← Mathematical compendium

Published equation contexts

F˙a(Ω)  ∝  Ωe2πcΩ/a−1,Ω>0\dot{\mathcal F}_a(\Omega) \;\propto\; \frac{\Omega}{e^{2\pi c \Omega / a} - 1}, \qquad \Omega > 0

Why this formula appears here

The first check any new object owes the literature is that it reproduce the literature’s own answer in the case the literature has already solved. For a worldline held at constant proper acceleration a forever, coupled linearly to a massless scalar field in the ordinary vacuum, with the switching left on for all time, the Wightman function depends only on the proper-time separation and the standard Fourier transform is exact and textbook: the excitation-branch rate per unit proper time takes the Bose-Einstein form F˙a(Ω)  ∝  Ωe2πcΩ/a−1,Ω>0\dot{\mathcal F}_a(\Omega) \;\propto\; \frac{\Omega}{e^{2\pi c \Omega / a} - 1}, \qquad \Omega > 0. with the proportionality constant fixed by the detector’s coupling strength and dimension — irrelevant here, since it cancels the instant the spectrum is…

Read the full article-specific guide →

Read the representative guide

F˙a\dot{\mathcal F}_a

Symbol dotmathcal F_a

dotmathcal FaF_a has a dot, marking the rate of change of the underlying indexed quantity with respect to the article’s time variable.

Read this term in its guide →
Ω\Omega

Symbol Omega

Omega is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Read this term in its guide →
e2πcΩ/ae^{2\pi c \Omega / a}

Symbol e^2pi c Omega / a

e2e^2pi c Omega / a occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Read this term in its guide →
e2πcΩ/a−1e^{2\pi c \Omega / a} - 1

Denominator: e^2pi c Omega / a - 1

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

Read this term in its guide →

How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction.

Research cited beside this formula

Learn the mathematics behind it

Published contexts (1)

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

F˙a(Ω)  ∝  Ωe2πcΩ/a−1,Ω>0,\dot{\mathcal F}_a(\Omega) \;\propto\; \frac{\Omega}{e^{2\pi c \Omega / a} - 1}, \qquad \Omega > 0,

Equation 50 · Evolutionary Physics

No Particle Without a Cosigner

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

The first check any new object owes the literature is that it reproduce the literature’s own answer in the case the literature has already solved. For a worldline held at constant proper acceleration a forever, coupled linearly to a massless scalar field in the ordinary vacuum, with the switching left on for all time, the Wightman function depends only on the proper-time separation and the standard Fourier transform is exact and textbook: the excitation-branch rate per unit proper time takes the Bose-Einstein form F˙a(Ω)  ∝  Ωe2πcΩ/a−1,Ω>0\dot{\mathcal F}_a(\Omega) \;\propto\; \frac{\Omega}{e^{2\pi c \Omega / a} - 1}, \qquad \Omega > 0. with the proportionality constant fixed by the detector’s coupling strength and dimension — irrelevant here, since it cancels the instant the spectrum is…

Equation guide → · Article →