Physical theory implies that the existence of astronomical entities above a certain mass is evidence for the existence of black holes. The Earth does not itself collapse upon itself under gravitational force because gravity is countered by the outward pressure generated by the electromagnetic repulsion between the atoms making up the planet. But if these forces are overpowered, gravity will always lead to the formation of a black hole. Assuming the validity of general relativity, we can calculate the upper bound for a star, the Tolman-Oppenheimer-Volkoff limit, to be 3.6 solar masses; any object heavier than this will be unable to resist collapse under its own mass and must be a black hole.
The search for entities more massive than the Tolman-Oppenheimer-Volkoff limit brings us to the examination of X-ray binary systems. In an X-ray binary, two bodies rotate around their center of mass, a point between them, while one component, usually a normal star, sheds matter to the other more massive component known as the accretor. The shedding matter is released as observable X-ray radiation. Since binary stars rotate around a common center of gravity, the mass of the accetor can be calculated from the orbit of the visible one. By 2004, about forty X-ray binaries that contained candidates for black holes had been discovered. The accretors in these binary systems did not appear visible, as is to be expected of black holes, but that fact alone does not distinguish them from very dense and hence less luminescent stars, such as neutron stars. More to the point is that these accretors were of mass far in excess of 3.6 solar masses. Famously, Cygnus X-1, an X-ray binary in the constellation Cygnus, has an accretor whose mass has been calculated to be 14 solar masses, plus or minus 4 solar masses. While it does not rule out other phenomena without further interpretation, it provides strong proof that black holes exist.
The conclusion that black holes exist depends on the reliability of the general-relativistic calculations involved. If more generous assumptions are made, the Tolman-Oppenheimer-Volkoff limit can be calculated to be as high as 10 solar masses. The finding also establishes plausibility, if not direct evidence, for the existence of supermassive black holes hypothesized to exist at the center of some galaxies.