Sharp bounds on $2m/r$ of general spherically symmetric static objects

TL;DR

This paper derives sharp bounds on the ratio 2m/r for static spherically symmetric objects under very general conditions, extending Buchdahl's classical inequality without restrictive assumptions.

Contribution

It generalizes the Buchdahl inequality by removing restrictive assumptions, providing sharp bounds for a wide class of matter models in static spherically symmetric spacetimes.

Findings

Derived a new upper bound on 2m/r for general matter models.

Showed the inequality is sharp and attainable.

Unified the bound with Buchdahl's original result when =1.

Abstract

In 1959 Buchdahl \cite{Bu} obtained the inequality $2M/R\leq 8/9$ under the assumptions that the energy density is non-increasing outwards and that the pressure is isotropic. Here $M$ is the ADM mass and $R$ the area radius of the boundary of the static body. The assumptions used to derive the Buchdahl inequality are very restrictive and e.g. neither of them hold in a simple soap bubble. In this work we remove both of these assumptions and consider \textit{any} static solution of the spherically symmetric Einstein equations for which the energy density $\rho\geq 0,$ and the radial- and tangential pressures $p\geq 0$ and $p_T,$ satisfy $p+2p_T\leq\Omega\rho, \Omega>0,$ and we show that $$\sup_{r>0}\frac{2m(r)}{r}\leq \frac{(1+2\Omega)^2-1}{(1+2\Omega)^2},$$ where $m$ is the quasi-local mass, so that in particular $M=m(R).$ We also show that the inequality is sharp. Note that when $\Omega=1$ the original bound by Buchdahl is recovered. The assumptions on the matter model are very general and in particular any model with $p\geq 0$ which satisfies the dominant energy condition satisfies the hypotheses with $\Omega=3.$