the Schwarzschild radius

How small would you have to crush an object before its gravity became so intense that not even light could escape? Squeeze the Sun down to a ball about 3 km across, or the whole Earth to the size of a marble, and it would become a black hole. The critical size is the Schwarzschild radius: the single length, set only by the mass, that marks the threshold between an ordinary body and one wrapped in an event horizon.

It is given by the beautifully simple formula r_s = 2 G M / c^2, growing in direct proportion to the mass M. A quick Newtonian heuristic, setting the escape speed equal to the speed of light, (1/2) v^2 = G M / r with v = c, gives r = 2 G M / c^2 and lands on the exact same number, though that derivation gets the right answer partly by luck. In the Schwarzschild metric r_s is where the time-time component 1 - r_s/r vanishes, marking the event horizon of a non-rotating black hole.

Every mass has a Schwarzschild radius, but for ordinary objects it lies absurdly deep inside them, far smaller than the object itself, so nothing dramatic happens. The radius becomes physical only when the mass is compressed within it. The caveat about the Newtonian derivation is worth keeping: light does not literally 'fail to reach escape velocity' like a thrown ball, and the coincidence of the factor of 2 hides genuinely relativistic physics. The correct meaning of r_s is geometric, the location where the horizon forms, not a Newtonian escape calculation.