Manuscript Hawk01-B@1 · Hawk01-B.tex · section · cfa976f1

Hawking-mode calibration and compatibility limits

Hawking-mode calibration and compatibility limits · lines 1239–1290

Hawk01-B@1#eq:Hawkoccupation, Hawk01-B@1#eq:thermalstate, Hawk01-B@1#eq:thermal01, Hawk01-B@1#eq:thermaltail

The source spectrum does not determine the transfer schedule $λ_{t}$ . We make only a declared comparison with selected outgoing occupation states.

For a neutral scalar stationary energy mode, the fixed-background outgoing occupation is

n (ω; M) = \frac{Γ _{0} ( ω ; M )}{e ^{ω / T_{H}} - 1}, q = \frac{n}{1 + n} .

This is the greybody-modified source interface developed in Hawk01-A and Em03-A CheungHawkA,CheungEm03A,Hawking1975. We do not repeat its field-theory derivation.

For $g$ identical degenerate scalar modes, define the product thermal state

σ_{g} (q) = (1 - q)^{g} n \in N^{g} \sum q^{∣ n ∣} ∣ n ⟩ ⟨ n ∣, 0 \leq q < 1.

Its normalized restriction to total occupation zero or one is

χ_{g} (q) = \frac{∣ vac ⟩ ⟨ vac ∣ + q I _{g} ^{(1)}}{1 + g q}, a_{g} (q) = \frac{g q}{1 + g q} .

The discarded thermal probability is

θ_{g} (q) = 1 - (1 - q)^{g} (1 + g q) .

Displayed formulas in this passage

Hawk01-B@1#eq:Hawkoccupation · line 1247
$n (ω; M) = \frac{Γ _{0} ( ω ; M )}{e ^{ω / T_{H}} - 1}, q = \frac{n}{1 + n} .$
Hawk01-B@1#eq:thermalstate · line 1263
$σ_{g} (q) = (1 - q)^{g} n \in N^{g} \sum q^{∣ n ∣} ∣ n ⟩ ⟨ n ∣, 0 \leq q < 1.$
Hawk01-B@1#eq:thermal01 · line 1272
$χ_{g} (q) = \frac{∣ vac ⟩ ⟨ vac ∣ + q I _{g} ^{(1)}}{1 + g q}, a_{g} (q) = \frac{g q}{1 + g q} .$
Hawk01-B@1#eq:thermaltail · line 1284
$θ_{g} (q) = 1 - (1 - q)^{g} (1 + g q) .$

Hawking-mode calibration and compatibility limits

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