Manuscript Hawk05-C@1 · Hawk05-C.tex · section · ba0719f1

A supplied semiclassical clock

Replacing the clock: compatibility, crossover, and cutoff › A supplied semiclassical clock · lines 940–976

Hawk05-C@1#eq:BH, Hawk05-C@1#eq:A, Hawk05-C@1#eq:envelope

Only in this subsection and its clock examples set $G = ℏ = c = k_{B} = 1$ . For a Schwarzschild mass label, the Bekenstein–Hawking entropy in bits is

B (M) = c_{BH} M^{2}, c_{BH} = \frac{4 π}{ln 2} .

This is a supplied semiclassical entropy law Bekenstein,Hawking. Identifying $B (M)$ with $lo g_{2} dim B_{proxy}$ is an additional modelling declaration.

Let $M_{0} > M_{cut} > 0$ . Supply a positive continuous power law $P (M)$ on $[M_{cut}, M_{0}]$ and define

M ˙ = - P (M), M (t_{0}) = M_{0} .

Its cutoff clock is the inverse of

t (M) - t_{0} = \int_{M}^{M_{0}} \frac{d u}{P ( u )} .

Continuity and positivity make this inverse unique.

Supply a positive continuous capacity-production ratio $ξ (M)$ and define the target, not the fine-grained entropy,

A (M) = \int_{M}^{M_{0}} ξ (u) B^{'} (u) d u .

The candidate envelope is

F (t) = min {A (M (t)), B (M (t))} .

Displayed formulas in this passage

Hawk05-C@1#eq:BH · line 946
$B (M) = c_{BH} M^{2}, c_{BH} = \frac{4 π}{ln 2} .$
Hawk05-C@1#eq:A · line 967
$A (M) = \int_{M}^{M_{0}} ξ (u) B^{'} (u) d u .$
Hawk05-C@1#eq:envelope · line 972
$F (t) = min {A (M (t)), B (M (t))} .$
unlabeled · line 956
$M ˙ = - P (M), M (t_{0}) = M_{0} .$
unlabeled · line 960
$t (M) - t_{0} = \int_{M}^{M_{0}} \frac{d u}{P ( u )} .$

Nearby in this section

theorem: Theorem: Capacity crossover before a positive cutoff