Schwarzschild Radius

Schwarzschild radius: r_s = 2GM/c²

The Schwarzschild radius defines the event horizon of a non-rotating black hole — the boundary inside which nothing, not even light, can escape. Karl Schwarzschild derived it in 1916 from Einstein's field equations, just months after general relativity was published. The formula r_s = 2GM/c² uses G = 6.674×10⁻¹¹ N·m²/kg² and c = 2.998×10⁸ m/s. Examples: the Sun (M = 1.989×10³⁰ kg) → r_s ≈ 2.95 km; Earth → 8.87 mm; a 70 kg human → r_s ≈ 10⁻²⁵ m (sub-Planck scale, physically meaningless). Sgr A* (4 million M☉) at the Milky Way's center → ~12 million km (≈17 solar radii); M87* (~6.5 billion M☉) → ~38 billion km, matching the Event Horizon Telescope image of 2019. Stephen Hawking (1974) showed black holes evaporate via Hawking radiation, with temperature inversely proportional to mass.

Applications

Astrophysics (X-ray binaries, accretion disks, jets), gravitational-wave detection by LIGO/VIRGO from black-hole mergers (the inspiral phase depends on the Schwarzschild radii of both components), cosmology (primordial black holes as dark-matter candidates), and science fiction grounded in real physics — Interstellar's Gargantua was rendered with equations supplied by Kip Thorne.

FAQ

Could the Sun become a black hole? No — its mass is well below the Tolman-Oppenheimer-Volkoff limit (~2-3 M☉) needed for a stellar-mass black hole. The Sun will end as a white dwarf.

What happens inside the event horizon? Classical GR predicts an unavoidable singularity at the center, where curvature diverges. A complete quantum theory of gravity is still missing, but inside r_s, all timelike paths point toward the singularity.

Does the formula apply to rotating black holes? Strictly no — rotating (Kerr) black holes have a more complex outer horizon depending on spin. But r_s still gives the right order of magnitude and is the standard reference scale.