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, these configurations suggest an underlying energetic balance. In this letter, we demonstrate, in a simple and general manner, that the corresponding equilibrium state always represents a minima of an appropriate quasi-local energy functional, thereby establishing energetic (variational) stability within the quasi-local configuration space.