Hawking Radiation and Black Hole Lifetimes

In classical general relativity a black hole is an eternal prison from which nothing escapes. The event horizon acts as a one-way membrane: matter and light may cross inward but can never leave. Quantum mechanics alters this picture in a remarkable way. By allowing vacuum fluctuations near the horizon, Stephen Hawking showed in 1974 that black holes should emit thermal radiation and consequently lose mass over time. The effect is extraordinarily small for stellar-mass or larger objects, yet the conceptual shift is profound—black holes are not truly black and may eventually evaporate away. The temperature of this radiation is inversely proportional to the mass, meaning tiny black holes radiate furiously while large ones are extremely cold. The calculator above illustrates this relationship by computing the Hawking temperature and the total time required for a non-rotating, uncharged black hole to evaporate.

The theoretical foundation of Hawking radiation arises from quantum field theory in curved spacetime. Near the event horizon vacuum fluctuations can create particle–antiparticle pairs. One member may fall inside the horizon while the other escapes to infinity, appearing as real radiation to a distant observer. Energy conservation requires the black hole to lose mass equivalent to the energy carried away. Although the calculation is subtle and involves Bogoliubov transformations between different vacua, the final result is surprisingly simple: the radiation spectrum is thermal with temperature given by , where is the reduced Planck constant, the speed of light, the gravitational constant, the black hole mass, and Boltzmann's constant.

Because the radiation carries away energy, the mass slowly decreases and the temperature rises, leading to a runaway process culminating in a final burst of high-energy particles. The timescale for complete evaporation, assuming no additional mass accretion, is determined by integrating the mass loss rate. Hawking's calculation gives . The cubic dependence on mass demonstrates that evaporation is incredibly slow for astrophysical black holes. A black hole with mass equal to our Sun would take roughly 10⁶⁷ years to vanish, dwarfing the current age of the universe (~10¹⁰ years). Conversely, hypothetical primordial black holes with asteroid-mass or smaller could evaporate within cosmic timescales and have been considered as sources of high-energy cosmic rays and gamma rays.

The calculator uses the inputs to determine the Hawking temperature and the total evaporation time. Mass is entered in units of solar masses, which are converted internally to kilograms. The constants used are , , , and . After computing the lifetime in seconds, the result is converted to years for readability. The temperature is reported in Kelvin. Results are purely theoretical and assume an isolated, non-rotating black hole.

To provide context, the following table lists approximate temperatures and lifetimes for a range of masses:

Mass (M☉)	Temperature (K)	Evaporation Time (years)
10−12	1.2×1011	3.5×105
10−6	1.2×105	3.5×1017
1	6.2×10−8	2.1×1067
10	6.2×10−9	2.1×1070
106	6.2×10−14	2.1×1085

These numbers underscore the immense range of scales involved. Primordial black holes lighter than 10¹² kg would have evaporated by now, while stellar and supermassive black holes are effectively eternal on cosmological times. The Hawking temperature for a solar-mass black hole is a minuscule 60 nanokelvin, colder than the cosmic microwave background. Only when the mass shrinks below about 10²³ kg does the temperature exceed that of interstellar space, allowing net evaporation to proceed. Large black holes instead gain mass by absorbing the ambient radiation.

Although Hawking radiation has not been observed directly, indirect evidence supports the underlying quantum theories, and analog systems such as sonic horizons in Bose–Einstein condensates have demonstrated similar effects. Detecting the final burst from an evaporating primordial black hole would be a landmark discovery, offering insights into both the early universe and quantum gravity. Future gamma-ray and cosmic-ray observatories continue to search for these signatures.

When using this calculator, remember that it assumes a Schwarzschild (non-rotating, uncharged) black hole isolated from external matter. Real astrophysical black holes may spin, carry charge, or accrete material, all of which modify the evaporation process. Nevertheless, the simple model captures the essential physics of quantum emission from curved spacetime and highlights the extraordinary longevity of most black holes. By exploring how evaporation time scales with mass, one gains appreciation for the delicate interplay between gravity and quantum mechanics that governs the fate of these exotic objects.