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

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.

💡 Research Summary

The paper investigates the orbital dynamics of a test particle moving in the gravitational field of a massive body (potentially a black hole) of mass m that is embedded in an expanding Friedmann‑Lemaître‑Robertson‑Walker (FLRW) universe, using the exact McVittie solution of Einstein’s equations. The authors argue that the standard global isotropic coordinates (t, r, θ, φ) employed in the original McVittie metric are unsuitable for interpreting local physical measurements because they mix genuine gravitational effects with coordinate artefacts associated with the Hubble flow. To obtain a physically meaningful description they construct a set of local coordinates (T, R, θ, φ) that are tied to the massive body and satisfy the Einstein equivalence principle (the metric reduces to Minkowski space at the origin when m → 0).

The coordinate transformation proceeds in two steps. First, the radial coordinate is re‑defined as
R = r e^{ν/2} = r a(t)