Schwarzschild Radius Calculator

The Schwarzschild radius marks where the time component of the Schwarzschild metric, the factor 1 minus 2GM/(rc squared), goes to zero, so a radially outgoing light ray can no longer make progress. It coincides numerically with the Newtonian result obtained by setting escape velocity equal to c, which is why the formula r_s = 2GM/c squared looks deceptively classical. Because the radius is strictly linear in mass, the scale is easy to memorize: about 2.95 km per solar mass. The Sun would need to be squeezed inside a 2.95 km radius, Earth inside about 8.87 millimeters, and a 70 kg person inside roughly 10^-25 meters. Sagittarius A* at 4.15 million solar masses has a horizon near 12 million km, about 0.08 AU, while M87* at roughly 6.5 billion solar masses spans about 19 billion km. Average density inside the horizon scales as mass to the minus two power, so supermassive holes are diffuse: a few billion solar masses gives a mean density below that of air, while a 10 solar mass hole exceeds 10^17 kg per cubic meter. The Hawking temperature, hbar c cubed divided by 8 pi G M k_B, is about 6.2 x 10^-8 K for one solar mass, far below the 2.725 K cosmic microwave background, so astrophysical black holes currently gain mass rather than evaporate. The main limitation is that this solution assumes zero rotation and zero charge; the Kerr horizon for a maximally spinning hole is only half the Schwarzschild value.

Karl Schwarzschild derived the first exact solution to Einstein's field equations within weeks of their publication in November 1915, working while serving with the German army on the Russian front. His paper on the gravitational field of a point mass appeared in the Prussian Academy proceedings in early 1916, and he died in May of that year from the autoimmune disease pemphigus. Johannes Droste, a student of Hendrik Lorentz, reached an equivalent result independently in 1916. The idea of an object whose escape speed exceeds light was far older: John Michell described such dark stars in a 1783 letter to the Royal Society, and Pierre-Simon Laplace raised the same possibility in 1796. For decades the surface at r = 2GM/c squared was misread as a physical singularity. Georges Lemaitre showed in 1933 that it is only a coordinate artifact, and coordinate systems developed by Arthur Eddington and later David Finkelstein in 1958 made the one-way character of the horizon explicit. J. Robert Oppenheimer and Hartland Snyder modeled continued gravitational collapse in 1939, Roy Kerr found the rotating solution in 1963, and John Wheeler popularized the term black hole in 1967. Stephen Hawking predicted horizon radiation in 1974. The Event Horizon Telescope published its image of M87* in April 2019 and of Sagittarius A* in May 2022.