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In classical physics a black hole is the one object that only takes. Anything that crosses its event horizon, light included, never comes back out. In 1974 Stephen Hawking published a short paper with a question for a title, Black hole explosions?, which argued that this picture is incomplete. Once quantum mechanics is included, black holes should glow. This post explains what he found, works through the temperature formula with real numbers, and draws out what a student can take from how he worked.
Who was Stephen Hawking?
The Royal Society's biographical memoir gives his dates as 8 January 1942 to 14 March 2018. It records that he was diagnosed with motor neurone disease during his early years as a postgraduate student in Cambridge and given two years to live, and that he lived a further 55 years. Gonville and Caius College, where he was a Fellow for over half a century, notes that he was Lucasian Professor of Mathematics at Cambridge for 20 years to 2009.
The memoir summarises his scientific path: singularity theorems in the 1960s, then the discovery that black holes have an entropy and therefore a finite temperature, then work on the early universe. His book A Brief History of Time made cosmology known to the general public.
What did Hawking find in 1974?
His first announcement was the letter Black hole explosions? (Nature, 1974). The full treatment followed in Particle creation by black holes (1975). Its abstract states the result plainly: quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies, with a temperature set by the black hole's surface gravity. The emission makes the black hole slowly lose mass, and the paper estimates that any primordial black hole lighter than about 10¹⁵ g would have evaporated by now.
The result also fixed a puzzle. A few years earlier Jacob Bekenstein had argued, in Black Holes and Entropy (1973), that the area of a black hole behaves like entropy, since both tend to increase irreversibly. If a black hole has entropy, thermodynamics says it should have a temperature, and anything with a temperature should radiate. Hawking's calculation showed that it does. His 1975 abstract notes that the radiation breaks the classical rule that horizon area can never decrease, but that a generalised second law still holds: matter's entropy plus a quarter of the horizon area never decreases.
How hot is a black hole?
For a non-rotating black hole of mass M, the Hawking temperature is
T = ħc³ / (8πGMk),
where ħ is the reduced Planck constant and k is Boltzmann's constant. The key feature is M in the denominator: heavier black holes are colder.
Worked example. Take a black hole with the Sun's mass, M = 1.989 × 10³⁰ kg.
Numerator: ħc³ = (1.055 × 10⁻³⁴) × (3.00 × 10⁸)³ = 1.055 × 10⁻³⁴ × 2.70 × 10²⁵ ≈ 2.84 × 10⁻⁹.
Denominator: 8π × G × M × k = 25.13 × 6.674 × 10⁻¹¹ × 1.989 × 10³⁰ × 1.381 × 10⁻²³ ≈ 4.61 × 10⁻².
T = 2.84 × 10⁻⁹ / 4.61 × 10⁻² ≈ 6.2 × 10⁻⁸ K.
That is 62 billionths of a kelvin. Compare it with the cosmic microwave background, which Fixsen (2009) measured at 2.72548 K. A solar-mass black hole is far colder than the sky around it, so today it absorbs more radiation than it emits and grows rather than shrinks.
Two follow-ups make good practice:
Size. The event horizon radius is 2GM/c² = 2 × 6.674 × 10⁻¹¹ × 1.989 × 10³⁰ / (9.0 × 10¹⁶) ≈ 2.95 km.
Peak wavelength. Wien's law, λmax = b/T with b = 2.898 × 10⁻³ m K, gives 2.898 × 10⁻³ / 6.2 × 10⁻⁸ ≈ 4.7 × 10⁴ m. The radiation peaks at a wavelength of about 47 km. Our guide to black-body radiation problems covers Wien's and Stefan's laws.
Hawking radiation itself is beyond the JEE and NEET syllabus, but every step above uses constants and laws you already know.
Has any of this been tested?
The tiny temperature in the worked example shows why Hawking radiation from a stellar black hole would be so hard to detect. The classical area theorem that Hawking's work extended has been checked. Isi and colleagues (2021) analysed the gravitational-wave signal GW150914 from two merging black holes and found that the final horizon area was larger than the starting total, in agreement with the area theorem with 97% probability in their main analysis.
What can a student take from how he worked?
Take a clash between theories seriously. Hawking's result came from putting quantum mechanics and general relativity side by side and asking what happens at the horizon.
Check with an analogy you trust. The link to thermodynamics (area like entropy) gave the idea a test: if it is entropy, there must be a temperature.
Estimate the numbers. A formula means little until you plug in real values. The worked example shows at once why the effect is invisible for stellar black holes.
Keep working when conditions are hard. The memoir notes how long he worked after a diagnosis that gave him two years.
If thermodynamics or gravitation is where the formula loses you, the Fermi Concept Library covers every concept in physics, each with its own lesson. You may also like our profile of S. Chandrasekhar and the limit on white dwarf mass.
Use the formula to find the mass of a black hole whose temperature equals 2.725 K, and compare it with the mass of the Moon.
