Calculating M from Charge & Angular Momentum in Black Hole

[michael879]

Ok so this has always confused me and I still can't seem to find an answer anywhere! A general black hole has parameters M, Q, and J which are given the meaning of mass, charge, and angular momentum. My question is what what is the contribution to M by Q and J?? Presumably if you start with a Schwarzschild black hole and drop "massless" charge (not possible in reality but electrons at rest come pretty close) into it, it will not only gain charge but M will be increased due to the compression of like-signed charges. The situation is similar for angular momentum where a rotating black hole partially repels objects with angular momentum in the same direction. The extra energy required to get the object into the black hole should add to M.

So how much of M is due to charge and angular momentum, and how would you go about calculating it? Naively I would guess that the mass gain is equal to the work required to bring the charge up to the event horizon (since no energy can escape whatever happens within the horizon can't increase the mass).

 

[WannabeNewton]

http://saj.matf.bg.ac.rs/178/pdf/039-044.pdf For background see ch12 of Wald "General Relativity". Equation (2) in the paper is what you would be immediately interested in, if time is a constraint.

 

[michael879]

Thanks, that's an interesting way to look at it. I'm a little curious why the entire concept would break down for naked black holes though. As far as I know GR allows for the existence of naked black holes. However if you look at the fraction of mass from the 3 "sources" as a black hole becomes extremal and then naked, you see some strange behavior.. An extremal black hole has 1/2 of its mass from charge/angular momentum and 1/2 from the "static" term. However once you lose the event horizon the entire equation breaks down and Q and J no longer appear to contribute to the mass at all!

 

[stevebd1]

WannabeNewton has probably already answered the question but another source for finding the irreducible mass (M_ir) is here-

http://www.ece.uic.edu/~tsarkar/Goodies/Black Hole.pdf page 12 eq. 10

and when the black hole is extremal (i.e. M²=a²+Q²), quantities for M, J and Q still hold (where J=aM)