Event horizon and quantum tunneling

[sevenperforce]

It's been a while since I took any QM so I'm fairly rusty...not even sure that I'm asking this in the right way.

How does one set up the equations to determine the characteristics of quantum tunneling if you have a particle with a particular energy inside an event horizon?

For example, suppose you have a particle with an energy of 7.311e8 J and a wavelength of 18.8 Planck lengths inside a spherical potential well with a radius of 1.5 Planck lengths. Is there any way (I may need to use a metric tensor) to figure out the tunneling probability or lifetime?

 

[jambaugh]

I don't think you can describe particles tunneling (backwards) across an event horizon any more than you can describe particle tunneling backward in time. At least not until you find that holy grail of physics, a quantum description of gravitation. The event horizon of a black hole is NOT the same thing as a spherical potential well.

Note that Hawking's paradigm for black hole emission of thermal radiation was not that particles "tunnel" out but that virtual particle pairs form outside but near the event horizon and there's a finite chance one of the virtual pair crosses into the event horizon effecting a promotion of its partner to a physical particle... which then might have a chance of escaping.

 

[sevenperforce]

I don't think you can describe particles tunneling (backwards) across an event horizon any more than you can describe particle tunneling backward in time. At least not until you find that holy grail of physics, a quantum description of gravitation. The event horizon of a black hole is NOT the same thing as a spherical potential well.

Note that Hawking's paradigm for black hole emission of thermal radiation was not that particles "tunnel" out but that virtual particle pairs form outside but near the event horizon and there's a finite chance one of the virtual pair crosses into the event horizon effecting a promotion of its partner to a physical particle... which then might have a chance of escaping.

Okay, thanks for the reply.

Yesterday, in a related thread, phinds suggested that this was not the case -- that the "virtual particle pair" explanation is a layman's convention for something that couldn't really be modeled. He didn't include any source for it, though.

A potential hypothesis for a black hole quanta, the minimum possible size of a black hole, is two gravitationally-bound relativistic particles which satisfy all the conditions for a Hawking-radiating black hole. Such a quanta would need to have a total mass lower than the Planck mass, a Schwarzschild radius greater than the Planck length, and decay into two particles which match the peak wavelength predicted by Hawking.

If these particles happen to have a wavelength of about 2.717e-34 m, a few interesting things happen. The total mass-energy is exactly 3/4 of the Planck mass, while the Schwarzschild radius is exactly 1.5 Planck lengths. This can alternatively allow us to model the system as two particles orbiting the photon sphere of a Planck-length black hole, which may allow a quantum tunneling representation of the system's decay. If there's any way to model the decay as a quantum tunneling event, then it will be interesting to see whether it matches the evaporation lifetime predicted by Hawking.

 

[mitchell porter]

sevenperforce, you have managed to touch on quite a few interesting and fundamental topics here, such as how Hawking radiation is supposed to work.

But black holes made of just a few elementary particles, which is one of those interesting and fundamental topics, is also the theme of some fringe physics which, if it shows up here in force, will get the thread closed by the moderators.

That would be unfortunate, because it is a topic that really is of fundamental significance, one that isn't yet figured out, and which has been the subject of work by some of the best physicists; and which also can be approached through some quite simple mathematical examples, so discussion doesn't have to be limited to just verbal pictures.

So, first of all, there is apparently quite a history of physicists seeking to model elementary particles, especially the electron, as black holes. The most recent work in that vein might be by Burinskii, whose papers are on arxiv... But when you examine an electron from this perspective, it can't be a Schwarzschild black hole, it has to be something more exotic.

I have to say that I have never seen the topic of Hawking radiation addressed in such models, and if that's your own original thought, sevenperforce, then I consider that admirable, even if you were just being logical... But Hawking radiation is indeed more complicated than just a particle tunneling across an event horizon. It's an interaction between the black hole and a field mode (e.g. electromagnetic field). Matt Visser has a paper which tries to identify the essence of the process, I don't know if he succeeds.

A place where the quest to understand black holes and elementary particles on a common footing has had some success is string theory, but it's only a limited success so far. String theory has produced microphysical models of black holes as bound states of branes, but only "eternal black holes" which are in an equilibrium - there is no exact string-theory model of black hole evaporation, i.e. progressive depletion of the black hole by Hawking radiation.

Also, there's no stringy model of black holes made of real-world particles. One of the things that distinguishes string theory, as an approach to quantum gravity, is that you can't just study gravity in isolation. You're always dealing with a setup where the graviton is just one string state among many. So it has a certain holism, but it also means that (until string theorists identify some specific calculable framework as The Real World), at best it provides models with a qualitative resemblance to reality.

Nonetheless, I am talking about the string theory approach at some length, because I think it really does represent progress. In particular, at Planck-scale distances, the gravitational force law will deviate from a simple inverse square law because of heavy string states in addition to the graviton, and it should mean that e.g. the usual reasoning about the "black hole electron" is invalid.

 

[mfb]

Nonetheless, I am talking about the string theory approach at some length, because I think it really does represent progress. In particular, at Planck-scale distances, the gravitational force law will deviate from a simple inverse square law because of heavy string states in addition to the graviton, and it should mean that e.g. the usual reasoning about the "black hole electron" is invalid.

Gravity is nonlinear and deviates from the inverse square law, and we see the effects in the solar system already (perihelion shift of Mercury). At the Planck scale, the deviations from Newtonian gravity should be massive.

 

[sevenperforce]

Black holes made of just a few elementary particles, which is one of those interesting and fundamental topics, is also the theme of some fringe physics which, if it shows up here in force, will get the thread closed by the moderators.

That would be unfortunate, because it is a topic that really is of fundamental significance, one that isn't yet figured out, and which has been the subject of work by some of the best physicists; and which also can be approached through some quite simple mathematical examples, so discussion doesn't have to be limited to just verbal pictures.

Well, I'll try to avoid that. :)

So, first of all, there is apparently quite a history of physicists seeking to model elementary particles, especially the electron, as black holes. The most recent work in that vein might be by Burinskii, whose papers are on arxiv... But when you examine an electron from this perspective, it can't be a Schwarzschild black hole, it has to be something more exotic.

I have to say that I have never seen the topic of Hawking radiation addressed in such models, and if that's your own original thought, sevenperforce, then I consider that admirable, even if you were just being logical... But Hawking radiation is indeed more complicated than just a particle tunneling across an event horizon. It's an interaction between the black hole and a field mode (e.g. electromagnetic field).

If elementary particle decay could be reliably modeled as the Hawking evaporation of an exotic black hole...yeah, that would be pretty awesome. Perhaps there is a way to mathematically show that certain field interactions (electromagnetic field for electrons, color field for quarks, etc) changes the shape and minimum mass of a Schwarzschild black hole. That would be far beyond my pay grade, though.

I'm kind of coming at it from the other direction, though, looking at possible ways of getting a quantized black hole. If it's possible to create a model which matches the macroscopic behavior of a black hole (including the characteristics of Hawking radiation) using the collective behavior of black hole quanta which satisfy both Hawking's predictions as well as quantum behavior, then I might be on to something. My first post here was a general question about stellar collapse in an attempt to see whether microscopic black hole creation would be possible in a collapsing star...it seems promising so far.

Hawking derived/discovered/proposed Hawking radiation as a solution to the interaction between a black hole and the field mode, but if quantum-level tunneling events in black hole quanta can collectively match the predicted characteristics of Hawking radiation, that would really be nice. Hence posting this thread to ask about how quantum tunneling is modeled in general.

 

[philton]

Things are becoming more and more interesting now! There have been quit a few possible similarities between the macro-scale black holes and micro-scale particles, from quantum wormhole, Hawking radiation to even quantum waves. And God knows what other similarities they may have in common between the two distinct scales.

Even if the none of the hypotheses could be justified in the end, the quests themselves might stir some inspirational ideas that may eventually shed a light on the emerging new physics.