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.