Calculating the Lifespan | AI Learning Guide

The cubic dependence on mass means that even small changes in mass drastically alter the lifespan. For a standard stellar-mass black hole, the number is astronomically large: ten to the sixty-seventh years. The universe is barely ten billion years old. So, while Hawking radiation is theoretically inevitable, it is practically undetectable for the black holes we see in the sky today.

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Evaporation Timescale

\(t_{evap} \approx \frac{5120 \pi G^2 M^3}{\hbar c^4}\)

Solar-Mass Black Hole Lifetime

For a black hole with the mass of our Sun (\(M_{\odot} \approx 2 \times 10^{30}\) kg), the lifetime is approximately \(10^{67}\) years. Compare this to the current age of the universe, which is only about \(1.4 \times 10^{10}\) years. The black hole will outlive the universe by many orders of magnitude.

Observational Reality

Because astrophysical black holes are so massive, their Hawking radiation is far too faint to detect against the Cosmic Microwave Background (CMB), which is at 2.7 Kelvin. We cannot currently observe Hawking radiation from any known black hole.