Exact solution for a black hole embedded in a nonstatic dust-filled universe

TL;DR

This paper presents an exact solution modeling a Schwarzschild-like black hole within a non-static dust-filled universe, bridging Schwarzschild and Friedmann solutions and analyzing test particle trajectories.

Contribution

It introduces a new exact solution for a black hole embedded in a dynamic dust universe, using the mass function method in comoving coordinates.

Findings

The solution includes Schwarzschild and Friedmann limits.

Test particle velocities and trajectories are characterized.

Near the center, observations match Schwarzschild predictions.

Abstract

An exact solution of the Lema\^{i}tre--Tolman--Bondi class is investigated as a possible model of the Schwarzschild-like black hole embedded in a non-static dust-filled universe for the three types of spatial curvature. The solution is obtained in comoving coordinates by means of the mass function method. It is shown that the central part of space contains a Schwarzschild-like black hole. The R-T-structure of the resulting spacetime is built. It is shown that the solution includes both the Schwarzschild and Friedmann solutions as its natural limits. The geodesic equations for test particles are analyzed. The particle observable velocities are found. The trajectories of the test particles are built from the point of view of both comoving and distant observers. For the distant observer, the results coincide with the Schwarzschild picture within a second-order accuracy near the symmetry center.