One more time on Hawking radiation

[jnorman]

i have posted a few questions regarding Hawking radiation in the past, regarding the assumption of "negative energy" of the particle which falls into the BH. i have tried to do as much reading as i can on the topic, since i have had a very hard time absorbing the concept of negative energy.

my question originally revolved around the idea that a photon is its own anti-particle, that all particles possesses energy, that the concept of negative energy is therefore moot, and thus HR is based on erroneous assumptions. further reading about HR which details the equations involved (no, i can't really follow it all the way) demonstrates that the equations require that the particle falling into the BH MUST have negative energy to provide for energy conservation since the particle pair was created from vacuum fluctuations. i can follow that.

however, when the particle pair is created, both the particle and anti-particle have positive energy. and while i have read comments to the effect that the negative energy state of the particle which falls in is in relation to infinity, that concept is a bit difficult. ie, at what point does the energy state of the particle which falls in change from a positive energy state to a negative energy state? well, apparently it doesn't change - the particle falling in remains positive energy from a local perspective. it is only recognized as negative energy from a universal perspective, ie, from the overall energy balance within the entire universe (have i stated that correctly?).

my current question is: why do we require this kind of total energy balance for BH's, when we accept the idea that for some reason in the early universe, the number of particles was no longer exactly balanced by the number of anti-particles (ie, where is all the antimatter?). if this imbalance (and the process which resulted in that imbalance) is part of our reality, for whatever reason, why can we not assume that something similar may explain that HR may not be correct?

sorry if this is confused - i am an idiot just trying to learn little by little. thanks.

 

[PAllen]

There are different ways to understand Hawking radiation. All must involve negative energy flux into the black hole, but that can simply be taken as equivalent to flux of positive energy out of the black hole. Here is an explanations that critiques the popular ones:

http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/hawking.html

and here is a paper that (while being a critical review) really clearly explains the various derivations and explanations of Hawking radiations:

http://xxx.lanl.gov/abs/gr-qc/0304042

 

[John232]

Hawking radiation requires the negative particle to fall in the black hole solely on energy conservation? Seems like that would be a big problem if white holes were shown to exist...

 

[jarlostensen]

It is my understanding that, since the wavelength of HR is comparable to the mass of the black hole itself, the whole idea of "particles" being created breaks down anywhere but at infinity (in the |out> state, with the |in> state on past-infinity being vacuum) - i.e. no particle detector will be able to detect distinct particles locally, near the horizon for example. Instead you have to go with the stress-energy-momentum tensor (which is local-ish) and determine that there is a energy flux.

So to speak of a "negative energy particle" falling into the black hole becomes a little bit moot, although I would agree that it is somewhat unsatisfactory to just dismiss it.
Only that I think you need to abolish the whole idea of particles (including photons) being localisable entities. Recall that Hawking and Bekenstein's original work considers the BH as a global object (and hence also the radiation effects.)

 

[IsometricPion]

It is my understanding that, since the wavelength of HR is comparable to the mass of the black hole itself

I guess you meant the wavelength of the average photon produced via hawking radiation is about the same as the radius of the black hole.

Seems like that would be a big problem if white holes were shown to exist

The wikipedia page on white holes says:

Since a thermal equilibrium state is time reversal invariant, Stephen Hawking argued that the time reverse of a black hole in thermal equilibrium is again a black hole in thermal equilibrium.[3] This implies that black holes and white holes are the same object. The Hawking radiation from an ordinary black hole is then identified with the white hole emission. Hawking's semi-classical argument is reproduced in a quantum mechanical AdS/CFT treatment,[4] where a black hole in anti-de Sitter space is described by a thermal gas in a gauge theory, whose time reversal is the same as itself.

the number of particles was no longer exactly balanced by the number of anti-particles (ie, where is all the antimatter?). if this imbalance (and the process which resulted in that imbalance) is part of our reality, for whatever reason, why can we not assume that something similar may explain that HR may not be correct?

The production of more matter than antimatter is thought to have been a result of charge symmetry violation, there are processes (that have been observed in particle accelerators) in which the production of matter is favored over its charge reversed complement. So, while magnitude of the charge symmetry violation in the early universe has not yet been explained (in a theory that has been experimentally tested), a similar effect has been observed. On the other hand, violation of conservation of energy has never been observed (in order for that to occur energy would have to be an ill-defined property, in the sense that some other quantity could be transformed into energy that was not initially what is considered to be energy or vice versa).

 

[Naty1]

"...why do we require this kind of total energy balance for BH's"

let's start there...it is NOT required...there is no "energy balance" required for a black hole...

But energy conservation IS required for the particle antiparticle pair production...if we see a real particle with positive energy on our side of the event horizon, then the other hidden particle of the pair must also be "real" and have negative energy. Total energy from the pair production must be zero according the conservation of energy.

In general black holes in our universe gain energy as they accrete mass or energy...that is, as mass or, say, radiation fall through the horizon...then they grow bigger..

Right now, no black holes are emitting observable Hawking radiation because they are colder than the sourrounding universe. It's a thermal radiation. At the end of the universe when things are REALLY cold they are expected to radiate, but very,very slowly...and to last on the order of maybe ten times the age of the universe...depending on size...

Baez: is right about the particle antiparticle pair production not following directly from the math...According to Roger Penrose in THE ROAD TO REALITY page 836 such a view was Hawking "intuitive" explanation..Penrse explains (and I do not understand this) that the production of real particles with positive and negative energy can be explained in terms the Killing vector K ...

 

[jarlostensen]

"...why do we require this kind of total energy balance for BH's"
Baez: is right about the particle antiparticle pair production not following directly from the math...According to Roger Penrose in THE ROAD TO REALITY page 836 such a view was Hawking "intuitive" explanation..Penrse explains (and I do not understand this) that the production of real particles with positive and negative energy can be explained in terms the Killing vector K ...

If my memory serves me right this has to do with the fact that in order to have particle creation in pairs of particle/anti-particle you need both positive and negative frequency solutions.
The negative frequency ones provide the negative energy particle.
Furthermore, to be able to define + and - frequency consistently you need a timelike vector field that is an isometry of the metric, i.e. a timelike Killing vector field.

However, in general space times, and even with a timelike Killing field, you can still find many different solutions to the field equations which, when you transform between them, causes a mix of + and - frequency modes to show up.

So, what looks like a perfectly well behaved vacuum to one set of solutions, has a mix of + and - modes in another; you have particle creation.

The transformations between these different bases of solutions are called http://en.wikipedia.org/wiki/Bogoliubov_transformation" transformations.

This mismatch between "no particles" in one vacuum state and another is how Hawking radiation shows up; In that case one looks at the vacuum in the remote past and compares it to the vacuum in the remote future with the collapse and formation of the BH happening in a (relatively short) interval in between, i.e. space is flat at both ends. Because of the fact that the geometry has changed in between, the in-vacuum and out-vacuum no longer agree and a relative mixing of + and - frequencies from the in-vacuum occurs which, in turn, is observed as particles by an observer who believes the out-vacuum is the "true" vacuum.

I think that the Killing vector field is just needed so that + and - frequencies make sense; i.e. so that we can define + and - frequencies terms the Lie derivative wrt to the Killing vector field K;

$$\Im_{K}u_{j}=-i \omega u_{j}$$

where

$$\omega>0$$

for a + frequency mode.