First, under the gravitational collapse condition, when the radius of the liquid sphere becomes equal to its gravitational radius
The condition for a trapping surface to be formed follows from the constraint that the gravitational radius
an external observer sees the star asymptotically shrinking to its gravitational radius
In addition we consider the range of angles for which light originating at the surface of such a collapsar crosses the photonsphere, at 1.5 times the gravitational radius
, and consequently may reach a terrestrial telescope.
Another interesting idea connects the classical proton radius and gravitational radius
of the electron by an exact equation as follows:
Planck's-length [L.sup.*] is a geometric-mean of: Compton-wavelength and Gravitational radius
of total-mass of the universe, because [R.sub.0] = G[M.sub.0]/[c.sup.2].
The observable Universe as a whole, being represented in the framework of the liquid model, is completely located inside its gravitational radius
. In other words, the observable Universe is a collapsar--a huge black hole.
In the case of regular cosmic bodies, the radius of the space breaking surface (known as the gravitational radius
, it is determined by the body's mass) is many orders smaller than the radius of such a body itself: it is 3 km for the Sun, and only 0.9 cm for the Earth.
Another derivation involves the consideration of a body whose Compton wavelength equals its Schwarzschild or gravitational radius
From this it is argued that the Universe was born from an Inversion Explosion of the primordial particle (pre-particle) whose outer radius was that of the classical electron, and inner radius was that of the gravitational radius
of the electron.