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Equation 52 · Part 5 · A Horizon Is a Toll Booth, Not a Loophole

Symbol f

N⊥=2πℏE∞ Δt=2πℏ(f E0′)Δτf=2πℏE0′ Δτ.N_\perp = \frac{2}{\pi\hbar}E_\infty\,\Delta t = \frac{2}{\pi\hbar}\left(\sqrt f\,E_0'\right)\frac{\Delta\tau}{\sqrt f} = \frac{2}{\pi\hbar}E_0'\,\Delta\tau.
ff

What this part means

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

Its job in the formula

f 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

…with coordinate time, never a local energy stapled to a coordinate-time integral or the reverse. That last error is precisely the one the naive “free thoughts” argument makes. It takes the huge blueshift factor 1/f\sqrt f multiplying local energy and pairs it with the huge 1/f\sqrt f multiplying elapsed coordinate time, double-counting the same factor as though it were two independent windfalls rather than one factor and its own reciprocal. Priced honestly, in whichever single currency the experimenter actually…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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