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

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Start with standard EM stress–energy:

TEMμν=FμαFνα−14 ημνFαβFαβ.T^{\mu\nu}_{\rm EM}
= F^{\mu\alpha}F^{\nu}{}_{\alpha}
-\frac{1}{4}\,\eta^{\mu\nu} F^{\alpha\beta}F_{\alpha\beta}.TEMμν=FμαFνα−41ημνFαβFαβ.
Model a thin interaction shell Σ\SigmaΣ of radius RRR by inserting a radial delta:

Tμν(x)=Tbulkμν(x)+Sμν(x) δ ⁣(ρ(x)−R),T^{\mu\nu}(x) = T^{\mu\nu}_{\rm bulk}(x) + S^{\mu\nu}(x)\,\delta\!\big(\rho(x)-R\big),Tμν(x)=Tbulkμν(x)+Sμν(x)δ(ρ(x)−R),
with ρ\rhoρ the geodesic radius from the nucleus/worldline and SμνS^{\mu\nu}Sμν the surface stress–energy induced by the net EM momentum flow across the shell (obtained by integrating the Poynting tensor through the pillbox).

Project SμνS^{\mu\nu}Sμν to the shell to get the intrinsic surface tensor SabS^{ab}Sab (indices a,ba,ba,b live on Σ\SigmaΣ, with induced metric habh_{ab}hab).

==== In GR, a thin shell source yields a jump in extrinsic curvature KabK_{ab}Kab across Σ\SigmaΣ: ====

[Kab−Khab]−+=8πGc4 Sab.\big[K_{ab} - K h_{ab}\big]^{+}_{-}
= \frac{8\pi G}{c^4}\, S_{ab}.[Kab−Khab]−+=c48πGSab.
This is a direct map from localized (photon-driven) surface energy–momentum to spacetime curvature. It bypasses the need to guess GμνG_{\mu\nu}Gμν: you compute SabS_{ab}Sab from EM on the shell, plug it into the junction condition, and you get the geometric response.

What to compute next (doable):
* Compute SabS_{ab}Sab from an isotropic train of spherical pulses with mean energy ⟨ΔE⟩\langle\Delta E\rangle⟨ΔE⟩ and event rate Γ\GammaΓ (your “ticks” of local time).
* Show that, in the weak-field limit, the junction equation reproduces the Newtonian potential with 1/r21/r^21/r2 and gives the right post-Newtonian correction signs.