Oct 1 '12 at 21:37
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I found this interesting bit of observation on the black hole in the center of our galaxy.

"Our infrared detection shows for the first time that the black hole's meal is more like the Grand Rapids, in which energetic glitches from shocked gas are occurring almost continually."

This theoretical speculation on interstellar lightning offers a mechanism how there could be charge separation in interstellar gas.

. Possible chemical and electromagnetic signatures of interstellar lightning are discussed.

If such a mechanism is detected and established it is conceivable that a black hole could eat a dust cloud with a, for example negative charge, while the positive charged cloud is attracted to another black hole :). Lo, charged black holes.

But I think theorists just solve and see that the solutions are physical, and then wait to see whether observations verify the solutions, by the work of hard pressed astronomers and astrophysicists :).

It is like the question of "why climb the mountain" answer: "because it is there."

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To throw also a slightly less mainstream suggestion out there (I don't know too much about it, but I know the field exists and is active), there is the proposal to model an electron as a tiny charged black hole. The main interest in that is, I think, born from some considerations by Einstein of the motion of a particle on a gravitating background (the geodesic hypothesis). The main papers were this one by Einstein, Infeld, and Hoffman and this follow-up by Einstein and Infeld, which modelled particles as little singularities in space-time.

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Let me add to the already many good answers, that from a string-theoretic perspective, (near)-extremal charged (Reissner-Nordstr\"om) black holes in various dimensions are important, because they are simpler to calculate with (mainly because of supersymmetry). They therefore serve as an important theoretical testbed for ideas concerning microscopic counting of entropy, holography, unitarity, etc.

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Yet another point (if someone do not know yet:) S. Chandrasekhar in "The Mathematical Theory of Black Holes" wrote, comparing the charged BH (i.e. Reissner-Nordstroem solution) and usual one:

And among the differences, the principal one -- from which all others derive -- is the fact that the Reissner-Nordstroem space-time has an inner Cauchy horizon which makes its maximal analytic extension reveal the possibility, in principle, of exploring other worlds and emancipating oneself from one's past.

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Technically, if you are suitably religious, on a Schwarzschild background you can also "explore other worlds and emancipate oneself from one's past". You are even guaranteed to do so in finite proper time. –  Willie Wong Jul 13 '11 at 20:03
    
"However, for one who has crossed the event horizon in the Reissner-Nordstroem geometry, there would appear to exist an infinite range of rich possibilities for experience, that are denied to one who crosses the event horizon in the Schwarzschild geometry. For the latter person, there is, as we have seen, no alternative to being inexorably propelled towards the singularity at r = 0." - ibid –  Alex 'qubeat' Jul 14 '11 at 18:41
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I would like to distinguish between large, astrophysical black holes resulting from stellar collapse and the more general notion of a black hole. Like others have pointed out, astrophyical black holes are certainly almost or exactly electrically neutral. But you could imagine that a microscopic black hole could be formed by colliding elementary particles with sufficient energy. Such a black hole would be charged if the constituents were, although if the theory of Hawking radiation is correct, this would be a rather ephemeral object. There are active searches being conducted for black hole formation at the LHC, so the concept is "practical" in that sense.

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