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That second identity is the entire resolution in miniature, and it is worth deriving $N_\perp$[$\Gamma$] both ways to see it operate. Hold the local energy budget fixed at $E_{\rm loc}$=$E_0$' (a battery physically at the processor, delivering a fixed amount of locally measured energy) over a proper interval $\Delta$$\tau$ : $N_\perp = \frac{2}{\pi\hbar}E_0'\,\Delta\tau$. Now hold the asymptotic Killing energy fixed instead, at $E_\infty$ = $\sqrt{f}$\,$E_0$' (the amount of energy it would cost a distant experimenter, at infinity, to deliver that same physical resource down to radius r ), and integrate over the corresponding coordinate interval $\Delta$ t = $\Delta$$\tau$/$\sqrt f$ :

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