A Validity of theoretical arguments for Unruh and Hawking radiation

[PeterDonis]

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.

 

[PeterDonis]

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.

 

[PeterDonis]

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.

 

[PeterDonis]

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

 

[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.

 

[PeterDonis]

Feynman diagrams describe accelerating electrons in collisions. There are no photons in a Feynman diagram which we would mark as Unruh photons.

That depends on which Feynman diagrams you choose to look at, which in turn depends on what set of external lines you choose to look at. Obviously a Feynman diagram for an accelerating electron in a collision is not going to include an extra external line for an Unruh photon, because you defined the process as a collision with a particular set of external lines, which simply rules out any diagrams for processes with different external lines. But that doesn't make other processes not happen; it just means you're ignoring the possibility of them happening in your analysis.

 

[PeterDonis]

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

Suppose there is friction present--say the rocket/electron setup is not in vacuum but in air. Then some of the energy from the rocket's fuel will not get transferred through the spring to the electron to accelerate it, but will be dissipated into heat, which raises the temperature of the electron.

Unruh radiation works the same way as friction from the rocket's viewpoint: the rocket/electron setup is not in vacuum, it's in a photon gas.

 

[Dale]

Under the accelerating coordinates of the surface of Earth, you cannot use a standard lagrangian.

You can if the standard Lagrangian is covariant.

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

All of which is dealt with by a covariant formulation.

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

Well, he does not make that argument in the updated paper posted by @PeterDonis. In fact, in the updated paper he repeatedly does QM in non inertial coordinate systems.

Frankly, I now think your claim is flat out wrong. Modern QM has a covariant formulation and is used in non-inertial coordinates by the very author you cite.

 

[Dale]

Just to be clear, I am agnostic about Hawking radiation and Unruh radiation. I would like to see some experimental evidence before making a conclusion.

However, it is a complicated topic, and I don’t believe that there are any easy proofs one way or the other. Both of the simple arguments presented here are clearly flawed. And most complicated arguments are probably flawed, but not clearly.

For now I would recommend that people stick with published proofs and counter-proofs, rather than personal ones. There are plenty of published ones to support either position, so let’s not speculate on personal theories.

 

[Heikki Tuuri]

You can if the standard Lagrangian is covariant.

All of which is dealt with by a covariant formulation.

Well, he does not make that argument in the updated paper posted by @PeterDonis. In fact, in the updated paper he repeatedly does QM in non inertial coordinate systems.

Frankly, I now think your claim is flat out wrong. Modern QM has a covariant formulation and is used in non-inertial coordinates by the very author you cite.

https://arxiv.org/abs/gr-qc/0607137
Thank you for pointing out the 2006 paper by Vladimir Belinski.

However, more thorough analysis shows that the scheme we have just described
is insolvable because the set of the left and right modes is incomplete for free
incident waves in Minkowski spacetime and such waves cannot be represented
as a linear superposition of these modes. This means that the equality (40)
in fact does not exist, and hence the relation (41) between ladder operators
also does not exist. Then for the real physical case, the expression (42) for the
particle number has no sense independent of whether it is finite or divergent.
...
mean that the quantization
...
(48)
...
(49)
...
is unitarily equivalent to the usual quantization in the plane wave basis and
the b-vacuum remains the same as the Minkowski vacuum:

Belinski claims that the accelerating frame quantization used by Unruh and others is wrong. That was his claim in the 1995 paper, too.

He goes on to say that the correct quantization is equivalent to the usual Minkowski quantization. That makes sense. You can, of course, use the inertial frame quantization in the accelerating frame. That is how we do in the Feynman diagram: the particles live in an inertial frame. The particles collide, that is, they experience huge accelerations. However, to analyze what happens, we use the quanta (= particles) of the inertial frame.

There are a lot of papers about doing quantum mechanics in curved spacetime, which includes accelerating frames. Haag and Fredenhagen are familiar names. No one claims it is easy.

In a non-uniform gravitational field we cannot work in inertial Minkowski coordinates. In the case of an accelerating Unruh rocket, we have the luxury of having inertial Minkowski coordinates available.

[Moderator's note: Personal theory content deleted.]

 

[Elias1960]

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

https://www.academia.edu/8281380/No_Hawking_Radiation_by_V.A._Belinski_Physics_Lett._A._1995_

 

[Dale]

Belinski claims that the accelerating frame quantization used by Unruh and others is wrong. That was his claim in the 1995 paper, too.

Yes. A much more reasonable claim than the claim that QM cannot be done in non-inertial frames.

 

[PeterDonis]

I believe I found the fundamental error of Unruh and others when I analyzed their papers a few years ago.

You have already been told once that personal theories are out of bounds here. Please limit discussion to actual published literature. If you mention a personal theory again you will receive a warning.

 

[PeterDonis]

You have already been told once that personal theories are out of bounds here.

I have edited the post to remove the personal theory content.

 

[Heikki Tuuri]

https://www.physicsforums.com/threads/is-it-true-that-qed-explains-the-reflection-of-light.978466/

This thread from yesterday turns out to be relevant here.

Suppose that we have a laser beam which hits a mirror orthogonally. The light is right-handed and circularly polarized.

If the mirror is traveling at a constant velocity toward the laser, then there is a blueshift in the reflected light.

We can understand it either

1. as a wave phenomenon where a wave is reflected having a higher frequency than the incoming wave, or

2. individual photons colliding to a field of electrons at the surface of the mirror and gaining energy and momentum. The handedness of the photons changes.

The picture is simple and clear in both the wave interpretation and the particle interpretation.

Suppose then that the mirror is in an accelerating motion toward the laser.

In the wave interpretation, the reflected laser pulse is an "up-chirp" where the frequency grows. The Fourier decomposition of a chirp in the laboratory frame contains both positive and negative frequencies, as observed by Hawking in his 1975 paper.

The wave interpretation is clear.

What about the particle interpretation?

If we say that the photons of the outgoing wave are quanta of the various waves in the Fourier decomposition in the laboratory frame, then we make the following observation: a small number of photons have a "wrong" handedness - they correspond to negative frequencies. The photons, overall, will have various frequencies.

https://en.wikipedia.org/wiki/Bogoliubov_transformation

The Bogoliubov transformation maps the modes in the Fourier decomposition of the incoming wave to modes of outgoing waves.

In the particle interpretation, a nice uniform flux of photons has turned into a collection which has a small number of strange wrong-handed photons. How were these odd photons born?

The process is simple if we interpret it as a classical wave hitting the accelerating mirror. A monochromatic wave reflects as a chirp.

If we try to quantize the wave as particles, the process gets conceptually more complicated.

Suppose then that we have a man sitting on top of the mirror with a polarizing filter. Will the man see a small number of wrong-handed photons?

 

[atyy]

[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.

This is an interesting argument.

Jacobson https://arxiv.org/abs/gr-qc/0308048 (p47)
"While the physical arguments for the Hawking effect do seem quite plausible, the trans-Planckian question is nevertheless pressing. Afterall, there are reasons to suspect that the trans-Planckian modes do not even exist. They imply an infinite contribution to black hole entanglement entropy from quantum fields, and they produce other divergences in quantum field theory that are not desirable in a fundamental theory."

Wikipedia has an article which links to Brout et al https://arxiv.org/abs/hep-th/9506121

 

[TeethWhitener]

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?

I’m not an expert, but the derivative here is gauge covariant: insensitive to gauge transformations. I don’t think this is the same as being covariant in the sense of a connection on a smooth space time manifold like you’d find in GR.

 

[Orodruin]

I’m not an expert, but the derivative here is gauge covariant: insensitive to gauge transformations. I don’t think this is the same as being covariant in the sense of a connection on a smooth space time manifold like you’d find in GR.

It is, in fact, exactly the same. The covariant derivative is a derivative that arises from connecting the fiber of a fiber bundle at different points. The only difference here with the affine connection of GR is that the fiber bundle is not the tangent bundle. The gauge field exactly corresponds to the connection coefficients of the tangent bundle.

 

[Heikki Tuuri]

It is, in fact, exactly the same. The covariant derivative is a derivative that arises from connecting the fiber of a fiber bundle at different points. The only difference here with the affine connection of GR is that the fiber bundle is not the tangent bundle. The gauge field exactly corresponds to the connection coefficients of the tangent bundle.

Thank you. Do I understand right that if we interpret the electron field and the EM field in the lagrangian as classical fields, then everything is ok and clear under a curved spacetime?

If yes, then the problem of quantum mechanics under a curved spacetime is in the quantization. What are the quanta and how do they behave?

Suppose that a freely falling laser emits 1,000 right-handed photons upward. An inertial observer far away in space may measure 1,000 right-handed photons and one left-handed photon.

How do we model this in quantum mechanics, in the particle interpretation? Was there some kind of scattering which caused one photon to split into two?

In the classical wave interpretation, the left-handed wave appears because the wave is a chirp, from the viewpoint of a distant observer. There is no problem in this interpretation.

But we know that a laser emits individual photons. How do we reconcile the particle interpretation with the classical wave interpretation?

 

[PeterDonis]

we know that a laser emits individual photons.

No, it doesn't. A laser emits a coherent state, which is very different from a Fock state (the latter is an eigenstate of photon number). A coherent state is the closest kind of quantum state to a classical EM field.

 

[Heikki Tuuri]

No, it doesn't. A laser emits a coherent state, which is very different from a Fock state (the latter is an eigenstate of photon number). A coherent state is the closest kind of quantum state to a classical EM field.

Humm... we could use a different source of photons than a laser. A device which emits exactly 1 right-handed photon at a time? Then it is kind of a coherent source, still.

The far-away observer will in rare cases detect a left-handed photon along with a right-handed one.

The particle interpretation is that the photon scattered from a graviton and split in two. The graviton hypothesis has the well-known problem that it is not renormalizable.

The classical wave interpretation is that a standard wave packet stretched into a chirp as it flew out of the gravitational field.

Why is there no renormalization problem in the classical interpretation?

 

[PeterDonis]

we could use a different source of photons than a laser. A device which emits exactly 1 right-handed photon at a time? Then it is kind of a coherent source, still.

Not coherent, no. "Coherent" means "coherent state". Which, as I said, is very different from a Fock state.

There are photon sources that emit Fock states, but they're very hard to set up experimentally.

 

[PeterDonis]

The particle interpretation is that the photon scattered from a graviton and split in two.

No, the particle interpretation would be that a photon of spin +1 (right-handed circular polarization) would absorb a graviton of spin -2 (or emit a graviton of spin +2) and become a photon of spin -1 (left-handed circular polarization).

 

[PeterDonis]

Why is there no renormalization problem in the classical interpretation?

Because there is no renormalization problem in classical GR, period.

 

[Dale]

A device which emits exactly 1 right-handed photon at a time? Then it is kind of a coherent source, still.

A coherent state is not a state with a definite number of photons. A state with a definite number of photons is not a coherent state.

A state with a definite number of photons has the property that if you annihilate a photon the state changes. A coherent state has the property that if you annihilate a photon it remains in the same state.

 

[Heikki Tuuri]

Because there is no renormalization problem in classical GR, period.

There might be a classical problem which resembles the renormalization problem. In the Navier-Stokes equation, a Millennium problem is to prove that solutions do not "blow up" because of turbulence.

In a realistic fluid there is a natural scale, the scale of molecules, at which the Navier-Stokes equation stops working. The blowup cannot happen. This sounds like an energy cutoff which is used to eliminate the divergence in renormalization.

The concept of an "effective theory" contains the idea that at very short distances there is new physics which provides the necessary cutoff.

If we try to model an electromagnetic field in a gravitational field, and consider the backreaction of the two fields when they interact, the renormalization problem may appear in the classical fields as a blowup problem. For example, the solution might not be stable under small perturbations.

After all, Feynman diagrams are perturbation calculations. If the perturbations diverge, then a classical solution might be unstable.

Which brings us to the old topic if General relativity has any solutions under realistic matter fields.

Christodoulou and Klainerman (1990) proved the "nonlinear stability" of the Minkowski metric under General relativity.

This is a very interesting question: if Feynman diagrams with gravitons diverge, how could Christodoulou and Klainerman show the stability in the very restricted case of the Minkowski metric?

In physics, if we are calculating with two fields, we usually ignore the backreaction. If we calculate the behavior of a laser beam which climbs out of a gravitational field, we assume that the backreaction on the gravitational field is negligible. But it might be that a precise calculation shows the the solution is not stable.

 

[PeterDonis]

the solution might not be stable

a classical solution might be unstable.

it might be that a precise calculation shows the the solution is not stable

These are all speculations which cannot be usefully discussed in the absence of some actual concrete examples. Lots of things "might" be the case.

 

[PeterDonis]

Klainerman et al. proved the "nonlinear stability" of the Minkowski metric under General relativity.

Can you give a reference?

 

[Heikki Tuuri]

http://www.numdam.org/item/?id=SEDP_1989-1990____A15_0

The global nonlinear stability of the Minkowski space
Christodoulou, D.; Klainerman, S.
Séminaire Équations aux dérivées partielles (Polytechnique),(1989-1990), Talk no. 13, 29 p.

 

[PeterDonis]

It is this famous paper from 1990. Christodoulou and Klainerman.

This paper looks like a purely classical analysis, so the properties of Feynman diagrams in a quantum field theory of gravitons are irrelevant.