Schwarzschild Radius
Reference for Schwarzschild radius r_s = 2GM/c² for black hole event horizons.
Returns r_s for the Sun, Earth, and supermassive black holes in meters.
Need to calculate, not just reference? Use the interactive version.
Open Schwarzschild Radius (Black Hole) Calculator →
The Formula
r = 2GM / c²
The Schwarzschild radius defines the boundary of a black hole's event horizon. If any mass is compressed within this radius, not even light can escape its gravitational pull.
Variables
| Symbol | Meaning |
|---|---|
| r | Schwarzschild radius (meters) |
| G | Gravitational constant (6.674 × 10⁻¹¹ N⋅m²/kg²) |
| M | Mass of the object (kg) |
| c | Speed of light (2.998 × 10⁸ m/s) |
Example 1
Find the Schwarzschild radius of the Sun
M = 1.989 × 10³⁰ kg
r = 2 × 6.674 × 10⁻¹¹ × 1.989 × 10³⁰ / (2.998 × 10⁸)²
r = 2.654 × 10²⁰ / 8.988 × 10¹⁶
r ≈ 2,953 meters ≈ 2.95 km
Example 2
Find the Schwarzschild radius of Earth
M = 5.972 × 10²⁴ kg
r = 2 × 6.674 × 10⁻¹¹ × 5.972 × 10²⁴ / (2.998 × 10⁸)²
r ≈ 0.00887 meters ≈ 8.87 mm
When to Use It
Use the Schwarzschild radius when:
- Determining the size of a black hole's event horizon
- Understanding what happens when mass is extremely compressed
- Studying general relativity and gravitational physics
- Comparing how different masses relate to black hole formation
Key Notes
- The event horizon is a mathematical boundary, not a physical surface — an infalling observer experiences nothing special crossing it; the extreme effects (tidal forces, time dilation) intensify inside
- The formula gives the right answer from Newtonian physics (setting escape velocity = c) by coincidence — the correct derivation comes from general relativity (Schwarzschild's solution to Einstein's field equations)
- Hawking radiation causes black holes to very slowly evaporate — a solar-mass black hole would take approximately 2×10⁶⁷ years to fully evaporate, far longer than the current age of the universe
- The Schwarzschild radius scales linearly with mass: doubling the mass doubles the event horizon radius — a supermassive black hole of 10⁹ M☉ has an event horizon ~3×10¹² m (≈20 AU) across