Manuscript Hawk01-A@1 · Hawk01-A.tex · proposition · 11979653

Proposition: Scalar quasistatic scaling

The four-dimensional semiclassical source › Quasistatic mass evolution and its limitations · lines 482–517 · proof included or linked

Hawk01-A@1#prop:mass, Hawk01-A@1#eq:masssolution, Hawk01-A@1#eq:cumulativecount, Hawk01-A@1#eq:adiabaticindicator

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If the sums in (eq:alpha)–(eq:nu) converge and the isolated quasistatic ledger is adopted, then

P_{em} (M) = \frac{α}{M ^{2}}, N ˙ (M) = \frac{ν}{M}, M (u)^{3} = M_{0}^{3} - 3 α (u - u_{0}) .

For $M (u) \geq M_{cut} > 0$ , the cumulative expected scalar count is

N (u_{0}, u) = \frac{ν}{2 α} [M_{0}^{2} - M (u)^{2}] .

The local surface-gravity adiabatic indicator is

\frac{∣ κ ˙ ∣}{κ ^{2}} = \frac{4 α}{M ^{2}} .

Proof. Substitute $z = M ω$ in the power and number integrals. This gives the first two identities. Integrating $M ˙ = - α / M^{2}$ gives the third. Using $d u = - M^{2} d M / α$ in the number integral gives (eq:cumulativecount). Finally, $κ = 1/ (4 M)$ gives $κ ˙ = - M ˙ / (4 M^{2})$ , and the displayed ratio follows.

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Hawk01-A@1#eq:masssolution · line 486
$P_{em} (M) = \frac{α}{M ^{2}}, N ˙ (M) = \frac{ν}{M}, M (u)^{3} = M_{0}^{3} - 3 α (u - u_{0}) .$
Hawk01-A@1#eq:cumulativecount · line 496
$N (u_{0}, u) = \frac{ν}{2 α} [M_{0}^{2} - M (u)^{2}] .$
Hawk01-A@1#eq:adiabaticindicator · line 502
$\frac{∣ κ ˙ ∣}{κ ^{2}} = \frac{4 α}{M ^{2}} .$

Proposition: Scalar quasistatic scaling

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