When is the radiation from an accelerated charge a matter of controversy?

[Stingray]

Perhaps it would be helpful to transform the Coulomb field into an accelerating frame. This should be very simple, and will answer any questions about uniform gravitational fields.

 

[pervect]

The equivalence principle suggests that a charge at rest in a uniform gravitational field radiates and it is a very good question as to where this energy comes from. There's been a lot of rubbish written about energy in general relativity and I am reluctant to add to it. But here goes. A universe contains only a charge sitting at rest near a planet's surface and radiates.

What makes you think that an observer at infinity would see any radiation? The charge is not accelerating relative to said observer.

 

[pmb_phy]

Hi Reilly

Perhaps you're forgetting one major point. Focus not on what exists. Focus on what one can detect. If you can't detect it then how can you really say that its there? I read an interesting article on this subject.

Radiation from an Accelerated Charge and the Principle of Equivalence, A. Kovetz and G.E. Tauber, Am. J. Phys., Vol. 37(4), April 1969

It may be interesting to discuss the foregoing results from a different view point - that of the photon picture: Is it possible that one of the two observers we have been considering counts a number of photons, while the other, looking at the same charge, does not encounter any of them? In order to answer this question we take the case of a supported charge charge and the supported observer. Projecting the four-potential of the Born field onto the orthonomal tetrad carried by the supported observer, we find that only the fourth component is different than zero. This means that a radiation detector carried by the observer will not record any transitions in which transverse photons are involved. This is the quantum electrodynamical explanation for the absence of radiation from the supported charge. It is not enough that photons are there; to be observable, they must be of the transverse kind, and this property (like the nonvanishing of a magnetic field) is not Lorentz invariant.

Pete

 

[Rob Woodside]

What makes you think that an observer at infinity would see any radiation? The charge is not accelerating relative to said observer.

It is the suggestion of the equivalence principle that a charge in a uniform gravitational field will radiate, if a uniformly accelerated charge does. I've been looking at some of the references supplied by Pete and Andrew, as well as arXiv, and are things all over the map! What makes you think that radiation is observer dependent? It seems that whatever position you take on these questions, you can find a like minded reference and a few diametrically opposite.

 

[Rob Woodside]

Perhaps it would be helpful to transform the Coulomb field into an accelerating frame. This should be very simple, and will answer any questions about uniform gravitational fields.

An excellent suggestion. I think this has been done (Fulton & Rohrlich?) and maybe the conclusion was that a charge uniformly accelerated along its velocity did not emit. I have yet to get the references together and study them. I notice there is no applet for this acceleration. If the radiation is "absolute" in the sense of an atom or charge, then the principle of general relativity will not allow you to create it by a mere coordinate change.

By the way there is a reference at arXiv to a paper that agrees with Rohrlich but claims it is the relative acceleration of the charge and the field lines that produces the radiation. I'm not sure I buy it yet, but it is similar to your observation that it is the extended nature of the field that is responsible for the radiation. One of the early objections to black holes was the question : What kind of boundary conditions are you going to put on the open set around the missing point? You can play the same game in flat space and excise the point charge to be left with a topological charge and an electromagnetic field whose Hodge dual is closed and not exact. It should be the inner boundary conditions that provide the acceleration and the radiation! Maybe!

 

[pervect]

Photon number is NOT conserved by arbitrary coordinate translations (such as acceleration). This is why I would say that the presence or absence of radiation is not absolute.

I haven't studied the mathematics in detail, but I believe that the transformations that are used are called Bogolibuov transformations.

In the language of quantum mechanics, one observer may see virtual particles, while an accelerating observer may see real particles. The same issue comes up with "Unruh" radiation.

A manifestly covariant treatment of electrostatics such as the Liénard-Wiechert potentials would probably be the best way to explore these issues.

 

[Andrew Mason]

I was talking about the assumption that charged objects interact with their own fields. By using the word 'know,' I wasn't implying any kind of consciousness. What I meant was that a classical charged object (which means it's not a point) in Maxwell's theory interacts via well-defined rules which do not decompose the fields at a point into 'self' and 'external.' I also don't think it would make any sense at all to modify this part of the theory.

It is not obvious that an electron can interact with itself. Feynman struggled with the question whether an electron could interact with its own field. He thought it could not. His views on this evolved but I think at the heart of it was his view that the principle of least action could be used to replace the concept of a field so the field was not a physically real thing (in the absence of another charge, in which case there would be force and energy). I don't think he ever came to a satisfactory conclusion on this, but I could be wrong.

In order for an electron's field to interact with itself, there has to be some physcial reality to the 'field' - something that gives it a separate existence (ie. it represents the past position of the electron) from the 'present' electron. It has to 'linger' around the electron's past position. Feynman thought this was absurd and provided his own analysis based on the least action principle (the principle is explained by Feynman in his Lectures on Physics, Vol II, Ch. 19). He produced valid solutions that did not require an assumption that the electron interacted with its 'past'.

Now I am not going to second guess Feynman. He was one of the great physicists of the last century. So if he disagreed that an electron interacts with its own field, I don't think we can say it is an obviously correct assumption.

AM

 

[Andrew Mason]

Had a thought in reading one of the previous posts - can't find it now - maybe I am merely restating someone else's idea - with that apology - the idea is that perhaps there is a fundamental difference between an electron in a G field and an acceleration field - when we derive the classical radiation formula we tacitly assume it is the electron as a particle that is rapidly accelerated - the field adjusts to the acceleration rather than being simultaneously wafted along by the accelerating force - thus causing a crowding of the lines and a discontinuity that is explained as radiation. But in a G field, not only would the electron itself be in free fall, but the field itself would also be acted upon equally (both in free fall) - so there is no field distortion since there is no relative acceleration between the particle and the field; ergo, no radiation.

This raises a very good question which we have tossed around in the Speed of Gravity Controversy thread. Does the field of a single charge represent anything physical? In the presence of another charge, it certainly represents something (force, energy). But alone, it doesn't. It is just a mathematical potential (not potential energy, - that only exists in the presence of another charge). If it doesn't represent some kind of self energy, and I don't see how it can, then what is the basis for it being affected by gravitation (ie. separate and apart from the charge itself)?

AM

 

[Stingray]

In order for an electron's field to interact with itself, there has to be some physcial reality to the 'field' - something that gives it a separate existence (ie. it represents the past position of the electron) from the 'present' electron. It has to 'linger' around the electron's past position. Feynman thought this was absurd and provided his own analysis based on the least action principle (the principle is explained by Feynman in his Lectures on Physics, Vol II, Ch. 19). He produced valid solutions that did not require an assumption that the electron interacted with its 'past'.

Your comments led me to a Google search which brought up Feynman's Nobel speech: http://nobelprize.org/physics/laureates/1965/feynman-lecture.html . It's quite interesting, and discusses much more clearly what you were saying (I've only read about half of it so far though).

Anyway, I think you are misunderstanding Feynman's position a little bit. His ideas on classical electrodynamics were largely motivated by the problems that come up when trying to incorporate point particles into classical electrodynamics. The main reason for this had to do with quantum field theory. It was hoped that the renormalization issues there could be understood and removed if an appropriate classical theory was found (Dirac was especially fond of this idea in his later years.).

The logic was that quantum field theory was too complicated to apply to extended bodies, so points were used instead. These points had the problem of having an infinite self-energy. The same problem occurs in the classical theory of point particles, so it was hoped that modifying the classical theory would allow one to construct a better quantum theory. The easiest way to do this is to say that point particles simply don't interact with themselves. To make this experimentally consistent requires the logical contortions found in e.g. the Wheeler-Feynman absorber theory (since electrons do experience what are apparently self-force effects, and it was assumed that classical point particles would have something to do with electrons, this is not simple to do).

This theory is very counter-intuitive, and it's sole purpose is to allow point particles to exist without any ad hoc assumptions. It is otherwise essentially identical to the usual electrodynamics with retarded fields. I think it is clear that if we agree that classical physics should only be applied to continuous distributions of matter, then Feynman's theory is just a mathematical curiosity. There are no problems with self-interaction for finite bodies with purely retarded fields. So Feynman was postulating the existence of new physical object - a classical point particle - and saying that there was no DIRECT self-interaction only for this new object. I think this is a considerably weaker statement than the one you have been trying to make. It is also irrelevant to answering whether a lump of charge radiates in a graviational field, since the answer to that question would be the same in both interpretations.

 

[Stingray]

This raises a very good question which we have tossed around in the Speed of Gravity Controversy thread. Does the field of a single charge represent anything physical? In the presence of another charge, it certainly represents something (force, energy). But alone, it doesn't. It is just a mathematical potential (not potential energy, - that only exists in the presence of another charge). If it doesn't represent some kind of self energy, and I don't see how it can, then what is the basis for it being affected by gravitation (ie. separate and apart from the charge itself)?

The field energy adds inertia to the object...

Also, a charge with internal structure could spontaneously shoot some light off in a particular direction. The charge will then recoil all on its own. This sort of effect requires that the field carry energy and momentum independently of the existence of any other charges in the universe.

 

[Andrew Mason]

The field energy adds inertia to the object...

Also, a charge with internal structure could spontaneously shoot some light off in a particular direction. The charge will then recoil all on its own. This sort of effect requires that the field carry energy and momentum independently of the existence of any other charges in the universe.

A charge distribution represents energy and, therefore, inertia. That inertia would be contained in the field within the volume of space that contains the charge distribution. But I don't see how the field of a single point charge (ie the electron) can have self-energy. The electron, representing a point charge, cannot represent a charge distribution. There is no volume to distribute the charge over.

If think of the electron as a sphere containing its charge as being made up of point charges and if you let the mass of the electron equal the energy of this notional charge distribution, you get the classical electron radius of 2.8 E-15 m. This makes the electron too large. If you work out the self energy of the electron as a point charge (0 radius), using classical or quantum principles, you get infinite self energy. This is still seems to be one of the a great unanswered (or unsatisfactorily answered) questions in physics. Perhaps it is the wrong question.

So, while a charge distribution has energy and can, therefore, emit photons all by itself, that is not the case with an electron. It requires interaction with something (other than itself) to emit a photon.

AM

 

[Rob Woodside]

A charge distribution represents energy and, therefore, inertia. That inertia would be contained in the field within the volume of space that contains the charge distribution. But I don't see how the field of a single point charge (ie the electron) can have self-energy. The electron, representing a point charge, cannot represent a charge distribution. There is no volume to distribute the charge over.

The horrible Dirac delta function is a point like distribution. For many years I thought it was crazy to have a point electron. After all every real thing has volume and as Stingray pointed out it it was a simplifying assumption. I believed that eventually scattering data would show some structure. However, my beliefs have changed from looking at charged black holes. These things are pure field with a singularity at r=0 (Swcharzchild or Boyer-Lindquist coordinate). There is no charge, no current and only a Maxwell stress-energy tensor. The charge is topological, there is no hard nut of a J anywhere.

If think of the electron as a sphere containing its charge as being made up of point charges and if you let the mass of the electron equal the energy of this notional charge distribution, you get the classical electron radius of 2.8 E-15 m. This makes the electron too large. If you work out the self energy of the electron as a point charge (0 radius), using classical or quantum principles, you get infinite self energy. This is still seems to be one of the a great unanswered (or unsatisfactorily answered) questions in physics. Perhaps it is the wrong question.

This classical radius of the electron is a nonsense. You integrate the energy density of the field down to a radius such that the field energy equals the rest energy of the electron. When you kick an electron it responds with inertia m and the field takes a long time to reorganize itself, spreading out at c. You aren't accelerating the field, you are accelerating the electron! So finding a radius such that the field energy OUTSIDE this radius equals mc^2, is cute and doable, but not physical. (When spin came along this classical radius was too small! If spin was rotational angular momentum and the electron a rotating fluid, then the equator would be moving at ~70c. This was why Lorentz told Uhlenbeck and Goodschmidt to publish in Dutch and not German and why Pauli initially called spin a joke. We now know better.)

Look at the spherical charged black hole (Riessner Nordstrom solution). Here there is a perfectly good gravitational potential energy. In geometric units it is 2M/r-Q^2/ r^2 which integrates to negative infinity. Without the charge it integrates to positive infinity. The energy density of the electromagnetic field is Q^2/r^4 which also integrates to positive infinity. Clearly we are integrating the wrong things. All these integrands have loose indices and are not well defined outside Swcharzchild coordinates. However there are integrands like *F that when integrated over a closed surface enclosing r=0 give finite answers. In the case of *F it is the topological charge Q. There are others that give the mass M and it too is topological. I suspect that if you could kick one of these charged black holes; it, like the electron, would respond with inertia M. Eventually when these ideas are further worked out and sink in this renormalizing problem might be solved.

 

[Rob Woodside]

Photon number is NOT conserved by arbitrary coordinate translations (such as acceleration). This is why I would say that the presence or absence of radiation is not absolute.

I haven't studied the mathematics in detail, but I believe that the transformations that are used are called Bogolibuov transformations.

In the language of quantum mechanics, one observer may see virtual particles, while an accelerating observer may see real particles. The same issue comes up with "Unruh" radiation.

A manifestly covariant treatment of electrostatics such as the Liénard-Wiechert potentials would probably be the best way to explore these issues.

Not conserving photon number is something I have to look at. Their Gibbs free energy is zero, so the is no hindrance to their production or destruction. I can certainly see excavating them out of a quantum backgound as you accelerate through it. But I am not yet sure how this relates to this classical radiation. I could totally wrong, but my prejudice is for an invariant charaterization of this radiation.

 

[Stingray]

A charge distribution represents energy and, therefore, inertia. That inertia would be contained in the field within the volume of space that contains the charge distribution. But I don't see how the field of a single point charge (ie the electron) can have self-energy. The electron, representing a point charge, cannot represent a charge distribution. There is no volume to distribute the charge over.

Well, if you insist on point particles, then you're right that things don't make any sense. That's why I think that they shouldn't be allowed into classical physics. For everything that the theory can honestly be applied to, it works out fine.

Electrons should be described by quantum physics, which has a very different notion of point particle. Of course there are still conceptual problems there, but it doesn't seem to affect any predictions as far as I know. So maybe it is the wrong question to ask. Self-energy certainly isn't the only place that renormalization is needed.

I suppose string theory grew out of a desire to incorporate extended objects into QFT. I don't know enough about it to say whether all the baggage of extra dimensions and so on are really necessary for an extended electron, or whether this solves everything.

By the way, point particles are much worse in GR than pure electrodynamics. There, it is possible to rigorously prove that there is no solution to Einstein's equation with a (0 or 1-dimensional) delta-function source. Or at least that solution does not exist as anything so 'simple' as a distribution. Topological charges, as Rob mentioned, still exist in that theory, but they are different. Their motion can be determined by the method of asymptotic matching (at least in principle). I think that there have been a few papers trying to describe electrons by spinning charged black holes, but I don't know how successful that's been.

 

[Rob Woodside]

By the way, point particles are much worse in GR than pure electrodynamics. There, it is possible to rigorously prove that there is no solution to Einstein's equation with a (0 or 1-dimensional) delta-function source. Or at least that solution does not exist as anything so 'simple' as a distribution. Topological charges, as Rob mentioned, still exist in that theory, but they are different. Their motion can be determined by the method of asymptotic matching (at least in principle). I think that there have been a few papers trying to describe electrons by spinning charged black holes, but I don't know how successful that's been.

How sweet, I wasn't aware of that. I have not heard of "asymptotic matching" could you please enlighten me? I am still worried about the inner boundary conditions near r=0, but no one else seems to be. It is tempting to think that a charged rotating black hole could be a model for an electron. The missing point in the black hole makes it look point like as suggested by the electron scattering data. It has the right gyromagnetic ratio. The mass, spin and charge are all topological and this is something new and different. However, the mass and charge are independently adjustable and one would rather have a solution that had only three possible charge to mass ratios with the right values to pick up the electron, muon and tauon. Lord knows what geometric structure lepton number could correspond to. Further the spin is so large that when the electron's mass, charge and spin are inserted the horizon vanishes and the singularity becomes naked. The glimmer of hope here is pretty faint, but it is a glimmer.

 

[pmb_phy]

Not conserving photon number is something I have to look at. Their Gibbs free energy is zero, so the is no hindrance to their production or destruction. I can certainly see excavating them out of a quantum backgound as you accelerate through it. But I am not yet sure how this relates to this classical radiation. I could totally wrong, but my prejudice is for an invariant charaterization of this radiation.

The term "conserved" is inaccurate here. I believe pervect means "invariant"?

Pete

 

[Stingray]

I have not heard of "asymptotic matching" could you please enlighten me?

The idea is that if the external perturbations are not too large on small wavelengths, then you can split up the system into three regimes. Call these the near, intermediate, and far zones. In the near zone (close to the black hole), your metric is just a perturbation of the stationary one. In the far zone, your metric acts like a perturbation of the background. Both of these are relatively well-understood problems.

Next, these two perturbation problems are compared in the intermediate zone where they are each approximately valid. They act sort of like boundary conditions for each other in this region, and end up giving a unique solution for the entire spacetime.

In practice, this method is quite difficult to use. The gravitational radiation reaction for a schwarzchild black hole was originally found this way, but generalizing that to charged holes is difficult. I know of someone who tried to do that (Eric Poisson - he has a few papers on the subject), but he gave up because the math got too messy. Still, the method should work if anyone wanted to spend enough time on it.

 

[Rob Woodside]

Thanks Stingray. That's a cute idea, but I can begin to imagine the calculational difficulty. I'll have a look for Poisson's papers at arXiv.

 

[pmb_phy]

Thanks Stingray. That's a cute idea, but I can begin to imagine the calculational difficulty. I'll have a look for Poisson's papers at arXiv.

Hi Rob - I had to wipe by system drive clean, reformat and reinstall my operating system. I lost your e-mail addess. Can you resend it? One of the authors of those papers is a friend of mine. You can talk to him directly since he spent a great deal of time during his career thinking about this.

Please e-mail me and let me know your desires. I can now scan that AJP article in e-mail it when you give me your e-mail address that you have ready access to.

Pete

 

[pmb_phy]

re "S. Parrott, "Radiation from Uniformly Accelerated Charge and the Equivalence Principle", http://arxiv.org/abs/gr-qc/9303025, (2001)"

I recall reading throught that article last week. Parrott assumes something which contradicts a calculation made by Cliff Will in article he wote regarding supporting a charged partilce in a Schwarzschild field.

Pete

 

[Rob Woodside]

re "S. Parrott, "Radiation from Uniformly Accelerated Charge and the Equivalence Principle", http://arxiv.org/abs/gr-qc/9303025, (2001)"

I recall reading throught that article last week. Parrott assumes something which contradicts a calculation made by Cliff Will in article he wote regarding supporting a charged partilce in a Schwarzschild field.

Pete

Thanks very much Pete. I found this and several others at arXiv by searching "accelerated charge". I'm putting together a collection of references for study in the new year.

Amusingly Parrot Starts off by saying it is well accepted that accelerating charges in Minkowski space radiate and it is well accepted that a charge at rest in a uniform gravitational field (Swcharzchild solution) does not radiate.
Ergo Problem Solved.

However. the charge at rest destroys the spherical symmetry of the Swcharzchild solution. Certainly a spherical shell of charge will give NO radiation in the Riessner Nordstrom electrovac around the charged planet. I would have thought that Rindler coordinates for flat space would do a better job of mimicking a uniform gravitational field and Parrot proceeds to use these in the body of his paper. He blames his disagreement with others on different definitions, especially for energy. He feels it is essentially an experimental problem answered by looking at the fuel consumption of a "fanciful rocket" that could either accelerate the charge in empty space or hold it at a fixed radius from a gravitating sphere. He also thinks the equivalence principle too ill defined to make any conclusions. As Andrew said the issue is very much alive.

I screwed up Claudio Teitelboim's name in previous posts and owe him an apology. The edit button on these posts disappears and I can't make the correction.