Raio de Schwarzschild

Schwarzschild radius: r_s = 2GM/c²

The Schwarzschild radius marks the event horizon of a non-rotating black hole, the boundary past which nothing escapes, not even light. Karl Schwarzschild derived it in 1916 from Einstein's field equations, only months after general relativity appeared in print. In the formula r_s = 2GM/c², G = 6.674×10⁻¹¹ N·m²/kg² and c = 2.998×10⁸ m/s. Plug in a few masses and the spread is remarkable. The Sun (1.989×10³⁰ kg) gives r_s ≈ 2.95 km. Earth comes in at 8.87 mm. A 70 kg person works out to ~10⁻²⁵ m, far below the Planck scale and physically meaningless. At the center of the Milky Way, Sgr A* (4 million M☉) reaches ~12 million km, roughly 17 solar radii, while M87* (~6.5 billion M☉) reaches ~38 billion km, which matches the 2019 Event Horizon Telescope image. Stephen Hawking (1974) later showed that black holes evaporate through Hawking radiation, at a temperature inversely proportional to their mass.

Applications

It shows up across astrophysics (X-ray binaries, accretion disks, jets) and in LIGO/VIRGO gravitational-wave detection from black-hole mergers, where the inspiral phase hinges on both components' Schwarzschild radii. You also find it behind the Event Horizon Telescope images of M87* (2019) and Sgr A* (2022), in cosmology (primordial black holes as dark-matter candidates), and even in physically grounded science fiction like Interstellar's Gargantua, which was rendered using equations Kip Thorne supplied.

FAQ

Could the Sun become a black hole? No. Its mass falls well short of the Tolman-Oppenheimer-Volkoff limit (~2-3 M☉) that a stellar-mass black hole requires, so the Sun will end its life as a white dwarf instead.

What happens inside the event horizon? Classical general relativity points to an unavoidable singularity at the center, where curvature blows up. Once you cross r_s, every timelike path bends toward that singularity.

Does the formula apply to rotating black holes? Strictly speaking, no. Kerr (rotating) black holes have a more complicated outer horizon that depends on spin. Even so, r_s still gives the right order of magnitude and remains the standard reference scale.