Motion of test particles around a black hole in the Hubble universe

We consider the Friedmann universe as a cosmological background manifold, with a point mass m embedded within it. The presence of this mass perturbs the gravitational field (spacetime geometry) of the Friedmann universe. To accurately describe this perturbation, it is necessary to construct a metric that seamlessly integrates elements of both the Schwarzschild and FLRW metrics. We explore two distinct approaches to formulating this model: one introduced by McVittie [1], which employs global coordinates to solve Einstein's field equations, and another developed by Lasenby and his collaborators [2], which utilizes the tetrad formalism. For a spatially flat universe ???? = 0, the expressions for density and pressure derived from McVittie's and Lasenby's approaches coincide under an appropriate coordinate transformation. However, for an open ???? = -1 or closed ???? = +1 perturbed Friedmann universe, they do not. Likewise, in the case of a spatially flat universe ???? = 0, the Lasenby metric can be mapped to McVittie's metric through the same coordinate transformation. However, for open and closed universes, the Lasenby metric cannot be transformed into McVittie's metric. Our objective is to analyze the mathematical differences between these two physically equivalent approaches and figured out whether the problem of a point mass in a cosmological manifold admits a unique solution. To this end, we establish a correspondence between the components of various geometric objects--such as the metric tensor, Ricci tensor, and energy-momentum tensor--defined in both coordinate and tetrad bases. We reformulate Einstein's field equations as a system of first-order partial differential equations for the tetrad components of a time-dependent, spherically symmetric gravitational field. As a subsequent step, we investigate 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 within an expanding cosmological manifold described by the McVittie metric. We introduce local coordinates attached to the massive body to eliminate nonphysical, coordinatedependent effects associated with Hubble expansion. The resultant equations of motion for the test particle are analyzed using the method of osculating elements, along with the time-averaging technique. We demonstrate that the orbit of the test particle is not affected by cosmological expansion up to terms of the second order in the Hubble parameter. However, cosmological expansion induces orbital precession over time and modifies the frequency of the mean orbital motion. We show that the direction of orbital precession depends on both the Hubble parameter and the deceleration parameter of the universe. Finally, we provide numerical estimates for the rate of orbital precession over time due to cosmological expansion in several astrophysical systems.