All black holes absorb mass attracted by gravity, and expel mass (Hawking Radiation). I've been led to believe, due to all popular representations of black holes, that astronomical (a.k.a. large) black holes grow. I've read that Micro-black holes shrink/evaporate.
My question is, at what mass and/or radius is a black hole a growing one, as opposed to a shrinking one?
Answer
If one assumes no other matter is supplied to the black hole (which is difficult to describe in a general manner as it depends on the details of the environment), the question if black holes evaporate depends on the difference between the emitted Hawking radiation and the absorbed cosmic microwave background radiation. According to the Stefan-Boltzmann law the Power emitted/absorbed by a black body is $$P=\sigma A T^4,$$ where $\sigma=\frac{\pi^2K_B^4}{60\hbar^3c^2}$ is just a constant, $A$ the surface of the body and $T$ the temperature. So the emitted power of a black hole is $$P=\sigma A \left(T_{H}^4-T_{CMB}^4\right),$$ where $T_H=\frac{\hbar c^3}{8\pi G M k_B}$ is the Hawking temperature and $T_{CMB}\approx 2.725\,\text{K}$ is the current temperature of the CMB. One can now simply check up to which mass $M$ the black hole's Hawking temperature is lower than the CMB temperature. One finds, with the formulas above, that $$ T_H>T_{CMB}\quad\Leftrightarrow\quad M<\frac{\hbar c^3}{8\pi G k_B T_{CMB}}=4.5 \cdot 10^{22}\,\text{kg}=0.075 M_\oplus,$$ where $M_\oplus$ is the mass of the earth. This mass correponds to a black hole radius of $6.5\cdot 10^{-5}m$ using $r_S=\frac{2GM}{c^2}$. Stellar black holes however, which are formed in gravitational collapse, have masses larger than $5M_\odot\approx 2\cdot 10^6M_\oplus$, where $M_\odot$ is the solar mass which corresponds to balck hole radii of above $70\,\text{km}$.
Notice however, that if the universe keeps on expanding then the CMB temperature will keep on dropping and even heavy black holes will evaporate eventually.
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