# Quantum tunneling into a black hole

The currently accepted answer to Throwing a micro black hole into the sun: does it collapse into a black hole or does it result in a supernova? states that a small black hole of mass approximately $$10^9$$ kg cannot accrete matter quickly, because radiation pressure would push particles away more strongly than gravity would pull them in. The argument given is that the micro black hole would exceed the Eddington limit. In other words, for a proton near the event horizon, the outward force due to radiation is

$$\frac{ L_\text{Hawking}\sigma_\text{Thomson} }{Ac} = \frac{\hbar c^6 \cdot \sigma_\text{Thomson} \cdot c^4}{M^2 15360 \pi G^2 \cdot 16\pi M^2 G^2 \cdot c} \approx 3 \times 10^{12} \,\text{N},$$

while (using the Newtonian approximation for simplicity), the gravitational force is

$$\frac{GMm_p}{R_\text{Schwarzschild}^2} = \frac{GMm_p}{(2GM/c^2)^2} = \frac{m_p c^4}{4GM} \approx 5 \times 10^7 \,\text{N}.$$

Thus, the proton is pushed away from the black hole rather than pulled towards it.

But since the proton's wave function is nonzero inside the black hole, isn't there a significant chance that the proton will simply tunnel into the black hole and never return?

I'm asking this question purely out of academic interest. I realize that it's irrelevant to the correctness of the answer linked above, since tunneling would presumably only affect a tiny number of protons near the event horizon. I also understand that giving a full answer would probably require a theory of quantum gravity. However, is there anything that can be said from semiclassical gravity alone?

• Could one view Hawking radiation as the inverse of this process? – Lewis Miller Feb 7 at 13:51

First of all, the first equation used in the question is not applicable for the black hole mass under consideration. The temperature of Hawking radiation for $$M=10^9 \,\text{kg}$$ exceeds $$10\,\text{GeV}$$, therefore there would be a lot more particles being radiated besides photons: electrons, muons, pions, … and their corresponding antiparticles and since the number of species of particles would be large the overall luminosity would be dozens of times greater than the value in the OP. Also, cross sections of interactions with protons for some of the species would also be larger than for photons.