Hawking Radiation: How Black Holes Evaporate

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T ≈ 10⁻⁷ K — a 10³⁰ kg black hole is astronomically cold

A black hole with mass 10³⁰ kg (about half a solar mass) has a Hawking temperature of roughly 10⁻⁷ Kelvin — far colder than the cosmic microwave background at 2.7 K. Such a black hole would take approximately 10⁵⁷ years to evaporate, vastly exceeding the current age of the universe.

Formula

T = ℏc³ / (8πGMkB)
t_evap = 5120πG²M³ / (ℏc⁴)
L = ℏc⁶ / (15360πG²M²)

Hawking's Revolutionary Insight

In 1974, Stephen Hawking made one of the most profound theoretical discoveries in physics: black holes are not perfectly black. By applying quantum field theory to the curved spacetime near an event horizon, Hawking showed that black holes must emit thermal radiation — now known as Hawking radiation — and slowly lose mass over time.

The mechanism arises from vacuum fluctuations near the event horizon. In quantum mechanics, the vacuum is not truly empty — virtual particle-antiparticle pairs constantly pop in and out of existence. Near the event horizon, one particle can fall into the black hole while the other escapes to infinity. The escaping particle carries positive energy, while the infalling particle effectively carries negative energy, reducing the black hole's mass.

Temperature and Mass

The temperature of Hawking radiation is given by T = ℏc³/(8πGMk_B). The key insight: temperature is inversely proportional to mass. A stellar-mass black hole has a temperature of about 10⁻⁸ K — unimaginably cold, far below the cosmic microwave background temperature of 2.7 K. This means astrophysical black holes actually absorb more energy from the CMB than they emit, and are currently growing rather than evaporating.

Only after the universe cools sufficiently (in roughly 10²⁰ years) will these black holes begin their slow evaporation. But microscopic black holes, if they exist, would be extraordinarily hot — a black hole with the mass of a mountain would shine with the luminosity of a small star and emit dangerous gamma radiation.

The Evaporation Endgame

As a black hole radiates, it loses mass, which increases its temperature, which accelerates the radiation — a runaway process. The evaporation time scales as M³, so the final moments are dramatic: the last fraction of the black hole's mass is released in an enormous burst of energy. This terminal explosion could in principle be detected, and searches for such events have been conducted, though none have been found.

The Information Paradox

Hawking radiation raises a deep puzzle: if a black hole evaporates completely, what happens to the information about everything that fell in? Quantum mechanics demands that information is never truly destroyed, but Hawking's original calculation suggested it was. This 'black hole information paradox' remains one of the central unsolved problems in theoretical physics, driving major advances in quantum gravity, string theory, and holographic principles.

FAQ

What is Hawking radiation?

Hawking radiation is thermal radiation predicted to be emitted by black holes due to quantum effects near the event horizon. Discovered theoretically by Stephen Hawking in 1974, it implies that black holes are not truly black — they slowly radiate energy and eventually evaporate completely.

What is the formula for Hawking temperature?

The Hawking temperature is T = ℏc³/(8πGMkB), where ℏ is the reduced Planck constant, c is the speed of light, G is the gravitational constant, M is the black hole mass, and kB is the Boltzmann constant. Temperature is inversely proportional to mass — smaller black holes are hotter.

How long does it take a black hole to evaporate?

The evaporation time scales as M³. A stellar-mass black hole would take approximately 10⁶⁷ years — enormously longer than the age of the universe (1.4 × 10¹⁰ years). A microscopic black hole of 10⁵ kg would evaporate in about 10⁻¹⁹ seconds.

Has Hawking radiation been observed?

No. Hawking radiation has never been directly detected because the effect is extraordinarily faint for astrophysical black holes. Their Hawking temperatures are far below the 2.7 K cosmic microwave background, making the radiation unmeasurable with current technology.

Sources

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