# Hawking Radiation and Shrinking Black Holes

1. May 5, 2013

### dsmikk

I was wondering whether anyone could provide an explanation as to why when two particles are created near the event horizon, only the negative energy particle is captured by the black hole, and the positive particle travels outward and is seen as Hawking Radiation.

Thanks!

2. May 5, 2013

### tom.stoer

The problem is that the explanation regarding pair-production at the horizon is missleading. It goes back to Hawking himself; he invented this explanation directly after his brilliant mathematical work, and from that time one one better stops reading his paper (and all following popular explanations). There is no pair-production at the horizon!!

The problem is the definition of vacuum. This requires to define positive and negative frequencies. But there is no unique and globally valid definition for positive and negative frequencies in curved spacetime and for distorted (= non-plane) waves. So one observer A defined vacuum as "no incoming waves with positive frequency from the infinite past". Another observer B far away from the black hole would define define vacuum as "no outgoing waves with positive frequency to the infinite future". In flat spacetime these two definitions are equivalent, but in curved spacetime the vacuum as defined by A is observed as thermal radiation by B (not as vacuum as defined by B).

This is a global effect for which no local expression can be given.
This is not pair creation (in the ordinary sense) b/c this would require an interaction.
And it's definitly not pair creation at the horizon!

3. May 5, 2013

### Naty1

This complements tom.stoer's post: [from earlier discussions on this subject]

Bapowell:
The idea of the Hawking effect being the result of virtual particles popping into existence (and, yes, this could be e+/e- pairs popping out of the vacuum) near the event horizon, one falling in and one flying off, is more an illustration for the sake of popular science than an accurate description of the process......

In the original paper Hawking writes:

4. May 6, 2013

### Demystifier

That is true!

Even though it is a global effect, a sort of a local description can still be given:
http://arxiv.org/abs/gr-qc/0111029
http://arxiv.org/abs/hep-th/0202204
http://arxiv.org/abs/hep-th/0205022

5. May 6, 2013

### sheaf

I fully accept that the best explanation of Hawking radiation is obtained by comparison of the vacua/Bogoliubov transformation, but there seems to have been a small industry which has grown up over the last decade or so concerned with deriving/analysing Hawking radiation in terms of tunneling of one member of a created pair through the "potential barrier". This sounds like an attempt to flesh out Hawking's heuristics.

One of the earliest references I'm aware of that takes this approach is here.

I was curious if anyone could shed some light on

(i) what the motivation for doing this is

(ii) what the mechanism is supposed to be for the virtual pair creation. I'm aware of a sort of virtual pair creation process in (say) QED, where you have a bubble diagram with an electron/positron and a photon - just a contribution to a vacuum-vacuum amplitude. But in the Hawking radiation treatments, they never mention these QED diagrams (maybe not so surprising since tunneling isn't perturbative), but talk of these spontaneously occuring virtual pairs seems a bit ad hoc.

6. May 6, 2013

### tom.stoer

for some reason my arxiv access is blocked currently

the motivation was - afaik - to find a mathematical approach which confirms Hawkings guesswork regarding tunneling

Hawking's original approach is based on definition of vacuum and does not involve any pair creation in the sense of QFT, simply b/c the definition of vacuum works in free field theory w/o any interaction and is zeroth order, whereas bubble diagrams contain two vertices and are at least second order.

So my expectation is that interaction terms (for massless fields) will add perturbative corrections to Hawking's zeroth order result.

7. May 6, 2013

### Naty1

from Wilczek!!.. THANKS will read that after this post.....

I prefer to keep an open mind....see below for some reasons....but I don't think that was
involved in Hawkings work. [edit: I see stoer posted while I was composing and he confirms this.]

In brief, event horizons produce particles!

If you accelerate [say, Unruh effect] or spacetime accelerates [say, inflationary expansion] particles are produced...because event horizons result. [Wilczak appears to say dynamic [evolving] spacetime produces particles...same thing.] There is no unique, no frame independent/invariant vacuum observation....you may see particles I don't. In fact even different inertial observers have different observations. Even the Hubble sphere has some of these characteristics. Closely related is the local versus global ambiguity in curved spacetime....there is no objective particle count, no GR global energy definition .... ..in short ambiguity, some subtlety, exists which is utterly fascinating.

Some views:

http://www.weburbia.com/physics/hawking.html

For example, Leonard Susskind describes such 'pair production' in his book the THE BLACK HOLE WAR from the perspective of string theory. He explains how lengthy strings would be 'smeared' on a stretched horizon, via the Holographic Principle, about a Planck Length outside the event horizon. vacuum energy fluctuations here and there would cause small portions of these strings to emerge and break loose...viola! Hawking radiation.

Also, From Leonard Susskind, THE BLACK HOLE WAR, almost an exact quote with some omissions for brevity.. Hawking radiation.

Emission process
Code (Text):
Hawking radiation is required by the Unruh effect and the equivalence principle applied to black hole horizons. Close to the event horizon of a black hole, a local observer must accelerate to keep from falling in. An accelerating observer sees a thermal bath of particles that pop out of the local acceleration horizon, turn around, and free-fall back in. The condition of local thermal equilibrium implies that the consistent extension of this local thermal bath has a finite temperature at infinity, which implies that some of these particles emitted by the horizon are not reabsorbed and become outgoing Hawking radiation
http://en.wikipedia.org/wiki/Hawking...ission_process [Broken]

[Note the similarity to bapowell's post and local/global difficulties from stoer....]

and I saved this from yet another discussion in these forums:

According to the book Quantum Fields in Curved Space by Birrell and Davies, pages 268-269,

Last edited by a moderator: May 6, 2017
8. May 6, 2013

### sheaf

@Tom Stoer:

Yes indeed, you're right, it's a key point that the usual calculations are done for a free field theory, so we wouldn't expect interaction vertices to play any role.

@Naty1: thanks for the quotes from Susskind

I struggle with phrases such as the one I highlighted. I don't see how vacuum fluctuations can cause anything, since they're not dynamic things. They're just uncertainties you get when you measure some quantities in the given state. Presumably in this case, the relevant measurement which "sees" the fluctuations is done by the particle detector.

9. May 6, 2013

### Naty1

well, I think they ARE 'dynamic'......zero point fluctuations ala Heinsenberg uncertainty....
if not, how did time,space,energy et al emerge from nothing to form a big bang....

Do you have a source that claims vacuums are not dynamic?? Do you have a special definition for 'dynamic'?

Last edited by a moderator: May 6, 2013
10. May 6, 2013

### Naty1

the following pretty much sums up my perspective that the vacuum IS pretty dynamic...in fact I wonder if it may not be one of the most 'dynamic' things in the universe...after the Big Bang, Quasars and maybe a few other things...

http://en.wikipedia.org/wiki/Vacuum_energy

In fact, isn't it SO dynamic that renormalization must be utilized to 'cancel infinities'.....

Last edited: May 6, 2013
11. May 6, 2013

### sheaf

I just meant that in a non interacting field theory (which applies to our HR scenario), the vacuum has no particles in it and stays that way. Hit the vacuum with the free Hamiltonian and ..... nothing happens. So it has no meaning to say that it's bubbling with activity, there's no time variability, the vacuum state stays the vacuum state. Consequently, in the tunneling treatments, when they say "a virtual pair pops into existence and one of the partners tunnels through the horizon" I'm not quite sure exactly what's meant. It's tolerable as a heuristic description, but Parrikh & Wilczek etc are trying to give a more rigorous treatment.

I am, however, happy that if I observe that vacuum state from the point of view of an asymptotic observer, I see a thermal state.

I don't really know how the universe started

Last edited by a moderator: May 6, 2013
12. May 6, 2013

### Naty1

that description has received severe criticism in these forums.

13. May 6, 2013

### Bill_K

sheaf, I agree one hundred percent with what you're saying. It's unfortunate that time-dependent terminology has become so second-nature that it's often used even when totally inappropriate. Introductory quantum books, for example, use the term "fluctuation" as if uncertainty implied time-variability. Given an observable A, we define its "fluctuation" as (ΔA)2 = <(A - <A>)2>. "When ΔA = 0, the fluctuations are absent and one may assert with certainty that A takes on a well-defined value."

No, in fact to turn it around, I can't help wondering what your definition is. Dynamic means changing with time, as opposed to static. Or at least steadily changing, as in dynamic equilibrium. It's quite picturesque to imagine the vacuum state as a "boiling cauldron of activity", but it's not - it's simply a static superposition of states having different particle numbers.

Now I know we're using different language. Renormalization has nothing even remotely to do with being dynamic.

Last edited by a moderator: May 6, 2013
14. May 8, 2013

### Naty1

To complete the record: Here is one additional perspective...

Carlo Rovelli:

http://arxiv.org/abs/gr-qc/0604045
Unfinished revolution