Equation 52 · Part 12 · A Horizon Is a Toll Booth, Not a Loophole

subscript

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.

subscript

What this part means

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Its job in the formula

A subscript distinguishes a version, component, step, or member of a quantity. It does not automatically mean multiplication.

Full expressionsubscriptArticle meaning

The passage around this formula

: $N_\perp = \frac{2}{\pi\hbar}E_\infty\,\Delta t = \frac{2}{\pi\hbar}\left(\sqrt f\,E_0'\right)\frac{\Delta\tau}{\sqrt f} = \frac{2}{\pi\hbar}E_0'\,\Delta\tau$ . The two calculations agree exactly. This is the covariance check the construction has to pass: $N_\perp$ [ $\Gamma$ ] 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/ $\sqrt f$ multiplying local energy and pairs it with the huge 1/ $\sqrt f$ multiplying elapsed coordinate time, double-counting the same factor as though it were two independent windfalls rather…

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Learn the underlying idea

A subscript is a label attached below a symbol. It often selects a time step, component, category, or member of a sequence.

Open the illustrated subscripts: which member of a family? guide →

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