(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

In a Penrose process, a mass [itex]\delta M < 0[/itex] and an angular momentum [itex]\delta J < 0[/itex] are added to a black hole. Given that

[tex]

\delta J \leq \frac{\delta M}{\Omega}

[/tex]

where [itex]\Omega[/itex] is the angular velocity of the horizon, show that the irreducible mass never decreases, i.e. [itex]\delta M_{irr} > 0[/itex].

2. Relevant equations

[tex]M_{irr}^2 = \frac{1}{2} \left[ M^2 + \sqrt{M^4 - J^2} \right][/tex]

3. The attempt at a solution

The function for the irreducible mass is weakly decreasing for decreasing [itex]M[/itex] and strongly increasing for decreasing [itex]J[/itex], so I imagine that if [itex]J[/itex] decreases enough compared to the decrease in [itex]M[/itex], you indeed get that the irreducible mass can only rise. That is, I can imagine that the given inequality might lead to the requested conclusion. I have not been able to prove that this particular inequality works. For one, I'm not sure how to relate the angular velocity to anything else, which I think might be crucial. Perhaps I should somehow relate it to the angular momentum, but how?

EDIT: I've found an expression saying

[tex]

\Omega = -\frac{g_{t\phi}}{g_{\phi\phi}}.

[/tex]

I don't understand this, can anyone explain why this is true?

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# Homework Help: Proof that black hole's irreducible mass never decreases

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