On the stability of the objects of limiting compactness: Black hole and Buchdahl star

TL;DR

The paper develops a non-perturbative, equation-of-state-independent stability criterion for objects at limiting compactness in General Relativity by leveraging Brown-York quasi-local energy. It shows that black holes and Buchdahl stars saturate the universal balance $E_{GF} = k E_M$ with $k=1$ and $k=1/2$, respectively, and that these configurations correspond to minima of a quasi-local energy functional $\mathcal{F} = E_{GF} - k E_M$, yielding dynamical stability. The authors extend the analysis from static spacetimes to slowly evolving exteriors and to charged, rotating, and Lovelock gravity, demonstrating the robustness of the energetic extremization principle. This framework links equilibrium to global spacetime energetics, providing a geometric mechanism for why limiting-compactness endpoints are dynamically selected in gravitational collapse and offering a practical diagnostic for stability across a broad class of relativistic objects.

Abstract

In General Relativity, there exist two objects of limiting compactness, one with a null boundary defining the horizon of a black hole and the other with a timelike boundary defining a Buchdahl star. The two are characterized by gravitational energy equal to or half the mass. Since non-gravitational mass-energy is the source of gravitational energy, both of these objects are manifestly stable. We demonstrate in this letter, in a simple and general way, that the equilibrium state defining the object is indeed stable, independent of the nature of the perturbation.