Adiabatic theory in Kerr spacetimes

TL;DR

This paper develops an adiabatic theory framework for analyzing test particle motion in Kerr spacetimes, deriving formulas for perihelion shifts and confirming the theory's accuracy in general relativity.

Contribution

It introduces a novel application of adiabatic invariants to Kerr spacetime, including a new harmonic coordinate representation and a general perihelion shift formula.

Findings

The adiabatic theory accurately predicts perihelion shifts in Kerr spacetime.

Superposition principle holds for mass and angular momentum effects up to second order.

Results agree with existing literature in limiting cases.

Abstract

We present the main aspects of the adiabatic theory and show that it can be used to study the motion of test particles in general relativity. The theory is based upon the use of vector elements of the orbits and adiabatic invariants. To prove the applicability of the adiabatic theory in Einstein's gravity, we derive a particular representation of the Kerr metric in harmonic coordinates, which allows us to obtain a general formula for the perihelion shift of test particles orbiting on the non-equatorial plane of a rotating central object. We show that the principle of superposition is fulfilled for the individual effects of the gravitational source mass and angular momentum up to the second order. We demonstrate that the adiabatic theory, along with its simplicity, leads to correct results, which in the limiting cases correspond to the ones reported in the literature.