From quasi-local definitions to a dynamical potential: A unified framework for evolving circular orbits in dynamical spacetimes

The study of circular orbits is fundamental in gravitational physics, yet their definition in dynamical spacetimes remains challenging due to the lack of temporal symmetry. In this work, we establish a unified framework by commencing from the geometrically invariant quasi-local definition of a particle surface. We demonstrate that this definition naturally leads to a set of conditions that can be recast into the language of a coordinate-dependent dynamical potential. This potential serves as a practical computational tool for locating evolving circular orbits within a specific coordinate system. We rigorously prove the equivalence between the quasi-local and dynamical potential approaches in dynamical spherically symmetric spacetimes. The efficacy and self-consistency of the dynamical potential method are explicitly verified through its application to the Oppenheimer-Snyder dust collapse model, where it correctly reproduces the established evolution equations for null and timelike circular orbits. This work bridges the gap between abstract geometric definitions and concrete calculations, providing a robust and adaptable framework for analyzing orbital dynamics in time-dependent gravitational fields.