Consider a pair of identical extremal black holes at rest. It is a fact that they exert no net force on each other and remain in static equilibrium. Now, clearly, each of the black holes is in equilibrium under the combined action of the gravitational and electromagnetic forces. Thus, none of the black holes is in a free fall. This implies that each of them should be emitting electromagnetic radiation in accordance with the equivalence principle--because they are accelerated with respect to the local inertial frames. But this would mean that they are losing mass without losing charge. And clearly, this means that they get spontaneously converted into naked singularities. Is there a flaw in this line of reasoning?

  • $\begingroup$ They would radiate only in the frame of a free falling observer, but not in the frame that is stationary relative to the black holes. $\endgroup$ – safesphere Sep 13 '17 at 8:48
  • $\begingroup$ How is this a different question than your earlier physics.stackexchange.com/q/356955/50583? it basically asks about the same issue, just in a slightly more detailed example. This should be an edit to your earlier question, not a new question, imo, since an answer to this question would also answer your earlier question. $\endgroup$ – ACuriousMind Sep 13 '17 at 10:39
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    $\begingroup$ @ACuriousMind This is a different question from my earlier question imo. In my previous question, I am puzzled about the absence of thermal Hawking radiation from extremal black holes. Here, I am talking about classical electromagnetic radiation from accelerated extremal black holes. Both are concerned about the violation of the censorship but I think the physics involved is different. Can you point out how an answer to this question would answer my previous question? $\endgroup$ – Dvij Mankad Sep 13 '17 at 10:48
  • $\begingroup$ Ah, I see. Are you aware that even without using black holes, the reconcilement of the radiation of accelerated charges with the equivalence principle is rather subtle, cf. e.g. Wikipedia? $\endgroup$ – ACuriousMind Sep 13 '17 at 10:59
  • $\begingroup$ @ACuriousMind Yes, I don't think I fully understand everything about that paradox but I am aware of it. As far as I understand, the amount of energy (mass) lost by the charge is a scalar. The static (which is actually accelerated wrt the inertial) frame doesn't see the radiation because the radiation is hidden behind the Rindler horizon. Please correct me if I am mistaken or wrong. But anyway, since the energy lost is a scalar, the change in the mass would also be a scalar (and non-zero). $\endgroup$ – Dvij Mankad Sep 13 '17 at 13:22

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