Schwarzschild

Karl Schwarzschild was a German physicist and astronomer (1873–1916) who, in 1915, obtained the first exact solution to Einstein's field equations of general relativity describing the spacetime outside a static, spherically symmetric mass. The result, published in 1916, provided the Schwarzschild solution, a cornerstone of modern gravitation theory and a foundation for the concept of a non-rotating black hole.

Mathematically, the Schwarzschild solution is a vacuum solution representing the spacetime around a non-rotating, uncharged body. In Schwarzschild coordinates it is described by the metric: ds^2 = -(1 - 2GM/rc^2) c^2 dt^2 + (1 - 2GM/rc^2)^{-1} dr^2 + r^2(dθ^2 + sin^2 θ dφ^2). This solution reduces to Newtonian gravity in the weak-field limit and makes precise predictions such as gravitational redshift, light bending, and perihelion precession.

An important feature is the Schwarzschild radius r_s = 2GM/c^2. If a mass lies within r_s, the region forms a black hole with an event horizon at r = r_s. At r = r_s the metric appears singular in these coordinates, but the spacetime is regular there; the apparent singularity is a coordinate artifact that can be removed by changing coordinates (to, for example, Eddington-Finkelstein or Kruskal-Szekeres coordinates). The interior of a Schwarzschild black hole contains a central singularity at r = 0.

Schwarzschild's solution remains a central tool in gravitational physics. It describes the exterior spacetime of any static, spherically symmetric mass and serves as the non-rotating black hole exemplar in astrophysics and theory. It provides a baseline against which rotating (Kerr) and charged (Reissner-Nordström) solutions are compared and is used in modeling gravitational lensing and relativistic orbits.