Buchdahl compactness limit for a pure Lovelock static fluid star

TL;DR

This paper derives the Buchdahl compactness limit for pure Lovelock static fluid stars, showing its universality across gravitational theories and identifying specific conditions where the limit is maximized or equivalent to energy criteria.

Contribution

It extends the Buchdahl limit to pure Lovelock gravity, revealing universal bounds and their relation to gravitational energy and spacetime dimensions.

Findings

Buchdahl limit holds universally in Lovelock gravity.

Maximum surface potential occurs at specific dimensions for each Lovelock order.

In Einstein gravity, the limit relates to gravitational field energy being less than half the mass.

Abstract

We obtain the Buchdahl compactness limit for a pure Lovelock static fluid star and verify that the limit following from the uniform density Schwarzschild's interior solution, which is universal irrespective of the gravitational theory (Einstein or Lovelock), is true in general. In terms of surface potential $\Phi(r)$, it means at the surface of the star $r=r_{0}$, $\Phi(r_{0}) < 2N(d-N-1)/(d-1)^2$ where $d$, $N$ respectively indicate spacetime dimensions and Lovelock order. For a given $N$, $\Phi(r_{0})$ is maximum for $d=2N+2$ while it is always $4/9$, Buchdahl's limit, for $d=3N+1$. It is also remarkable that for $N=1$ Einstein gravity, or for pure Lovelock in $d=3N+1$, Buchdahl's limit is equivalent to the criteria that gravitational field energy exterior to the star is less than half its gravitational mass, having no reference to the interior at all.