Observables for bound orbital motion in axially symmetric space-times

TL;DR

This paper derives analytical formulas for orbital observables like periastron shift and Lense-Thirring effect in axially symmetric space-times, enabling characterization of various space-time parameters.

Contribution

It provides explicit hyperelliptic integral and Lauricella function expressions for observables in Plebański-Demiański space-times, expanding analytical tools for orbital analysis.

Findings

Analytical expressions for periastron shift and Lense-Thirring effect in terms of special functions.

Post-Newtonian expansions reveal parameter influences on observables.

Characterization of different space-times like Kerr and Taub-NUT using derived formulas.

Abstract

The periastron shift and the Lense-Thirring effect of bound orbital motion in a general axially symmetric space-time given by Pleba\'nski and Demia\'nski are analyzed. We also define a measure for the conicity of the orbit and give analytic expressions for all three observables in terms of hyperelliptic integrals and Lauricella's $F_D$ function. For an interpretation of these analytical expressions, we perform a post-Schwarzschild and a post-Newton expansion of these quantities. This clearly shows the influence of the different space-time parameters on the considered observables and allows to characterize Kerr, Taub-NUT, Schwarzschild-de Sitter, or other space-times.