Black Hole Lifetimes

Black holes are not totally black! They will evaporate by Hawking radiation. This is required by Thermodynamics and Quantum Mechanics. All properties of a Schwarzschild Black are determined by its mass, so if you know the mass the lifetime and other properties follow automatcally. Or you can start with the lifetime and determine the initial mass. Or the Schwarzschild radius, or the temperature, or the entropy, etc. For black holes comparable in mass to “normal” astronomical objects this lifetime is much longer than the current age of the universe. Viktor Toth’s Hawking radiation calculator is a convenient tool for such calculations. Here are some results:

Object Mass Lifetime
Sun 1 Solar Mass = 1.99E30 kg 1.16E67 years
Earth 5.97E24 kg 3.14E50 years
Me 86 kg 2.96E-11 seconds
Carbon atom 1.99E-26 kg 3.66 E-94 seconds
Hydrogen atom 1.67E-27 kg 2.17 E-97
Maximum collision energy of the LHC 14 TeV = 2.24 E-6 joules =
2.49 E-23 kg (E = mc2)
7.18E-85 seconds
Minimum mass for a black hole that existed at the Big Bang
(Primordial Black Hole)
2.11E11 kg 13.8 billion years
Black Hole with a temperature of the Cosmic Microwave
Background (2.73oK)
4.49E22 kg = 0.007257 Earth masses 1.34E44 years

Let us consider the 14TeV blackhole, supposing all the collision energy of the LHC went into producing a black hole. That black hole would have a lifetime tl. During that lifetime it could travel a distance of at most d = ctl = 2.15 E-76 meters. That is not far. The Bohr radius for a Hydrogen atom is 5.29 E-11 meters, enormously larger. Even the radius of a proton is about 0.85 fm = 0.85E-15 meters, 61 orders of magnitude greater than the distance that black hole could travel before evaporating. When the LHC was first turned on there was some uninformed critcism that it might create a black hole that would swallow the Earth. This is clearly impossible: Such a black hole would not live long enough to find any other matter to eat. This will hold for any future human made particle accelerator for a very long time.

As noted at the bottom of Mr. Toth’s page, high mass (>0.07257 Earth masses) black holes will actually live longer than the equation behind this calculator indicates. That is because they are currently colder than the Cosmic Microwave Background, and so are absorbing energy and growing. This will not last forever. Eventually the CMB will cool below the temperature of any such black hole and at thet time in the very far future it will start to radiate, and far beyond that time, disappear.

[Edit 12/17/2021] Atomic-sized primordial black holes: what new experimental evidence suggests puzzled me for a while. How could such small black holes survive from the big bang, 13.8 billion years ago. The answer is that while I was reading “atomic-sized” my mind was substituting “atomic mass,” Correcting this by interpreting “atomic-sized” to mean “having a Schwarzschild radius the size of atom” clears this up. A black hole with a Schwarzschild radius of 5.3E-11 meters, the Bohr radius, will have a mass of 3.6E16 kg and a life time of 6.7E25 years, far beyond the age of the universe.

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