Schwarzschild metric

The Schwarzschild metric is the unique static, spherically symmetric, asymptotically flat vacuum solution of Einstein's field equations. Written in Schwarzschild coordinates (t, r, θ, φ), its line element is ds² = −(1 − r_s/r) c² dt² + (1 − r_s/r)⁻¹ dr² + r²(dθ² + sin²θ dφ²), where r_s = 2GM/c² is the Schwarzschild radius of the central mass M. It describes the gravitational field outside any non-rotating, uncharged spherical body — a star, a planet, or a black hole.

Every physical feature of the geometry is carried by the single factor (1 − r_s/r). It divides the time component, so a static clock at radius r ticks at a rate √(1 − r_s/r) relative to a clock at infinity (gravitational time dilation). It inversely multiplies the radial component, stretching proper radial distances. Far from the mass the factor approaches 1 and the metric reduces to flat Minkowski spacetime; the curvature is entirely an effect of the mass, switched off at large distance.

The metric was derived by Karl Schwarzschild in December 1915, within weeks of Einstein publishing the field equations, while Schwarzschild was serving on the Russian front. By Birkhoff's theorem (1923), it is the only spherically symmetric vacuum solution, so it governs the classical tests of general relativity — Mercury's perihelion precession, the deflection and lensing of light, and the Shapiro time delay — and, when a mass is compressed inside its own Schwarzschild radius, it becomes the spacetime of a non-rotating black hole.

Karl Schwarzschild communicated the exterior solution to Einstein in a letter dated December 22, 1915, and the interior fluid-sphere solution in February 1916. Einstein presented both to the Prussian Academy on Schwarzschild's behalf. Schwarzschild died of disease contracted at the front in May 1916. The metric was initially regarded as a mathematical idealization; its black-hole interpretation matured only over the following decades, with the coordinate pathology at r = r_s resolved by Eddington (1924), Lemaître (1933), Finkelstein (1958), and Kruskal (1960).