How are black-hole evaporations possible?

Question

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.

Answer 1

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.

Answer 2

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.