Does a stationary charge in a gravitational field emit radiation?

[GRDixon]

An isolated charge, accelerated by a constant force, theoretically radiates (Larmor). Does the same charge, held at rest in a gravitational field, constantly radiate?

 

[Dale]

How do you intend to detect the radiation?

 

[Count Iblis]

It does not radiate because the large distance behavior of the fields in the latter case do not correspond to what you would get from the former case by applying the equivalence principle.

 

[atyy]

The EP does not apply to charged particles. Nonetheless: http://arxiv.org/abs/physics/9910019

 

[GRDixon]

How do you intend to detect the radiation?

If you refer to the radiation emitted (or not emitted) in the gravitational field, I envisioned placing the charge in a blackened container and monitoring to see if the container's temperature increases (up to a limit).

 

[sylas]

An isolated charge, accelerated by a constant force, theoretically radiates (Larmor). Does the same charge, held at rest in a gravitational field, constantly radiate?

Great question! [strike]The equivalence principle DOES apply, and the charge does radiate.[/strike] Reference given by atyy.

• Radiation from a Charge in a Gravitational Field by Amos Harpaz, Noam Soker (1999) in Gen.Rel.Grav. 36 (2004) 315-330. (arXiv:physics/9910019v1)

Caveat: This paper is new to me. I am going to check up further and make sure I understand it better; hence this answer should be understood simply as the answer of the reference, and not of me personally. I do not have the expertise to confirm it independently. But I'll check around.

Cheers -- sylas

Postscript. I am speaking above out of turn; and so I have struck out the sentence above for which I do not have sufficient confidence to give such a definite answer. Apologies, I shall focus for now on reading rather than giving answers.

 

[Count Iblis]

I think this paper is better:

http://arxiv.org/abs/gr-qc/9303025

Put simply, if you're going to apply the equivalence principle, you have to do it correctly, and thus also look at the proper boundary conditions/initial conditions on the fields.

 

[Count Iblis]

And that paper can now be updated as the ancient problem of the electromagnetic self force which is responsible for the radiation of accelerated charges, has now been solved rigorously:

http://arxiv.org/abs/0905.2391

 

[atyy]

Yes, I like Parrott's paper, which is one of those I had in mind when I said the EP does not apply to charged particles (but you seem to have drawn a different conclusion?). In my understanding the Harpaz and Soker paper is just an amusing case where by accident the EP "applies" to a charged particle.

 

[sylas]

I think this paper is better:

http://arxiv.org/abs/gr-qc/9303025

Put simply, if you're going to apply the equivalence principle, you have to do it correctly, and thus also look at the proper boundary conditions/initial conditions on the fields.

Thank you! I withdraw my previous answer, and shall look further.

Cheers -- sylas

 

[atyy]

Some more references, in addition to Parrott's given above by Count Iblis.

The qualification "uncharged" in the statement of the EP in http://arxiv.org/abs/0707.2748

The EP does not apply to charged particles (unless it's a gravitational charge):
http://relativity.livingreviews.org/Articles/lrr-2004-6/
http://arxiv.org/abs/0806.0464
http://arxiv.org/abs/gr-qc/0008065

Also in Rindler's and J L Martin's GR texts (though Fredrik has now made me suspicious of Rindler ... )

 

[Bob S]

An isolated charge, accelerated by a constant force, theoretically radiates (Larmor). Does the same charge, held at rest in a gravitational field, constantly radiate?

If the charge is held at rest, meaning mgh = constant, then where would the radiated energy come from? Wouldn't conservation of energy apply?
Bob S

 

[heldervelez]

Why the restriction 'isolated' charge?

I think it does the same for all charges in any configuration subject to acceleration.

And as all matter of the Universe is accelerated at all times ...

 

[heldervelez]

the conservation of energy...
the energy has to be of such a tiny value that I think is not measurable, and all instruments of measure also vary in the same way (the reference atom at lab) giving a null result.

 

[GRDixon]

If the charge is held at rest, meaning mgh = constant, then where would the radiated energy come from? Wouldn't conservation of energy apply?
Bob S

Those were questions I had in mind when I submitted the thread. The majority consensus seems to be that the EP doesn't apply to electric charge. I'll give some thought to an analogous situation that doesn't involve electric charge. Thanks to all for responding, and for the links.

 

[yuiop]

Those were questions I had in mind when I submitted the thread. The majority consensus seems to be that the EP doesn't apply to electric charge. I'll give some thought to an analogous situation that doesn't involve electric charge. Thanks to all for responding, and for the links.

Is it just coincidence that the article http://www.maxwellsociety.net/Charge%20and%20the%20Equivalence%20Principle.html" that you linked to in the other thread on the same subject, was authored by someone called G.R.Dixon?

 

[bcrowell]

I've spent some time studying this, and although the mathematical treatment of charged particles moving in curved spacetime is heinously complex, and I haven't dug into it, I think I've gotten to the point where I understand the issue reasonably well at the conceptual level. Of course this stuff is very subtle, so most likely I'm making mistakes :-), but I felt confident enough to incorporate an explanation in my book, http://www.lightandmatter.com/html_books/genrel/ch01/ch01.html#Section1.5 .

Here's a slightly edited version of what I'm saying in the book:

----------------

The equivalence principle is not a single, simple, mathematically well defined statement. As an example of an ambiguity that is still somewhat controversial, 90 years after Einstein first proposed the principle, consider the question of whether or not it applies to charged particles. Raymond Chiao http://arxiv.org/abs/quant-ph/0601193v7 proposes the following thought experiment. Let a neutral particle and a charged particle be set, side by side, in orbit around the earth. If the equivalence principle applies regardless of charge, then these two particles must go on orbiting amicably, side by side. But then we have a violation of conservation of energy, since the charged particle, which is accelerating, will radiate electromagnetic waves (with very low frequency and amplitude). It seems as though the particle's orbit must decay.

The resolution of the paradox, as demonstrated by detailed calculations by Gron and Naess http://arxiv.org/abs/0806.0464 is interesting because it exemplifies the local nature of the equivalence principle. When a charged particle moves through a gravitational field, in general it is possible for the particle to experience a reaction from its own electromagnetic fields. This might seem impossible, since an observer in a frame momentarily at rest with respect to the particle sees the radiation fly off in all directions at the speed of light. But there are in fact several different mechanisms by which a charged particle can be reunited with its long-lost electromagnetic offspring. An example (not directly related to Chiao's scenario) is the following.

Bring a laser very close to a black hole, but not so close that it has strayed inside the event horizon at rH. It turns out that at r=(3/2)RH, a ray of light can have a circular orbit around the black hole. Since this is greater than RH, we can, at least in theory, hold the laser stationary at this value of r using a powerful rocket engine. If we point the laser in the azimuthal direction, its own beam will come back and hit it.

Since matter can experience a back-reaction from its own electromagnetic radiation, it becomes plausible how the paradox can be resolved. The equivalence principle holds locally, i.e., within a small patch of space and time. If Chiao's charged and neutral particle are released side by side, then they will obey the equivalence principle for at least a certain amount of time --- and "for at least a certain amount of time" is all we should expect, since the principle is local. But after a while, the charged particle will start to experience a back-reaction from its own radiated electromagnetic fields. Since Chiao's particles are orbiting the earth, and the Earth is not a black hole, the mechanism clearly can't be as simple as the one described above, but Gron and Naess show that there are similar mechanisms that can apply here, e.g., scattering of light waves by the nonuniform gravitational field.

[later...]

The equivalence principle says that electromagnetic waves have gravitational mass as well as inertial mass, so it seems clear that the same must hold for static fields. In Chiao's paradox (p. 26), the orbiting charged particle has an electric field that extends out to infinity. When we measure the mass of a charged particle such as an electron, there is no way to separate the mass of this field from a more localized contribution. The electric field "falls" through the gravitational field, and the equivalence principle, which is local, cannot guarantee that all parts of the field rotate uniformly about the earth, even in distant parts of the universe. The electric field pattern becomes distorted, and this distortion causes a radiation reaction which back-reacts on the particle, causing its orbit to decay.

---------------

So if the question is whether the equivalence principle applies to charged particles or not, I think the answer is not a yes/no answer. You have to keep in mind that the equivalence principle only applies locally, and this can be tricky to translate into actual experiments. A lot of thought experiments that might seem purely local are actually nonlocal, so they appear to violate the equivalence principle. If the e.p. were a mathematically well defined statement, then we'd be able to define "local" and "nonlocal" in a rigorous way, and probably everyone would have to agree on a yes/no answer. However, nobody has ever succeeded in stating the e.p. in a mathematically rigorous way.

I like Chiao's thought experiment a lot better than the ones involving linear acceleration. I think it's conceptually simpler. For one thing, there's been some debate over the significance of horizons in the linear case. Boulware says that the radiation disappears behind the event horizon of the accelerated observer, where it can't be observed, but Parrott argues that this is wrong.

 

[Dale]

If you refer to the radiation emitted (or not emitted) in the gravitational field, I envisioned placing the charge in a blackened container and monitoring to see if the container's temperature increases (up to a limit).

Well, I don't know about using heat to measure it. I was thinking about measuring it with an antenna. My guess is that a charge accelerating past a stationary antenna will induce the same signal in the antenna as an antenna falling past a stationary charge.

 

[Bob S]

Well, I don't know about using heat to measure it. I was thinking about measuring it with an antenna. My guess is that a charge accelerating past a stationary antenna will induce the same signal in the antenna as an antenna falling past a stationary charge.

Very interesting comment. My guess is that a charge moving past a stationary antenna at constant velocity will induce the same signal in the antenna as an antenna falling past a stationary charge at constant velocity. Both are Faraday induction of an electric field.

So how does the charge then radiate energy?

My guess is that a charge accelerating past a stationary antenna will radiate the same signal in the antenna as an antenna accelerating past a stationary charge.

Bob S

 

[pellman]

I have not read the linked papers yet. (I'm in the middle of "A Rigorous Derivation of Electromagnetic Self-force". So far, so good.) But here is my question.

If we are asking, "Does a charged particle radiate when it accelerates?" we have to specify: accelerate relative to what?

If it has to be relative to an inertial frame, ... well, the only (locally) inertial frame we have in a (non-uniform) gravitational field is the freely falling one. So I would say that the equivalence principle has to hold because .. what else is there?

This business about local vs non-local.. the equivalence principle is local, Maxwell's equations are local (curved spacetime or flat), so what is there that is nonlocal?

 

[Dale]

My guess is that a charge accelerating past a stationary antenna will radiate the same signal in the antenna as an antenna accelerating past a stationary charge.

Well, that is not exactly what I was saying, but that may be correct also. I really don't know.

I was thinking of a non-inertial charge and an inertial antenna in both cases, one where the charge was accelerating in the absence of gravity (e.g. on a rocket), and the other where the charge was stationary on the Earth and the antenna was in free fall. But it could be that for EM it doesn't matter which is inertial and which is non-inertial. I honestly don't have a good enough feel for it without working out the equations in detail.

So how does the charge then radiate energy?

In the absence of gravity case you would probably say that the energy comes from the work done by the rocket, and in the gravity case you would probably say that the energy comes from the lost gravitational potential energy of the antenna. But again, I don't have a good feel for it.

I expect that in both cases a co-accelerating antenna will detect no radiation. That should be easier to analyze than a relatively moving antenna.

 

[atyy]

The EP does not apply to charged particles (unless the charge is a gravitational charge). Is there such a thing as a freely falling charge? No. Because the charge is in its own electric field (roughly speaking).

 

[GRDixon]

Is it just coincidence that the article http://www.maxwellsociety.net/Charge%20and%20the%20Equivalence%20Principle.html" that you linked to in the other thread on the same subject, was authored by someone called G.R.Dixon?

No, it's not a coincidence. I authored the article and linked to it rather than type in a voluminous reply here. I hope that's an OK thing to do. George.

 

[atyy]

I like Chiao's thought experiment a lot better than the ones involving linear acceleration. I think it's conceptually simpler. For one thing, there's been some debate over the significance of horizons in the linear case. Boulware says that the radiation disappears behind the event horizon of the accelerated observer, where it can't be observed, but Parrott argues that this is wrong.

Yes, the orbiting case is simpler, and Rindler uses that in his textbook too. I think Boulware is right about the horizons and the detectability of the radiation. Gron, with Eriksen, is one of those who agree with Boulware's conclusion (ref [7] in Gron and Naess), also Almeida and Saa's http://arxiv.org/abs/physics/0506049. Nonetheless, I think the horizons are irrelevant to saving the EP for charged particles. I would agree with Gron and Naess's "As measured by this observer the charge is at rest in an inertial frame, and hence it does not radiate. However, a thorough analysis proves that this is not the solution to the paradox. It has been shown that, as measured by an observer that is not falling freely, a freely falling charge radiates with a power given by Larmor’s formula... The principle of equivalence has a local character. The mentioned equivalence is only valid as far as the measurements does not reveal a possible curvature of space." Try also section 5.5.3 in http://relativity.livingreviews.org/Articles/lrr-2004-6/ "In the scalar and electromagnetic cases, the picture of a particle interacting with a radiative field removes any tension between the nongeodesic motion of the charge and the principle of equivalence."

 

[bcrowell]

This business about local vs non-local.. the equivalence principle is local, Maxwell's equations are local (curved spacetime or flat), so what is there that is nonlocal?

There are different ways of defining "local." After all, you can determine the Riemann tensor by "local" measurements, but the e.p. is specifically supposed to be talking about things that are "more local" than that. The e.p. basically says you can do a local linear approximation to the structure of spacetime, just like the freshman calculus idea of approximating a function with a line. But that doesn't mean that the second derivative isn't also a "local" quantity.

AFAICT the reason that we see this long string of papers all disagreeing with one another is completely because nobody knows the right way to define "local" where it occurs in the equivalence principle. A good example of this is Parrott's criticism of Boulware's idea about how the radiation gets dumped behind a horizon where you can't see it. He says that the distinction between the near-field region and the radiation region is only an approximate one. He argues that any analytic function can be determined everywhere from knowledge of all its derivatives at a point, so an observer in the "near field" region can actually extrapolate to find the radiation field. IMO this is kind of over-reaching, because by the same argument you could do local measurements of all the derivatives of the metric (well, modulo gauge) and determine the metric everywhere in space. The equivalence principle would then be meaningless, because local experiments would all come out different from one another. In real measurements, if you try to determine the 37th derivative of something by local measurements, you're going to get killed by random errors. If this general idea were easy to formalize in the case of the e.p., I think someone would have done it by now, and we wouldn't see people like Parrott politely saying that people like Boulware are idiots, and vice versa.

If we are asking, "Does a charged particle radiate when it accelerates?" we have to specify: accelerate relative to what?

I think the basic answer is: accelerate relative to what Isaac Newton would have considered to be a good, God-fearing inertial frame. This seems to give the right answer for all the cases I have in mind (charge lying on a table, charge in an accelerating elevator, charge orbiting the earth). The other rule of thumb that seems to work is just to check conservation of energy. All of these answers are completely inconsistent with how we usually think about the e.p., but I think they just go to show that the e.p. has fuzzy boundaries.

 

[atyy]

A good example of this is Parrott's criticism of Boulware's idea about how the radiation gets dumped behind a horizon where you can't see it. He says that the distinction between the near-field region and the radiation region is only an approximate one. He argues that any analytic function can be determined everywhere from knowledge of all its derivatives at a point, so an observer in the "near field" region can actually extrapolate to find the radiation field. IMO this is kind of over-reaching, because by the same argument you could do local measurements of all the derivatives of the metric (well, modulo gauge) and determine the metric everywhere in space.

But the argument from derivatives is not so different from Chiao's argument that radiation can only be detected non-locally. Basically Parrott's, Gron and Naess, Chiao's etc resolutions are either EP does not hold/does not apply because it's intrinsically non-local/non-free-falling etc. I suppose there are subtle differences between "does not hold" and "does not apply", but I'd say those are all quite different from the resolution of say Almeida and Saa, or Peierl's (one of his surprises in theoretical physics) which says the EP does apply and hold, and the horizons save it.

 

[bcrowell]

But the argument from derivatives is not so different from Chiao's argument that radiation can only be detected non-locally. Basically Parrott's, Gron and Naess, Chiao's etc resolutions are either EP does not hold/does not apply because it's intrinsically non-local/non-free-falling etc. I suppose there are subtle differences between "does not hold" and "does not apply", but I'd say those are all quite different from the resolution of say Almeida and Saa, or Peierl's (one of his surprises in theoretical physics) which says the EP does apply and hold, and the horizons save it.

To me, the take-home thought from all this is that when we try to understand how gravity works, we understand it in terms of ideas like the e.p. and Mach's principle, but that those ideas are not crisp, well-defined mathematical statements. In general, all you can say is that our universe is highly e.p.-ish and fairly non-Mach-ish (in the sense that the Brans-Dicke $\omega$ parameter, which measures non-Mach-ish-ness, is constrained by the latest measurements to be at least 40000).

 

[atyy]

To me, the take-home thought from all this is that when we try to understand how gravity works, we understand it in terms of ideas like the e.p. and Mach's principle, but that those ideas are not crisp, well-defined mathematical statements. In general, all you can say is that our universe is highly e.p.-ish and fairly non-Mach-ish (in the sense that the Brans-Dicke $\omega$ parameter, which measures non-Mach-ish-ness, is constrained by the latest measurements to be at least 40000).

Yeah, I think so. Another reference I found very helpful was section 24.7 "Curvature Coupling Delicacies in the Equivalence Principle" of http://www.pma.caltech.edu/Courses/ph136/yr2004/ .

 

[yuiop]

If the equivalence principle applies regardless of charge, then these two particles must go on orbiting amicably, side by side. But then we have a violation of conservation of energy, since the charged particle, which is accelerating, will radiate electromagnetic waves (with very low frequency and amplitude). It seems as though the particle's orbit must decay.

You seem to be assuming that an orbiting charged particle must be accelerating, but that is not necessarily the case. The orbiting particle is in free fall and there is no force due to gravity in GR. The particle simply follows a geodesic which is in some sense equivalent to a straight line in curved space and therefore the particle is not accelerating and is equivalent to an inertial frame and so there is no reason why the particle should radiate. Therefore the EP is not violated and conservation of energy is not violated and no paradox. Bear in mind that a charged particle on the rotating disc in flat space, is a very different situation from an orbiting charged particle in a gravitational field.

The resolution of the paradox, as demonstrated by detailed calculations by Gron and Naess http://arxiv.org/abs/0806.0464 is interesting because it exemplifies the local nature of the equivalence principle. When a charged particle moves through a gravitational field, in general it is possible for the particle to experience a reaction from its own electromagnetic fields. This might seem impossible, since an observer in a frame momentarily at rest with respect to the particle sees the radiation fly off in all directions at the speed of light. But there are in fact several different mechanisms by which a charged particle can be reunited with its long-lost electromagnetic offspring.

Here you seem to be saying that the orbiting charged particle can radiate energy and later recover its energy. This in turn seems to suggest that the EP can saved because the particle loses no energy and therefore there is no need for the orbit to decay relative to an uncharged particle. However, your further statements indicate this is not the line of thought you are pursuing.

An example (not directly related to Chiao's scenario) is the following.

Bring a laser very close to a black hole, but not so close that it has strayed inside the event horizon at rH. It turns out that at r=(3/2)RH, a ray of light can have a circular orbit around the black hole. Since this is greater than RH, we can, at least in theory, hold the laser stationary at this value of r using a powerful rocket engine. If we point the laser in the azimuthal direction, its own beam will come back and hit it.

Since matter can experience a back-reaction from its own electromagnetic radiation, it becomes plausible how the paradox can be resolved. The equivalence principle holds locally, i.e., within a small patch of space and time. If Chiao's charged and neutral particle are released side by side, then they will obey the equivalence principle for at least a certain amount of time --- and "for at least a certain amount of time" is all we should expect, since the principle is local. But after a while, the charged particle will start to experience a back-reaction from its own radiated electromagnetic fields. Since Chiao's particles are orbiting the earth, and the Earth is not a black hole, the mechanism clearly can't be as simple as the one described above, but Gron and Naess show that there are similar mechanisms that can apply here, e.g., scattering of light waves by the nonuniform gravitational field.

[later...]

The equivalence principle says that electromagnetic waves have gravitational mass as well as inertial mass, so it seems clear that the same must hold for static fields. In Chiao's paradox (p. 26), the orbiting charged particle has an electric field that extends out to infinity. When we measure the mass of a charged particle such as an electron, there is no way to separate the mass of this field from a more localized contribution. The electric field "falls" through the gravitational field, and the equivalence principle, which is local, cannot guarantee that all parts of the field rotate uniformly about the earth, even in distant parts of the universe. The electric field pattern becomes distorted, and this distortion causes a radiation reaction which back-reacts on the particle, causing its orbit to decay.

OK, you have solved the paradox by explaining that the EP is not violated because the EP applies locally and the radiation of a charged particle being accelerated by a gravitational field is a non local effect, but now you seem to have to introduced a new paradox. You stated earlier that the orbiting charged particle has mechanisms to recover it energy and yet its orbit still decays. Why go to all the trouble of explaining complex mechanisms to recover energy if you want to conclude the orbit decays. Why not just say the charged particle radiates energy and as a result of the loss of energy, its orbit decays?

So if the question is whether the equivalence principle applies to charged particles or not, I think the answer is not a yes/no answer.

Actually the question posed by the OP can be paraphrased as "Does a charged particle that is stationary in a gravitational field (GF) radiate or not? Yes or no?" and you seem to have avoided that question. However, you are not alone, as most of the papers linked to in this thread avoid answering that direct question too. I hope you do not think I am being too critical of your book which I think is generally well written and presented and you can not be blamed for being a bit sketchy on this issue, as most of the literature on this isssue is also sketchy or downright contradictory.

I would answer the OP's question with another question, which is "From whose point of view?" The question should actually be split into two questions:

Q1) Does a charged particle that is stationary in a GF radiate, from the point of view of an observer that is also stationary in the GF?
Q2) Does a charged particle that is stationary in a GF radiate, from the point of view of an observer that free falling in the GF?

The answer to those 2 questions is not necessarily the same.

This becomes clearer when you consider a “well known paradox”which is nicely described by Chiao in this paper: http://arxiv.org/abs/quant-ph/0601193v7

However, a well-known paradox now arises when we ask the following
question: Is it the falling charged object, or is it the stationary charged object
at rest on the ground, that radiates electromagnetic waves?
On the one hand, a freely-falling observer, who is co-moving with the freely
falling neutral and charged objects, sees these two objects as if they were freely
floating in space. The falling charged object would therefore appear to him
not to be accelerating, so that he would conclude that it is not this charge
which radiates. Rather, when he looks downwards at the charged object which
is at rest on the ground, he sees a charge which is accelerating upwards with
an acceleration −g towards him. He would therefore conclude that it is the
charged object at rest on the ground, and not the falling charge, that is radiating
electromagnetic radiation.
On the other hand, an observer on the ground would come to the opposite
conclusion. She sees the falling charge accelerating downwards with an acceleration
g towards her, whereas the charged object at rest on the ground does not
appear to her to be undergoing any acceleration. She would therefore conclude
that it is the falling charge which radiates electromagnetic radiation, and not the charge which is resting on the ground. Which conclusion is the correct one?

Unfortunately he does not explicitly solve the paradox in his paper and digresses off into a new design for a gravitational wave detector. Nevertheless, one interesting possibiltiy is to declare both points of view as valid and not mutually exclusive. In other words radiation may be relative. What appears to be radiating to one observer, is not necessarily radiating according to another observer. If this seems too radical, consider the Unruh effect. An accelerating observer in flat space sees radiation in the form of photons and massive particles coming from an event horizon behind him, while an observer in the same flat space does not see or detect any of the radiation particles. That might seem paradoxical in itself, but the Unruh effect is considered to be a serious possibility by many physicists.

Here is an extended list of closely related questions:

Q1) Does a charged particle that is stationary in a GF radiate, from the point of view of an observer that is also stationary in the GF?
Q2) Does a charged particle that is stationary in a GF radiate, from the point of view of an observer that is free falling in the GF?
Q3) Does a charged particle that is free falling in a GF radiate, from the point of view of an observer that is also free falling in the GF?
Q4) Does a charged particle that is free falling in a GF radiate, from the point of view of an observer that is stationary in the GF?
Q5) Does a charged particle that is accelerated in flat space radiate, according to an inertial observer?
Q6) Does a charged particle that is accelerated in flat space radiate, according to an co-accelerating observer?
Q7) Does the orbit of a charged particle in a gravitational field decay faster than the orbit of an uncharged particle?

I would be interested in what the members of this forum think the answers to those 7 (yes/no) questions are. That might put us some solid ground, before we move on to what the implications for the EP are.

 

[bcrowell]

I hope you do not think I am being too critical of your book which I think is generally well written and presented and you can not be blamed for being a bit sketchy on this issue, as most of the literature on this isssue is also sketchy or downright contradictory.

Hi, kev -- No need to apologize; I explicitly asked for comments, so constructive criticism is most welcome :-)

You seem to be assuming that an orbiting charged particle must be accelerating, but that is not necessarily the case.

I'm basically taking Gron and Naess's analysis at face value. They calculate the whole thing in gory detail, and they find that the charge does radiate. I probably didn't make it clear enough that I thought the charge *did* radiate. I've added the following text to clarify that: "But after a while, the charged particle will start to experience a back-reaction from its own electromagnetic fields, and this causes its orbit to decay, satisfying conservation of energy."

Here you seem to be saying that the orbiting charged particle can radiate energy and later recover its energy.

That wasn't what I intended. Maybe the edit above will clarify that. The back-reaction is a mechanism that allows it to *lose* energy.

Why not just say the charged particle radiates energy and as a result of the loss of energy, its orbit decays?

Hmm...from the point of view of nonrelativistic mechanics, we expect that given an initial-value problem, there is only one possible motion that simultaneously satisfies all the conservation laws. This comes up when I teach freshman engineering physics, because I use an unusual order of topics that does conservation laws first, and then Newton's laws. For the most part uniqueness holds pretty straightforwardly, but you do get some pathological cases that are harder to rule out. For instance, if you release a rock above the ground, you can satisfy conservation of energy and momentum in all inertial frames of reference if the rock simply hovers. I think the question of radiation by an orbiting particle is similar. It could satisfy all the conservation laws while failing to radiate, and yet it would fail to satisfy the field equations.

Actually the question posed by the OP can be paraphrased as "Does a charged particle that is stationary in a gravitational field (GF) radiate or not? Yes or no?" and you seem to have avoided that question.

The answer is clearly no in the lab frame of reference, based on conservation of energy. It's conceivable that in other frames of reference, the answer is different, because of horizons.

However, you are not alone, as most of the papers linked to in this thread avoid answering that direct question too.

As far as I can tell, all the papers agree that a charge sitting on my desk does not radiate. I think they also all agree that if you see a charge whiz by you, being accelerated by a rocket engine, it does radiate. These are both 100% solid conclusions based on classical electromagnetism, and they're consistent with conservation of energy. I think the real disagreement comes when you talk about an observer aboard the rocket.

Q1) Does a charged particle that is stationary in a GF radiate, from the point of view of an observer that is also stationary in the GF?
Q2) Does a charged particle that is stationary in a GF radiate, from the point of view of an observer that free falling in the GF?

Q1 - No. Conservation of energy forbids it from radiating.

Q2 - This thought experiment has the advantage of getting rid of the horizon that occurs in the version with the constant-acceleration observer, but the disadvantage of introducing curved spacetime. I'm not sure it's a well posed question. The distinction between a radiative field and a nonradiative field is normally stated in the context of Maxwell's equations in an inertial frame in flat space. It's not obvious to me whether that can be generalized in a useful way to this context. The observer only has a finite amount of time to observe the fields while free-falling, and has no choice of what distances to observe at; it's not obvious to me whether that's sufficient to distinguish a radiative from a nonradiative field. It's conceivable that the observer could see very weak radiation, but that the source of energy for the radiation would be the observer's own kinetic energy (but maybe this could be ruled out by considering how the effect would scale with the size of the observer's antenna).

Unfortunately he does not explicitly solve the paradox in his paper and digresses off into a new design for a gravitational wave detector.

But Gron and Naess do explicitly solve it.

Nevertheless, one interesting possibiltiy is to declare both points of view as valid and not mutually exclusive. In other words radiation may be relative. What appears to be radiating to one observer, is not necessarily radiating according to another observer. If this seems too radical, consider the Unruh effect. An accelerating observer in flat space sees radiation in the form of photons and massive particles coming from an event horizon behind him, while an observer in the same flat space does not see or detect any of the radiation particles. That might seem paradoxical in itself, but the Unruh effect is considered to be a serious possibility by many physicists.

This possible frame-dependence of the radiative character of a wave is sort of what I had in mind in my answer to Q2 above. However, I don't think the Unruh effect is the right way to view this, because that's a purely quantum-mechanical effect (and far too weak to observe in any practical experiment anyone has been able to devise), whereas Q2 is posed as a classical problem.

 

[atyy]

my book, http://www.lightandmatter.com/html_books/genrel/ch01/ch01.html#Section1.5

How did you get the photo of the butterfly on your forehead?

 

[bcrowell]

How did you get the photo of the butterfly on your forehead?

Here: http://www.calacademy.org/

 

[atyy]

Here: http://www.calacademy.org/

Wow, I used to run by there all the time (their old buildings), but somehow never stopped to take a look inside.

 

[yuiop]

The Gron and Naess paper http://arxiv.org/abs/0806.0464 states

"It has been shown that, as measured by an observer
that is not falling freely, a freely falling charge radiates
with a power given by Larmor’s formula, and it extension
to a non-inertial reference frame. No generalization to curved
spacetime has been made, but since the above result shows
that it does not matter for the radiated effect whether a
charge is accelerated by gravitational or normal forces in flat
spacetime, we will assume that this also holds.."

My interpretation of the above paragraph and their conclusions about an orbiting charge would lead me to answer the extended question list like this:

Q1) Does a charged particle that is stationary in a GF radiate, from the point of view of an observer that is also stationary in the GF? N
Q2) Does a charged particle that is stationary in a GF radiate, from the point of view of an observer that is free falling in the GF? N
Q3) Does a charged particle that is free falling in a GF radiate, from the point of view of an observer that is also free falling in the GF? N
Q4) Does a charged particle that is free falling in a GF radiate, from the point of view of an observer that is stationary in the GF? Y
Q5) Does a charged particle that is accelerated in flat space radiate, according to an inertial observer? Y
Q6) Does a charged particle that is accelerated in flat space radiate, according to an co-accelerating observer? N
Q7) Does the orbit of a charged particle in a gravitational field decay faster than the orbit of an uncharged particle? Y
Q8) Does a charged inertial particle in flat space appear to radiate from the POV of an observer that is accelerating towards the charge? N
Q9) Does a charged particle "fall" at the same rate as a neutral particle, as measured by an observer in an accelerating rocket in flat space? Y
Q10) Does a charged particle fall vertically at the same rate as a neutral particle in a gravitational field? N

Do my answers (N,N,N,Y,Y,Y,Y,N,Y,N) seem correct in the context of that particular paper?

Anyone else want to have a stab at the answers in the context of their favorite paper or book?

 

[heldervelez]

...

Does a charged particle ... radiate, from the point of view of an observer that ...?
...

No to all answers.

by simple reasoning:

radiation is a form of energy release into space.
then at anyone instant of time the particle radiate or-exclusive do not radiate.
it can not be the observer that determine the radiative state of the particle
Why?
1 - I is not accepted the existence of a simultaneous multiple reality
2 - the observer is not at the particle position than only allowing action at a distance the observer can notify the particle of its presence (and state of moving)

The whole story of radiation deserves a neat clarification (not now)
because I see everywhere a lot of misconceptions

I leave a question:
If a particle sometimes radiate energy then... when does the particle refuel ?