Beyond the Separatrix: Analytic Continuation of Darwin Variables for Plunging Geodesics in Schwarzschild Spacetime

TL;DR

This paper develops an analytic extension of Darwin variables to provide a unified, real parametrization of all Schwarzschild geodesics, including plunging trajectories, and demonstrates its application in a toy model of orbit transition.

Contribution

It introduces a novel analytic continuation of Darwin variables that seamlessly describes bound, scattering, and plunging geodesics in Schwarzschild spacetime.

Findings

Unified parametrization of all geodesics including plunges.

Application of variables to a toy model of orbit transition.

Extension of Darwin variables across the separatrix.

Abstract

We study geodesic motion of a test particle in Schwarzschild spacetime. Bound and scattering geodesics are commonly described using Darwin variables, which provide a convenient parametrization of the radial motion. However, this description breaks down at the separatrix and does not extend straightforwardly to plunging trajectories. We construct an analytic continuation of Darwin variables that yields a real parametrization of bound, scattering, and plunging Schwarzschild geodesics, thereby providing a unified kinematical description of all types of test-mass motion. As a proof of concept, we then apply these variables to a simple non-geodesic evolution in which the energy and angular momentum are driven by a constant external force. This toy model is not intended to represent a physical radiation-reaction model, but rather to illustrate how the extended variables can be used to follow an orbit through a transition to plunge using a single orbital phase variable across the separatrix.