Schwarzschild Radius

Karl Schwarzschild solved Einstein's field equations in 1915, within weeks of their publication and while serving on the Russian front, and found that any mass M compressed inside the radius rₛ = 2GM/c² wraps spacetime so tightly that nothing — not even light — can leave. Curiously, the formula is exactly what naive Newtonian reasoning gives by setting escape velocity equal to c. For the Sun (M = 1.989 × 10³⁰ kg): rₛ = 2 × 6.674 × 10⁻¹¹ × 1.989 × 10³⁰ / (2.998 × 10⁸)² ≈ 2.95 km; squeeze Earth to its Schwarzschild radius and it would be a marble 8.9 mm across.

Because rₛ grows linearly with mass, big black holes are surprisingly dilute: the four-million-solar-mass hole at the Milky Way's centre, Sagittarius A*, has rₛ ≈ 1.2 × 10¹⁰ m — inside Mercury's orbit — while the M87* giant imaged by the Event Horizon Telescope in 2019 spans about 250 AU, with an average density below that of air. The M-rearrangement turns a measured horizon size directly into a mass.