A particle near an event horizon

[Jeff Rosenbury]

Homework Statement

A sub-atomic particle is near the event horizon of a black hole. Due to the nearby gravitational field, the Ricci Curvature Tensor is changing rapidly. The particle then performs quantum tunneling.

Homework Equations

Which version of spacetime does the tunneling particle use? The two most obvious are following the changing curve, and following the curve fixed at its current (start of tunneling) position.

The Attempt at a Solution

This isn't a homework problem, but rather self study. I'm looking for a qualitative answer rather than an analytic one. I hope this is the right forum. If not could someone direct me?

The proper answer would of course depend on experimental measurement which, lacking a tabletop black hole, would require some subtle experimental work. I have no idea if some experiment has been done. I hope so. If not, it's an area which could yield some deep insights into how our universe works.

Some conundrums: Particles don't have fixed positions due to uncertainty. I'm weak in linear algebra (I can grind equations, but lack deep understanding). The two obvious solutions are not the only possible solutions. Further given the incompatibility of Quantum Theory and General Relativity, it is unlikely the answer will be one of those two. Finally I'm weakish in Quantum Theory and just plain weak in General Relativity.

Some further information: I was considering hyper-massive black holes. I noticed someone claimed (on wikipedia) that the radius of a black hole is linearly proportional to its mass. Thus the density goes down with the square of the mass. As the mass grows much larger the density should approach that of empty space. Also naively the surface gravity should drop as the black hole grows. But of course the actual equations are much more complex. I took this to mean the curvature got more severe near the event horizon, but that's a very shaky assumption. Still, assuming that to be the case, the curvature near the event horizon should get below the tunneling distance of sub-atomic particles (being more or less flat near the particle). At that point the black hole could speculatively no longer interact with our world except in ephemeral ways. Particles would tunnel across the event horizon (most of the time) and pop out the other side (where the black hole's density would be so low it might seem as empty space).

Before this happened though there would be a time when the black hole would eat particles randomly, passing through most, and occasionally gobbling some. This might explain quasars.

Of course this speculation is based at it's inception on many unfounded assumptions. Wikipedia must be right (it does happen sometimes). Particles must tunnel more or less according to the space time curvature of their origin (thus my question). The high curvature region must get smaller, and fast enough to get really small. The inside of a low density black hole must appear as empty space.

I'm likely wrong about most of these assumptions. But I don't know. I do want to know the answer to the original question because it is fundamental to how the universe fits together. I almost posted the rest of it in the science fiction section, but the homework faq wanted me to show my thinking.

 

[wabbit]

I would suggest reading http://en.m.wikipedia.org/wiki/Hawking_radiation, and the article about black holes first - tunnelling at the horizon is one way to describe Hawking radiation.
Note : yes, the proportionality of mass and radius is not just a claim, it's a fundamental property Schwarzschild black holes. And you are correct, gravity near the horizon of a large black hole is weak, and so is the tunnelling.

 

[Jeff Rosenbury]

I would suggest reading http://en.m.wikipedia.org/wiki/Hawking_radiation, and the article about black holes first - tunnelling at the horizon is one way to describe Hawking radiation.
Note : yes, the proportionality of mass and radius is not just a claim, it's a fundamental property Schwarzschild black holes. And you are correct, gravity near the horizon of a large black hole is weak, and so is the tunnelling.

Thank you for replying.

Hawking radiation leaves the event horizon. But then it needs to climb the gravity well which, if I understand it, is a long distance from a local perspective but short from the perspective of someone at the top of the well. This could take a large (infinite?) amount of time since the slope is a much greater distance inside than outside.

I forget the formula for tunneling distance except that it's short and distance tunneled (probably speaking) falls off rapidly with the distance. Basically it's on the order the distance of whatever particle. The covalent bond distance for an electron, the proton/neutron radius for a quark, etc.

Since a large black hole has a low gravity near the event horizon, but a large gravity at the event horizon, it must be very steep (my assumption?). Could a particle theoretically tunnel right past the event horizon and "miss" the black hole? Or would it tunnel down the slope following the stretched coordinate system?

So whether the particle keeps the Rici curvature of its start point and skips the event horizon or gains the Ricci curvature of the spacetime and tunnels a little way toward the event horizon seems an open question to me.

The answer to this question might be important in reconciling quantum mechanics with general relativity.

I can think of a couple of ways of answering it such as looking at decay rates for quasars (assuming quasars are driven by large black holes) or looking at decaying black holes from the large hadron collider. But I don't know if anyone has done so.

 

[wabbit]

Just to clarify, a large black hole has weak gravity at the horizon, as well as near the horizon. There is no Ricci curvature discontinuity, in fact the Ricci curvature is zero everywhere for a black hole.

 

[Jeff Rosenbury]

So how is this black hole different from empty space? It's density is effectively zero. It has no effective gravity. Is there any reason matter would interact with it at all?

 

[wabbit]

It is empty space. The difference with flat empty space is that there is a central mass ("at" the singularity), and this creates curvature - just not Ricci curvature. Still, this curvature is low (and continuous) at the horizon of a large black hole.

 

[PeterDonis]

Due to the nearby gravitational field, the Ricci Curvature Tensor is changing rapidly.

Not necessarily. If the hole is large enough, the curvature at the horizon can be small, and changing slowly.

Which version of spacetime does the tunneling particle use?

I don't understand. There is only one "version" of spacetime. The spacetime geometry of the black hole is what it is; there aren't different "versions" of it.

The density goes down with the square of the mass.

No. A black hole is a vacuum spacetime; its density is zero.

I would recommend taking some time to study GR and how it models black holes; you appear to have a number of misunderstandings. A good online source is Sean Carroll's lecture notes on GR:

http://arxiv.org/abs/gr-qc/9712019

 

[Jeff Rosenbury]

Thank you.