Sharp Tangential NEC Minimization and Israel Surface Layers in Schwarzschild Mass Interpolations

Abstract

We study static, spherically symmetric metrics in Schwarzschild gauge whose mass function increases smoothly from $M_1$ to $M_2>M_1$ across an annulus $[R_1,R_2]$ outside the larger Schwarzschild radius. An elementary obstruction shows that tangential NEC violation is unavoidable in this class. This reduces the physical question to a quantitative one: what is the least possible violation, and what geometry is selected by near-minimization? We first determine the exact $L^1$-relaxation of the resulting nonlocal weighted positive-variation functional and solve the relaxed problem explicitly. Its unique minimizer is a normalized box profile, which yields the infimum of violations: the relaxed minimum is attained, while the smooth infimum is not. An exact deficit decomposition yields sharp quantitative stability, with a square-root rate at a nondegenerate critical optimizer and a linear rate at a strict constrained boundary optimizer; the same rates control the mass function and metric coefficients. We identify a variational phase transition between these regimes and prove reciprocal-width divergence in the thin-annulus limit. Every smooth minimizing sequence converges to one locally Lipschitz metric with a constant-density $p=-\rho$ bulk and two timelike surface layers, its Einstein tensors converge distributionally, and the singular terms agree with the Israel surface stress tensors. The negative tangential null energy concentrates on the inner layer, whose integrated negative pressure equals the sharp variational cost.

2 figures