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dτ=f(r) dtd\tau = \sqrt{f(r)}\,dt

Why this formula appears here

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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dτ=f(r) dtd\tau = \sqrt{f(r)}\,dt

Equation 40 · Evolutionary Physics

A Horizon Is a Toll Booth, Not a Loophole

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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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