Timelike Thin Shells: Local Lorentzian Geometry from Timelike Boundaries

Timelike boundaries provide a natural setting for organizing geometric, quasilocal, and coarse-grained information in general relativity. This work develops a cut-level reference framework for finite-radius timelike interfaces in Lorentzian spacetime. Starting from a timelike boundary, a tangent observer field, and observer-adapted spatial cuts, the construction assigns selected boundary quantities, coarse-grained reference structures, channel-specific comparison values, resolved deviations, local event closure, and cut-level response terms to the same geometric surface. The framework is local in its physical reading. The coarse-grained reference structure is not treated as a single resolved boundary record, but as the macroscopic comparison structure relative to which local deviations are defined. A local boundary event is represented by a boundary-relative deviation that becomes resolvable at the candidate event. The causal condition fixes the Lorentzian admissibility domain; it does not by itself define a resolved trajectory or microscopic propagation history between spacetime points. In the classical realization developed here, the selected variables are supplied by the Brown--York cut-level dictionary. Observer-adapted projections of the boundary stress tensor define surface energy density, momentum density, spatial cut stress, and isotropic pressure. A coarse-grained boundary reference package specifies which variables are resolved, on which cut they are evaluated, and which reference structure serves as their comparison level. The corresponding deviation map and channel-dependent resolution norms identify the locally resolved boundary content. The same cut-level variables also enter a classical balance structure in which cut-energy variation separates into normal exchange and tangential mechanical response. In isotropic spherical symmetry, this response reduces to the pressure--area form, linking cut-level stress to the area-response channel of a timelike shell. Timelike thin-shell dynamics and macroscopic shell-balance laws then appear as concrete realizations of the general reference-cut structure. The resulting formulation provides a classical boundary-reference language for finite-radius timelike systems, relating local Lorentzian geometry, quasilocal stress, coarse-grained reference structure, resolved deviations, causal admissibility, and area response within one common cut-level framework.