# Validity of theoretical arguments for Unruh and Hawking radiation

• A
Elias1960
[Moderator's note: New thread spun off from previous discussion due to more advanced subject matter being discussed.]

There is, in fact, a quite good argument that Hawking radiation cannot be derived by semiclassical theory.

It is the comparison with the scenario where the collapse stops some ##\epsilon## above the Schwarzschild radius. In this case, Hawking radiation stops once the collapsing star becomes stable. And the time for this is very short.

Now the question is how all this depends on the ##\epsilon##. Let's assume it is quite small, say, ##\epsilon=10^{-100} l_{Planck}##. The number does not really matter, the time is quite short because the dependence is logarithmic. Let's make it even smaller, ##\epsilon=10^{-1000} l_{Planck}##. This will add some seconds, but not more.

Now, where does the last Hawking radiation particle come from? Once it would not have been created for ##\epsilon=10^{-100} l_{Planck}##, it has been created during the collapse from ##\epsilon=10^{-100} l_{Planck}## to ##\epsilon=10^{-1000} l_{Planck}##. And what has caused the creation was the difference between the two solutions. But this difference is localized completely below ##\epsilon=10^{-100} l_{Planck}## from the Schwarzschild radius. Which is, therefore, the region where it has been created.

What is the time dilation, and, therefore, the corresponding redshift the particle obtains moving from that place? There will be a ##10^{100}## factor for moving up to Planck length and then even more from Planck length to infinity. So, what resulted in Hawking radiation of some ##10^{-8}K## has, at the origin at the surface, a quite solid energy. That for such energies the semiclassical approximation is applicable is not plausible at all.

Last edited by a moderator:

Mentor
It is the comparison with the scenario where the collapse stops some ϵ above the Schwarzschild radius.
But that means ##\epsilon## cannot be smaller than ##R_s/8##. I am not sure how that is even relevant for investigating Hawking radiation.

Mentor
It is the comparison with the scenario where the collapse stops some ##\epsilon## above the Schwarzschild radius

If the collapse stops and the object becomes stable, ##\epsilon## cannot be smaller than ##1/8## of the Schwarzschild radius, since ##9/8## of the Schwarzschild radius is the smallest possible radius for a stable object by Buchdahl's Theorem.

• Dale
Elias1960
You have forgotten to mention that there are assumptions for this theorem. These assumptions have been described in the literature in the following words: "The assumptions used to derive the Buchdahl inequality are very restrictive and for instance neither of them hold in a simple soap bubble." And while one can prove some similar bounds using weaker energy conditions, I would refer to arxiv:gr-qc/0001099 for a general criticism of such energy conditions.

Then, the consideration is obviously not about proposing realistic matter which would lead to GR solutions with such configurations. It is about what the derivation of Hawking radiation is worth. It is derived for semiclassical gravity, and the classical solution serves simply as a background. If this background is a GR solution, or a solution of some other metric theory of gravity, like a theory of massive gravity which has stable stars very close to the Schwarzschild radius, or GR with some exotic matter forming gravastars, is irrelevant for the further derivation. The article

Paranjape, A., Padmanabhan, T. (2009). Radiation from collapsing shells, semiclassical backreaction and black hole formation, Phys.Rev.D 80:044011, arxiv:0906.1768v2,

where the scenario I have considered here has been studied, also found it unnecessary to discuss Buchdahl.

Mentor
And while one can prove some similar bounds using weaker energy conditions, I would refer to arxiv:gr-qc/0001099 for a general criticism of such energy conditions.
Although you can find here or there someone who dislikes energy conditions, they are by far accepted and used by more authors than they are rejected.

In particular, I use and accept energy conditions and find proofs, like yours, that blatantly violate them entirely unconvincing.

Elias1960
My posts here are not intended to prove something, but to illustrate that there is no proof that Hawking radiation exists.

I would, similar to you, not take seriously claims about the existence of, say, traversable wormholes if they depend on violations of reasonable energy conditions.

But once the aim is to show (moreover, in a popular forum, not a scientific paper) that the derivation of Hawking radiation has a serious problem (namely the trans-Planckian one), then I'm essentially free to use whatever background I like to illustrate the problems related with that proof. Because it is that proof, if it is worth to be named a proof, which has to cover everything which is not excluded by the conditions of the proof. And this clearly includes matter which does not fulfill particular energy conditions - even if these energy conditions are reliable.

• weirdoguy
Elias1960
If you are comfortable with a "proof" as long as it fails only for backgrounds which violate some energy conditions (even if these energy conditions are not listed among the assumptions of the theorem), I give up attempts to convince you.

As mentioned earlier, there are many proofs, so your argument likely does not apply to all of them anyway.

First, let's quote about the reliability of these proofs.

None of the derivations that have been given of the prediction of radiation from black holes is convincing. All involve, at some point, speculations of what physics is like at scales which are not merely orders of magnitude beyond any that have so far been investigated experimentally (##\sim 10^3## GeV), but at and increasing beyond the Planck scale (##\sim 10^{19}## GeV), where essentially quantum–gravitational effects are expected to be dominant. (In Hawking’s treatment, this increase occurs exponentially quickly.) Some of these speculations may be plausible, but none can be considered reliable.
(A.D. Helfer, Do black holes radiate? Rept. Prog. Phys. 66, 943-1008 (2003), arxiv:gr-qc/0304042)
Then, it applies to a lot of them.

For example, the argument of analogy with Unruh radiation. The observer at rest is in some acceleration, thus, will similarly observe something like Unruh radiation. But why there is no such effect if the star is stable? The acceleration of the observer is the same for a stable star and a black hole with equal mass. But only the BH gives Hawking-radiation.

Moreover, good counterexamples have the property that they suggest where one has to look for the weak points of "proofs". Simply look what happens in this case. Here, one starts with the fact that a stable star does not radiate. The "proof" has to give this result too but to predict Hawking radiation for the continuing collapse. This is not at all a trivial job, especially if the region where both differ is one with trans-Planckian time dilation.

For example, there are approaches which allow for modified dispersion relations depending on some preferred frame. It works nicely, but, strangely, not if the preferred coordinates are those of observers at rest. Some consider lattices - again, everything nice, except for lattices at rest. In the "dumb hole" analogs there is nothing at rest. The flow may be stationary, but it is a flow, and without a flow there would be no Hawking-like radiation.

Not that I do not question the effect that a changing gravitational field creates radiation. So, I expect a lot of such radiation, given that changes appear quite often. The stable situations are the exceptions. The BH fits into this only because it is changing forever - we always "see" a collapsing surface. If some trans-Planckian effect would stop the collapse, there would be no Hawking radiation.

Mentor
If you are comfortable with a "proof" as long as it fails only for backgrounds which violate some energy conditions (even if these energy conditions are not listed among the assumptions of the theorem), I give up attempts to convince you.
I am not sure what you are complaining about here. Any logical reasoning is based on some assumptions, and any person has a set of assumptions that they accept. Obviously, reasoning based on assumptions that a person does not accept is going to be unconvincing to that person.

Your reasoning is based on an assumption that I do not share, so I do not find your reasoning convincing. My objection has nothing directly to do with the proofs that you are arguing against, it is an objection to your reasoning. The validity of the proofs is open to debate, but I do not accept your reasoning about their validity because your reasoning is based on a premise I do not hold.

In my opinion your argument is not “in fact, a quite good argument” as you claim.

• weirdoguy
Mentor
I use and accept energy conditions and find proofs, like yours, that blatantly violate them entirely unconvincing.

This statement is too strong. There are at least three known violations of energy conditions that play a role in our current physical models:

(1) A scalar field, such as the one that drives inflation in inflationary cosmology;

(2) A cosmological constant, such as the one that drives the current acceleration of the universe's expansion;

(3) Some quantum field states in curved spacetime, such as at or near the horizon of a black hole, which play a role in the standard derivation of Hawking radiation.

If you are going to take the position that no violation of energy conditions is ever acceptable, then you have to reject the standard derivation of Hawking radiation, since if energy conditions are satisfied the Hawking area theorem applies and no black hole can ever emit radiation or decrease its mass. (You also, of course, have to reject inflationary cosmology and the current accelerated expansion of the universe.)

Mentor
The article

Paranjape, A., Padmanabhan, T. (2009). Radiation from collapsing shells, semiclassical backreaction and black hole formation, Phys.Rev.D 80:044011, arxiv:0906.1768v2

On an initial skim, it doesn't look like this paper is saying that Hawking radiation doesn't exist. It's just considering a wider class of models that could possibly produce similar observations at large distances and late times, in order to see how viable they are. I'll take a closer look when I get a chance, I haven't been able to read it in detail yet.

Elias1960
On an initial skim, it doesn't look like this paper is saying that Hawking radiation doesn't exist. It's just considering a wider class of models that could possibly produce similar observations at large distances and late times, in order to see how viable they are. I'll take a closer look when I get a chance, I haven't been able to read it in detail yet.
My claim is also not that it does not exist. There exists radiation as long as something changes in the geometry. And a BH created by a collapse is, if one uses Schwarzschild time, formally changing (collapsing) forever. The problem is that after a short time, this change happens only in an extremely trans-Planckian region, with surface time dilation factors becoming arbitrary large.

So, the point is that if the collapse stops because of some trans-Planckian effect, say, as a surface time dilation factor ##z=10^{100}## or whatever one accepts as unreasonably large, then Hawking radiation stops after a very short time (order of seconds). Stable configurations do not Hawking-radiate. t

The question is not if the Hawking derivation is mathematically wrong, it isn't. The question is if it is reasonable to accept that semiclassical gravity remains applicable if the collapsing surface increases its time dilation from ##z=10^{100}## to ##z=10^{200}## so that a photon with Hawking temperature far away would have on that surface a momentum increased by a factor ##10^{100}## or ##10^{200}##.

The paper simply shows that the "stable stars do not Hawking-radiate" principle holds close to the horizon too.

I am not sure what you are complaining about here. Any logical reasoning is based on some assumptions, and any person has a set of assumptions that they accept.
As a mathematician, I accept theorems if they really prove what they claim, and if I like the assumptions made, or believe that they hold in reality, is irrelevant. And if the theorem makes claims about some objects which are not forbidden in principle, but where it is extremely implausible that they exist in reality, this will not prevent me from studying what the theorem claims about these objects.

What we can extract from this is that some trans-Planckian physics simply stops the collapse, then there will be no Hawking radiation. That means, there cannot be a derivation of Hawking radiation which does not have to make assumptions about trans-Planckian things. If these assumptions are plausible or not is nothing that matters. What matters is that you will not get rid of the trans-Planckian problem.

And, BTW, if I simply give up to convince you, this is not a complaint.

Mentor
As a mathematician, I accept theorems if they really prove what they claim, and if I like the assumptions made, or believe that they hold in reality, is irrelevant.
As physicists the assumptions and the experimental reality are paramount. Physics is a branch of science, not a branch of mathematics.

• Dragrath
Staff Emeritus
Homework Helper
Gold Member
You can never prove that something exists in Nature based on a model and logical reasoning alone. What you can do is to prove that a model of your specification predicts it to exist. It is then up to empirical experimentation to find evidence for whether this something is found in Nature or not and thereby for or against the model.

• Dragrath, JD_PM and Motore
Heikki Tuuri
@Elias1960,

you are right that all the "proofs" of semiclassical Hawking or Unruh radiation are unsatisfactory.

Belinski, V. A. Phys. Lett. (1995) A209,13

Vladimir Belinski published in 1995 a paper where he claims to show that no Hawking or Unruh radiation exists.

https://arxiv.org/abs/1212.2409

Detlev Buchholz has published a refutation of Unruh radiation.

I have not seen any refutation of the papers of Belinski or Buchholz.

My own argument against Unruh radiation is very simple. Suppose that a detector in an accelerating rocket heats up. From where does the energy come?

It has to come from the kinetic energy of the rocket. But if the rocket loses kinetic energy, it loses also a lot of momentum. Where does the momentum go?

It cannot go to the photons which are radiated away, because momentum/energy is much smaller for a photon than the rocket.

The extra momentum should be absorbed by the jet of the rocket. But there is no obvious mechanism how the extra momentum can end up there.

---

As an aside, if I wave an electric charge, how can it emit photons? The energy comes from the kinetic energy of my hand, and my hand loses momentum. Where does the extra momentum go?

In this case the solution is that the electromagnetic field stores the momentum temporarily, and returns it to my hand when the hand is moving back.

---

The "trans-Planckian" problem in Hawking radiation is mentioned in many sources. Hawking assumes an extreme blueshift of the order 10^10^100 in his 1975 paper. We do not know if semiclassical reasoning works under such conditions.

https://arxiv.org/abs/hep-th/9907001

Parikh and Wilczek derived Hawking radiation from a tunneling argument. But there are lots of assumptions in their paper.

We would like to have Hawking radiation since it fits nicely with thermodynamics. There is no satisfactory proof yet.

Last edited:
Mentor
My own argument against Unruh radiation is very simple. Suppose that a detector in an accelerating rocket heats up. From where does the energy come?

It has to come from the kinetic energy of the rocket.
The KE of the rocket is 0 in the accelerating frame (which is where the Unruh radiation exists).

Heikki Tuuri
The KE of the rocket is 0 in the accelerating frame (which is where the Unruh radiation exists).

Yes. A major complaint of Belinski is that we do not know if we can meaningfully do quantum mechanics in an accelerating frame. Quantum mechanics was built for inertial frames.

The derivation of Unruh radiation should be given in an inertial frame. Belinski and Buccholz tried to do that and found out that there is no Unruh radiation.

I have checked the book of Birrell-Davies as well as papers of Unruh and Hawking. They fail to consider momentum and energy conservation.

• weirdoguy
Mentor
Yes. A major complaint of Belinski is that we do not know if we can meaningfully do quantum mechanics in an accelerating frame. Quantum mechanics was built for inertial frames.
I am not a QM expert, but isn't modern QM formulated in terms of tensors? Then it should work in any frame.

The derivation of Unruh radiation should be given in an inertial frame.
That doesn't make any sense to me. It is claimed to exist in the accelerating frame, so how could it be derived in an inertial frame. That is like requesting that the derivation for an inertial force should be given in an inertial frame where it doesn't exist.

• weirdoguy
Heikki Tuuri
In quantum mechanics, we work in an inertial frame. Particles accelerate relative to that frame. For example, a Feynman diagram resides in an inertial frame, though the colliding particles accelerate violently.

Hawking, Unruh, Davies, and others in the 1970s thought that we can quantize the electromagnetic field in an accelerating frame. They derived surprising results, like an accelerated electron heating up in empty space, and a black hole evaporating.

If we look at classical mechanics, we can calculate in an accelerating frame. For example, we can work in the rotating frame of the surface of Earth. But calculations in an accelerated frame are error-prone.

That is one of the reasons why we should work in an inertial frame: we understand it much better and will make less errors.

Thus, the most reliable way to derive Unruh radiation is to look at an accelerating electric charge in an inertial frame. Detlev Buchholz did the calculations and found no Unruh radiation.

In the past 30 years, several people have tried to derive the results of Unruh in other ways. I have not seen any successful derivation. There are a number of publications where the author states that there is no Unruh radiation.

Testing experimentally Unruh radiation requires us to accelerate an electron by some 10^22 m/s^2. We have not yet achieved that, and distinguishing Unruh radiation from other types of radiation would be a challenge.

Hawking radiation is assumed to arise close to a black hole horizon. In the Hawking case, energy and momentum might be conserved through some unknown process with the black hole. Hawking himself thought that an unknown mechanism sucks energy from the black hole to the outgoing radiation. He realized that the information in the black hole would be lost. That was the beginning of the famous black hole information paradox.

Hawking never considered momentum conservation. That might be called the black hole momentum paradox.

• weirdoguy
Mentor
Suppose that a detector in an accelerating rocket heats up. From where does the energy come?

It has to come from the kinetic energy of the rocket.

No, it comes from the rocket's engine, which is burning fuel to produce energy.

Mentor
Hawking never considered momentum conservation.

Hawking radiation is emitted isotropically, so there is no issue with momentum conservation.

Mentor
Detlev Buchholz has published a refutation of Unruh radiation.

Do you have a reference?

Heikki Tuuri
Do you have a reference?

Conclusions

In the present article we have studied the macroscopic effects of acceleration on equilibrium states,
as seen by an observer in a rigid, spatially extended laboratory. The macroscopic properties of these
states are determined by local observables in the respective laboratory system, which form central
sequences at asymptotic times. These sequences have sharp limits, hence quantum fluctuations are
suppressed. It turned out that acceleration does not affect the macroscopic properties of an inertial
vacuum state. Irrespective of the accelerated, possibly erratic motion of the laboratory, the observer
will find the same macroscopic properties of the vacuum as an inertial observer. In particular, he will
not find himself immersed in a thermal gas, respectively heat bath.

https://arxiv.org/abs/1412.5892

Heikki Tuuri
No, it comes from the rocket's engine, which is burning fuel to produce energy.

The rocket has an electron attached to a spring. The acceleration of the rocket causes a force in the spring, which in turn accelerates the electron. We may well say that the energy comes from the rocket kinetic energy. The spring slows down the rocket.

The energy does not come directly from the fuel combustion.

You may accelerate an electron also by other ways besides fixing it to a rocket. You can shoot photons at the electron. Then it is the scattering of photons which accelerates the electron. We handle that in QED, and there is no Unruh radiation in QED.

Heikki Tuuri
Hawking radiation is emitted isotropically, so there is no issue with momentum conservation.

Yes. If we assume that the gravitational field or something else in the black hole can absorb the momentum when a quantum of Hawking radiation is born, then there is no problem.

But I am not aware of a mechanism which would do that absorption of momentum. When a photon collides to a mirror and is reflected back, the momentum goes to electrons.

The same problem is for the energy: we do not know the mechanism which would suck energy from the black hole and give it to the quantum.

• weirdoguy
Mentor
The spring slows down the rocket.

No, it doesn't. Internal motions in a system can't change the motion of the system's center of mass.

The energy does not come directly from the fuel combustion.

Yes, it does. All your talk about an electron attached to a spring simply obfuscates the ultimate source of the energy. If there is no fuel combustion, there is no acceleration of the rocket and therefore no temperature rise in the detector.

You may accelerate an electron also by other ways besides fixing it to a rocket. You can shoot photons at the electron. Then it is the scattering of photons which accelerates the electron.

Yes, and the energy in the photons, which gets transferred to the electron when the photons scatter off it and accelerate it, comes from the photon source, which will be a laser or some similar device that is powered by some kind of fuel combustion.

there is no Unruh radiation in QED.

Sure there is. Unruh's original argument can be formulated perfectly well using QED as the quantum field theory. Put the QED EM field in its vacuum state for an inertial observer. Then an accelerated photon detector will have a nonzero amplitude to transition to an excited state.

Last edited:
• Dragrath and weirdoguy
Mentor
I am not aware of a mechanism which would do that absorption of momentum.

we do not know the mechanism which would suck energy from the black hole and give it to the quantum

In other words, we don't know what the microscopic degrees of freedom of a black hole are, since it's ultimately those microscopic degrees of freedom which would have to absorb momentum and energy in order to maintain the conservation laws. That's true, but it's not an argument against the existence of Hawking radiation; it's an argument for figuring out a more complete theory of black holes. That's a major part of what theorists working on quantum gravity are trying to do.

Mentor
In quantum mechanics, we work in an inertial frame.
I don't buy that claim. The QED Lagrangian, ##\mathcal L = \bar{\psi}(i\gamma ^{\mu} D_{\mu}-m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}## certainly looks to me like it should be valid in any coordinates.

Do you have a reference which explains the failure of QM in non-inertial frames? I am skeptical. Perhaps you are referring to an older formalism before these issues had been worked out?

• Dragrath, Heikki Tuuri and weirdoguy
Heikki Tuuri
There is another simple argument against Unruh radiation. Support an electron at a fixed position in a gravitational field. It is then in a constant acceleration relative to an inertial observer.

Unruh radiation would mean that the electron heats up, as observed by other supported observers. Where does the energy come from?

---

Concerning the spring mechanism, we can treat the electron and the rocket as separate objects.

The electron gains kinetic energy as well as momentum from the rocket. If the electron does not start to lag behind the movement of the rocket, then the electron has to use for its own needs all the energy and the momentum it gains from the rocket.

This analysis opens a possibility for emitting photons: if the electron or its field would lag more and more behind the rocket, then there would be extra energy available which the electron could emit as photons.

It is like hanging a weight from the rocket from a tether and letting the tether to slide through your fingers. There will be heat from friction.

However, the electron travels with the rocket, and I do not see how its field could lag behind more and more.

---

Feynman diagrams describe accelerating electrons in collisions. There are no photons in a Feynman diagram which we would mark as Unruh photons. The accelerations are so huge that Unruh radiation should appear there, but there is none.

Heikki Tuuri
I don't buy that claim. The QED Lagrangian, ##\mathcal L = \bar{\psi}(i\gamma ^{\mu} D_{\mu}-m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu}## certainly looks to me like it should be valid in any coordinates.

Do you have a reference which explains the failure of QM in non-inertial frames? I am skeptical. Perhaps you are referring to an older formalism before these issues had been worked out?

Under the accelerating coordinates of the surface of Earth, you cannot use a standard lagrangian. There are Coriolis forces, and, of course, the gravitational force.

Quantum mechanics is formulated without gravity. If you use an accelerated coordinate system, it is equivalent to working under gravity.

The Belinski paper of 1995 is not free on the Internet, I think. I recall Belinski criticized quantization under an accelerating coordinate system.

We can trust an inertial frame much more. If there is Unruh radiation, there must be a derivation under an inertial frame.

Unruh was not able to refute the arguments by Buchholz. Buchholz mentions that somewhere.

• weirdoguy
weirdoguy
Quantum mechanics is formulated without gravity. If you use an accelerated coordinate system, it is equivalent to working under gravity.

No it is not. Acceleration and gravity are different things. Special Relativity can deal with acceleration without any problem.

• Dragrath
Mentor
Support an electron at a fixed position in a gravitational field. It is then in a constant acceleration relative to an inertial observer.

Unruh radiation would mean that the electron heats up, as observed by other supported observers. Where does the energy come from?

From the source of the gravitational field, whose mass will decrease. This scenario is more like Hawking radiation than Unruh radiation, since in the presence of a gravitational field spacetime is not flat and the source of the field has to be taken into account.

Concerning the spring mechanism, we can treat the electron and the rocket as separate objects.

Which simply further obfuscates what is going on. See below.

If the electron does not start to lag behind the movement of the rocket, then the electron has to use for its own needs all the energy and the momentum it gains from the rocket.

But the rocket does not have to put all of the energy produced from its engine into the electron's (and its own) proper acceleration. Some of it can be put into raising the electron's temperature, which is what will happen when Unruh radiation is taken into account.

Mentor
Under the accelerating coordinates of the surface of Earth, you cannot use a standard lagrangian. There are Coriolis forces, and, of course, the gravitational force.

You are missing the point @Dale was making. He wrote down a tensor equation which is valid in any coordinates; that's how tensor equations work. If you expand out the terms in that equation in non-inertial coordinates, the terms corresponding to the "forces" you refer to will appear in the expansion.

• Dragrath, Dale and weirdoguy
Mentor
Acceleration and gravity are different things.

Acceleration and spacetime curvature are different things. But "gravity" can just mean "acceleration due to gravity", which can't be distinguished from acceleration due to a rocket in flat spacetime because of the equivalence principle.

• weirdoguy
Mentor
Belinski, V. A. Phys. Lett. (1995) A209,13

I can't find this paper online except behind paywalls, but it looks like Belinsky published a further update in 2006:

https://arxiv.org/pdf/gr-qc/0607137.pdf

• Dale
Heikki Tuuri
But the rocket does not have to put all of the energy produced from its engine into the electron's (and its own) proper acceleration. Some of it can be put into raising the electron's temperature

The rocket transfers the energy to the electron through the spring.

This is more intuitive if we just assume that the rocket stands still on the surface of Earth. Can you somehow sap the perceived acceleration of the rocket and make the electron to heat up? No.

Unruh believes that there is "negative frequency" radiation coming from empty space, and the electron turns it into real, positive frequency radiation.

Paul Davies considers an accelerating mirror. The mirror converts some of the negative frequency radiation into real positive frequency radiation.

When I started studying Unruh radiation a few years back, I was astounded to find out that the authors ignored energy and momentum conservation. Maybe I should write Dr. Unruh and ask about momentum.

Last edited: