 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-311/4/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

Classically, the laws of black-hole mechanics have the same form as the laws of thermodynamics, but this could have been only a formal analogy. Hawking radiation makes it physical: a stationary black hole emits approximately thermal radiation with

T_{H} = \frac{ℏ κ}{2 π k _{B} c},

so surface gravity is proportional to genuine temperature.

The correspondence is:

constancy of $κ$ on a stationary horizon becomes constancy of equilibrium temperature;
the First law of black-hole mechanics

d M = \frac{κ}{8 π G} d A + Ω_{H} dJ + Φ_{H} d Q

S_{B H} = \frac{k _{B} c ^{3} A}{4 G ℏ};

Hawking's area theorem becomes a second law, replaced in the quantum theory by the generalized second law for $S_{B H} + S_{outside}$ ;
the impossibility of reaching $κ = 0$ by a finite process parallels the unattainability of absolute zero.

A real Hawking temperature therefore identifies horizon area as thermodynamic entropy and turns the geometric laws into thermodynamic laws.

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