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Thermodynamics
Bekenstein–Hawking entropy of a black hole
Entropy of a black hole in terms of the area A of its event horizon.
- Ref
- THE-25
- Rev
- v1
- Field
- Thermodynamics
- Result
- S [J/K]
A black hole’s entropy is proportional to the area A of its event horizon: S = k_B c³A/(4Għ), where ħ = h/2π. For a non-rotating hole A = 16πG²M²/c⁴, so the entropy grows as the square of the mass. In units of k_B, the entropy is a quarter of the horizon area measured in Planck areas.
Entropy of a black hole in terms of the area A of its event horizon. It is proportional to area, not volume, and connects gravity, quantum theory and thermodynamics. A stellar-mass black hole has far more entropy than the star that formed it.
Calculator
- Entropy in units of k_B1.049×1077
- Horizon radius, √(A/4π)2,953 m
- Mass of the black hole, c²r/(2G)1.988×1030 kg
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A horizon area of 1.096×108 m² gives an entropy of 1.448×1054 J/K, about 1.05×1077 k_B. For the default solar-mass hole that is roughly twenty orders of magnitude more than the Sun’s own entropy.
- Entropy grows with area, so doubling a black hole’s radius quadruples its entropy.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (J/K) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Event-horizon area | 1 to 1e20 m² | 1.321e46 | 1.321e66 | 1.321e66 | +10% | +10 % |
Sweep of Event-horizon area
| Event-horizon area (m²) | Black-hole entropy (J/K) |
|---|---|
| 1 | 1.321e46 |
| 316.2 | 4.178e48 |
| 100,000 | 1.321e51 |
| 3.162e7 | 4.178e53 |
| 1e10 | 1.321e56 |
| 3.162e12 | 4.178e58 |
| 1e15 | 1.321e61 |
| 3.162e17 | 4.178e63 |
| 1e20 | 1.321e66 |
Sweep one variable
Variables and how each one contributes
Black-hole entropy
What the equation gives you. Everything else feeds it.
Event-horizon area
The area of the event horizon, in m². Raises the entropy in direct proportion. Raising it by 10 % (to 1.206×108) moves the result by +10 %, from 1.448×1054 to 1.593×1054 J/K.
Worked example with the values above
- Constant: k_B c³/(4Għ) = 1.32131×1046 J/(K·m²).
- Multiply by the area: A = 1.096×108 m².
- S = 1.44815×1054 J/K.
Assumptions and limits
- A non-rotating, uncharged black hole in empty space.
- The semiclassical Bekenstein–Hawking result, which ignores quantum-gravity corrections.
Common mistakes
- Using h in place of ħ = h/2π (an error of 2π).
- Using the area of the star’s surface instead of the horizon area.
- Treating the entropy as proportional to volume.
Statistics and uncertainty
The formula is theoretical rather than measured, so the uncertainty is that of the constants (about 2 × 10⁻⁵, mostly from G). The result is very sensitive to the area, and therefore to the square of the mass.
Where it is used
- A standard example linking thermodynamics, gravity and quantum theory.
- Comparing the entropy of black holes of different mass.
Maths you need
- Entropy: S = k_B ln Ω
- Schwarzschild radius: r = 2GM/c²