Schwarzschild Radius Calculator
Squeeze any mass small enough and it becomes a black hole. Enter a mass to see how small: the Schwarzschild radius is the size of the event horizon of a non-rotating black hole with that mass.
The Schwarzschild radius formula
The Schwarzschild radius is \( R_s = \dfrac{2GM}{c^2} \), where \(G\) is the gravitational constant, \(M\) the mass and \(c\) the speed of light. It grows in direct proportion to mass: double the mass and the black hole is twice as wide. For the Sun it is about 2.95 km; for the Earth about 8.9 mm, roughly the size of a marble.
Karl Schwarzschild found this solution of Einstein's field equations in 1915. You can follow the full derivation of the Schwarzschild metric in Chapter 3, or read the plain-English version in Schwarzschild radius explained.
What the results mean
- Event horizon: the boundary from inside which nothing, not even light, can get back out. For a non-rotating black hole it sits at the Schwarzschild radius.
- Photon sphere (1.5 Rs): where light can orbit the black hole in an unstable circle. It is why black hole images show bright, thin rings. See event horizon vs photon sphere.
- Innermost stable circular orbit (3 Rs): the closest that matter can orbit without spiralling in. Accretion discs end around here.
- Shadow diameter (≈ 5.2 Rs): gravity bends light so strongly that the dark region we see is about 2.6 times wider than the horizon itself. If you enter a distance, the calculator converts this into the angle the shadow covers on the sky, which is what the Event Horizon Telescope measures. More in black hole shadow explained.
- Average density: the mass divided by a sphere of radius Rs. Because density falls as 1/M², small black holes are absurdly dense but supermassive ones can be less dense than air.
- Tidal stretch at the horizon: the difference in gravity between the head and feet of a 2 m person at the horizon. Small black holes would tear you apart well outside the horizon; for supermassive ones you would barely notice crossing it.
- Hawking temperature and evaporation time: quantum effects make black holes glow very faintly. The evaporation time shown is the simplest estimate (photons only). Any black hole heavier than about 60% of the Moon's mass is currently colder than the 2.7 K cosmic microwave background, so it is absorbing more energy than it radiates. See Hawking radiation explained.
FAQ
- What would happen if Earth became a black hole? It would be about 1.8 cm across. The Moon would keep orbiting exactly as before, because from far away gravity depends only on mass, not size.
- Does a bigger black hole have stronger gravity at its horizon? No. The gravitational pull and the tidal forces at the horizon get weaker as the black hole gets bigger.
- Is the Schwarzschild radius a solid surface? No. It is a point of no return, not a physical surface; nothing would mark it as you fall through.
- Does this work for spinning black holes? Spinning (Kerr) black holes have a smaller horizon, down to half the Schwarzschild radius at maximum spin. Try the time dilation calculator to explore spin.
More tools
- Schwarzschild radius calculator: how big a black hole of any mass would be
- Black hole time dilation calculator: hovering, orbiting and Miller's planet
- Black hole image simulator: warp a photo with gravitational lensing
- Gallery: ray-traced black holes and wormholes