Geodesic dynamics in Chazy–Curzon spacetimes

Abstract

In the last decades, the dynamical studies around compact objects became a subject of active research, partially motivated by the observed differences in the profiles of the gravitational waves depending on the dynamics of the system. In this work, via the Poincaré section method, we conduct a thorough numerical analysis of the dynamical behavior of geodesics around Chazy-Curzon metrics. As the main result, we find only regular motions for the geodesics in all cases, which suggest the existence of the so-called Carter's constant in this kind of exact solutions. Moreover, our simulations indicate that in the two-particle Chazy-Curzon solution, some oscillatory motions take place as in the classical MacMillan problem.

Non-integrability of geodesic dynamics of Chazy-Curzon space-time

“…The question of the integrability of the geodesic equation has been investigated by several in the authors context of Stationary Axially-symmetric Vacuum (SAV) space-time in [2], [3], [8] and [9]. The properties of stationarity and axial-symmetry suggest that every SAV spacetime possesses a pair Killing vector fields ∂ t and ∂ φ , and a pair first integrals p t = −E, and p φ = L, corresponding to energy E and azimuthal angular momentum L. The problems of geodesic motion reduces to two dimensional Hamiltonian system in the meridian plane (ρ, z).…”