It is possible to find the relationship between the mass and the radius of a spherical black hole keeping in mind that the maximum speed that could reach an object, according to the theory of relativity, it is the speed of light.
The scape velocity in the surface of a spherical star will be the maximum velocity that could reach an object in order to stay in orbit around the star. This will happen when the kinetic energy of one object is equal to the potential energy due to the star gravitational attraction.
The kinetic energy (Ec) according to the classical mechanics is
Ec=½ mv² (1)
and the potential energy is Ep=GmM/r (2)
being v the speed of the object in orbit, m the mass of the object in orbit, M the mass of the star, r the distance from the center of the star until the point where it is the object in orbit and G the universal gravitation constant.
Equaling the potential energy with the kinetic energy and isolating the velocity we obtain the equation of the speed of escape:
then for a scape velocity equal to the speed of the c light and isolating M/r in the previous formula we can obtain
as c=2.99793 x 108 m/s and G=6.6732 x 10-11 Nm²/kg² we obtain
M/r=6.734 x 1026 kg/m
this is the relationship between the mass and the radius of a spherical body in order to be a black hole. With this relationship we could find the radius that must have diverse stellar objects in order to be a black hole.
CHART of RADIUS CORRESPONDING TO BLACK HOLES
1 sun (2 x 1030 Kg) 3 Km
25 suns (blue giants) 75 Km
1000 suns 3000 Km
107 suns (galactic nucleus) 3 x 107 Km
1011 suns (galaxy) 3 x 1011 Km
Thus we can see that if the Sun could be compressed until be a 3 Km radio sphere it would become a black hole.
But this reasoning is mixing the theory of relativity with the classical mechanics, since the equation of the kinetic energy of a body according to the special relativity is different to the classical
Thus a relativistic escape velocity is obtained (Ver):
Thus you can observe that the escape velocity will never reach the speed of light except in a star of infinite mass or radius zero.
But this is considering the theory of the special relativity only. If we keep in mind the theory of the general relativity of Einstein, some new very interesting consequences appear.