# Hawking radiation and BH evaporation time

1. Mar 20, 2005

### nightcleaner

My understanding is that black holes evaporate by Hawking radiation. Hawking radiation, by my reading, occurs when a pair of virtual particles emerges very close to but just outside the event horizon of the black hole. When they emerge (from quantum foam, not from the BH) one of the pair may occur closer to the event horizon, the other further from it. In some cases, this difference is just right to cause one of the virtual particles to enter the event horizon, while the other escapes into our universe.

Normally virtual particles immediately anihiliate each other, but in this case, as they are seperated, they do not immediately anihiliate. In fact, the particle which is free to enter our universe is really no different than any other particle in this regard, and may expect to have the same half-life other particles of its kind enjoy. This would be a cause for concern, since it appears to violate the conservation of mass, since a "new" particle is created and enters our universe.

To explain this apparent violation of conservation of mass, one only has to realize that the particle which enters our universe is matched by an anti-particle which goes into the closed and very small region of the BH. Because this region is closed and very small, particles entering it soon encounter the other particles that are trapped in there. On average, any particle resulting from the stripping of a virtual pair will soon encounter its anti-particle which has been stripped from another pair, and these two will anihiliate. When they do, conservation of mass is restored, since the infall anihiliation accounts for two particles radiated.

Last night while boiling out the deep fryer I was thinking about this, from the point of view of infall. Now it happens that the information ie mass, contained by a BH is proportional to its surface area, not its volume as one might assume. This is because we cannot know what goes on inside a black hole, but we can have an idea anyway of conditions on its surface. The seeming contradiction here is due to the distortion of time near an event horizon.

If we hover outside the event horizon and lob rocks into it, we could watch the rocks as they fall in. But we don't see them enter the event horizon. Instead, they seem to go slower and slower as approach the horizon, and at the same time they grow dimmer and dimmer, and their escaping photons become weaker and weaker, the energy waves longer and longer, until after a while we do not have any quanta from the rocks at all. But during the time that we can watch them, they do not enter the horizon at all, but seem to us to slow down and stop right at the horizon.

So, from our perspective outside the horizon, everything that goes into the event horizon seems to just hang there until we can't see it any more. This is why we can surmize that all the information that goes into a black hole is right there on the surface, and, as far as we are concerned, does not proceed any deeper.

Another way to look at this is to consider that time, viewed from the outside, seems to stop for the infalling object. The infalling object wouldn't see it that way, but from the outside, looking in, that is what we see. In a sense, as far as we are concerned, the infalling object becomes eternal and no longer changes in time as we do. It no longer shows any indication of ageing, moving or reacting.

So that's the puzzle. If, as far as we are concerned, the object as it infalls attains an eternal state, then it cannot, as far as we are concerned, react with its antiparticle on the inside to produce the required mass loss to counter the mass gain resulting from the radiated partner.

Our universe, then, must experience a net gain of mass and energy, upsetting the mass conservation applecart. We now have to look for a sink somewhere else where mass is lost from the universe to restore our precious conservation.

But this is not what concerns me immediately. My immediate concern has to do with the supposed loss of mass of the Black Hole to evaporation, which seems to depend upon the anihiliation of particles enclosed within the horizon. This loss, when calculated, yields an evaporation time for small black holes which is pretty fast, say about 10^-23 seconds. A small black hole doesn't have enough time, at that rate, to interact with much else in the universe. It is not likely, for example, that a small black hole would suck up the earth's atmosphere, or fall to the center of the earth and give another revival to the hollow earth theories. It just can't last long enough to get the matter it needs to stay alive.

However, the analysis above may change that scenario back again to threat status. If my reading is correct, the balanceing act of particle extinction within the black hole does not happen until sometime in the extremely distant future. If that is so, the black hole will not be seen to evaporate, but will continue to have opportunities to encounter some mighty tastey bananas at our expense for a long, long time. The first black hole we create on the planet will indeed have a chance to swallow us all up.

I should very much like to hear an argument that counters this unfortunate scenario.

Thanks,

nc

2. Mar 20, 2005

### Chronos

The evaporation time is given by:
$$t_e_v = \frac{5120\pi G^2M\odot ^3}{\hbar c^4}$$

While there is some question as to whether this is the correct way to do the calculation, there is considerably less doubt that black holes do radiate. A rather serious violation of thermodynamics otherwise arises.

There is also room to question whether of not microscopic black holes can be created. If the lower size limit is the planck mass [which would give it an event horizon of one planck length], no accelerator imaginable will have anywhere near the power required to create that much mass. Even the most powerful cosmic rays ever detected are many OOM short of that energy level.

Last, but not least, cosmic ray energies are far above even those that will be possible with the LHC. If the LHC can make them, cosmic rays already have. Since the earth is still here, and we are pretty sure it's not hollow , we probably don't have much to worry about.

3. Mar 20, 2005

### marcus

often poetry is exactly not what is wanted so excuse me in advance, Richard, but it just occurred to me that your writing or else your intelligence sometimes "shines by its uncertainty" and in a poetical sense that is also what a black hole does

it shines (I shall suppose, dont really know and maybe no one does) because the location of the boundary is itself uncertain and a particle can "tunnel" out at the very moment that the boundary itself shrinks
the whole process being quantum-fuzzy.

remember that the RADIUS horizon (uncertain) of the horizon is proportional to the MASS (uncertain) of the hole. famous formula 'bout that.

the location of the horizon (which is a mathematical construct, not a physical shell) is just uncertain enough that a particle can appear outside it and it can simultaneously shrink (proving that the hole is less massive than you thought) thus balancing the books! the radiation is permitted by the fuzziness.

John Baez said on SPR that he didnt really understand the popular explanation of the pair of virtual particles where one falls in and the other gets away. He said that when experts give a seminar on hawking radiation they dont use that explanation. there is another quite different treatment of hawking radiation that however will not work in casual conversation or in the media. So I have been keeping a lookout for alternative explanations of hawkingradiation (since the virtual pair one doesnt really seem to work) and I came across a "tunneling" one by someone at princeton published in 2004. I gather there is not yet one right way to discuss it. After all hawking analysis is "semiclassical" meaning a provisional stopgap analysis combining some classical and some quantum steps that people use because they dont yet have a quantum theory of spacetime. And hawkingradiation is definitely a place you absolutely need QM combined with GR to analyze it!
So the hawkingradiation denoument has to wait for a quantum theory of gravity! and in the meantime take people's stories about it with grano salis.

4. Mar 20, 2005

### marcus

this looks correct and prompts me to show how nice natural units are
because they make the calculation so easy

you are talking about a solar mass hole. solarmass is what M with a circledot means. so i will follow suit.
the solar mass is some 4E38 mass units
and a year is E50 time units

working this way to find the evaporation time all you need to do is
CUBE THE MASS AND MULTIPLY BY 80/pi.

So you just cube 4E38 which gives 64E114
and multiply by 80/pi which gives 1630E114 = 1.6E117
and interpreting that in years (E50 time units) you get
1.6E67 years.

it really streamlines the calculation, and it is algebraically precisely equivalent.

let's do it for an Earth mass black hole too. Earth mass is about E33 natural. So cube that and get E99
multiply by 80/pi and get 25E99

convert to years and get 25E49 years, or 2.5E50 years.

could be more accurate and use 1.38E33 for earth mass, but no big deal if only want an order of magnitude handle on the evaporation time.

really like these units. sweet to use. and I see them (or close relatives) used increasingly in the QG technical literature.

5. Mar 21, 2005

### nightcleaner

Hi Marcus and Chronos

Thanks very much for the sharp information. I really like the quantum tunneling and uncertain horizon idea much better that the strange accounting involved in split virtual pairs. Honestly I don't know where the split pair idea comes from and only vaguely even remember hearing about it. Maybe it does happen sometimes but it does not seem then to be the driving reason behind the idea of evaporating black holes.

So Chronos you mentioned that there are thermodynamic arguments. Where might one start to look for those?

I think the idea of BH surface being proportional to mass comes from information theory somehow, flip-flopping qubits, I suppose SetA1 might say. Of course the surface of the black hole and the surface of the event horizon are not the same thing. Presumably the black hole singularity occurs in a much smaller volume that its event horizon shell. The event horizon merely marks the region of space where the gravity of the BH overcomes the power of light to escape it.

So the radius of the event horizon and the mass of the hole itself are uncertainties, making a mass to surface area ratio rather meaningless. And then the time and space distortions due to the singularity make our notions of black hole interior geometry rather shadowy. Well I might have known that, since horizons in general are not easy to post as a boundary.

Sun's up and I have to sleep. more later.

Be well,

Richard

6. Mar 21, 2005

### marcus

hello Richard,
pardon me if Ive said something unclearly and misled you.
when I was talking about the radius of a BH, I meant the Schw. radius which is the radius of the event horizon

when I was talking about the surface area, I meant the area of the event horizon.

when I was saying the radius is proportional to the mass I meant

I definitely do not understand the theoretical mechanism that is supposed to produce Hawking radiation and, as I indicated, I have found that there is even some difference among people I consider experts in how they discuss the details. But they all seem to agree on the overall outcome which is the temperature and the fact that there is radiation. So I take their general conclusion on faith.

What I told you about was an alternative explanation of how hawking radiation arises (which I do not think is a standard explanation, just one somebody proposed in a paper last year) where IIRC you focus on the fact that the mass M is uncertain
so the Schw. radius, 2GM/c2, and therefore actual location of the event horizon, is uncertain
and there is a possibility that a particle can tunnel out while the mass, the radius, and the event horizon simultaneously contract just a tiny bit

EDIT: WELL I WENT AND FOUND THE PAPER and it does seem to be nicely written. the author's name is Maulik Parikh and it won first prize in the 2004 Gravity essay contest.
http://arxiv.org/abs/hep-th/0405160
A Secret Tunnel Through The Horizon
7 pages
Gen.Rel.Grav. 36 (2004) 2419-2422

"Hawking radiation is often intuitively visualized as particles that have tunneled across the horizon. Yet, at first sight, it is not apparent where the barrier is. Here I show that the barrier depends on the tunneling particle itself. The key is to implement energy conservation, so that the black hole contracts during the process of radiation. A direct consequence is that the radiation spectrum cannot be strictly thermal. The correction to the thermal spectrum is of precisely the form that one would expect from an underlying unitary quantum theory. This may have profound implications for the black hole information puzzle."

I had forgotten where Parikh is, turns out he is at Columbia. and he has co-authored with Frank Wilczek (whom I much admire from way before he got the Nobel prize). Also the essay goes unusually light on the math and tries to explain some in mental pictures. See what you think

Last edited: Mar 21, 2005
7. Mar 21, 2005

### Chronos

Oh well, the worst I can do is further confuse matters, Richard. The idea of Hawking radiation came from a rather simple notion. If you take an object with lots of entropy, say a cloud of uniformly diffuse gas and toss it into a black hole, what happens to the entropy? If it merely vanishes, a rather egregious violation of the 2nd law occurs - the total entropy of the universe decreases!. The obvious way out of this mess is to conclude the black hole contains the missing entropy. This same conclusion has since been reached using other approaches. We are also pretty confident that any object with entropy has a temperature. The method and accuracy of calculating that temperature is rather messy, but the important matter is the discovery that black holes radiate. We would, of course, love to have an itty bitty one and measure its actual temperature. That would give us an important clue as to which theoretical approach is correct.

There has been a lot of discussion whether black hole entropy resides outside or inside the event horizon [or both]. It's hard to say. There are good arguments on both sides. For an entertaining and technically light discussion of this see:

Black hole entropy: inside or out?
Ted Jacobson, Donald Marolf, and Carlo Rovelli
http://www.arxiv.org/abs/hep-th/0501103

The tunneling concept is also interesting. Yan has been among the more active researchers. Here is one of his recent entries:

http://www.arxiv.org/abs/gr-qc/0406017:
Quantum horizon and thermodynamics of black hole

8. Mar 22, 2005

### nightcleaner

Hi Chronos and thanks for this reply. I finally have some time and a clear enough head to consider it.

Did you read about the night sky survey which recorded an event, over Nevada, if memory serves, in which a proton moving at near light speed was observed skimming our atmosphere? The article I read (sorry, did not think to save reference, but will go look for it) said the particle had momentum equal to a golf ball in mid-flight.

Now a golf ball, even at rest, has a mass equal to lots of fleas. In your opinion, would this be an example of a cosmic ray event with sufficient energy to whip up a small black hole?

I very much like your argument that if such cosmic rays could make small black holes they already would have, and if such small black holes can live long enough to swallow the earth, they already would have, and since we are still here, we may presume no such threat exists. Thanks, I am much relieved.

However, I still have to try to convince Goat boy's gramma. She is a rather superstitious person who still spits at gypsies, and who is not convinced the Earth is not hollow. In fact, I already know what she will say. She will say that in the many worlds interpretation, black-hole-swallows-Earth events have already happened countless times, but we continue to exist because we inhabit the region of the multiverse in which these events have always been, however narrowly, averted. So by the hair on the skin of her tooth, we hang.

Gramma's argument may seem ridiculuous to some who read PF, but perhaps it is really no more riduculous than some of the anthropic principle appeals made by people here who claim to care if they are taken seriously. Alpha is what Alpha is because that's what Alpha has to be for us to be here to see it. Hmmmmmm. I'm sorry. That just doesn't seem to work for me. Might as well just say G-d made it that way and be done with thinking about it.

Ahh, Spring. Now perhaps my black hole doldrums will melt like the sparkle of ice on Lake Superior.

Be well,

Richard

quick search of Google turned up 75 entries for "golf ball" Nevada Proton. More reading! And thanks also for the material in your last post. I will have to digest for a while. R

Last edited by a moderator: Mar 22, 2005
9. Mar 22, 2005

### nightcleaner

Thanks Marcus. I have added this, along with so many other excellent references you have provided, to my reading list.

Richard

10. Mar 22, 2005

### Billy T

I just want to note that, as some of you alredy know, a very related discussion is taking place in the general astronomy/cosmology section thread "First Stars - how big - Black Holes Now?" which I started and also state that if other desire, I would think it wise, if possible, to merge them in to one. Stellar core BH holes certainly are not much of a source of Hawkings radiation (too big) but that thread has drifted to be focused on this radiation now. Perhaps only the recent posts should move.

11. Mar 22, 2005

### Chronos

I remember reading about that as well. Just your typical ultra-high energy cosmic ray. These sizzling furies weigh in at as much as 10^22 electron volts - and that in itself is a real head scratcher. There is nothing anywhere near energetic enough in our neighborhood to generate particles this energetic, and current theory says particles from sources distant enough to produce such critters can't make it this far packing that kind of punch. This is called the GZK paradox. Here is some good stuff about that:
http://www.physics.rutgers.edu/hex/HIRES.html
I like the juiced neutrino explanation myself, but that's just a matter of taste. Doubly special relativity is also an interesting candidate.

But even these impressive energies fall far short of a Planck mass - 10^28 electron volts - and there is good reason to suspect black holes smaller than this cannot form. See here for that discussion:
http://zebu.uoregon.edu/~imamura/209/apr5/planck.html