Subrahmanyan Chadrashekhar was the first person who prove that Black Holes like solutions can be real. In 1931 he calculated that a stationary astrophysical object above a certain limiting mass has no stable solutions. At that time this idea was appreciated by many scientist but Eddington opposed his idea. Eddington suppose in his mind that some new physical principle must come in to play to save the star from collapsing. ( Eddington, A.S 1946, Fundamental Theory, Cambridge University Press). 

In 1939, Oppenheimer and others predicted that neutron stars above another certain limit (Oppenheimer limit )would collapse further for the reasons presented by Chandrasekhar. He concluded that no law of physics was to stop at least some stars from collapsing to black holes.

Yet Eddington had a point that impossibility of an equilibrium state would lead to a star unstoppable collapse. Suppose a star reaches its Schwartzschild radius, so it will further collapse till than space time become singular. Is it possible?  Because star continuously collapsing inwards so it should reach a state where matter density becomes infinite and space time structure becomes singular from Einstein General Theory of Relativity.

During the 1930s, most of the astrophysicist was not used general relativity so much, similarly, S Chandrasekhar did not directly attack on the problem of gravitational collapse.  He entered into a study of General Relativity around 1960 after completion of his work on stellar structure and stellar dynamics. He did not directly discuss the issue of collapsing star. His concern was the effects of GTR on the stability of gravitational astrophysical objects.