← All parts of this equation

Equation 52 · Part 16 · A Horizon Is a Toll Booth, Not a Loophole

Denominator: pihbar

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.
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

​ : N⊥=2πℏE∞ Δt=2πℏ(f E0′)Δτf=2πℏE0′ ΔτN_\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. The two calculations agree exactly. This is the covariance check the construction has to pass: N⊥N_\perp[Γ\Gamma] does not care which energy convention is used, provided the convention is applied consistently — local energy paired with proper time, or Killing energy paired 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…

Read this part in the article →

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.

Open the illustrated fractions: division written vertically guide →

The article lists its research sources here.