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

Yong Song

TL;DR

This paper addresses the challenge of defining and locating circular orbits in dynamical spacetimes lacking a timelike Killing vector. It develops a unified framework that starts from a geometrically invariant quasi-local particle-surface definition and introduces a coordinate-dependent dynamical potential $V_{+}(t,r)$ and $V_{-}(t,r)$ to identify evolving circular orbits, proving their equivalence to the quasi-local conditions. The authors also show an angular-momentum conservation formulation as an interchangeable perspective and validate the approach in three dynamical spacetimes—Vaidya, Oppenheimer-Snyder, and LTB—recovering known evolution equations. The framework extends the traditional effective-potential method to fully dynamical contexts and provides practical tools for studying orbital dynamics in time-dependent gravitational fields, with potential applications to gravitational collapse, accretion processes, and gravitational-wave source modeling.

Abstract

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.

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

TL;DR

This paper addresses the challenge of defining and locating circular orbits in dynamical spacetimes lacking a timelike Killing vector. It develops a unified framework that starts from a geometrically invariant quasi-local particle-surface definition and introduces a coordinate-dependent dynamical potential and to identify evolving circular orbits, proving their equivalence to the quasi-local conditions. The authors also show an angular-momentum conservation formulation as an interchangeable perspective and validate the approach in three dynamical spacetimes—Vaidya, Oppenheimer-Snyder, and LTB—recovering known evolution equations. The framework extends the traditional effective-potential method to fully dynamical contexts and provides practical tools for studying orbital dynamics in time-dependent gravitational fields, with potential applications to gravitational collapse, accretion processes, and gravitational-wave source modeling.

Abstract

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.

Paper Structure

This paper contains 10 sections, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Effective Potential in Static Spacetimes: A Review
  4. Quasi-Local Definition
  5. Dynamical Potential Formalism
  6. Examples
  7. Vaidya spacetime
  8. Oppenheimer-Snyder model
  9. Lemaître-Tolman-Bondi (LTB) model
  10. Conclusions and discussions

Figures (1)

  • Figure 1: The red point represents a moving point particle. $r_1$ and $r_2$ are the radii of its two circular orbits at $t=t_1$ and $t=t_2$, respectively. When $t$ is constant, e.g., $t=t_1$, the radius $r(\lambda,t_1)=r_1$ remains unchanged, and thus does not vary with $\lambda$. This indicates that for an evolving circular orbit in dynamic spacetime, $r$ does not explicitly depend on $\lambda$.