Dynamical potential and evolving circular orbits in dynamical spacetimes

Abstract: In this work, we extend the effective potential method to dynamical spacetimes by introducing a novel concept of dynamical potential and establishing three fundamental conditions that govern the evolution of circular orbits in such spacetimes. By generalizing the framework from static to dynamical spacetimes, we propose three conditions for the evolution equation of circular orbits: (1) The orbital radius of the evolving circular orbits does not explicitly depend on the affine parameter; (2) the dynamical potential does not change along the evolving circular orbits; and (3) the angular momentum of the evolving circular orbits is conserved. The dynamical potential conditions yield evolution equations for circular orbits that are shown to be equivalent to the quasi-local method. This approach provides a unified framework for analyzing orbital dynamics in time-dependent gravitational fields, with direct implications for black hole dynamics. The results bridge traditional effective potential methods with dynamical spacetime studies, enhancing the tools available for probing black hole mergers and evolving relativistic systems.