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

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.

💡 Research Summary

The paper addresses a long‑standing problem in relativistic gravitation: the equilibrium and stability of the most compact objects allowed by four‑dimensional General Relativity. Two distinct configurations saturate the compactness bound: (i) a black hole, whose horizon is a null surface, and (ii) a Buchdahl star, a static fluid sphere whose surface is timelike and whose compactness reaches the Buchdahl limit. The authors propose a unified, equation‑of‑state‑independent stability criterion based on the Brown‑York quasi‑local energy.

First, the Brown‑York energy (E_{BY}) is evaluated for a spherical 2‑surface of radius (r) in the exterior Schwarzschild spacetime, giving (E_{BY}=r\bigl