Schwarzschild radius

The Schwarzschild radius is the size of the event horizon of a non-rotating black hole, given by the clean formula r_s = 2 G M / c^2. It depends only on the mass M: double the mass and you double the radius. It tells you how small you would have to squeeze an object before its own gravity sealed it off behind a horizon and turned it into a black hole.

The numbers are surprising. For the Sun the Schwarzschild radius is only about 3 kilometres, smaller than many cities, even though the Sun is more than a million kilometres across. For the whole Earth it is about 9 millimetres, the size of a grape. Nothing forces ordinary objects to collapse to these sizes, but the formula marks the threshold where, if it did happen, a black hole would be born.

Because r_s grows in proportion to mass while a normal object's volume grows much faster, supermassive black holes are, paradoxically, not very dense when averaged over the horizon. A black hole as heavy as a galaxy's central monster can have an average density lower than air, even though its core hides a singularity. The radius simply sets the line of no return, not how packed the matter is.