A Universal Master Relation For Static Thin Shells In The Electrovacuum–Λ Class

Static, spherically symmetric thin shells governed by the Darmois–Israel junction conditions are studied for the family of metrics with g tt g rr = −1 and f(r) = 1 − 2M/r + Q 2 /r 2 − Λ r2 /3, which contains the Minkowski, Schwarzschild, Reissner–Nordström, and (anti-)de Sitter geometries and their combinations. In terms of the shell mass m s = 4πR 2 Σ, every static shell of the family obeys 2w = −d ln m s /d ln R, where w = P/Σ is the ratio of the tangential pressure to the surface energy density. We show that this relation is equivalent to a single algebraic condition in which a first-order Euler operator acts diagonally on the powers of r in f, the mode r n carrying the weight 1 + 2w + n/2. Three consequences follow: a resonance rule, by which the mode r n drops out of the static condition exactly when w = −(2 + n)/4; an orientation sum rule relating the equations of state of the two possible gluings; and an identity expressing Σ + P through the photon-sphere functions of the two sides. The framework is applied to shells around charged and de Sitter black holes, gravastars, and thin-shell wormholes, for which the known equation of state is recovered, and the null energy condition at the throat is found to be saturated exactly at the photon sphere, independent of Λ. For a Minkowski interior and Schwarzschild exterior it yields a closed-form classification of static puretension shells, with a unique, unstable equilibrium of compactness 8/9. The relationship between each result and earlier work is explicitly stated.