Last updated: 2026-06-16

The Schwarzschild Radius

The Schwarzschild radius is the radius of the event horizon of a non-rotating, uncharged black hole. It is the distance from the centre of a mass at which the escape velocity reaches the speed of light:

$R_s = \frac{2GM}{c^2}$

where $G = 6.674 \times 10^{-11}\ \text{m}^3\text{kg}^{-1}\text{s}^{-2}$ is the gravitational constant, $M$ is the mass, and $c = 2.998 \times 10^8\ \text{m/s}$ is the speed of light. The result was first derived by Karl Schwarzschild in 1916, only months after Einstein published the field equations of general relativity.

What the Radius Means

If all of an object's mass is compressed inside its Schwarzschild radius, the object becomes a black hole and the sphere of radius $R_s$ becomes the event horizon — the boundary beyond which no light or signal can escape. Because $R_s$ is directly proportional to mass, doubling the mass doubles the horizon; a black hole is not necessarily dense, only sufficiently compact.

Worked Example

Find the Schwarzschild radius of the Sun ($M = 1.989 \times 10^{30}\ \text{kg}$).

The Sun's actual radius is about 696,000 km, so it is nowhere near becoming a black hole. To collapse the Sun into one, its entire mass would have to be squeezed into a sphere less than 3 km across.

A Sense of Scale

The linear scaling makes the numbers easy to remember: roughly 3 km per solar mass. A 10-solar-mass stellar black hole has a horizon about 30 km wide, while the supermassive black hole at the centre of the Milky Way, around 4 million solar masses, has a Schwarzschild radius of roughly 12 million km — about 17 times the radius of the Sun, yet still smaller than Mercury's orbit. For everyday objects the radius is microscopic: the Earth's is about 9 mm, and a typical person's is around $10^{-25}$ m, far smaller than an atomic nucleus.

Related Calculations

The Schwarzschild radius is closely tied to escape velocity: setting the escape-velocity formula equal to the speed of light reproduces $R_s = 2GM/c^2$ exactly. The surface gravity of a compact object and the gravitational force between masses use the same gravitational constant, and the event horizon itself sets the scale for phenomena such as gravitational time dilation and Hawking radiation.