Bounds on the mass-to-radius ratio for non-compact field configurations

TL;DR

This paper derives bounds on the mass-to-radius ratio for non-compact, slowly decaying matter configurations in static, spherically symmetric spacetimes, extending classical bounds known for compact stars.

Contribution

It introduces new upper bounds on the mass-to-radius ratio applicable to non-compact matter distributions with power-law decay, generalizing previous results for compact objects.

Findings

Bound depends on decay rate of matter fields

Increased radial pressure can raise the ratio limit

For non-decaying fields, the bound reduces to 2/3

Abstract

It is well known that a spherically symmetric compact star whose energy density decreases monotonically possesses an upper bound on its mass-to-radius ratio, $2M/R\leq 8/9$. However, field configurations typically will not be compact. Here we investigate non-compact static configurations whose matter fields have a slow global spatial decay, bounded by a power law behavior. These matter distributions have no sharp boundaries. We derive an upper bound on the fundamental ratio max_r{2m(r)/r} which is valid throughout the bulk. In its simplest form, the bound implies that in any region of spacetime in which the radial pressure increases, or alternatively decreases not faster than some power law $r^{-(c+4)}$, one has $2m(r)/r \leq (2+2c)/(3+2c)$. [For $c \leq 0$ the bound degenerates to $2m(r)/r \leq 2/3$.] In its general version, the bound is expressed in terms of two physical parameters: the spatial decaying rate of the matter fields, and the highest occurring ratio of the trace of the pressure tensor to the local energy density.