Manuscript Hawk02-E@1 · Hawk02-E.tex · proposition · 4841592b

Proposition: Integrated gravitational energy allowance

Gravitational correspondence and a common validity interval › Integrated energy comparison is safer than an unsupported derivative · lines 1818–1856 · proof included or linked

Hawk02-E@1#prop:integrated-energy

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Suppose a future correspondence proves, on $[t_{0}, t_{1}]$ ,

∣ E_{grav} (t) - E_{mic} (t) ∣ \leq η_{E} (t),

and the microscopic balance satisfies

E_{mic} (t_{1}) - E_{mic} (t_{0}) = - \int_{t_{0}}^{t_{1}} F_{mic} (s) d s .

If it also proves

\int_{t_{0}}^{t_{1}} (F_{out} - F_{mic}) (s) d s \leq η_{F},

then

E_{grav} (t_{1}) - E_{grav} (t_{0}) + \int_{t_{0}}^{t_{1}} F_{out} (s) d s \leq η_{E} (t_{0}) + η_{E} (t_{1}) + η_{F} .

Proof. Insert $E_{mic}$ at both endpoints and insert $F_{mic}$ in the integral. Use the exact microscopic balance and the triangle inequality.

Displayed formulas in this passage

unlabeled · line 1821
$∣ E_{grav} (t) - E_{mic} (t) ∣ \leq η_{E} (t),$
unlabeled · line 1826
$E_{mic} (t_{1}) - E_{mic} (t_{0}) = - \int_{t_{0}}^{t_{1}} F_{mic} (s) d s .$
unlabeled · line 1832
$\int_{t_{0}}^{t_{1}} (F_{out} - F_{mic}) (s) d s \leq η_{F},$
unlabeled · line 1840
$E_{grav} (t_{1}) - E_{grav} (t_{0}) + \int_{t_{0}}^{t_{1}} F_{out} (s) d s \leq η_{E} (t_{0}) + η_{E} (t_{1}) + η_{F} .$

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section: Integrated energy comparison is safer than an unsupported derivative