Questions: Hawking Radiation

Question 1

A black hole with twice the mass of another has a Hawking temperature that is:

• A. Twice as high
• B. Four times as high
• C. Half as high
• D. The same

Answer: (C)

The Hawking temperature T_H = ħc³/(8πGMk) is inversely proportional to the mass M. Doubling M halves the temperature. This is the opposite of ordinary thermodynamic objects, where adding energy (mass-energy) increases the temperature. Black holes have negative heat capacity: as they radiate and lose mass, they get hotter, which causes them to radiate faster, which causes them to lose mass faster — a runaway process that ends in complete evaporation.

Question 2

The Hawking temperature of a solar-mass black hole is about 60 nanokelvin. This is far colder than the cosmic microwave background temperature of 2.7 K.

• T. True
• F. False

Answer: True

For M = M_☉ ≈ 2 × 10³⁰ kg, T_H = ħc³/(8πGMk) ≈ 6 × 10⁻⁸ K. Since this is far below the CMB temperature (2.725 K), a solar-mass black hole actually absorbs more radiation from the CMB than it emits, causing it to grow rather than evaporate. Only black holes lighter than about 10²² kg (roughly the mass of the Moon) would currently have T_H > T_CMB and be actively evaporating. No black hole evaporation has been observed, and astrophysical black holes will not begin to evaporate until the universe cools below their Hawking temperature — trillions of years from now.

Question 3

Explain the information paradox that arises from black hole evaporation.

Model answer: If a black hole forms from matter in a specific quantum state (a pure state) and then evaporates completely via Hawking radiation, the radiation appears to be exactly thermal — determined only by the black hole's mass, spin, and charge, with no information about the initial state encoded in the radiation. A pure state has evolved into a mixed (thermal) state, violating unitarity — the quantum mechanical principle that time evolution preserves information. Either: (1) information is truly lost (violating quantum mechanics), (2) information is encoded in subtle correlations in the Hawking radiation (violating the semiclassical approximation), (3) a remnant retains the information, or (4) the semiclassical picture breaks down in unexpected ways. Recent progress via the Page curve and quantum extremal surfaces suggests option (2), but a complete resolution requires a full theory of quantum gravity.

The information paradox, posed by Hawking in 1976, has driven four decades of theoretical progress. It is not merely a technicality — it forces a confrontation between general relativity and quantum mechanics at a fundamental level. The resolution likely requires understanding quantum gravity, making it one of the sharpest theoretical constraints on any quantum gravity candidate.

Question 4

Describe the relationship between Hawking radiation and the Unruh effect, and explain why they are considered manifestations of the same physics.

Model answer: The Unruh effect states that a uniformly accelerating observer in flat Minkowski spacetime perceives the vacuum as a thermal bath at temperature T_U = ħa/(2πck), where a is the proper acceleration. Hawking radiation can be understood as a gravitational analog: by the equivalence principle, a stationary observer near a black hole horizon is equivalent to an accelerating observer in flat spacetime. The surface gravity κ plays the role of the acceleration a, and the Hawking temperature T_H = ħκ/(2πck) has the same form as the Unruh temperature. Both effects arise from the same underlying physics: the observer-dependence of the particle concept in quantum field theory — different observers in relative acceleration disagree about the vacuum state and hence about the number of particles present.

The connection between Hawking and Unruh effects illustrates a deep principle: the particle content of a quantum field is observer-dependent. There is no absolute notion of 'empty space' in quantum field theory on curved backgrounds. This observer-dependence is the conceptual core of Hawking radiation.