Dynamics of a spherical object of uniform density in an expanding universe

TL;DR

This paper derives Newtonian and general-relativistic models for the evolution of a uniform-density spherical object in an expanding universe, analyzing special cases and providing new equations for force and structure.

Contribution

It introduces a generalized Oppenheimer-Volkov equation for time-dependent spherically symmetric systems and explores specific cases of embedded spherical objects in cosmology.

Findings

Derived Newtonian and relativistic solutions for spherical objects in expanding universes.

Obtained force expressions for particles inside and on the boundary of the object.

Formulated a generalized Oppenheimer-Volkov equation for dynamic spherically symmetric systems.

Abstract

We present Newtonian and fully general-relativistic solutions for the evolution of a spherical region of uniform interior density \rho_i(t), embedded in a background of uniform exterior density \rho_e(t). In both regions, the fluid is assumed to support pressure. In general, the expansion rates of the two regions, expressed in terms of interior and exterior Hubble parameters H_i(t) and H_e(t), respectively, are independent. We consider in detail two special cases: an object with a static boundary, H_i(t)=0; and an object whose internal Hubble parameter matches that of the background, H_i(t)=H_e(t). In the latter case, we also obtain fully general-relativistic expressions for the force required to keep a test particle at rest inside the object, and that required to keep a test particle on the moving boundary. We also derive a generalised form of the Oppenheimer-Volkov equation, valid for general time-dependent spherically-symmetric systems, which may be of interest in its own right.