A POINT MASS PLACED IN COSMOLOGICAL BACKGROUND

We consider the Friedmann universe which is taken as a cosmological background manifold, with a point mass m placed on the manifold. The placement of the point mass perturbs the gravitational field (spacetime geometry) of the Friedmann universe. To describe such perturbation of space-time one needs to build a metric which should be a general metric able to reduce to the Schwarzschild metric and Friedmann–Lemaˆıtre-Robertson–Walker (FLRW) metric. We focus on two different approaches to build this model: one made by McVittie and the other one made by Lasenby and his collaborators. McVittie uses the power series method of solving Einstein's field equations and Lasenby uses tetrad formalism. We develop McVittie's metric in tetrad formalism (that is in local coordinates). With the McVittie metric in tetrad formalism we derive equations of motion for a test particle moving around the point mass m placed in the cosmological background. The resultant differential equations with respect to time are for each of six osculating elements: semi-major axis a, eccentricity e (0 ≤ e < 1), the inclination i, the longitude of the ascending node Ω, the argument of pericenter ω, and the mean anomaly at the epoch l_0. The solutions of these differential equations are the first-order perturbations that explain in detail the orbit of the test particle due to: General Theory of Relativity, Hubble parameters and the curvatures of the universe.