Black Hole Thermodynamics: Entropy, Hawking Radiation & Evaporation

Hawking's discovery: black holes are not entirely black

Stephen Hawking showed in 1974 that black holes are not entirely black. Quantum field theory in curved spacetime predicts that a black hole emits thermal radiation with a temperature set entirely by its mass:

Hawking temperature:  T_H = ħc³ / (8πGM k_B)

For a solar-mass black hole: T_H ≈ 6 × 10⁻⁸ K — colder than the CMB.
Smaller black hole → higher T_H → faster evaporation.

Because T_H scales inversely with mass, a solar-mass black hole's Hawking temperature is far colder than the 2.7 K cosmic microwave background it sits in — meaning it currently absorbs more energy than it radiates. Only much smaller black holes, with far higher Hawking temperatures, would be net emitters and shrink over time. This inverse mass-temperature relationship is the opposite of everyday thermodynamics, where bigger objects are usually hotter, and it is the key to why evaporation eventually runs away at the very end of a black hole's life.

Bekenstein entropy and the area law

Before Hawking's calculation, Jacob Bekenstein had already argued that a black hole must carry entropy proportional to the area of its event horizon, not its volume — a radical departure from ordinary thermodynamic systems where entropy scales with volume. Combined with Hawking's temperature, this gives the black hole a full set of thermodynamic laws: the first law relates changes in mass (energy) to changes in horizon area (entropy) and angular momentum; the second law says the total horizon area of a system of black holes never decreases in any classical process, an echo of the ordinary second law of thermodynamics applied to gravity itself. This area-entropy relationship later became a cornerstone of the holographic principle, the idea that all the information inside a volume of space can be encoded on its boundary.

The M³ evaporation law

As a black hole radiates, it loses mass, and losing mass raises its temperature, which increases its radiation rate further — evaporation accelerates as the black hole shrinks. The evaporation timescale scales as the cube of the initial mass, M³. A solar-mass black hole would take roughly 10⁶⁷ years to evaporate completely, vastly longer than the ~1.4×10¹⁰-year age of the universe. Only hypothetical primordial mini black holes, formed from density fluctuations in the very early universe and small enough from the start, could be finishing their evaporation now — and their possible final burst of radiation is one of the proposed observational signatures researchers have searched for.

The information paradox and Kerr black holes

Hawking radiation, as originally calculated, is purely thermal — it carries no information about what fell into the black hole. If a black hole evaporates completely, that information appears to be destroyed, directly conflicting with quantum mechanics, where information must always be preserved. This black hole information paradox remains one of the deepest open problems in theoretical physics; Hawking himself conceded in 2004 that information is likely preserved, though the precise mechanism by which it escapes is still debated. Real astrophysical black holes also rotate, described by the Kerr metric rather than the simpler non-rotating Schwarzschild solution; a spinning black hole's ergosphere allows energy extraction via the Penrose process, and its thermodynamic laws depend on the Kerr parameter a/M in addition to mass, linking angular momentum directly to the horizon's entropy and temperature.

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