# Can't figure out how hawking radiation fixes thermodynamic problem

1. May 10, 2013

### Stonius

Hi,

I've tried several times to get my head around this, but every way I look at it Hawking Radiation doesn't seem to fix the thermodynamic issues it was supposed to solve. People who are a lot smarter than me seem to believe it's real, so can someone please point out where I'm going wrong?

From Wikipedia;
My understanding is that antiparticles share the same mass as their matter counterparts (ie, not negative mass). So when one of those particles fall into the black hole it thereby increases the total mass, instead of diminishing it. In which case there is no eventual evaporation of the black hole because the vacuum actually helps it grow, rather than diminish.

On the other hand, if the mass of the trapped particle somehow doesn't contribute to the mass of the black hole, then the universe now has one more particle in it that it didn't have before, violating the first law of thermodynamics by creating energy from nothing in a closed system.

I assume Hawking Radiation must somehow account for the trapped particle not adding mass to the black hole while the radiated particle does have mass which is somehow drawn from the inside of the event horizon which it doesn't have access to since, by definition the radiated particle is on the wrong side of the event horizon and cannot access any information from inside it.

Thanks for any answers, and I apologise for my proliferation of posts lately. This is the last of my three burning questions I really wanted to ask.

Markus

2. May 10, 2013

### Bill_K

Stonius, Virtual particles are not restricted to the mass-shell condition that real particles must obey, namely E2 = p2c2 + m2c4. Virtual particles have the usual rest mass, but not the usual energy and momentum.

On the other hand, energy must always be conserved, and when a virtual particle pair is created, their energies must add up to zero.

In this model of Hawking radiation, the positive energy particle goes off to infinity, while the negative energy particle falls into the hole, decreasing its mass. (Note it doesn't matter which one is the antiparticle. The antiparticle could just as well be the one that escapes.)

3. May 10, 2013

### Charles Wilson

When Dirac shook the world with his equation for the electron, he discovered that there were two solutions, one for the "standard" electron but also a solution that implied an "Opposite Electron" - a "Negative Energy Electron", if you will.

No such animal had ever been seen and to compress the story a bit too much, it could not have been a proton, though that was the only candidate at the time.

"[[Dirac]]...was focusing mainly on solving the mystery of the negative energy electrons. Why had no one observed jumps of the familiar, positive-energy electrons into negative energy states? After a few weeks, has had found an answer. He imagined all of the electrons in the universe gradually filling up the energy states: the states with the negative energy will be populated first, because they have the lower energies. Only when they are full will electrons occupy positive energy states. Because the negative energy states are full, there are no vacancies into which these positive-energy electrons can jump...

"Only a disturbance in Dirac's sea - a bursting bubble, for example - would be observable. He envisaged just this when he saw that there would be some vacant states in the sea of negative-energy electrons, causing tiny departures from the otherwise perfect uniformity. Dirac called these unoccupied states, "holes". They would be observed , he reasoned, only when they are filled by an ordinary electron, which would then emit radiation as it make the transition."

Graham Farmelo, _The Strangest Man_, ISBN; 978-0-465-02210-6, pp 166 - 167.

Dirac's Sea was, at the time, a Philosophical Dead End. However, as we are seeing on some of the Posts on this site, a contradiction in language may not lead to to an outright contradiction if the physical states (Ugh!) may be Re-Described to eliminate the contradictions.

Dirac's Sea has found great use in Low Temperature Physics and the development of Theories of Semi-conductors. From my favorite article from SciAm, "Helium3 Superfluids":

"Atoms in Superfluid He3 are bound in Cooper pairs..." [[ A wire device which acts as a thermometer also illuminates Helium Cooper Pairs by breaking apart these Bosonic Pairs.]] Pair breaking also provides a controllable source of artificially produced quasiparticles and quasiholes...
"Some puzzles remain. Because the wire is moving back and forth. the pulsed beam of excitations emitted should consist of alternating bursts of quasiparticles and quasiholes..."
"The concept of a hole arises from the nature of the lowest energy-level, or ground state, of a system of particles. In the lowest-energy state of a system of fermions, for example, particles fill all the states up to a certain energy level, determined by the number of particles, because each fermion must be in a different state. All higher-energy levels are empty. Such a ground state is what field theorists call a vacuum, because, as long as all the low-level states remained filled and no particle is boosted to an unfilled level, nothing can interact with it.
"If a single fermion is removed from among the filled states and put in some higher-energy state, the situation changes markedly. The particle in the higher-energy level can now interact with various forces , and it leaves behind an empty quantum state - a hole. The particle and the hole behave in substantially different ways. Push a particle, and it moves away. It's momentum and energy increase or decrease together. Holes, however, do the opposite. Push a hole, and it approaches you. The momentum of a hole decreases as its energy increases, and vice versa. A hole behaves as if it had negative mass - indeed , it is a missing particle, so in a sense it does indeed have negative mass."

With this mind, you may put together some understanding of what happens.

Notice also that the analyses seen above occur in positive time. With the "Hole Theory" and "Negative Mass", there is more to this story that hasn't been touched yet, as might be seen in a reverse analysis of a "Particle" exchanging places with a "Hole" at the event horizon. The particle adds mass to the Black Hole. What happens to the Hole? Where does it go? Do Black Holes ever run out out of Holes to exchange? What do we find sometimes at the poles of a rapidly spinning Black Hole?

Oops! Times up!

CW

4. May 10, 2013

### Bill_K

Thanks, CW, you've mentioned several things here... but I don't agree they are related!

The negative energy particles said to be involved in Hawking radiation are not the same as Dirac's sea of negative energy solutions. Dirac's sea was just a tentative idea. It was full of problems, and was shown to be essentially incorrect by Heisenberg almost as soon as it was proposed. Dirac himself quickly abandoned it.

The Fermi Sea used in many-body theory does have a valid basis, and is similar to the Dirac Sea but again not the same. The Fermi Sea is populated with real particles occupying positive energy states.

Last edited: May 10, 2013
5. May 10, 2013

### The_Duck

On this particular point, I think quantum mechanics and antimatter are distractions. It is possible even in classical GR to extract mass from a black hole by dropping into it a body with positive mass but negative energy: see the Penrose process (the black hole has to be rotating in this case).

How can a body with positive mass have negative energy, since the body's mass energy is always positive? Roughly speaking, it's because you also have to take into account the body's gravitational potential energy in the gravitational field of the black hole. This potential energy is always negative. Near a rotating black hole, the gravitational potential energy can become greater than minus the mass-energy of the body. Then the total energy is negative. If a body with negative total energy falls into the black hole, the total energy of the black hole--that is, its total mass--decreases.

Caveats:
(a) This is an oversimplification of the Penrose process and the simplification may not be quite correct.
(b) I don't know much about Hawking radiation and so I have no idea if the same ideas that show up in the Penrose process really apply to Hawking radiation, or whether it is just similar language.

6. May 10, 2013

### Charles Wilson

I'm not surprised. When I announce that the sky is blue, people rush to the windows to see if I was right this time, at this place, with this particular patch of sky.

The short answer then, is that I'm probably wrong. The longer answer may be that Hawking is considering the decay of A single particle and the results thereof. Again, see the SciAm article on Helium. The Helium3 atoms are bound in Cooper Pairs. When pair breaking occurs, one of the particles is a "Quasiparticle" and the other is a "Quasihole". Which is which? In Positive Space (There we go again...) each particle registers as a Positive in a process named "Andreev Reflection".
What happens if this understanding of single particle behavior is transferred to a "Fermionic, Statistical" treatment of the entire Black Hole? The short sentence is, "Every Black hole radiates energy". Remember, one interpretation states that the created virtual pair of particles becomes actual by using the gravitational energy of the Black Hole.

Yep. See what I wrote in the Big Post on this, above. Still, even when Dirac was wrong he was correct. That's the nature of Pure, Raw genius. I listen to Copland and I just stop sometimes: "How could ANYONE see so far?" Same with Einstein, A N Whitehead. I read some of Shimony's stuff. "Where do I even start?"

Tell me more.

My only problem here is "occupying positive energy states". Are there "Negative Energy States"? I think there are and I doubt that a TOE can be completed without some version of what we end up perceiving as Time Reversed Sequences. The nature of these Negatives (Energy, Time and others) may be EXTREMELY narrow in allowed scope but it may allow us to make progress.

CW

Last edited: May 10, 2013
7. May 11, 2013

### Stonius

Ah, bingo, it's the virtual particle bit that messed me up. I hadn't heard of them before and just looking them up, the whole argument became a lot clearerer.

Please correct me if I'm wrong;

Heisenberg/'s uncertainty means you can't know both the lifetime of a particle and its energy. Since the energy can be anything at all, a negative value is a possibility. The total energy must be 0, so there are a positive and negative virtual particle pair. The negative energy particle is trapped by the event horizon. The positive energy particle is left hanging long enough that the probability wave of whether the positive virtual particle actually exists or not, collapses, turning it into a real particle free to radiate out into the universe.

So if I have that right, perhaps I can ask a few more questions?

1) Why does the negative energy virtual particle always fall into the black hole?

2) What of the the particle falling into the black hole? Does it become a real particle too? If so, it would have to conform to the mass shell relation and thereby have positive mass, like normal matter, right? If it doesn't, then why not? Do we assign it negative mass because time on the other side of the event horizon theoretically runs backwards?

3)Hmmm this seems to lead me back to the original question, which is whether the particle falling in has mass. Case 1; the black hole never evaporates. Case 2; energy is not conserved in the closed system of the universe (IOW, when the black hole finally evaporates, there will be more matter total in the universe).

I think more reading may be required on this point. Any suggestions?

8. May 11, 2013

### Charles Wilson

Your are onto something here. Take a moment and consider what you've asked:
1. Two quantities sum to zero. If one is positive, the other is negative. Which is which? The first half of the question is in #1. You ask that "other half" of the question in the first sentence of #2: "What of the the particle falling into the black hole?"
2. In #1, you have assigned the "Negative Particle" to the object falling into the Black Hole. In #2, you have assigned the "Positive Particle" to the object falling into the Black Hole.

3. This leads to to your great question at the end of #2: "Do we assign it negative mass because time on the other side of the event horizon theoretically runs backwards?"

No.
The "Particle/Antiparticle" question is concerned with the "Closed System" of the "Two Objects with Quantities that Sum to Zero". The Black Hole may or may not be a Quantum Object but the scope of the particles in the Black Hole compare, I would think, with looking at a freestanding grain of sand and its reflection in a particular type of mirror and hoping to determine how the oceans and the shores act over billions of years.

I'm sorry here but I'm gonna plug "The Helium3 Superfluids" from SciAm, June 1990 again. Just above absolute zero, 2 Helium3 atoms are bound in a Cooper Pair arrangement and are broken apart. "One of the Helium3 atoms is a "quasiparticle" and the other is a "quasihole"." Which one is the positive and which one is the negative?

"Well, let's check!" And what do we find? Both impinge on our instruments as "Positives". Which we should expect. After all, both Helium3 atoms are Helium3 atoms. Once again, QM seems just out of reach.

It's not!

CW

9. May 11, 2013

### Charles Wilson

Here's a diagram from the SciAm, June 1990 article of the Helium3 Superfluids.
Many will recognize the "Mexican Hat" form also seen in articles on Higgs Symmetry Breaking and the like.
Two things that are absolutely startling in this diagram.
The gray parabola of the "Unbroken Symmetry", identified as atoms bound in Superfluid Cooper Pairs.

This makes the main part of the graph (Something like an "aX^4 - bX^2 + C = 0") truly worth studying.
If this is a Symmetry Break function, notice the appearance of the quasiparticles/quasiholes.

"The particle and the hole behave in substantially different ways. Push a particle, and it moves away. Its momentum and energy increase together. Holes, however, do the opposite. Push a hole and it approaches you. The momentum of a hole decreases as its energy increases and vice versa. A hole behaves as if it had negative mass - indeed, it is a missing particle, so in a sense it does indeed have negative mass."

"Helium3 Superfluids", p. 111

This should add meaning to understanding not just Symmetry Breaks but what happens to particles as they "Become Aware" of energies supplied by surrounding particles and energy events. Give the Hole a small push and it approaches. Add energy and it slowly converts into a positive particle.

Great Stuff!

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10. May 13, 2013

### Stonius

Okay Charles, I think i get an inkling of what you're talking about. I've heard of holes being discussed in (IIRC) semiconductor physics.

Its an interesting theory, but I may have it wrong, so I'll repeat what I understand to check I'm not garbling it.

Essentially the universe is full of holes that are all filled by electrons to create a nominally neutral void. The extra electrons have no hole to fill and are therefore observed as free electrons (or part of a system, atom, whatever).

The vacuum energy then, is an electron being disturbed out of its hole before falling into it again.

So would that mean that the gravitational attraction of the black hole would repel the hole, and as it gathered speed it would become a positive mass particle? The only particle left for the black hole to absorb is the positive particle with a positive mass which contributes to the mass of the black hole!

So we are still left with the problem of how exactly the Hawking radiation manages to reduce the mass of a black hole by dropping stuff into it. There must be a method for ensuring that the negative energy particle (or hole, in your theorem) falls in to the black hole >50% of the time. If not I don't get how energy is conserved.

Cheers
Markus

11. May 14, 2013

### Charles Wilson

Stonius-
1. The first 100 times I read about Dirac's Sea, I went, "Huh?!??". When he proposed it, a lot of real scientists said the same, only they understood it. The positron had not been found yet!!! Dirac's Sea might have been seen, on the Physics Blunder Poll, second only to Einstein's Cosmological Constant. Funny how Genius works sometimes, isn't it?
2. The article I keep pushing, "The Helium3 Superfluids", SciAm, June 1990, is The Gold Standard for me in understanding the interplay between fermions and bosons and QM in general. It took several years before I even realized that they were talking about a modern iteration of Dirac's Sea. (Note: Someone tell me how I can get "^3Helium" to appear in the proper notation.)
3. We are approaching a conceptual problem - several actually. An "Electron" and a "Positron" and any other set of particles/antiparticles are "real, stand alone" things. So Dirac's insight has to be made "more real" to be scientifically descriptive. Let's start with "Fermions".
4. In a manner of speaking, Fermions cannot be "stacked like pancakes". They each have their own "space". "Three tennis balls to a can" means that if you ship a year's supply of tennis balls to Wimbledon, the shipment will take up lots of room. You cannot compress "Fermionic Tennis Balls" to save space for the big tournament. Bosons can be squished together all kinds of ways. I can get 100 movie projectors and shine 100 pictures of cans of tennis balls and line 'em up so that it appears as if there is still one can of tennis balls. (Maybe someone can come up with a better idea to explain this. This is what insomnia does to you...)
If the Big Bang started out in "Bosonic Unification", with all that energy compressed into a small point (Like maybe all the current matter in the universe compressed into a space smaller than the dot of this "i"), when fermions came into being, for whatever reason, each fermion had to have its space, or more accurately, it's own State. *BOOM!*
5. Dirac doesn't have anti-particles. No one has ever seen one. Yet, the math unambiguously shows that the negative solution to his equation cannot be thrown out. I can have an "aX^2 + bX + C = 0" equation and model a diver on a diving board: "When does the diver enter the water after diving from such and such a height?" One of my solutions is negative: "The diver hit the water 3 seconds BEFORE she began her dive." NONSENSE! Throw that solution out.
Dirac cannot do this.
6. So Dirac has to find a reason for not seeing "Opposite Electrons". "Let's see, this thing is the Opposite in charge to an electron. Maybe if everything is "Opposite", mass is as well."

7. Let's change our "fermionic tennis balls" to "Quantum States". Every fermion will associate with another fermion only to a certain extent because each fermion must have its own state and these fermions cannot settle into the same "Ground State". "Opposite Electrons" have "Opposite Mass" and since the electron has positive mass, "Opposite Electrons" must have negative mass and therefore negative energy (Note: This reasoning is used by Guth and Steinhardt with the "False Vacuum" in their Inflationary Universe: "Because the pressure of the true vacuum is zero, the pressure of the false vacuum must be negative..." Guth and Steinhardt, "The Inflationary Universe", SciAm, May 1984. Now, back to the confusion...). See how an "Opposite Particle" is different from a "Hole"? At the time Dirac is using every available thought to figure this out! Don't feel guilty because it doesn't make sense for the first 100 times of thinking about it!
8. Now we cover the older ground again. All of the States for all of the particles get filled. Bosons - like photons - can pile onto each other and have a good ol' time. Bosons can achieve a lowest Quantum State. Here is the transition: Our fermions cannot "sink" into a Lowest Quantum State together. So Dirac sees this: Since the "Opposite Electrons" have "Negative Energy" they fill up all of the best seats in the house first. Here is our friendly Helium3 explanation again. The slow moving (Just above Absolute Zero) Helium3 atoms cannot sink into a lowest ground state - Like the Dirac Sea. It is a vacuum. The unpaired fermions, like lonely hippies at a love-in, wander around looking for someone to hug. When they find someone else who needs a hug, they can sit down together.
9. Here's the next confusion: Dirac introduces the idea of a "Hole" to take care of the messy details. The particle in the positive universe has positive mass. The "Opposite Electron" is negative. "Push a particle and it moves away from you. Push a hole and it approaches you. This doesn't sound like a positron does it?

This is where the Black Hole misunderstanding occurs.

10. Virtual particles appear, interact and then disappear "all the time in QM". Empty space is seething with them all the time (I'm trying to use all the lingo I've ever read about this...). The particle and anti-particle can be created by "Borrowing" the energy from the vacuum and then "repaying it" almost immediately, determined by our old friend, the Uncertainty Principle.
If enough energy is brought in, the virtual particle/anti-particle pair become real.
11. Where would the energy come from? From the Black Hole. Remember, in the Dirac Sea/Helium3 model, the particles only interact when they can be boosted out of the background. It's called "Andreev Reflection" and it turns "quasiholes" smoothly into "quasiparticles".

12. So the Black Hole provides energy into the conversion of virtual particles into real ones that do not have to repay the debt to the vacuum. At the Event Horizon, one particle spirals into the Black Hole. The other goes off into space. This is another way of stating that "The Black Hole radiates its energy away".
13. So, now there are anti-particles for particles and holes that match up with unpaired fermions. Dirac's Sea has been modified to allow for the enormous possibilities that "Opposite Electrons" offered.

Don't expect to see holes walking around very much. Liquid helium shows how they are conceptually useful. I believe Galactic Jets show this as well but that's for another day. Electronics couldn't get by without 'em.

"How'd I do, Coach?"

CW

12. Jan 22, 2014

### Dumbledore211

@Stonius Okay what I understand about how the Hawking radiation seems to preserve the 1st law of thermodynamics is the fact that nothing not even light can escape a black hole and any particle getting trapped within the event horizon of the black hole will have a huge amount of gravitational potential energy since the gravitational field of a black hole is strong to the point that even our understanding breaks at it's center. This gravitational potential energy is greater than the energy of the particle that gets trapped. Then the total energy becomes negative which is why the particle falling into the black hole decreases the black hole's mass owing to it's negative energy. It is worth noting that the pairs of particle-antiparticle's total energy quantities sum up to zero which are often generated near the event horizon. It also doesn't really matter which particle gets trapped within the hole because the positive particle will eventually end up gaining negative energy while the virtual particle escapes to infinity. So I think the first law of thermodynamics is preserved because when the pair is created their energy quantities sum up to 0

13. Jan 22, 2014

### Stonius

That's what I don't get.

Two virtual particles form near a black hole. Being virtual particles, one of them will have negative mass (unlike normal particles or anti particles). The sum of the two, as you mention is zero.

If the choice of which particle fall in is completely random, there is no net gain or loss and therefore the black hole never evaporates. When considering the entire universe as a very large closed system, this violates conservation of energy because the black hole never evaporates, and the matter and energy that caused it to become a black hole in the first place are removed from the universe forever.

On the other hand, if the black hole somehow has a tendancy to 'select' the negative mass particle, I can see how this would lessen the mass of the black hole over time as only negative mass is absorbed, and positive mass particles are radiated.

But, why would the black hole 'select' negative mass particles? They should be *repelled by gravity, rather than drawn towards it, right?

Even then, a virtual particle only exists over very short time spans. Once it is around long enough to become a real particle, it can't have negative mass any more. maybe the time dilation near the event horizon allows virtual particles to exist for much longer as viewed from outside the black hole.

But if it's time dilation that causes the effect, then as far as any observer *not near the event horizon (ie, the rest of the universe) is concerned, the anti particle is never observed to cross the event horizon anyway, so the thermodynamic question isn't really solved as far as the rest of the universe is concerned.

As you say, anything falling into a black hole has a lot of gravitational potential energy. But if I drop something from 1 m above the earth's surface it has x kinetic energy. Same item dropped from 1km above the earth's surface has a lot more than x. Wouldn't something much further from the black hole that crosses the event horizon far in the future have a greater gravitational potential than something close to the event horizon that exists only for a short time before crossing?

Why does the infalling particle become negative energy? If I understand CPT symmetry correctly on the inside of a black hole event horizon mass is preserved (even relativistic mass which is essentially energy anyway). I didn't think it became negative energy/mass.