I Quasi-Static Change of Event Horizon Area

ergospherical
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Let ##\mathscr{H}## be a constant-##v## cross-section of the event horizon (area ##A##). The expansion is the fractional rate of change of the surface element, ##\theta = \frac{1}{\delta S} \frac{d(\delta S)}{dv}##. The problem asks to prove the formula ##\frac{dA}{dv} = \frac{8\pi}{\kappa} \oint_{\mathscr{H}} (\frac{1}{8\pi} \sigma^2 + T_{ab} \xi^a \xi^b) dS## where ##\xi## is the tangent to the null generators.

I used the Raychaudhuri equation to write down\begin{align*}
\oint_{\mathscr{H}} \left( \frac{1}{8\pi} \sigma^2 + T_{ab} \xi^a \xi^b \right) dS &= \frac{1}{8\pi} \oint_{\mathscr{H}} \left(\kappa \theta - \frac{1}{2} \theta^2 - \frac{d\theta}{dv} \right) dS \\ \\
&= \underbrace{\frac{\kappa}{8\pi} \frac{d}{dv} \oint_{\mathscr{H}} dS}_{= \frac{\kappa}{8\pi} \frac{dA}{dv} } - \frac{1}{8\pi} \oint_{\mathscr{H}} \left( \frac{1}{2} \theta^2 + \frac{d\theta}{dv} \right) dS
\end{align*}I suppose the quasi-static approximation is supposed to kill the other term but I'd like to justify it properly?
 
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As I suggested in another one of your questions, you should really provide a reference.
Your homework question is
not self-contained.

For the possibly interested reader, what is \kappa, \sigma,…,etc?

If the question was only intended for those already familiar with the situation, then this should be classified as A-level, not I-level.
 
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Sorry, the question is the last one in the Black Hole section of E. Poisson's relativist's toolkit. ##\kappa## is the surface gravity, ##\sigma^2 = \sigma^{ab}\sigma_{ab}## the square of the shear, ##v## the parameter along the null generators and ##dS = \sqrt{^2g} d^2 \theta## the surface element on ##\mathscr{H}## with ##(^2g)_{ab} = \frac{\partial x^{c}}{\partial \theta^a} \frac{\partial x^d}{\partial \theta^b} g_{cd}## the pull-back of ##g## onto ##\mathscr{H}##.
 
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