Schwarzschild metric

Karl Schwarzschild discovered the metric in 1916, barely a month after Einstein published the final form of general relativity, while serving on the Eastern Front during World War I.¹ He solved the vacuum Einstein equations for a static, spherically symmetric mass distribution. The metric in Schwarzschild coordinates (t, r, θ, φ) is given by ds² = −(1 − 2GM/c²r)c²dt² + (1 − 2GM/c²r)⁻¹dr² + r²(dθ² + sin²θ dφ²). The parameter M is the mass, G the gravitational constant, and c the speed of light. The solution is valid for r greater than the radius of the mass; the interior solution for a fluid sphere is a separate problem.²

The Schwarzschild metric exhibits two singularities: one at r = 2GM/c², called the Schwarzschild radius or event horizon, and one at r = 0, the curvature singularity. The event horizon is a coordinate singularity—it can be removed by a change of coordinates (e.g., Eddington-Finkelstein or Kruskal coordinates). At r = 0, the curvature becomes infinite, indicating a breakdown of classical general relativity. For a non-rotating black hole, the event horizon is a sphere of radius Rₛ = 2GM/c². No information or matter can escape from within this radius.

The Schwarzschild metric correctly predicts the anomalous precession of Mercury's perihelion, one of the classic tests of general relativity.³ It also describes the bending of light by gravity, confirmed during the 1919 solar eclipse. In modern astrophysics, the Schwarzschild metric is the foundation for studying non-rotating black holes, such as those formed by the collapse of massive stars. The Event Horizon Telescope's image of M87*'s shadow is consistent with the Schwarzschild (and Kerr) predictions.⁴

Although the Schwarzschild metric is often presented as a black hole solution, Schwarzschild himself did not interpret the event horizon as a physical boundary; he considered the singularity at r = 2GM/c² as a coordinate artifact he called the 'Schwarzschild singularity.'¹ The first complete understanding of the event horizon as a one-way membrane came decades later, with work by Oppenheimer and Snyder in 1939. The metric also admits a maximal analytic extension—the Kruskal-Szekeres coordinates—which reveals a white hole region and a parallel universe, though these are not physically realized for realistic stellar collapse. Another obscure feature: the Schwarzschild solution is the unique static, spherically symmetric vacuum solution (Birkhoff's theorem), meaning any spherical mass distribution, even if pulsating, has an exterior Schwarzschild geometry.²

The Schwarzschild metric is the simplest of a family of exact solutions. The Reissner-Nordström metric adds electric charge, while the Kerr metric describes rotating black holes. The Kerr-Newman metric combines both rotation and charge. All these solutions reduce to the Schwarzschild metric when charge and angular momentum are zero.⁵ The Schwarzschild metric also serves as a starting point for studying perturbations around black holes, essential for gravitational wave astronomy.

Event horizon

The boundary in spacetime beyond which events cannot affect an outside observer; for Schwarzschild at r = 2GM/c².

Schwarzschild radius

The radius of the event horizon of a Schwarzschild black hole, proportional to its mass.

Coordinate singularity

A singularity in the metric that can be removed by a change of coordinates, not a true curvature singularity.

Birkhoff's theorem

Any spherically symmetric vacuum solution of Einstein's equations is static and given by the Schwarzschild metric.