Black hole growth and evaporation

[egcavalcanti]

What is the maximum approximate black hole size that would have negative growth rate as a function of average local mass density (and/or any other relevant parameters)? In other words, when does an evaporating black hole become a growing black hole and vice-versa?

Cheers
Eric

 

[Mordred]

Blackholes evaperation is an incredibly slow process, for a Schwarzschild black hole of 1 solar mass
the evaperation process by Hawking radiation is 10⁶⁷ years. Thats for an uncharged non rotating black hole. I have no idea on the rates of Kerr black holes. Some key points on Hawking radiation which is a blackbody temperature. The smaller the BH the faster it will evaperate. So micro black holes will evaperate faster. Also evaperation does not occur unless the blackbody temperature is greater than the surrounding temperature. So the blackbody temperature must be greater than 2.7 kelvin. For growth rate that depends on its feeding rates of infalling materials , I know of no limit on BH size.

If your interested in the accretion disk and jet measurements and structure this article has a large collection of formulas for measurements etc. There is some discussion on portions of the accretion disk and it does have a brief mention of Hawking radiation including the formula. Its near the beginning of the 91 page article. Its math intensive however.

http://arxiv.org/abs/1104.5499

by the way welcome to the forum

 

[George Jones]

Expanding on the 2.7 k point:

Suppose a black hole is completely isolated from everything except the Cosmic Microwave Background (CMB) radiation at 2.7 K. Let's find the Schwarzschild radius $r$ for a black hole that is in equilibrium with the CMB, i.e., that has a Hawking temperature of 2.7 K. If a black hole is larger than this, its Hawking temperature will be less 2.7, and the black hole will grow and keep growing (neglecting the drop in the CMB temperature caused by the expansion of the universe), because it will absorb energy from the CMB. If a black hole is smaller than this, its Hawking temperature will be more than 2.7, and the black hole will shrink and keep shrinking, because it will radiate energy.

The Schwarzschild radius of a black is given by $r =2GM/c^2$. Combining this with its Hawking temperature

$$T = \frac{\hbar c^3}{8 \pi k G M}$$
gives

$$r = \frac{\hbar c}{4 \pi k T}$$
Using T = 2.7 K gives a radius of 0.000067 metres and a mass of $4.6 \times 10^{22}$ kg (a little less than the mass of the Moon).

This is much smaller than black holes formed from astrophysical processes.