Quasi-Static Change of Event Horizon Area

[ergospherical]

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?

 

[robphy]

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.

 

[ergospherical]

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 $\mathscr{H}$.