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Equation 40 · Part 3 · A Horizon Is a Toll Booth, Not a Loophole

Symbol f

dτ=f(r) dtd\tau = \sqrt{f(r)}\,dt
ff

What this part means

f is an input to the expression that computes the quantity on the left.

Its job in the formula

f is an input to the expression that computes the quantity on the left.

The passage around this formula

For a static observer at fixed areal radius r in a static, spherically symmetric spacetime, proper time and coordinate time are related by dτ\tau = f(r)\sqrt{f(r)}\,dt , with f(r) = 1-2GM/(rc2c^2) for Schwarzschild. A locally conserved (Killing) energy E∞E_\infty — the energy the same quantity would be assigned by a distant static observer — and the energy measured by the local static observer, ElocE_{\rm loc} , are related by the standard blueshift-on-the-way-down relation

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Learn the underlying idea

A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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