What metric describes a non-rotating, uncharged black hole?

The Schwarzschild metric describes a non-rotating, uncharged black hole. It is an exact solution of Einstein's field equations for the spacetime outside a spherical mass when angular momentum, electric charge, and the cosmological constant are taken to be zero.

Karl Schwarzschild found the solution in 1916, shortly after Einstein published general relativity. The metric predicts a spherical event horizon at the Schwarzschild radius, which depends only on the object's mass: doubling the mass doubles the radius. The horizon is not a solid surface; it is a boundary beyond which signals cannot reach a distant observer.

The most common mix-up is with the Kerr metric, which describes a rotating, uncharged black hole. The Kerr–Newman metric includes both rotation and electric charge, while the Reissner–Nordström metric describes a non-rotating charged black hole. These solutions have different spacetime structures and, for rotating cases, an ergosphere.

By Birkhoff's theorem, the Schwarzschild solution is the most general spherically symmetric vacuum solution. It also approximates the exterior gravitational field of slowly rotating bodies such as the Sun and Earth, provided their rotation and charge can be neglected.