Geodesic motion in the vacuum C-metric

TL;DR

This paper derives and analyzes the geodesic equations in the vacuum C-metric, revealing stability conditions for orbits around accelerating black holes and providing new insights into their dynamics.

Contribution

It provides a detailed derivation and solution of geodesic equations in the vacuum C-metric, including stability analysis of circular orbits under acceleration.

Findings

Circular Schwarzschild geodesics with radius >6m are stable under small accelerations.

Orbits exhibit harmonic oscillations due to perturbations.

New algebraic stability condition for orbits parallel to acceleration direction.

Abstract

Geodesic equations of the vacuum C-metric are derived and solved for various cases. The solutions describe the motion of timelike or null particles with conserved energy and angular momentum. Polar, nearly-circular orbits around weakly accelerated black holes may be regarded as a perturbation of circular Schwarzschild geodesics. Results indicate that circular Schwarzschild geodesics of radius $r_0>6m$ are stable under small uniform accelerations along the orbital plane. These stable orbits undergo small oscillations around $r_0$, behaving like a harmonic oscillator driven by a periodic force plus another constant force. Circular orbits with axis parallel to the direction of black hole acceleration are also considered. In this case an algebraic relation expressing the condition of stability is obtained. This refines the stability analysis done in previous literature. We also present an analysis of radial geodesics along the poles. There exist a solution where a particle remains at unstable equilibrium at a fixed distance directly behind the accelerating black hole. Examples of numerical solutions are presented for other more general cases.