On the active gravitational mass of a non-spherical source leaving hydrostatic equilibrium

TL;DR

This paper derives an expression for the active gravitational mass of a non-spherical, initially hydrostatic source, showing small deviations from sphericity can significantly affect stability during collapse or expansion.

Contribution

It provides a new formula for active gravitational mass immediately after departure from equilibrium for non-spherical sources, highlighting the impact of shape deviations on stability.

Findings

Small non-spherical deviations significantly alter active gravitational mass.

Decreasing mass favors collapse, increasing mass favors expansion.

Results emphasize the role of shape in gravitational stability.

Abstract

We obtain an expression for the active gravitational mass (Tolman) of a source of the $\gamma$ metric, just after its departure from hydrostatic equilibrium, on a time scale of the order of (or smaller than) the hydrostatic time scale. It is shown that for very compact sources, even arbitrarily small departures from sphericity, produce significant decreasing (increasing) in the values of active gravitational mass of collapsing (expanding) spheres, with respect to its value in equilibrium, enhancing thereby the stability of the system.