# Quantum tunneling through the event horizon (EH): Is the EH a potential barrier for Quantum tunneling?

This is not a duplicate. I am not asking whether anything can escape a BH, I understand nothing can. My question is whether the potential barrier in the definition of Quantum tunneling can be a BH EH? Does the EH qualify as a potential barrier?

I have read this question:

But there is no answer.

Now the answers mention Hawking radiation. Not the type where particle-antiparticle gets created, one inside the EH, one outside. I am talking about the Hawking radiation where it is about Quantum tunneling.

Yet, I read everywhere on this site that nothing can escape the BH.

https://arxiv.org/abs/hep-th/9907001

From wiki:

Quantum tunnelling or tunneling (see spelling differences) is the quantum mechanical phenomenon where a subatomic particle passes through a potential barrier. Quantum tunneling is not predicted by the laws of classical mechanics where surmounting a potential barrier requires enough potential energy.

Now the trick here is, that even though the EH would be a potential barrier, still, nothing can tunnel through it from inside, because nothing can move outwards from inside. Am I correct? Is the EH a potential barrier? Is it just that the direction of the particles can never be outwards?

Question:

1. Can this potential barrier be the EH of a BH?

2. If not, why is the EH different then a potential barrier in the definition of Quantum tunneling?

A hand waving answer refers once more to the basic tunneling phenomenon as described here

Please note the dark energy level line. Tunneling happens at the same energy level, it is only probabilities that allow the existence of the wavefunction at a measurable level beyond the barrier.

Gravity has not been quantized , only effective quantizations exist, but the arguments should work. The black hole at the quantum level by definition is a potential well and the deeper in the well the higher the binding energy , in quantum mechanics.

By construction of the horizon, there are no energy levels outside the horizon at the same level as the energy levels inside the horizon, so the probability of tunneling is zero by mathematical arguments. There is no energy level outside the "horizon barrier" to which a particle inside the barrier could tunnel, as per the figure above.

Hypothetically: tunneling should be a part of the merging of the horizons in the merging of black holes, as in the LIGO event. There should be similar energy levels in the approaching black holes , and if quantum mechanics holds there would be tunneling during the approach to merge.

• Thank you! Can you please elaborate on this: "The black hole at the quantum level by definition is a potential well and the deeper in the well the higher the binding energy" and "By construction of the horizon, there are no energy levels outside the horizon at the same level as the energy levels inside the horizon" Commented May 28, 2019 at 16:18
• As nothing can get our of the black hole, an analogue is a square well with infinite walls phys.libretexts.org/Bookshelves/… . . When a particle falls in a potential well it loses energy, it goes to a lower energy level. These levels do not exist outside the region of the potential well. That is why we have stable nuclei and exist here. Nothing tunnels out of a stable nucleus.because there is no energy level outside the nucleus to which it could go. Commented May 28, 2019 at 16:54
• So the energy levels inside the nucleus or inside the black hole are lower then outside? All the available energy levels outside are higher? Commented May 28, 2019 at 16:57
• Yes, unless there is another black hole in the vicinity with its own potential well, then a particle can have a probability to tunnel Commented May 28, 2019 at 17:01

An event horizon is not a classical potential barrier. Photons and other particles can exist perfectly fine at and inside of the event horizon, however they will only be traveling in one direction. Gravity can with a bit of handwaiving be seen as space itself moving towards the center of the potential well, dragging anything that is in that space along.

Your question is equivalent to asking why a particle can not travel faster than the speed of light. If I create a photon in position $$x$$ at time $$t_0$$, there won't be any wave amplitude/probability of that photon at position $$y$$ at the same time $$t_0$$. Only at time $$t_1$$, when the photon has had time to travel from $$x$$ to $$y$$, will there be any wave amplitude of this photon at $$y$$. Now, can you describe the space between $$x$$ and $$y$$ as a potential barrier because it prevents the photon wave at $$x$$ from also existing at $$y$$ at $$t_0$$? As far as I know you can't, because they are just different concepts. The same holds for a point $$x$$ inside a black hole and a point $$y$$ outside of it, except that there is no $$t_1$$ where the wave has had enough time to travel from $$x$$ to $$y$$.

I've been pondering the same question, from the perspective of atomic and molecular orbitals. A ground-state hydrogen atom has a single electron in a spherically symmetrical orbital, which is a probability distribution that rapidly decays with distance from the proton, but is never non-zero. There is accordingly a small probability that the electron can be found at a large distance from the proton. Now put this hydrogen atom in orbit within the event horizon of a black hole. If the wavefunction of the electron extends beyond the event horizon, it means that there is some non-zero probability that the electron can be found outside, i.e., having escaped the black hole. The shape of the orbital, i.e. the probability function as a function of distance, is going to be distorted by gravity along with the distortion of space itself, but should still permit non-zero values outside of the event horizon.
I'll concede that calculations of the wavefunction that explicitly take gravitation into account might give zero probability outside of the event horizon, but they're beyond my pay grade.

• If you have a new question, please ask it by clicking the Ask Question button. Include a link to this question if it helps provide context. - From Review Commented Oct 11, 2022 at 15:06