Questions: Black Hole Thermodynamics

Question 1

The Bekenstein-Hawking entropy of a black hole is proportional to:

• A. The volume enclosed by the event horizon
• B. The mass of the black hole
• C. The area of the event horizon
• D. The surface gravity of the event horizon

Answer: (C)

S_BH = kA/(4l_P²), where A is the horizon area and l_P = √(ħG/c³) is the Planck length. This area scaling is deeply surprising — in ordinary thermodynamics, entropy is an extensive quantity proportional to volume, not area. The area law suggests that the degrees of freedom of a black hole live on its boundary (the horizon), not in its interior. This is one of the key motivations for the holographic principle, which conjectures that the information content of any region is bounded by its surface area in Planck units.

Question 2

Hawking's area theorem states that the total horizon area of black holes can never decrease in any classical process. This is violated by Hawking radiation.

• T. True
• F. False

Answer: True

The classical area theorem (proved by Hawking in 1971) assumes the null energy condition, which is satisfied by all classical matter. Hawking radiation is a quantum effect that violates the null energy condition near the horizon (through the creation of negative-energy partners that fall into the black hole). As the black hole radiates, it loses mass and its horizon area decreases, violating the classical area theorem. This is consistent because the area theorem's assumptions (classical GR + null energy condition) do not hold when quantum effects are included. The generalized second law — that the sum of black hole entropy and ordinary entropy never decreases — remains valid.

Question 3

Explain why Bekenstein's area-entropy relation was initially controversial, and how Hawking's radiation calculation resolved the controversy.

Model answer: Bekenstein proposed in 1972 that black holes carry entropy proportional to their horizon area, to prevent violations of the second law of thermodynamics (dropping a hot object into a black hole would otherwise destroy entropy). The controversy was that if black holes have entropy, they must have temperature, and if they have temperature, they must radiate — but classically, nothing escapes a black hole. Hawking's 1974 calculation of quantum particle creation near the horizon showed that black holes do radiate thermally at temperature T_H = ħκ/(2πck), with the entropy coefficient being exactly S = kA/(4l_P²). This resolved the controversy by showing that the thermodynamic analogy is exact: black holes are genuine thermodynamic objects.

Hawking's calculation was originally intended to disprove Bekenstein's proposal — he expected to show that black holes do not radiate. When the calculation showed they do, it was one of the most important surprises in theoretical physics, connecting gravity, quantum mechanics, and thermodynamics.

Question 4

A Schwarzschild black hole of mass M has entropy S = 4πGM²k/(ħc). Compare this with the entropy of the Sun.

Model answer: For a solar-mass black hole (M ≈ 2 × 10³⁰ kg): S_BH = 4πG M²k/(ħc) ≈ 4π(6.67×10⁻¹¹)(2×10³⁰)²(1.38×10⁻²³)/((1.05×10⁻³⁴)(3×10⁸)) ≈ 1.5 × 10⁵⁴ J/K, or about 10⁷⁷ in units of k. The Sun's thermodynamic entropy is approximately 10⁵⁸ k. A solar-mass black hole has about 10¹⁹ times more entropy than the Sun — an enormous factor. This reflects the fact that a black hole is the highest-entropy state for a given mass and size, which is why gravitational collapse is thermodynamically irreversible (in the classical limit).

The vast entropy of black holes dominates the entropy budget of the observable universe. The total entropy of all black holes in the universe (~10¹⁰⁴ k) vastly exceeds the entropy of all other sources combined (~10⁸⁸ k for CMB photons, the next largest contributor). This makes black holes the thermodynamic 'endpoint' of gravitational evolution.