Hawking Radiation and Shrinking Black Holes

[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.

I have read numerous posts about this phenomenon but none seemed too helpful in my understanding.

Thanks!

 

[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 definitely not pair creation at the horizon!

 

[Naty1]

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

Bapowell:

Hawking's original derivation made no mention of virtual particles slipping over event horizons; he instead examined the asymptotic transformation of a field fluctuation brought in from far away, past a black hole, and back out to very far away. If the initial field fluctuation far away was in its vacuum state, the final fluctuation far away generally is not (as a result of its gravitational interaction with the black hole). John Baez has a nice write up about this:

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

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:

"One might picture this negative energy flux in the following way. Just
outside the event horizon there will be virtual pairs of particles, one with negative
energy and one with positive energy. The negative particle is in a region which
is classically forbidden but it can tunnel through the event horizon to the region
inside the black hole where the Killing vector which represents time translations
is spacelike. In this region the particle can exist as a real particle with a timelike
momentum vector even though its energy relative to infinity as measured by the
time translation Killing vector is negative. The other particle of the pair, having
a positive energy, can escape to infinity where it constitutes a part of the thermal
emission described above. The probability of the negative energy particle tunnelling
through the horizon is governed by the surface gravity K since this quantity
measures the gradient of the magnitude of the Killing vector or, in other words,
how fast the Killing vector is becoming spacelike. Instead of thinking of negative
energy particles tunnelling through the horizon in the positive sense of time one
could regard them as positive energy particles crossing the horizon on pastdirected
world-lines and then being scattered on to future-directed world-lines by
the gravitational field. It should be emphasized that these pictures of the mechanism
responsible for the thermal emission and area decrease are heuristic only
and should not be taken too literally."

 

[Demystifier]

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!

That is true!

This is a global effect for which no local expression can be given.

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

 

[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 occurring virtual pairs seems a bit ad hoc.

 

[tom.stoer]

for some reason my arxiv access is blocked currently

the motivation was - afaik - to find a mathematical approach which confirms Hawking 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.

 

[Naty1]

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

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

There is no pair-production at the horizon!

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

...what the motivation for doing this is... what the mechanism is supposed to be for the virtual pair creation.

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:

...If we are using really different co-ordinate systems, we'll have really different notions of time, hence really different notions of energy - since energy is defined in quantum theory to be the operator H such that time evolution is given by exp(-itH).

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

If we could hang a thermometer on a long string and dangle it just outside the horizon of a black hole, it would register incredibly high perhaps approaching infinite temperature. On the other hand a free falling observer falling towards the black hole registers no such increase in temperature and passes harmlessly through the mathematical (theoretical) horizon without any immediate effect except increasing gravitational acceleration and tidal effects...So thermal and quantum radiation become two sides of the same coin near a horizon.

Emission process

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

[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,

These consideration resolve an apparent paradox concerning the Hawking effect. The proper time for a freely-falling observer to reach the event horizon is finite, yet the free-fall time as measured at infinity is infinite. Ignoring back-reaction, the black hole will emit an infinite amount of radiation during the time that the falling observer is seen, from a distance to reach the event horizon. Hence it would appear that, in the falling frame, the observer should encounter an infinite amount of radiation in a finite time, and so be destroyed. On the other hand, the event horizon is a global construct, and has no local significance, so it is absurd too conclude that it acts as physical barrier to the falling observer.

The paradox is resolved when a careful distinction is made between particle number and energy density. When the observer approaches the horizon, the notion of a well-defined particle number loses its meaning at the wavelengths of interest in the Hawking radiation; the observer is 'inside' the particles. We need not, therefore, worry about the observer encountering an infinite number of particles. On the other hand, energy does have a local significance. In this case, however, although the Hawking flux does diverge as the horizon is approached, so does the static vacuum polarization, and the latter is negative. The falling observer cannot distinguish operationally between the energy flux due to oncoming Hawking radiation and that due to the fact that he is sweeping through the cloud of vacuum polarization. The net result is to cancel the divergence on the event horizon, and yield a finite result, ...

 

[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

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

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

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.

 

[Naty1]

I don't see how vacuum fluctuations can cause anything, since they're not dynamic things.

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'?

 

[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

Quantum field theory states that all fundamental fields, such as the electromagnetic field, must be quantized at each and every point in space. A field in physics may be envisioned as if space were filled with interconnected vibrating balls and springs, and the strength of the field were like the displacement of a ball from its rest position. The theory requires "vibrations" in, or more accurately changes in the strength of, such a field to propagate as per the appropriate wave equation for the particular field in question. The second quantization of quantum field theory requires that each such ball-spring combination be quantized, that is, that the strength of the field be quantized at each point in space. Canonically, if the field at each point in space is a simple harmonic oscillator, its quantization places a quantum harmonic oscillator at each point. Excitations of the field correspond to the elementary particles of particle physics.

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

 

[sheaf]

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'?

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

 

[Naty1]

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.

that description has received severe criticism in these forums.

 

[Bill_K]

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.

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.

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."

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

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.

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

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

 

[Naty1]

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

Carlo Rovelli:

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

Roughly speaking, we learn from GR that spacetime is a dynamical field and we learn from QM that all dynamical field are quantized. A quantum field has a granular structure, and a probabilistic dynamics, that allows quantum superposition of different states. Therefore at small scales we might expect a “quantum spacetime” formed by “quanta of space” evolving probabilistically, and allowing “quantum superposition of spaces”. The problem of quantum gravity is to give a precise mathematical and physical meaning to this vague notion of “quantum spacetime”.