Hawking Radiation Temperature Formula
Calculate the temperature of Hawking radiation emitted by a black hole.
Derived by Stephen Hawking in 1974, linking quantum mechanics and general relativity.
The Formula
In 1974, Stephen Hawking showed that black holes are not perfectly black — they emit thermal radiation due to quantum effects near the event horizon. This "Hawking temperature" is the temperature of that radiation. It is one of the most profound results in theoretical physics, uniting quantum mechanics, thermodynamics, and general relativity.
Variables
| Symbol | Meaning | Value / Unit |
|---|---|---|
| T | Hawking temperature of the black hole | Kelvin (K) |
| ħ | Reduced Planck constant (h/2π) | 1.0546 × 10⁻³⁴ J·s |
| c | Speed of light | 2.998 × 10⁸ m/s |
| G | Gravitational constant | 6.674 × 10⁻¹¹ N·m²/kg² |
| M | Mass of the black hole | kg |
| k_B | Boltzmann constant | 1.381 × 10⁻²³ J/K |
The key insight: smaller black holes are hotter. As a black hole radiates energy, it loses mass, which makes it hotter, causing it to radiate even faster — a runaway process that ends in a final burst of radiation when the black hole evaporates completely.
Example 1 — Solar Mass Black Hole
Calculate the Hawking temperature of a black hole with the mass of the Sun (M = 2.0 × 10³⁰ kg).
Numerator: ħc³ = 1.0546 × 10⁻³⁴ × (2.998 × 10⁸)³
ħc³ = 1.0546 × 10⁻³⁴ × 2.694 × 10²⁵ = 2.840 × 10⁻⁹ J³·s/m³ × m³/s
Denominator: 8π × 6.674 × 10⁻¹¹ × 2.0 × 10³⁰ × 1.381 × 10⁻²³
= 8π × 1.843 × 10⁻³ = 4.634 × 10⁻²
T ≈ 6.2 × 10⁻⁸ K — about 60 nanokelvin, far colder than the cosmic microwave background (2.73 K)
Example 2 — Black Hole Evaporation Time
The evaporation time of a black hole scales as t_evap ∝ M³. How long would a solar-mass black hole take to evaporate?
t_evap = 5120πG²M³ / (ħc⁴)
For M = M_Sun = 2.0 × 10³⁰ kg:
t_evap ≈ 2.27 × 10⁶⁷ years
This is vastly longer than the age of the universe (~1.38 × 10¹⁰ years). Stellar black holes will not evaporate for an unimaginably long time.
When to Use It
The Hawking radiation formula is used when:
- Calculating the theoretical temperature of black holes of various masses
- Estimating evaporation timescales for primordial black holes (possibly formed shortly after the Big Bang)
- Studying the black hole information paradox in theoretical physics
- Exploring the thermodynamics of black holes (entropy = area/4 in Planck units)
For comparison: the cosmic microwave background has a temperature of 2.73 K. Any black hole with mass above about 4.5 × 10²² kg (roughly the mass of the Moon) will be colder than the CMB and will actually absorb more radiation than it emits — meaning real black holes in the universe today are still growing, not shrinking.