Coarse-Grained Vacuum Boundaries in Gravitational Collapse: A Timelike Thin-Shell Balance Framework

A macroscopic boundary-balance framework for gravitational collapse is developed for a timelike thin shell separating an effective interior vacuum reference sector from an exterior Schwarzschild or Schwarzschild–de Sitter region. The interior sector is represented by a coarse-grained reference density ρref(χ) and the associated reference energy Eref=(4π/3)R3ρref. Along the shell history, reference-energy descent occurs when the decrease of ρref dominates the geometric increase of the enclosed volume. This condition defines the effective quasilocal input Φeff=−AΣ−1E˙ref, which is positive precisely on the descending-reference branch. The timelike shell converts this input into a finite boundary response. The central balance law, E˙Σ=AΣ(Φeff−Φout)−PA˙Σ, partitions reference-sector input into quasilocal shell storage, exterior release, and pressure–area work. A trajectory-dependent response coefficient Ceff=dEref/dTΣ parametrizes the local boundary-temperature response; on a negative-response branch, reference-energy descent increases the shell temperature. Local shell temperatures and near-boundary mode frequencies are mapped to exterior static observers by the exterior lapse, with spatial infinity recovered only in the asymptotically Schwarzschild limit. The resulting timelike thin shell is a finite-radius quasilocal boundary that organizes reference-state change through surface stress, flux balance, area response, and redshifted observables. The entropy-like variable SΣ=αAΣ records the macroscopic area response and enters the same balance through the pressure–area work term. The framework identifies the classical boundary variables and closure conditions required for perturbative stability analyses, finite-thickness response models, and microscopic boundary descriptions.