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

==== Two steps give a rigorous link: ====

(A) Build a surface stress–energy SabS_{ab}Sab from photon–electron boundary interactions (energy per unit area σ, tangential pressure p, and flow j):

Sab=σuaub+p hab+jaub+jbua,S_{ab} = \sigma u_a u_b + p \, h_{ab} + j_{a}u_{b} + j_{b}u_{a},Sab=σuaub+phab+jaub+jbua,
where habh_{ab}hab is the induced metric on the 2-surface and uau^aua the local 4-velocity.

(B) Use the Israel junction condition to relate a thin shell’s surface stress tensor to the jump in extrinsic curvature KabK_{ab}Kab and thus to spacetime curvature:

[Kab]−hab [K]=−8πG Sab.\big[ K_{ab} \big] - h_{ab}\, \big[ K\big] = -8\pi G \, S_{ab}.[Kab]−hab[K]=−8πGSab.
This is a standard GR result for surface layers. Therefore if QAT yields a well-defined SabS_{ab}Sab coming from area-based photon energy, it will generate curvature (gravity) exactly as GR prescribes for thin shells. The effective long-range metric (Schwarzschild-like) can then be found by solving Einstein’s equations outside the shell.

How to connect to standard stress-energy Tμν: smear the surface into a thin volume with thickness δ and define

Teffμν(x)≈σ(θ,ϕ) δ(r−r0)δ uμuν+⋯ ,T^{\mu\nu}_{\rm eff}(x) \approx \sigma(\theta,\phi)\, \frac{\delta(r-r_0)}{\delta}\, u^\mu u^\nu + \cdots,Teffμν(x)≈σ(θ,ϕ)δδ(r−r0)uμuν+⋯,
then plug into Rμν−12Rgμν=8πGTμνR_{\mu\nu}-\tfrac12 R g_{\mu\nu}=8\pi G T_{\mu\nu}Rμν−21Rgμν=8πGTμν.

Status: the mathematical toolset is standard (Israel junctions); QAT's job is to produce explicit SabS_{ab}Sab from microphysics (photon exchange rates → σ and p). That is the critical technical step left undone.