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Equation 54 · Part 2 · A Horizon Is a Toll Booth, Not a Loophole

√

1/f1/\sqrt f
√

What this part means

Take a square root.

Its job in the formula

Take a square root.

The passage around this formula

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 than one factor and its…

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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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