$$D_{\rm det}(a,\, a+\delta a) \;\approx\; \left[\max_{u>0} u\,g(u)\right] \cdot \frac{|\delta a|}{a}$$

Equation 82 · Evolutionary Physics

No Particle Without a Cosigner

This equation gives an approximation: it relates the quantities while allowing an approximation.

Every accelerated detector’s normalized spectrum is the same universal curve g , only rescaled by a/c . For two accelerations separated by a small increment, a and a+$\delta$ a , standard calculus for a scale family gives, to leading order in $\delta$ a , $D_{\rm det}(a,\, a+\delta a) \;\approx\; \left[\max_{u>0} u\,g(u)\right] \cdot \frac{|\delta a|}{a}$. This is the same trick used to derive Wien’s displacement law: because u\,g(u) rises from zero, passes through a single interior maximum, and falls back to zero, the integral of the absolute value of its derivative over all u is exactly twice that maximum, with no need to evaluate the integral directly. Finding the maximum of u\,g(u) = 24\,$u^2$/($e^{2\pi u}$-1) means solving $e^{2\pi u}(1-\pi u)$=1 , the same class of…

Meanings in this article

• $D_{\rm det}$: well defined, it can be evaluated, and the cleanest case where it is well defined and non-trivial is a comparison between two accelerated trajectories, both with finite normalizable spectra, differing only slightly in acceleration.
• $u$: exactly twice that maximum.

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