Spherically Symmetric Gravitational Collapse of General Fluids

TL;DR

This paper formulates Einstein's equations for spherically symmetric fluid collapse, unifying fluid, Vaidya, and Schwarzschild spacetimes in a single coordinate system, facilitating initial value problem analysis without complex interface matching.

Contribution

It introduces a unified coordinate framework for modeling fluid collapse, Vaidya, and Schwarzschild spacetimes, simplifying the analysis of gravitational collapse and spacetime matching.

Findings

Unified equations for fluid, Vaidya, and Schwarzschild spacetimes.

Reduction to static cases and generalization of TOV equations.

Demonstrates no need for complex interface matching schemes.

Abstract

We express Einstein's field equations for a spherically symmetric ball of general fluid such that they are conducive to an initial value problem. We show how the equations reduce to the Vaidya spacetime in a non-null coordinate frame, simply by designating specific equations of state. Furthermore, this reduces to the Schwarzschild spacetime when all matter variables vanish. We then describe the formulation of an initial value problem, whereby a general fluid ball with vacuum exterior is established on an initial spacelike slice. As the system evolves, the fluid ball collapses and emanates null radiation such that a region of Vaidya spacetime develops. Therefore, on any subsequent spacelike slice there exists three regions; general fluid, Vaidya and Schwarzschild, all expressed in a single coordinate patch with two free-boundaries determined by the equations. This implies complicated matching schemes are not required at the interfaces between the regions, instead, one simply requires the matter variables tend to the appropriate equations of state. We also show the reduction of the system of equations to the static cases, and show staticity necessarily implies zero ``heat flux''. Furthermore, the static equations include a generalization of the Tolman-Oppenheimer-Volkoff equations for hydrostatic equilibrium to include anisotropic stresses in general coordinates.