# How are black-hole evaporations possible?

It is believed that if, $$R \leq \frac {2Gm}{c^2} \; \;\;\;\; (1)$$ is black hole and due to Hawking radiation it losses mass and subsequently when, $$R \geq \frac {2GM}{c^2} \;\;\;\;\;\; (2)$$ black hole will blast.

Now, if that is so, then from Eq. (1) it is clear that the mass and radius of the black hole (Schwarzschild radius) both are interdependent; however, if there is mass loss due to Hawking radiation, proportionally the radius of the black hole will shrink and it will always retain the Schwarzschild radius, thus it always remains a black hole; the idea of a black hole explosion is not possible then? please clarify it.

• Hawking paper nature.com/articles/248030a0 Mar 12 at 5:38
• Most of what you wrote isn't true. The mass and radius of a black hole aren't independent, and black holes don't explode when their radius exceeds a certain value. Mar 12 at 6:31

## 2 Answers

Black holes do not abruptly explode when they reach a certain size. What happens is that the power of the emitted radiation is proportional to $$1/M^2$$. Specifically the equation is:

$$P = \frac{\hbar c^6}{15360 \pi G^2 M^2}$$

When the black hole is large the factor of $$1/M^2$$ is very small so the emitted power is very small. However as the black hole evaporates its mass $$M$$ decreases so $$1/M^2$$ increases and the power increases. The increased power makes the mass decrease faster, which makes the power even greater, and as the black hole becomes very small the power is so large it exceeds the power of an exploding atomic bomb (by many orders of magnitude).

This is a continuous process so there is no point at which the black hole suddenly goes BOOM, but in the last stages of the evaporation the power increases so rapidly with time that it resembles an explosion like a supernova.

• It would be good to note that the formula given for the power is an approximation. It ignores things like greybody factors. The real formula would differ by a factor “a few”. Mar 12 at 13:05

A simple reply

For a given mass one can always calculate a Schwarzschild radius, see the one for the earth here .

whereas Earth's is only about 9 mm (0.35 in) and the Moon's is about 0.1 mm (0.0039 in).

To make a black hole of the earth there should be a force that presses it down to such small dimensions.

The speculation of Hawking was if in primordial times there existed small black holes created by the enormous energies of that time, what would be effect of Hawking radiation, a quantum effect. He came up with the article in Nature , where , assuming a quantization of gravity,

Any such black hole of mass less than $$10^{15}$$ g would have evaporated by now. Near the end of its life the rate of emission would be very high and about $$10^{30}$$ erg would be released in the last 0.1 s.

Here is the summary from the original paper

So the basic clarification needed is that Hawking radiation depends on quantization of gravity, so just the general relativity mathematics arguments are not enough to model the decay of small mass primordial black holes. We need to model them , because there are no small black holes whizzing around the current universe , enlarging themselves.

• --Anna, I'm reading Merali's 2017 pop-sci book titled "A Big Bang in a Little Room", which includes interviews with Guth, Vilenkin, Linde, & other well-known physicists, all tending toward acceptance of the possibility that the local universes of a multiverse might each have had an artificial origin in laboratories on sequentially-decreasing scales of spacetime, whose occupants would, consequently, have appeared submicroscopic to their predecessors. You're working at CERN, so I'm wondering what you think of this conjecture: Many laypeople (maybe including the OP) have had the opposite notion. Mar 19 at 12:39
• Influenced by your sensible opening comment on the Schwarzschild radius, I've deleted an earlier (& idiotic) comment that I'd made about the OP's question, but, in view of the possibility that the magnification energies required to discern primordial BH's might be sufficient to result in their collapse into a smaller size even less accessible, I'm still wondering about the possibility that such hyper-minimal LU's might exist. (Verification of their existence might have to await some larger-scaled extra-terrestrial version of CERN, but might not remain "unscientific" forever....) Mar 19 at 19:20
• @Edouard as an experimental physicist I do not have the background to answer hypothetical scenaria. I know that in order to resolve the hypothesis I quote one has to have a definitive quantization of gravity , to be checked against the data. Mar 20 at 4:42