how black holes were first calculated

The Mechanism

*Karl Schwarzschild* (born Frankfurt am Main, 9 October 1873; died Potsdam, 11 May 1916, aged 42) was a German astronomer, director of the *Astrophysical Observatory of Potsdam* (1909-1916), and a Prussian artillery lieutenant who, between November 1915 and February 1916, *while serving on the Russian Front of the First World War*, produced *the first two exact solutions of the Einstein field equations of general relativity* — exterior and interior — for a static spherically-symmetric mass. The exterior solution, known today as the *Schwarzschild metric*, is the mathematical description of spacetime around any non-rotating uncharged mass, and it contains the *Schwarzschild radius* `r_s = 2GM/c²`, the radius at which an inward-falling light ray cannot escape — the *event horizon* of what was later called a *black hole*. Schwarzschild had volunteered for the German Army at the outbreak of war in August 1914 despite being 40 years old and an established academic; he served first in Belgium, then on the Western Front in northern France, then from late summer 1915 on the Eastern Front in *Russian Poland* (the area east of Warsaw, north of the Pripyat marshes) as a lieutenant in an artillery unit computing trajectories for long-range guns. On 18 November 1915 in Berlin, *Albert Einstein* presented the final form of his *field equations* in a lecture at the *Prussian Academy of Sciences*; the paper appeared as *Einstein, A., "Die Feldgleichungen der Gravitation," Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften* (Berlin) 844-847 (25 November 1915). Schwarzschild — by then suffering from *pemphigus vulgaris*, a rare and at the time invariably-fatal autoimmune blistering disease of the skin and mucous membranes, contracted in the trenches — read the paper on the front. Within four weeks, working in an artillery dugout and then a field-hospital cot, he produced the exterior solution. He sent it to Einstein in a letter dated *22 December 1915*. Einstein replied on *9 January 1916*: *"I had not expected that one could formulate the exact solution of the problem in such a simple way."* Einstein presented Schwarzschild's solution to the *Prussian Academy* on *13 January 1916* — Schwarzschild was an Academy member but could not attend; the paper appeared as *Schwarzschild, K., "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie," Sitzungsber. Preuß. Akad. Wiss.* (Berlin) 189-196 (3 February 1916). A second paper, the *interior* solution for a uniform-density sphere, followed on *24 February 1916*. Two weeks later Schwarzschild's pemphigus had spread; the army discharged him on medical grounds in *March 1916*; he returned to Potsdam and was hospitalised. He died on *11 May 1916*, two months after his 42nd birthday. The Schwarzschild radius was, in 1916 and for the following half-century, regarded as a *mathematical pathology* rather than a physical feature of the universe — a *coordinate singularity* where the metric components blew up, presumed to mark the breakdown of the theory rather than the description of any real object. *Arthur Eddington* discussed the question in his 1922 book *The Mathematical Theory of Relativity* and concluded that no actual star could ever be compact enough to approach `r_s`. *J. Robert Oppenheimer* and *Hartland Snyder* in 1939 (*Physical Review* 56: 455-459) showed that a sufficiently massive star whose nuclear fuel was exhausted *would inevitably collapse past its Schwarzschild radius* and form what they called a *"frozen star"* — but their paper appeared on 1 September 1939, the day Germany invaded Poland, and went largely unread. The astrophysical importance of Schwarzschild's solution was not appreciated until the late 1950s, when *David Finkelstein* (1958) clarified that the singularity at `r_s` was a feature of the coordinate system, not the physics; *Roger Penrose* (1965) proved that gravitational collapse generates a *real* singularity inside the event horizon; and the term *"black hole"* was popularised by *John Archibald Wheeler* in a 1967 lecture. The *first stellar-mass black hole candidate* — *Cygnus X-1*, identified as an X-ray binary in *1964* and confirmed as containing a compact object more massive than the maximum-mass neutron star in *1972* — vindicated the prediction physically. The 2019 Event Horizon Telescope image of *M87\** and the 2022 image of *Sagittarius A\** showed, for the first time, what the geometry around `r_s` actually looks like; both images are direct visualisations of the Schwarzschild geometry (modified for rotation by the Kerr 1963 generalisation). The fact that *every black-hole physics paper ever written rests on a calculation a dying astronomer carried out in a field hospital on the Russian Front in the winter of 1915* is one of the strangest facts in the history of physics. Schwarzschild had no idea what his solution would mean; he was a careful classical astronomer who had been working on photographic photometry and the structure of the sun's atmosphere before the war. He did the calculation because Einstein had just published the equations and he wanted to see if he could solve them. He could. He sent the solution in. He went home to die.