The Schwarzschild radius and event horizon
The Schwarzschild radius rs = 2GM/c² is the radius at which the escape velocity equals the speed of light. Below this radius, nothing — not even light — can escape.
- For the Sun: rs ≈ 3 km (the Sun is much larger, so it is not a black hole).
- For Earth: rs ≈ 9 mm.
- The event horizon is a one-way boundary in spacetime curvature, not a physical surface.
Hawking radiation and black hole temperature
Stephen Hawking showed in 1974 that quantum effects cause black holes to emit thermal radiation with temperature T_H = ħc³/(8πGMkB). Smaller black holes are hotter.
- Stellar-mass black holes have T_H ≈ 60 nK — far colder than the CMB (2.7 K).
- A black hole with rs = 1 mm would have T_H ≈ 1.2 × 10²³ K — explosively hot.
- Evaporation timescale scales as M³: stellar-mass black holes last longer than the age of the universe.
Gravitational time dilation
Clocks closer to a massive object run slower. GPS satellites must account for relativistic time dilation (both special and general) to maintain meter-level accuracy.
- GPS satellites run ~45 µs/day fast (GR effect) minus ~7 µs/day slow (SR effect) = net +38 µs/day.
- Time dilation near a Schwarzschild radius approaches infinite slowing as seen by a distant observer.
- Gravitational redshift causes photons escaping a gravity well to lose energy (longer wavelength).