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When a dying star's core exceeds roughly 33 solar massesβ€”surpassing the Tolman-Oppenheimer-Volkoff limitβ€”even neutron degeneracy pressure collapses under gravity. With nothing left to resist the inward crush, the entire core implodes into an infinitely dense region called a singularity.

Spacetime Curvature

In 1915, Albert Einstein published his theory of General Relativity, showing that gravity is not an invisible pulling force, but rather the curvature of spacetime caused by mass and energy.
A stellar-mass black hole concentrates so much mass into zero volume that it bends local spacetime to an extreme degree, creating a bottomless gravitational well.
How close can you get to this singularity before escape becomes physically impossible?

The Schwarzschild Radius

In 1916, while serving on the Russian front during World War I, German physicist Karl Schwarzschild solved Einstein's field equations to find the exact boundary where the escape velocity equals the speed of light, cc.
This boundary is the event horizon, and its spherical radius from the singularity is the Schwarzschild radius (RsR_s).

Why the Formula Works

To escape any gravitational body, an object's kinetic energy ( rac{1}{2}mv^2) must balance its gravitational potential energy ( rac{GMm}{r}).
Setting escape speed v=cv = c and solving for radius gives r = rac{2GM}{c^2}. While derived here using classical energy, full General Relativity yields this exact same radius for a non-rotating (Schwarzschild) black hole.