# Why would a black hole explode?

It is common in popular science culture to assume that Hawking radiation causes black holes to vaporize. And, in the end, the black hole would explode. I also remember it being mentioned in A Brief History of Time.

Why would a black hole explode? Why can't it gradually vanish to zero? What is the exact mechanism or theory which causes a black hole to explode?

• What is an explosion? What is not an explosion? How do one recognize an exploding thing? Jan 19, 2015 at 11:57
• @Mindwin i.word.com/idictionary/explosion Sudden and violent release of energy should do, but I am unable to find precise numerical boundary. Good question, though. Ask it on the site. Jan 19, 2015 at 15:18

The expression for the power emitted as Hawking radiation is $$P = \frac{\hbar c^6}{15360 \pi G^2 M^2} = 3.6\times10^{32} M^{-2}\ \text{W} = -c^2 \frac{dM}{dt},$$ where the term on the far right hand side expresses the rate at which the black hole mass decreases due to the emission of Hawking radiation.

You can see that what happens is that the power emitted actually increases as $M$ decreases. At the same time, the rate at which the mass decreases also increases.

So as the black hole gets less massive, the rate at which it gets less massive increases rapidly and hence the power it emits increases very, very rapidly.

By solving this differential equation it can be shown that the time to evaporate to nothing is given by $$t = 8.4\times10^{-17} M^3\ \text{s},$$ so for example a 100 tonne black hole would evaporate in $8.4 \times10^{-2}\ \text{s}$, emitting approximately $E = Mc^2 = 9\times 10^{21}$ joules of energy as it does so – equivalent to more than a million megatons of TNT. I guess you could call this an explosion!

This will be the fate of all evaporating black holes, but most will take a very long time to get to this stage (even supposing they do not accrete any matter). The evaporation time is only less than the age of the universe for $M <$ a few $10^{11}\ \text{kg}$. A 1 solar mass black hole takes $2\times10^{67}$ years to evaporate.

EDIT: The Hawking radiation temperature is given by $$kT = \frac{\hbar c^3}{8 \pi GM}.$$ Unless the temperature is well above the ambient temperature (at a minimum the cosmic microwave background temperature), the black hole will always absorb more energy than it radiates, and get bigger. i.e. to evaporate $$\frac{\hbar c^3}{8 \pi GM} > kT_{\rm ambient}$$ $$M < \frac{1.2\times10^{23}}{T_{\rm ambient}}\ {\rm kg}$$

Therefore unless I've made a mistake, this proviso is of no practical importance other than for evaporating black holes (i.e. those with $M<10^{11}$ kg) in the early universe.

The temperature of a black hole goes as its evaporation timescale as $t_{\rm evap}^{-1/3}$. The temperature of the early, radiation-dominated, universe scales as $t^{-1/2}$. Thus it appears to be the case that at some point in the past, a black hole that might have had an evaporation timescale shorter than the age of the universe is incapable of doing so.

• It should be noted that the evaporation time does not take into account the fact that the cosmic microwave background radiation is adding mass even to black holes that aren't accreting matter. A black hole will not start to shrink until the temperature of the CMB becomes less than the temperature of the Hawking radiation.
– Gabe
Jan 19, 2015 at 4:15
• @Mindwin $E=mc^2$, i.e. all its mass prior to that is radiated away as well, but at a slower rate. I picked the 100 tonne figure as an arbitrary example as the lifetime is very short from there. Jan 19, 2015 at 12:06
• if you take the same 100T black hole and calculate its mass $8.4 \times10^{-2}\ \text{s}$ before the moment it reaches 100T, what would be its mass? What is the energy emitted in those previous $8.4 \times10^{-2}\ \text{s}$? Jan 19, 2015 at 12:11
• @Gabe Would you look at my edit - I think I've got this right. Any black hole evaporating in the next billion years or so will have a temperature much higher than the CMB. Jan 19, 2015 at 12:45
• Your 7.7 should be 1.2, but otherwise I would agree -- only very small, hot black holes could possibly evaporate in the near future. Any black hole created as a result of a star collapsing will be way to big to even start to evaporate any time soon.
– Gabe
Jan 19, 2015 at 14:07

Unlike most objects, a black hole's temperature increases as it radiates away mass. The rate of temperature increase is exponential, with the most likely endpoint being the dissolution of the black hole in a violent burst of gamma rays. A complete description of this dissolution requires a model of quantum gravity, however, as it occurs when the black hole approaches Planck mass and Planck radius.

Wikipedia

All black holes are theorized to emit Hawking radiation at a rate inversely proportional to their mass. Since this emission further decreases their mass, black holes with very small mass would experience runaway evaporation, creating a massive burst of radiation at the final phase, equivalent to a hydrogen bomb yielding millions of megatons of explosive force.

Wikipedia

• Jan 17, 2015 at 17:31
• I'd say this is a perfectly valid answer, though it usually makes for a better answer if you add some of your own words to the quote - even if it's just summarizing or restating the main point from the quoted passage in a way that makes it explicit how it answers the question. If you really have nothing to add, that may be a sign of a bad question. (I'm not saying this needs to be edited; just food for thought.) Jan 17, 2015 at 20:14
• @DavidZ Sure. This answer was the first answer (and was even accepted for a short time). Rob's expert answer wasn't there yet. I didn't know the answer to this (for me) interesting question, but the quick Wikipedia finds were quite satisfactory for me, and I suspected, to more people. (I already chose to quote the relevant bits instead of commenting with the links only.) Don't expect an edit (from me). Feel free to remove. Jan 17, 2015 at 20:24
• @GlenTheUdderboat oh, don't worry, I certainly don't think this qualifies for deletion. It does answer the question, after all. Jan 17, 2015 at 21:03
• The only thing is that the black holes are not exploding, in the same sense a grenade does. A grenade is inert and suddenly emits energy and matter (explodes) in a discrete moment. The black hole is always emitting Hawking radiation, but the rate of the emission over time increases exponentially, allowing the phenomena to be compared to an explosion. Jan 19, 2015 at 12:00