📝 SummarXiv

Abstract

We show that any one-parameter family of static, spherically symmetric spacetime metrics can be embedded as the unique vacuum solution of a generally covariant theory of pure gravity. These theories, involving at most second-order derivatives of metrics, are uniquely determined -- up to two arbitrary functions that leave the metric unchanged -- by a framework that resolves the problem of restoring general covariance in the Hamiltonian formulation. This result solves the inverse problem of Birkhoff's theorem in full generality and establishes a systematic approach for reconstructing covariant dynamics from a given geometry. The framework applies broadly to quantum-gravity-inspired metrics, including regular black hole models, and provides a new tool for probing their dynamical and observational foundations.