I was doing some basic algebra to estimate the range of possible masses M and electric charge Q for a nearly extremal Reissner-Noström black hole. I want to see if the logic is correct
the electric field near the event horizon of such a black hole would be
E=14πϵ0qr2=14πϵ0qc4G2M2=1066Volt Kg2m CoulqM2
since Schwinger-pair vacuum breakdown occurs at 1018Voltm, it implies that
qM2<10−48CoulKg2
Would be a bound that guarantees electromagnetic vacuum stability
Question: can a charged black hole that is stable against this breakdown still be nearly extremal?
well, it seems so, because the extremality condition:
rQ=rs2
Implies that
Q√G4πϵ0c4=GMc2
QM≈10−16CoulKg
So the above two relations would imply that a black hole with a mass
M>1032Kg≈100 Solar masses
Can be both stable (against the specific Schwinger-pair decay) and extremal.
In particular, if the mass is 1032 kilograms, a charge around 1016 Coulomb would make the black hole arbitrarily close to extremal as one might want, and is not at all clear that it would decay from the extremality state
But probably i'm overlooking other ways than the extremality of the black hole can decay (in particular, i'm ignorant about how Hawking radiation will contribute in here, specially if the temperature is below 511 KeV). Any insights into this would be greatly appreciated
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