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Published equation contexts

Eloc=E∞f(r),so thatEloc dτ=E∞ dtE_{\rm loc} = \frac{E_\infty}{\sqrt{f(r)}}, \qquad\text{so that}\qquad E_{\rm loc}\,d\tau = E_\infty\,dt

Why this formula appears here

​ d t , 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 Eloc=E∞f(r),so thatEloc dτ=E∞ dtE_{\rm loc} = \frac{E_\infty}{\sqrt{f(r)}}, \qquad\text{so that}\qquad E_{\rm loc}\,d\tau = E_\infty\,dt. That second identity is the entire resolution in miniature, and it is worth deriving N⊥N_\perp[Γ\Gamma] both ways to see it operate. Hold the local energy budget fixed at ElocE_{\rm loc}=E0E_0' (a battery physically at the processor, delivering a fixed amount of locally measured energy) over a proper interval Δ\Deltaτ\tau :

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ElocE_{\rm loc}

Symbol E_rm loc

ErE_rm loc is part of the quantity the equation computes from the expression on the right.

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E∞E_\infty

Symbol E_infty

EiE_infty occurs above the fraction bar. The numerator is divided by the entire denominator below it.

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rr

Symbol r

r occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

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f(r)\sqrt{f(r)}

Denominator: sqrtf(r)

The complete quantity below the fraction bar; it must be nonzero for this division.

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How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

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Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

Eloc=E∞f(r),so thatEloc dτ=E∞ dt.E_{\rm loc} = \frac{E_\infty}{\sqrt{f(r)}}, \qquad\text{so that}\qquad E_{\rm loc}\,d\tau = E_\infty\,dt.

Equation 44 · 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.

​ d t , 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 Eloc=E∞f(r),so thatEloc dτ=E∞ dtE_{\rm loc} = \frac{E_\infty}{\sqrt{f(r)}}, \qquad\text{so that}\qquad E_{\rm loc}\,d\tau = E_\infty\,dt. That second identity is the entire resolution in miniature, and it is worth deriving N⊥N_\perp[Γ\Gamma] both ways to see it operate. Hold the local energy budget fixed at ElocE_{\rm loc}=E0E_0' (a battery physically at the processor, delivering a fixed amount of locally measured energy) over a proper interval Δ\Deltaτ\tau :

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