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The Schwarzschild metric ds² = -(1 - 2GM/rc²)c²dt² + (1 - 2GM/rc²)⁻¹dr² + r²dΩ² is the unique spherically symmetric vacuum solution to Einstein's field equations (Birkhoff's theorem). It describes the spacetime geometry outside any non-rotating, uncharged, spherically symmetric mass M. The metric has a coordinate singularity at the Schwarzschild radius r_s = 2GM/c² (the event horizon for a black hole) and a true curvature singularity at r = 0. In the weak-field limit (r >> r_s), it reduces to Newtonian gravity, but near r_s it predicts qualitatively new phenomena: extreme gravitational time dilation, the bending of light, the precession of orbits, and the existence of event horizons. The Schwarzschild solution is the foundation for understanding black holes, gravitational redshift, and the classic tests of GR.
Karl Schwarzschild found the first exact solution to Einstein's field equations in 1916, just weeks after Einstein published the final form of general relativity. The solution describes the spacetime outside a spherically symmetric, non-rotating mass — a star, a planet, or a black hole. In Schwarzschild coordinates (t, r, θ, φ), the line element is ds² = -(1 - r_s/r)c²dt² + (1 - r_s/r)⁻¹dr² + r²(dθ² + sin²θ dφ²), where r_s = 2GM/c² is the Schwarzschild radius. The angular part r²dΩ² is the metric of a standard 2-sphere, reflecting the assumed spherical symmetry. The two metric functions (1 - r_s/r) in g_{tt} and g_{rr} encode all the gravitational physics.
Birkhoff's theorem guarantees that this solution is unique: any spherically symmetric vacuum solution must be the Schwarzschild metric. This is a remarkably strong result. It means the exterior spacetime of a spherically symmetric star is Schwarzschild regardless of the star's internal dynamics — even during radial pulsation or spherically symmetric collapse. No gravitational radiation escapes, and distant observers see a static gravitational field. This is the general-relativistic generalization of Newton's shell theorem. For a non-black-hole object (r_s well inside the body), the Schwarzschild solution applies only to the vacuum exterior; the interior is described by a different solution matched at the surface.
Far from the mass (r >> r_s), the Schwarzschild metric approaches the flat Minkowski metric, with small corrections of order r_s/r. The leading correction in g_{00} is exactly 2Φ/c², where Φ = -GM/r is the Newtonian potential. This is the regime in which Newtonian gravity is an excellent approximation. The geodesic equation for slow-moving particles reproduces Newton's inverse-square law. The corrections to Newtonian gravity — perihelion precession, gravitational time dilation, light deflection — are of order r_s/r compared to the Newtonian terms, which for the Sun at the Earth's orbit is about 10⁻⁸. These effects are small but measurable, and their observation constitutes the classic tests of general relativity.
At r = r_s, the metric components have apparent singularities: g_{tt} → 0 and g_{rr} → ∞. For decades, this was confused with a physical singularity, but it is actually a failure of the coordinate system — like the coordinate singularity at the North Pole in latitude-longitude coordinates. The curvature invariant K = R_{μνρσ}Rμνρσ = 48G²M²/(c⁴r⁶) is finite and well-behaved at r = r_s. Coordinate systems that are regular at the horizon — Eddington-Finkelstein coordinates, Kruskal-Szekeres coordinates — show that spacetime is smooth there and that freely falling observers cross the horizon in finite proper time, experiencing finite (though potentially large) tidal forces. The true singularity is at r = 0, where K → ∞ and the curvature is genuinely infinite. The physics of this singularity and the event horizon at r = r_s are explored in the black hole topic.