The radius of a sphere such that, if all the mass of an object were compressed within it, the escape velocity would equal the speed of light.The Schwarzschild radius is the critical radius of a sphere around a mass such that, if all that mass were compressed within it, the escape velocity at the surface would equal the speed of light. For any object, this defines the size of the event horizon of a non-rotating black hole: nothing, not even light, can escape from within this radius. It is a purely geometric quantity, proportional to the mass, and for the Sun it is about 3 kilometers, while for Earth it is about 9 millimeters.
Karl Schwarzschild derived the solution in 1916, just months after Einstein published general relativity, while serving on the Russian front in World War I. The concept of an object so dense that light cannot escape had been speculated earlier by John Michell (1783) and Pierre-Simon Laplace (1796), but Schwarzschild's work placed it on a rigorous relativistic footing.
The Schwarzschild radius arises from the exact solution to Einstein's field equations for a spherically symmetric, non-rotating mass, derived by Karl Schwarzschild in 1916. In general relativity, mass curves spacetime, and at the Schwarzschild radius the curvature becomes so extreme that all future-directed light cones point inward. For a non-rotating black hole, the event horizon is precisely this radius; for rotating black holes, the horizon is smaller and the geometry is more complex, but the Schwarzschild radius still sets the characteristic scale.
The Schwarzschild radius is used to estimate the size of black holes from their mass, to interpret observations of black hole shadows (as in the Event Horizon Telescope images), and to calibrate dynamical mass measurements of black holes in galaxies.
If you compressed the entire Earth to the size of a marble, it would become a black hole with a Schwarzschild radius of about 9 millimeters.










