  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 !
