On the equilibrium of the Buchdahl star

Abstract

The Buchdahl star is the limiting compactness (which is indicated by sturation of the Buchdahl bound) object without horizon. It is in general defined by the potential felt by radially falling timelike particle, $\Phi(R) = 4/9$, in the field of a static object. On the other hand black hole is similarly characterized by $\Phi(R)=1/2$ which defines the horizon. Further it is remarkable that in terms of gravitational and non-gravitational energy, the Buchdahl star is alternatively defined when gravitational energy is half of non-gravitational energy while the black hole when the two are equal. When an infinitely dispersed system of bare mass $M$ collapses under its own gravity to radius $R$, total energy encompassed inside $R$ would be $E_{tot}(R)=M-E_{grav}(R)$. That is, energy inside the object is increased by the amount equivalent to gravitational energy lying outside and which manifests as internal energy in the interior. If the interior consists of free particles in motion interacting only through gravity as is the case for the Vlasov kinetic matter, internal (gravitational) energy could be thought of as kinetic energy and the defining condition for the Buchdahl star would then be kinetic (gravitational) energy equal to half of non-gravitational (potential) energy. Consequently it could be envisaged that equilibrium of the Buchdahl star interior is governed by the celebrated Virial theorem like relation (average kinetic energy equal to half of average potential energy). On the same count the black hole equilibrium is governed by equality of gravitational and non-gravitational energy !