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

Occupation and flux are not dimensions

Greybody weighting: what is calculable and what is declared › Occupation and flux are not dimensions · lines 1206–1245

Hawk05-C@1#eq:power

In the stationary massless scalar setup, with the additional time–frequency counting convention, the supplied occupation is

n_{α} (ω; M) = \frac{Γ _{α} ( ω ; M )}{e ^{8 π M ω} - 1} .

Its flux is

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

Here $α$ denotes the declared modes or species and $g_{α}$ their counted multiplicities. The stationary field derivation and its validity hypotheses are external physical inputs Hawking,PageEmission,CheungEm03A. Using them at an evolving $M (t)$ additionally assumes a quasistatic approximation.

If $Γ_{α} (ω; M) = γ_{α} (M ω)$ for a fixed massless mode set, and the following integrals converge, then

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

This change of variables explains the inverse-square clock. It says nothing by itself about $r (t)$ or $b (t)$ .

Displayed formulas in this passage

Hawk05-C@1#eq:power · line 1217
$P (M) = α \sum g_{α} \int_{0}^{\infty} \frac{d ω}{2 π} ω n_{α} (ω; M) .$
unlabeled · line 1210
$n_{α} (ω; M) = \frac{Γ _{α} ( ω ; M )}{e ^{8 π M ω} - 1} .$
unlabeled · line 1236
$P (M) = \frac{λ}{M ^{2}}, λ = α \sum g_{α} \int_{0}^{\infty} \frac{d x}{2 π} \frac{x γ _{α} ( x )}{e ^{8 π x} - 1} .$

Occupation and flux are not dimensions

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