AI 中文总结
该研究针对施瓦西规范下质量平滑变化的球对称度量，确定了切向零能量条件违反的最小程度及对应几何，揭示了变分相变与表面层特性，推导了稳定性速率及相关收敛结果。
AI 中文摘要
我们研究施瓦西规范下的静态球对称度量，其质量函数在较大的施瓦西半径外的环带[R₁,R₂]上从M₁平滑增加至M₂（M₂>M₁）。一个基本阻碍表明，此类结构中不可避免地存在切向零能量条件（NEC）违反，这将物理问题简化为定量问题：最小可能的违反程度是多少？近最小化会选择何种几何结构？我们首先确定所得非局部加权正变分泛函的精确L¹松弛解，并显式求解该松弛问题。其唯一极小值为归一化箱型轮廓，对应违反的下确界：松弛最小值可达到，而平滑下确界无法达到。精确亏空分解可得到尖锐定量稳定性，在非退化临界优化器处为平方根速率，在严格约束边界优化器处为线性速率；相同速率控制质量函数与度量系数。我们识别出这些 regime 间的变分相变，并证明在薄环带极限下存在倒数宽度发散。每个平滑极小序列收敛至一个具有常密度p=-ρ体的局部利普希茨度量，以及两个类时表面层，其爱因斯坦张量按分布收敛，且奇异项与以色列表面应力张量一致。负切向零能量集中在内层，其积分负压等于尖锐变分代价。
英文摘要
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 $p=-ρ$ 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.
Comments2 figures
