Schwarzschild Radius Calculator
Enter mass m to compute the Schwarzschild radius R_s=2GM/c², escape velocity and surface gravity. Sun 1.989e30 kg → R_s≈2.95 km; Earth ≈8.87 mm.
Input Data
Results
At a glance:The Schwarzschild radius R_s=2GM/c² is the critical radius for any mass m: if compressed within it, the mass becomes a black hole (light cannot escape). Found by Karl Schwarzschild in 1916 from Einstein's field equations of general relativity — the first exact solution, the Schwarzschild metric. R_s=2GM/c² with G=6.6743e-11 N·m²/kg² and c=2.998e8 m/s. The event horizon is the sphere of radius R_s; its escape velocity equals c. Under the Schwarzschild metric the surface gravity is g=GM/R_s²=c⁴/(4GM). Classic values: the Sun (1.989e30 kg) has R_s≈2.95 km; Earth (5.972e24) ≈8.87 mm; Jupiter (1.898e27) ≈2.82 m; a 1 kg object ≈1.48e-27 m (far below a proton). Applications: (1) the collapse condition for stellar black holes — a neutron star exceeding the Tolman–Oppenheimer–Volkoff limit (~2–3 solar masses) collapses to a black hole; (2) supermassive black holes (e.g. Sgr A* at the Milky Way centre, ~4e6 solar masses) with R_s≈1.2e10 m; (3) primordial black-hole theory; (4) the first gravitational-wave detection (LIGO, 2015) of black-hole mergers. Note: this is for a non-rotating, uncharged black hole; rotating ones use the Kerr metric, charged ones the Reissner–Nordström metric.
Formula
Schwarzschild radius: R_s = 2GM/c²
Escape velocity at horizon: v_e = √(2GM/R_s) = c
Surface gravity: g = GM/R_s² = c⁴/(4GM)
$$R_s = \frac{2GM}{c^2}, \quad v_e = c, \quad g = \frac{c^4}{4GM}$$How to Use
- Enter mass m (kg).
- The tool gives R_s (m, km), escape velocity at the horizon (=c) and surface gravity g.
- m≤0 gives R_s=0 and g=0.
Case Studies
Sun collapsing to a black hole
Solar mass M=1.989e30 kg.
R_s = 2×6.6743e-11×1.989e30/(3e8)² ≈ 2954 m ≈ 2.95 km.
If compressed within 2.95 km the Sun becomes a black hole; but it lacks the mass to collapse naturally — that needs >~2–3 solar masses.
Supermassive black hole Sgr A*
Milky Way centre Sgr A* ≈ 4e6 solar masses ≈ 7.96e36 kg.
R_s ≈ 2×6.6743e-11×7.96e36/9e16 ≈ 1.18e10 m ≈ 0.079 AU.
Surface gravity g≈1.5e13 m/s², yet tidal force at the horizon is surprisingly weak due to the huge mass.
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.