Adiabatic Deformations of Black Hole Moduli: I. Canonical Slices and Fredholm Solvability
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.