Local coordinates and motion of a test particle in the McVittie spacetime

TL;DR

This paper investigates how cosmological expansion affects the orbital motion of a test particle around a massive body in McVittie spacetime, revealing orbit precession influenced by the universe's expansion parameters.

Contribution

It introduces local coordinates to analyze test particle motion in McVittie spacetime and demonstrates the impact of cosmological expansion on orbital precession and frequency.

Findings

Orbit not affected by expansion up to second order in Hubble parameter.

Cosmological expansion causes orbital precession and frequency changes.

Precession direction depends on Hubble and deceleration parameters.

Abstract

We consider the orbital motion of a test particle in the gravitational field of a massive body (that might be a black hole) with mass $m$ placed on the expanding cosmological manifold described by the McVittie metric. We introduce the local coordinates attached to the massive body to eliminate nonphysical, coordinates-dependent effects associated with Hubble expansion. The resultant equation of motion of the test particle are analyzed by the method of osculating elements with application of time-averaging technique. We demonstrate that the orbit of the test particle is not subject to the cosmological expansion up to the terms of the second order in the Hubble parameter. However, the cosmological expansion causes the precession of the orbit of the test particle with time and changes the frequency of the mean orbital motion. We show that the direction of motion of the orbital precession depends on the Hubble parameter as well as the deceleration parameter of the universe. We give numeric estimates for the rate of the orbital precession with respect to time due to the cosmological expansion in case of several astrophysical systems.