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 hybrid of the Schwarzschild metric and 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.

The expressions for density and pressure of the perturbed universe in Lasenby’s and McVittie’s approaches are equal for the spatially-flat universe (k = 0) under certain coordinate transformation. However, they do not match for the open or closed perturbed Friedmann’s universe. Similarly, for a flat universe (k = 0), the Lasenby metric can be transformed to McVittie’s metric by making use of the same coordinate transformation. However, the Lasenby metric can not be transformed to McVittie’s metric for the other two types of universe (closed, k = +1, and open, k = -1, universes).

Our goal is to analyze and resolve this mathematical contradiction between the two physically equivalent approaches and to find out if the problem of the point mass in the cosmological manifold has a unique solution. To this end we have established a correspondence between the components of various geometric objects (metric tensor, Ricci tensor, TEM, etc.) defined in the coordinate and tetrad bases. We presented Einstein’s field equations in the form of a system of the first-order partial differential equations for the tetrad components of time-dependent and spherically-symmetric gravitational field. Our next step is to find out the general solution of these equations in terms of the generalized hypergeometric functions and to solve the uniqueness problem of gravitational field of the point mass in the non-asymptotically flat cosmological manifold.