Manuscript Hawk02-A@1 · Hawk02-A.tex · proposition · 7fbae745

Proposition: Adiabatic diagnostic

Stage 1: the fixed-background comparison object › Power and the quasistatic ledger · lines 443–469 · proof included or linked

Hawk02-A@1#prop:adiabatic, Hawk02-A@1#eq:adiabatic

For a differentiable mass ledger satisfying $M ˙ = - P_{sc} (M)$ ,

χ_{ad} := \frac{∣ κ ˙ ∣}{κ ^{2}} = 4 G P_{sc} (M) = \frac{κ ^{- 1}}{M / P _{sc} ( M )} .

If $P_{sc} (M) = a / (G^{2} M^{2})$ for a fixed $a > 0$ , then

χ_{ad} = \frac{4 a}{G M ^{2}}, M (u)^{3} = M (0)^{3} - \frac{3 a}{G ^{2}} u,

only up to the declared stopping time.

Proof. Differentiate $κ = (4 G M)^{- 1}$ and substitute the ledger equation. The cubic formula follows by differentiating $M^{3}$ .

Displayed formulas in this passage

Hawk02-A@1#eq:adiabatic · line 447
$χ_{ad} := \frac{∣ κ ˙ ∣}{κ ^{2}} = 4 G P_{sc} (M) = \frac{κ ^{- 1}}{M / P _{sc} ( M )} .$
unlabeled · line 458
$χ_{ad} = \frac{4 a}{G M ^{2}}, M (u)^{3} = M (0)^{3} - \frac{3 a}{G ^{2}} u,$