In 1783 John Michell asked what would happen if a star were so compact that its escape speed exceeded the speed of light. Using escape speed, $v = \sqrt{2GM/r}$, the answer is easy to find. In 1916 Karl Schwarzschild found the same radius in Einstein's general relativity, where it marks the event horizon of a black hole (see Black Holes).
Why it works
Key idea
| Object | Mass | $r_s$ |
|---|---|---|
| Earth | $6.0 \times 10^{24}$ kg | 8.9 mm |
| Sun | $1\,M_\odot$ | 2.95 km |
| Stellar black hole | $10\,M_\odot$ | 30 km |
| Sagittarius A* | $4.3 \times 10^{6}\,M_\odot$ | $1.3 \times 10^{7}$ km (0.085 AU) |
| M87* | $6.5 \times 10^{9}\,M_\odot$ | $1.9 \times 10^{10}$ km (about 130 AU) |
Worked examples
Worked example: A stellar black hole
- $$r_s = \frac{2(6.67 \times 10^{-11})(1.99 \times 10^{31})}{(3.00 \times 10^{8})^2}$$
- $= 2.65 \times 10^{21}/9.00 \times 10^{16}$ m.
Answer: About 29.5 km, the size of a city.
Worked example: How dense is a black hole?
- $r_s = 2.95 \times 10^{9}$ km $= 2.95 \times 10^{12}$ m.
- $\bar\rho = 1.99 \times 10^{39}/(\tfrac43\pi(2.95 \times 10^{12})^3)$.
Answer: About 18 kg/m³, less dense than water. Since $r_s \propto M$, mean density falls as $1/M^2$; a solar-mass black hole averages about $2 \times 10^{19}$ kg/m³.
Worked example: The Sun as a black hole
- Outside a spherical mass, gravity is $GM/r^2$, whatever the size (the shell theorem).
- At 1 AU, $M$ and $r$ are unchanged.
Answer: Earth would keep orbiting exactly as before, only in the dark.
Common mistake