# Buchdahl compactness limit and gravitational field energy

## TL;DR

This paper links the Buchdahl compactness limit to gravitational field energy, showing it is constrained by the exterior solutions and providing bounds on charge and surface red-shift for static objects.

## Contribution

It introduces a novel interpretation of the Buchdahl limit based on gravitational energy and derives bounds on charge and red-shift without interior distribution details.

## Key findings

- Gravitational field energy must be less than or equal to half of matter energy for the Buchdahl limit.
- Surface red-shift is bounded above by 3.
- Charge-to-mass ratio cannot exceed rac{3}{2}.

## Abstract

The main aim of this paper is essentially to point out that the Buchdahl compactness limit of a static object is given by \it{gravitational field energy being less than or equal to half of its non-gravitational matter energy}. It is thus entirely determined without any reference to interior distribution by the exterior unique solutions, the Schwarzschild for neutral and the Reissner-Nordstr{$\ddot o$}m for charged object. In terms of surface potential, it reads as $\Phi(R) = (M-Q^2/2R)/R \leq 4/9$ which translates to surface red-shift being less than or equal to $3$. It also prescribes an upper bound on charge an object could have, $Q^2/M^2 \leq 9/8 > 1$.