Schwarzschild Radius Calculator
Calculate the Schwarzschild radius (event horizon) of a black hole from its mass. Enter values for instant results with step-by-step formulas.
Reviewed by Daniel Agrici, Founder & Lead Developer
Formula
r_s = 2GM / c^2
Where r_s = Schwarzschild radius, G = gravitational constant (6.674 x 10^-11 m^3 kg^-1 s^-2), M = mass of the object, c = speed of light (2.998 x 10^8 m/s). This gives the radius at which escape velocity equals the speed of light.
Worked Examples
Example 1: Stellar Black Hole (10 Solar Masses)
Problem:Calculate the Schwarzschild radius, density, and Hawking temperature of a black hole with 10 times the mass of our Sun.
Solution:Mass = 10 x 1.989 x 10^30 kg = 1.989 x 10^31 kg\nr_s = 2 x 6.674e-11 x 1.989e31 / (2.998e8)^2\nr_s = 2.954e20 / 8.988e16 = 29,543 meters = 29.54 km\nDensity = M / (4/3 x pi x r_s^3) = 1.84 x 10^14 kg/m^3\nHawking Temp = hbar x c^3 / (8pi x G x M x k_B) = 6.17 x 10^-9 K
Result:Schwarzschild radius: 29.54 km | Density: 1.84 x 10^14 kg/m^3 | Temp: 6.17 nK
Example 2: Sagittarius A* (4 Million Solar Masses)
Problem:Calculate the event horizon size of the Milky Way's central supermassive black hole at approximately 4 million solar masses.
Solution:Mass = 4 x 10^6 x 1.989e30 = 7.956 x 10^36 kg\nr_s = 2 x 6.674e-11 x 7.956e36 / (2.998e8)^2\nr_s = 1.062e27 / 8.988e16 = 1.181 x 10^10 m = 1.181 x 10^7 km\nIn AU = 1.181e7 / 1.496e8 = 0.0789 AU\nDensity = 1.14 x 10^6 kg/m^3 (about the density of gold)
Result:Schwarzschild radius: 11.81 million km (0.079 AU) | Density: ~gold
Frequently Asked Questions
What is the Schwarzschild radius and what does it represent?
The Schwarzschild radius defines the size of the event horizon of a non-rotating, uncharged black hole. Named after Karl Schwarzschild, who derived this solution to Einstein's field equations in 1916, it represents the critical radius at which the escape velocity equals the speed of light. Any object compressed within its own Schwarzschild radius becomes a black hole from which nothing, not even light, can escape. The formula is remarkably simple: r_s = 2GM/c^2, where G is the gravitational constant, M is the mass, and c is the speed of light. For our Sun, the Schwarzschild radius is approximately 2.95 kilometers, meaning if you compressed the entire mass of the Sun into a sphere less than 3 km in radius, it would become a black hole. For Earth, this critical radius is only about 8.87 millimeters.
How are real black holes different from the Schwarzschild solution?
The Schwarzschild solution describes an idealized, non-rotating, electrically neutral black hole in a vacuum. Real astrophysical black holes almost certainly rotate, making the Kerr metric a more accurate description. Rotating black holes have two important differences: they possess an ergosphere outside the event horizon where spacetime itself is dragged along with the rotation, and their singularity is ring-shaped rather than a point. The event horizon of a Kerr black hole is smaller than the Schwarzschild radius for the same mass, with maximum rotation reducing it by half. Charged black holes are described by the Reissner-Nordstrom metric (non-rotating) or the Kerr-Newman metric (rotating and charged), though astrophysical black holes likely carry negligible net charge. Despite these differences, the Schwarzschild radius remains a useful first approximation and upper bound on the event horizon size.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy