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

=== If the shell is not just passive charge but carries energy and stress (EM energy + any mechanical tension that keeps the shell from flying apart), the shell is described by a surface stress-energy tensor SabS_{ab}Sab (indices a,ba,ba,b tangential to the 2+1D worldsheet). Two important facts: ===

• The surface energy per unit area σsurf\sigma_{\rm surf}σsurf integrates to a total mass/energy:

M  =  1c2∫Σσsurf dA.M \;=\; \frac{1}{c^2}\int_{\Sigma} \sigma_{\rm surf}\,dA.M=c21∫ΣσsurfdA.
• In General Relativity a thin shell is coupled to geometry by the Israel junction conditions. If KabK_{ab}Kab is the extrinsic curvature of the surface, the jump [Kab][K_{ab}][Kab] across the shell is related to the surface stress-energy:

  [ Kab ]  −  hab [ K ]  =  −8πG Sab,  \boxed{\;
\big[\,K_{ab}\,\big] \;-\; h_{ab}\,\big[\,K\,\big] \;=\; -8\pi G\, S_{ab},
\;}[Kab]−hab[K]=−8πGSab,
where habh_{ab}hab is the induced 3-metric on the shell and K=habKabK = h^{ab}K_{ab}K=habKab. (Units: geometric units or insert ccc’s as needed.)

Interpretation: a surface energy density and tangential stresses produce a discontinuity in extrinsic curvature → curvature of spacetime → Newtonian gravity emerges in the weak limit. In the nonrelativistic (weak field) limit the energy density component of SabS_{ab}Sab acts as a source for the Newtonian potential Φ\PhiΦ via Poisson’s equation:

∇2Φ  ≃  4πGρeff,ρeff≈S00c2.\nabla^2 \Phi \;\simeq\; 4\pi G \rho_{\rm eff},\qquad \rho_{\rm eff}\approx \frac{S_{00}}{c^2}.∇2Φ≃4πGρeff,ρeff≈c2S00.
So you can turn boundary (2-D) energy into inertial/gravitational mass.