Schwarzschild radius

The Schwarzschild radius is the quantity r_s = 2GM/c², the single length scale appearing in the Schwarzschild metric for a mass M. It is the radius at which the metric factor (1 − r_s/r) vanishes — heuristically, the radius at which the Newtonian escape velocity reaches the speed of light. It scales linearly with mass: double the mass, double the radius.

For ordinary bodies the Schwarzschild radius lies far inside the physical surface and the body never approaches it. The Sun's Schwarzschild radius is about 2.95 km, buried roughly 700,000 km beneath the photosphere; the Earth's is about 8.9 mm, the size of a marble. Because r_s is so much smaller than the body's actual radius, the relativistic corrections to Newtonian gravity at the surface are tiny — though, near the Sun, large enough to measure as Mercury's perihelion shift, light deflection, and the Shapiro delay.

Only when an object is compressed inside its own Schwarzschild radius does r = r_s become an exposed surface: the event horizon of a non-rotating black hole, a one-way causal boundary from which not even light escapes. The apparent singularity of the metric at r = r_s is a coordinate artifact, regular in infalling coordinates; the genuine curvature singularity sits at r = 0.

The radius is named for Karl Schwarzschild, whose 1916 solution introduced it, though Schwarzschild himself worked with a shifted radial coordinate. The notion that a sufficiently compact body could trap light has eighteenth-century antecedents in John Michell (1783) and Pierre-Simon Laplace, who computed the same 2GM/c² threshold from Newtonian mechanics and the corpuscular theory of light.