Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== 1. Asymptotic energy definition ===

(γM0c2)∞=lim⁡T→∞1T∫0T ⁣[E(t)+Lrad(t)−∑i:ti≤tΔEi] ⁣dt.(\gamma M_0 c^2)_\infty
= \lim_{T\to\infty}\frac{1}{T}\int_0^T\!\left[E(t)+\mathcal{L}_{\rm rad}(t)-\sum_{i:t_i\le t}\Delta E_i\right]\!dt.(γM0c2)∞=T→∞limT1∫0T[E(t)+Lrad(t)−i:ti≤t∑ΔEi]dt.
# Weak-field redshift for exchange/clock rate

Γ(r)Γ(∞)=1−2GMrc2⇒δΓΓ≈−GMrc2.\frac{\Gamma(r)}{\Gamma(\infty)}=\sqrt{1-\frac{2GM}{rc^2}}
\quad\Rightarrow\quad
\frac{\delta \Gamma}{\Gamma}\approx -\frac{GM}{rc^2}.Γ(∞)Γ(r)=1−rc22GM⇒ΓδΓ≈−rc2GM.
# Optical-metric time and variation with radius

t(R)≈R−rac+GMc3ln⁡ ⁣Rra,δtt≈δRR+GMRc21ln⁡(R/ra).t(R)\approx \frac{R-r_a}{c}+\frac{GM}{c^3}\ln\!\frac{R}{r_a},
\qquad
\frac{\delta t}{t}\approx \frac{\delta R}{R}
+\frac{GM}{Rc^2}\frac{1}{\ln(R/r_a)}.t(R)≈cR−ra+c3GMlnraR,tδt≈RδR+Rc2GMln(R/ra)1.
# Israel junction (shell curvature sourcing)

[Kab−Khab]−+=8πGc4Sab,Sab from EM surface stress (Poynting) + boundary terms.\big[K_{ab}-Kh_{ab}\big]^{+}_{-}=\frac{8\pi G}{c^4}S_{ab},
\quad
S_{ab}\ \text{from EM surface stress (Poynting) + boundary terms}.[Kab−Khab]−+=c48πGSab,Sab from EM surface stress (Poynting) + boundary terms.
# Effective spin-2 from coarse-grained photons

hμν=N(⟨AμAν⟩−14ημν⟨A2⟩),□hμν≃κ(JμJν−14ημνJ2).h_{\mu\nu}=\mathcal{N}
\left(\langle A_\mu A_\nu\rangle-\tfrac14\eta_{\mu\nu}\langle A^2\rangle\right),
\quad
\square h_{\mu\nu}\simeq \kappa\left(J_\mu J_\nu-\tfrac14\eta_{\mu\nu}J^2\right).hμν=N(⟨AμAν⟩−41ημν⟨A2⟩),□hμν≃κ(JμJν−41ημνJ2).

==== - A clean, checkable meaning for your (⋅)∞(\cdot)_\infty(⋅)∞. ====
* A metric-level derivation of δt/t\delta t/tδt/t (and when it reduces to δR/R\delta R/RδR/R).
* A direct, GR-exact bridge from spherical EM exchange to curvature via the junction conditions.
* A credible spin-2 mechanism framed in QFT language.

If you want, I can now do a compact worked example: steady spherical shell with power PPP ⇒ compute SabS_{ab}Sab ⇒ apply Israel junction ⇒ recover Newtonian potential and estimate post-Newtonian size. That would be a perfect, bite-sized “unit test” for QAT.