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

An additional coarse entropy-rate model

Greybody weighting: what is calculable and what is declared › An additional coarse entropy-rate model · lines 1282–1325

Hawk05-C@1#eq:xi-greybody

If one separately assumes independent geometric outgoing modes and the same counting convention, define

g_{2} (u) = (u + 1) lo g_{2} (u + 1) - u lo g_{2} u, g_{2} (0) = 0,

and a coarse entropy flux

A ˙_{cg} (M) = α \sum g_{α} \int_{0}^{\infty} \frac{d ω}{2 π} g_{2} (n_{α} (ω; M)) .

The geometric-state entropy is $g_{2} (u)$ , obtained by summing its number probabilities. Mode independence makes these entropies additive. This is a coarse state model, not the fine-grained entropy of a variable-code evaporation channel.

Under the same massless scaling,

A ˙_{cg} (M) = \frac{σ}{M}, σ = α \sum g_{α} \int_{0}^{\infty} \frac{d x}{2 π} g_{2} (\frac{γ _{α} ( x )}{e ^{8 π x} - 1}) .

Since $- B ˙ = 8 π λ / (M ln 2)$ , its declared ratio is

ξ = \frac{σ ln 2}{8 π λ} .

Equations (eq:power) and (eq:xi-greybody) require different state and counting assumptions. If this ratio is not one, using $A_{cg}$ as $lo g_{2} r$ changes the fixed-dimensional proxy, by Proposition prop:xi.

A greybody factor suppresses a selected transmitted occupation. It does not universally specify an “early-time suppression” of a Page curve, nor prove typicality of the source code.

Displayed formulas in this passage

Hawk05-C@1#eq:xi-greybody · line 1313
$ξ = \frac{σ ln 2}{8 π λ} .$
unlabeled · line 1286
$g_{2} (u) = (u + 1) lo g_{2} (u + 1) - u lo g_{2} u, g_{2} (0) = 0,$
unlabeled · line 1291
$A ˙_{cg} (M) = α \sum g_{α} \int_{0}^{\infty} \frac{d ω}{2 π} g_{2} (n_{α} (ω; M)) .$
unlabeled · line 1304
$A ˙_{cg} (M) = \frac{σ}{M}, σ = α \sum g_{α} \int_{0}^{\infty} \frac{d x}{2 π} g_{2} (\frac{γ _{α} ( x )}{e ^{8 π x} - 1}) .$