Constant-$r$ geodesics in the Painleve-Gullstrand form of Lense-Thirring spacetime

TL;DR

This paper investigates non-equatorial constant-$r$ geodesics in a Painleve-Gullstrand form of Lense-Thirring spacetime, revealing complex orbital behaviors including precession, nutation, and surface-filling trajectories, with implications for slow-rotation models of gravity.

Contribution

It provides a detailed analysis of constant-$r$ geodesics in a novel Lense-Thirring spacetime variant, highlighting their existence, stability, and complex orbital dynamics.

Findings

Constant-$r$ geodesics exist in non-spherical Lense-Thirring spacetime.

Orbits exhibit precession and nutation with incommensurate frequencies.

Surface-filling orbits cover a spherical zone segment.

Abstract

Herein we explore the non-equatorial constant-$r$ ("quasi-circular") geodesics (both timelike and null) in the Painleve-Gullstrand variant of the Lense-Thirring spacetime recently introduced by the current authors. Even though the spacetime is not spherically symmetric, shells of constant-$r$ geodesics still exist. Whereas the radial motion is (by construction) utterly trivial, determining the allowed locations of these constant-$r$ geodesics is decidedly non-trivial, and the stability analysis is equally tricky. Regarding the angular motion, these constant-$r$ orbits will be seen to exhibit both precession and nutation -- typically with incommensurate frequencies. Thus this constant-$r$ geodesic motion, though integrable in the precise technical sense, is generically surface-filling, with the orbits completely covering a symmetric equatorial band which is a segment of a spherical surface, (a so-called "spherical zone"), and whose latitudinal extent is governed by delicate interplay between the orbital angular momentum and the Carter constant. The situation is qualitatively similar to that for the (exact) Kerr spacetime -- but we now see that any physical model having the same slow-rotation weak-field limit as general relativity will still possess non-equatorial constant-$r$ geodesics.