Relativistic Virial theorem, limiting compactness, and the end state of gravitational collapse

It is noteworthy that limiting compactness of a static bounded configuration is characterized by a general principle: one, by equipartition of mass between inside and outside, and the other by vanishing of energy inside. The former implies gravitational energy being half of mass leading to limiting compactness [Formula: see text] of Buchdahl star while for the latter, the two are equal giving [Formula: see text] of black hole with horizon. This is the relativistic Virial theorem respectively for massive and massless particles. It is remarkable that it prescribes that there can exist only two equilibrium states which also define limiting compactness of the object. Consequently, it leads to a profound prediction that the ultimate endproduct of gravitational collapse could only be one of the two, Buchdahl star or black hole.