On the asymptotic behavior of static perfect fluids

TL;DR

This paper analyzes the long-term behavior of static, spherically symmetric perfect fluid solutions in Einstein's equations, revealing how different equations of state influence their asymptotic structure and introducing a new concept of scaled quasi-asymptotic flatness.

Contribution

It provides a comprehensive geometric description of the asymptotic behavior for solutions with linear and polytropic equations of state with index n>5, and introduces scaled quasi-asymptotic flatness.

Findings

Most solutions are not asymptotically flat with finite ADM mass.

Solutions with certain equations of state exhibit conical asymptotics.

The new notion captures a form of asymptotic conicality and simplicity.

Abstract

Static spherically symmetric solutions to the Einstein-Euler equations with prescribed central densities are known to exist, be unique and smooth for reasonable equations of state. Some criteria are also available to decide whether solutions have finite extent (stars with a vacuum exterior) or infinite extent. In the latter case, the matter extends globally with the density approaching zero at infinity. The asymptotic behavior largely depends on the equation of state of the fluid and is still poorly understood. While a few such unbounded solutions are known to be asymptotically flat with finite ADM mass, the vast majority are not. We provide a full geometric description of the asymptotic behavior of static spherically symmetric perfect fluid solutions with linear and polytropic-type equations of state with index n>5. In order to capture the asymptotic behavior we introduce a notion of scaled quasi-asymptotic flatness, which encodes a form of asymptotic conicality. In particular, these spacetimes are asymptotically simple.