Schwarzschild Radius
Event horizon radius of a black hole.
Formula
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Example
Sun ≈ 3 km
How it works
The Schwarzschild radius is the critical radius at which the escape speed from a mass equals the speed of light. If all of an object’s mass were compressed inside that radius, not even light could escape, so the object would appear as a black hole from the outside. It comes from combining Newtonian escape-speed reasoning with relativity’s limit that nothing can travel faster than light, which gives the characteristic scale for an uncharged, non-rotating black hole.
When to use it
Use this when estimating whether a mass compressed into a given size would become a black hole, or when calculating the event-horizon size of an idealized spherical black hole. It is also useful in astrophysics for comparing black-hole sizes across different masses.
Worked example, step by step
Find the Schwarzschild radius of Earth. Given: G = 6.674×10^-11 m^3·kg^-1·s^-2, M = 5.97×10^24 kg, c = 3.00×10^8 m/s r_s = 2GM/c^2 r_s = 2(6.674×10^-11)(5.97×10^24) / (3.00×10^8)^2 r_s = 7.96×10^14 / 9.00×10^16 r_s = 8.85×10^-3 m r_s ≈ 8.9 mm
Common mistakes to avoid
- Using kilograms for mass but forgetting to keep G and c in SI units
- Confusing the Schwarzschild radius with the actual radius of a star or planet
- Treating the formula as exact for spinning or charged black holes, where corrections are needed
Did You Know?
Squeeze Earth to the size of a marble and it would become a black hole — that's its Schwarzschild radius.
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