Bounds on 2m/R for static spherical objects

TL;DR

This paper investigates bounds on the compactness of static spherical objects, extending classical results by relaxing assumptions and deriving new bounds based on stress and density profiles.

Contribution

It generalizes the classical 8/9 bound on 2m/R for static spheres by relaxing restrictive assumptions and establishing bounds based on stress and density variations.

Findings

The 8/9 bound holds for decreasing density and non-negative tangential stress when using quasi-local mass.

If stresses are anisotropic or density is non-monotonic, the bound can be violated but remains strictly less than unity.

Explicit upper bounds are derived depending on stress ratios and density variation.

Abstract

It is well known that a spherically symmetric constant density static star, modeled as a perfect fluid, possesses a bound on its mass m by its radial size R given by 2m/R \le 8/9 and that this bound continues to hold when the energy density decreases monotonically. The existence of such a bound is intriguing because it occurs well before the appearance of an apparent horizon at m = R/2. However, the assumptions made are extremely restrictive. They do not hold in a humble soap bubble and they certainly do not approximate any known topologically stable field configuration. We show that the 8/9 bound is robust by relaxing these assumptions. If the density is monotonically decreasing and the tangential stress is less than or equal to the radial stress we show that the 8/9 bound continues to hold through the entire bulk if m is replaced by the quasi-local mass. If the tangential stress exceeds the radial stress and/or the density is not monotonic we cannot recover the 8/9 bound. However, we can show that 2m/R remains strictly bounded away from unity by constructing an explicit upper bound which depends only on the ratio of the stresses and the variation of the density.