Precise relativistic orbits in Kerr and Kerr-(anti) de Sitter spacetimes

TL;DR

This paper derives and solves the exact equations for timelike geodesics in Kerr and Kerr-(anti) de Sitter spacetimes, enabling precise modeling of particle motion in these gravitational fields, including applications to satellite and stellar orbits.

Contribution

It provides the first exact solutions for timelike geodesics in Kerr and Kerr-(anti) de Sitter spacetimes, incorporating the cosmological constant and using elliptic and theta functions.

Findings

Exact solutions for non-spherical geodesics in Kerr geometry.

Precise calculations of frame dragging effects for satellites and stars.

Application of solutions to astrophysical scenarios with observational relevance.

Abstract

The timelike geodesic equations resulting from the Kerr gravitational metric element are derived and solved exactly including the contribution from the cosmological constant. The geodesic equations are derived, by solving the Hamilton-Jacobi partial differential equation by separation of variables. The solutions can be applied in the investigation of the motion of a test particle in the Kerr and Kerr-(anti) de Sitter gravitational fields. In particular, we apply the exact solutions of the timelike geodesics i) to the precise calculation of dragging (Lense-Thirring effect) of a satellite's spherical polar orbit in the gravitational field of Earth assuming Kerr geometry, ii) assuming the galactic centre is a rotating black hole we calculate the precise dragging of a stellar polar orbit aroung the galactic centre for various values of the Kerr parameter including those supported by recent observations. The exact solution of non-spherical geodesics in Kerr geometry is obtained by using the transformation theory of elliptic functions. The exact solution of spherical polar geodesics with a nonzero cosmological constant can be expressed in terms of Abelian modular theta functions that solve the corresponding Jacobi's inversion problem.