Absence of the barotropic shell-matter instability for σ = 0 thin-shell junctions

Pitre, Schneider, and Poisson recently established that self-gravitating thin shells with nonvanishing surface energy density σ > 0 and barotropic matter are dynamically unstable on all even-parity angular scales ℓ ≥ 2. We show that their instability mode is absent for thin-shell junctions with σ = 0. For generalized-Vaidya interiors matched to Vaidya exteriors, the first junction condition forces M(v) = m(v, r_Σ), giving σ = 0 identically. The linearized angular extrinsic curvature trace jump δ