Equation 40 · A Horizon Is a Toll Booth, Not a Loophole
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
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Symbol d
d is part of the quantity the equation computes from the expression on the right.
Symbol τ
τ is part of the quantity the equation computes from the expression on the right.
Symbol f
f is an input to the expression that computes the quantity on the left.
Symbol r
r is an input to the expression that computes the quantity on the left.
Symbol t
t is an input to the expression that computes the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
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What the article says around this equation
For a static observer at fixed areal radius r in a static, spherically symmetric spacetime, proper time and coordinate time are related by d = \,dt , with f(r) = 1-2GM/(r) for Schwarzschild. A locally conserved (Killing) energy — the energy the same quantity would be assigned by a distant static observer — and the energy measured by the local static observer, , are related by the standard blueshift-on-the-way-down relation
For background, read the article’s source list.
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