Buchdahl stars and bounds with cosmological constant

TL;DR

This paper extends the Buchdahl bound, which limits the mass-to-radius ratio of compact objects, to include the effects of a cosmological constant, showing the bound's universality and proposing a new related inequality.

Contribution

It generalizes the Buchdahl compactness bound to incorporate a cosmological constant and explores its implications and variations using different approaches.

Findings

The Buchdahl bound remains universal with a cosmological constant.

Different bounds emerge depending on the approach used.

A new inequality related to the Buchdahl bound is proposed.

Abstract

The Schwarzschild interior solution, when combined with the assumption of a finite central pressure, leads to the well-known Buchdahl bound. This bound establishes an upper limit on the mass-to-radius ratio of an object, which is equivalent to imposing an upper limit on the gravitational potential. Remarkably, this limit exhibits considerable universality, as it applies to a broader class of solutions beyond the original Schwarzschild interior metric. By reversing this argument, one can define the most compact horizonless object that satisfies this gravitational bound. Intriguingly, the same bound arises when applying the virial theorem to an appropriately chosen combination of gravitational and potential energy. In this work, we explore the generalised Buchdahl compactness bound in the presence of a cosmological constant. We investigate its implications, define a suitable gravitational energy and an associated potential energy that incorporate the cosmological term, and demonstrate that the universality of the Buchdahl bound persists. However, we also observe that different bounds emerge depending on the chosen approach. We discuss these findings in detail and conclude by proposing a new inequality.