Angular extrinsic curvature jumps for spacelike thin shells in generalized Vaidya spacetimes: a closed-form classification

For any spacelike hypersurface r = R(v) separating two regions of a D-dimensional generalized Vaidya spacetime (D ≥ 4), with Ṙ > f/2, continuous Misner–Sharp mass at the boundary, and a common orientation branch on the two sides, the angular extrinsic curvature jump vanishes identically: [K^θ̂_θ̂] = 0. The result follows from a general warped-product identity: on any spherically symmetric non-null hypersurface, K^θ̂_θ̂ = (n^a ∂_a r)/r, so the angular jump is fixed by the boundary value of f and