SchwarzschildRadius

The Schwarzschild radius, denoted r_s, is a characteristic radius associated with a mass in general relativity. For a non-rotating, uncharged, spherically symmetric body of mass M, r_s = 2GM/c^2. It corresponds to the radius of the event horizon for the Schwarzschild black hole of mass M: if the body's physical radius is smaller than r_s, it would be a black hole; if larger, the exterior spacetime is described by the Schwarzschild solution.

The Schwarzschild solution, derived by Karl Schwarzschild in 1916, is the simplest exact solution to Einstein's field equations in vacuum outside a spherical mass. In Schwarzschild coordinates, r is the circumferential radius, and the surface r = r_s marks the boundary beyond which light and information cannot escape for a non-rotating, uncharged black hole.

Numerical scales illustrate the concept: for the Sun, r_s is about 3 kilometers; for the Earth, it is roughly 9 millimeters. The Schwarzschild radius scales linearly with mass, so more massive objects have proportionally larger r_s values.

Extensions to other cases exist: for charged bodies the relevant solution is Reissner–Nordström, and for rotating bodies it is Kerr (or Kerr–Newman when charged). These more general spacetimes can involve multiple horizons or different causal structures, while the Schwarzschild radius remains the reference radius for the simple, non-rotating, uncharged case.

In astrophysics, the concept helps describe black holes, gravitational lensing near compact objects, and the limits of stellar collapse. It does not imply a physical surface at r_s for most objects, but rather a geometric boundary in spacetime associated with mass.