黑洞模的绝热形变：I. 类空切片与Fredholm可解性
Adiabatic Deformations of Black Hole Moduli: I. Canonical Slices and Fredholm Solvability
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中文总结 AI 辅助
该研究探讨静态黑洞对缓慢演化渐近模的绝热响应，通过显式标量准则刻画几何与泛函分析问题，确定类空切向方向，推导Fredholm可解性条件，为黑洞模绝热形变提供理论框架。
中文摘要 AI 辅助
静态黑洞对缓慢演化的渐近模的主导绝热响应可分为几何问题与泛函分析问题：前者确定黑洞在静态解流形上实际遵循的切向方向，后者判定所得形变是否存在唯一解。我们证明两者均可由显式标量准则表征。对静态解的光滑族求导可在其切空间上定义一个固有余矢量；当该族由渐近模与静态视界-质量参数参数化时，此余矢量的核确定一个类空切向方向，其与缓慢变化的外模所选方向一致。将静态族推广为缓慢演化的瞬时代表并不能满足精确场方程，所得偏差定义为滞后场，在度规微扰消除后，该滞后场由单个约化径向算子支配。一种表示约定将精确动力学视界与瞬时代表的视界区分开；局部切片则消除剩余的切向歧义，并诱导约化算子的有限秩扰动。Fredholm指数保持不变，且当有限秩更新非平凡时，类空切向零模会从约化算子的核中位移。在横截性与Fredholm假设下，完整的主导阶绝热边值问题的存在性与唯一性可简化为对所得核生成元的单个标量匹配条件。该构造仅依赖光滑非极端静态解流形和约化算子的Fredholm性质；一篇姊妹论文将该框架应用于磁GHS族。
英文摘要
The leading adiabatic response of a static black hole to a slowly evolving asymptotic modulus separates into a geometric problem and a functional-analytic one: the former identifies the tangent direction on the manifold of static solutions actually followed by the black hole, and the latter determines whether the resulting deformation admits a unique solution. We show that both can be characterized by explicit scalar criteria. Differentiating a smooth family of static solutions defines an intrinsic covector on its tangent space. When the family is parameterized by the asymptotic modulus and the static horizon-mass parameter, the kernel of this covector identifies a canonical tangent direction that coincides with the one selected by a slowly varying exterior modulus. Promoting the static family to a slowly evolving instantaneous representative does not solve the exact field equations; the resulting discrepancy defines a lag field governed, after elimination of the metric perturbations, by a single reduced radial operator. A representation convention separates the exact dynamical horizon from that of the instantaneous representative. A local slice then removes the remaining tangential ambiguity and induces a finite-rank perturbation of the reduced operator. The Fredholm index is preserved, and, whenever the finite-rank update is nontrivial, the canonical tangent zero mode is displaced from the kernel of the reduced operator. Under the transversality and Fredholm hypotheses, existence and uniqueness of the completed leading-order adiabatic boundary-value problem reduce to a single scalar matching condition on the resulting kernel generator. The construction depends only on a smooth non-extremal static solution manifold and the Fredholm properties of the reduced operator. A companion paper specializes the framework to the magnetic GHS family.
