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Equation 48 · Part 10 · A Horizon Is a Toll Booth, Not a Loophole

Denominator: pihbar

N⊥=2πℏE0′ Δτ.N_\perp = \frac{2}{\pi\hbar}E_0'\,\Delta\tau.
πℏ\pi\hbar

What this part means

The complete quantity below the fraction bar; it must be nonzero for this division.

Its job in the formula

pihbar occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

The passage around this formula

That second identity is the entire resolution in miniature, and it is worth deriving N⊥N_\perp[Γ\Gamma] both ways to see it operate. Hold the local energy budget fixed at ElocE_{\rm loc}=E0E_0' (a battery physically at the processor, delivering a fixed amount of locally measured energy) over a proper interval Δ\Deltaτ\tau : N⊥=2πℏE0′ ΔτN_\perp = \frac{2}{\pi\hbar}E_0'\,\Delta\tau. Now hold the asymptotic Killing energy fixed instead, at E∞E_\infty = f\sqrt{f}\,E0E_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 Δ\Delta t = Δ\Deltaτ\tau/f\sqrt f :

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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