Retarded Time Solutions for Non-Homogeneous Wave Equations

[Nusc]

What is the significance of solutions to non-homogeneous wave equations for the scalar and vector potentials that utilize retarded time?

 

[lbrits]

Causality?

 

[Chrisc]

Nusc, I have to say you are the incredibly fortunate recipient of the most succinct answer in the history of this forum. I chuckle every time I think this post and I can't resist the temptation to add . . . "42".

 

[Nusc]

What are you talking about?

 

[JesseM]

I think the retarded waves are usually discarded as unphysical, but for another perspective see the Wheeler-Feynman absorber theory. The philosopher of science Huw Price argues that there would be no way to distinguish an advanced wave from a retarded wave and that therefore they are just two different descriptions of the same thing (see http://www.jstor.org/pss/687619 and http://www.jstor.org/pss/687664 for debate).

 

[DaveLush]

I think the retarded waves are usually discarded as unphysical, but for another perspective see the Wheeler-Feynman absorber theory. The philosopher of science Huw Price argues that there would be no way to distinguish an advanced wave from a retarded wave and that therefore they are just two different descriptions of the same thing (see http://www.jstor.org/pss/687619 and http://www.jstor.org/pss/687664 for debate).

I think you meant, the advanced waves are usually discarded as unphysical.

Personally, I don't think the advanced waves can be discarded so easily, as the advanced solutions are required if there is to be absorption, which is to say that if charged particles are to move in response to electromagnetic fields, then the time-advanced solutions are at work.

I tried to start a thread on this a while ago but nobody replied. Can anybody refute this argument? I believe it is supported by the Green's function analysis for advanced and retarded solutions as per Jackson, for example.

Also, I may have picked up the general idea from reading Huw Price's excellent book.

Seems to me nobody has a clue how causality really works so to wave away the advanced solutions as non-causal is a mistake.

 

[lbrits]

Seems to me nobody has a clue how causality really works so to wave away the advanced solutions as non-causal is a mistake.

I don't think that's an accurate statement.

In any case, these are all solutions to the PDE, and the way you build up your wanted solution is by looking at boundary conditions (same is true of any PDE).

 

[DaveLush]

I don't think that's an accurate statement.

In any case, these are all solutions to the PDE, and the way you build up your wanted solution is by looking at boundary conditions (same is true of any PDE).

To the second statement, yes and what I said is that the boundary condition of charged particle moving in response to an electromagnetic field (more specifically that there will be a force on the particle that causes it to move) requires that the time-advanced solution be non-zero.

To the first statement, you are welcome to your opinion, and perhaps you can refer me to where there is a description of how causality works. This question is a big part of Price's book, "Time's Arrow and Archimedes' Point". I found it quite convincing on this point that the pat explanations for causality are generally based on circular or otherwise fallacious reasoning. Most importantly, this applies to the Second Law of Thermodynamics.

Well I don't want to go down that road here, at least not alone. Read Price if you are interested. For now I just want to say that if you think that by waving away time-advanced solutions to the EM wave equation that causality is explained then you are laboring under a vast misconception.

 

[JesseM]

To the second statement, yes and what I said is that the boundary condition of charged particle moving in response to an electromagnetic field (more specifically that there will be a force on the particle that causes it to move) requires that the time-advanced solution be non-zero.

To the first statement, you are welcome to your opinion, and perhaps you can refer me to where there is a description of how causality works. This question is a big part of Price's book, "Time's Arrow and Archimedes' Point". I found it quite convincing on this point that the pat explanations for causality are generally based on circular or otherwise fallacious reasoning. Most importantly, this applies to the Second Law of Thermodynamics.

Well I don't want to go down that road here, at least not alone. Read Price if you are interested. For now I just want to say that if you think that by waving away time-advanced solutions to the EM wave equation that causality is explained then you are laboring under a vast misconception.

But Price didn't say that advanced waves are needed in addition to retarded waves to explain any physical behavior, he just said that an advanced wave is nothing more than a different sort of description of a retarded wave, so retarded waves can be redescribed as advanced waves and vice versa.

 

[DaveLush]

But Price didn't say that advanced waves are needed in addition to retarded waves to explain any physical behavior, he just said that an advanced wave is nothing more than a different sort of description of a retarded wave, so retarded waves can be redescribed as advanced waves and vice versa.

Yes, I remember Price saying that, and I hope you remember him saying that the standard reasoning behind the Second Law is circular and that Boltzmann realized this.

About the obvious physical necessity of the time-advanced solutions, I only said I might have gotten this idea from Price. It's also possible I thought it up myself. I'm perfectly comfortable with that too.

In any case it fits in perfectly with what you attribute to Price. When a charge moves it radiates and the Green's function for this is retarded, but when the fields reach another charge and the charge moves then from the point of view of the second charge these are time-advanced Green's function solutions.

This might seem like a mostly semantic distinction but it is actually quite a distinct case when you try to solve, say, the electromagnetic two-body problem. The equations of motion are different depending on whether you include the time-advanced terms. Eliezer showed this in the 40s, working with the 1938 Dirac classical electron theory, and at about the same time as Wheeler and Feynman were doing the first Absorber theory paper. It's Rev. Mod. Phys. 19, 3, July 1947. Del Luca also uses them generally in his 2006 Phys Rev. E paper, and then provides a special case of retarded-only fields.

 

[JesseM]

Yes, I remember Price saying that, and I hope you remember him saying that the standard reasoning behind the Second Law is circular and that Boltzmann realized this.

Well, he said that the time-asymmetric version of the second law involves circular reasoning. You can show that starting from any given low-entropy state, entropy is likely to increase in the future, but the same proof should also show that entropy is likely to have been higher in the past of that state as well.

In any case it fits in perfectly with what you attribute to Price. When a charge moves it radiates and the Green's function for this is retarded, but when the fields reach another charge and the charge moves then from the point of view of the second charge these are time-advanced Green's function solutions.

I don't have any knowledge of QED, but I'm a little skeptical--if this is true, why do most physicists think that they can discard the advanced solutions and still explain electromagnetic behavior?

 

[Phrak]

Retarded waves and retarded time are two completely different kinds of retardation. In fact, they have been given different names, in order to better tell them apart.

 

[DaveLush]

I don't have any knowledge of QED, but I'm a little skeptical--if this is true, why do most physicists think that they can discard the advanced solutions and still explain electromagnetic behavior?

It's not necessarily QED as it originates in classical ED. In my 2nd edition Jackson it's section 12.11 Solution of the Wave Equation in Covariant Form, Invariant Green's Functions.

About what physicists think, I think it hardly ever comes up. Anyhow they shouldn't be thinking that because the advanced solutions are not discardable in that Jackson chapter. They are used to define the formal radiation field, which is the retarded minus the advanced fields (see 12.136). I'm not sure where that originates but it is also in the 1938 classical electron theory of Dirac.

Being an EE, I never took a course using Jackson. I'd be interested to know if anybody covered that bit in their physics-department electrodynamics, and what was said about the reality of the advanced solutions. I took a lot of electromagnetics in my EE department though and have that idea in at least one of my EE textbooks. It was a very advanced EE course, though. In introductory EM I remember when the prof said there are two solutions but we were only interested in one 'cause the other is nonphysical and that was that.

 

[Stingray]

When a charge moves it radiates and the Green's function for this is retarded, but when the fields reach another charge and the charge moves then from the point of view of the second charge these are time-advanced Green's function solutions.

I don't know what you mean by this. If you have electromagnetically interacting charges, their motion can be found by solving the relevant PDEs. Whether or not advanced fields are involved depends entirely on how your boundary conditions are specified. If the fields and matter distributions are all specified at an initial time, the future behavior of the system is uniquely determined by acting on this initial data with retarded propagators. There is nothing being thrown out in this procedure. It is the unique solution (see Kirchhoff representations). Advanced Green functions can appear when the initial data isn't given in this form or isn't so complete.

As you mentioned, it can also be convenient in some contexts to break up a retarded field into "radiative" (F_{ret}-F_{adv})/2 and "singular" (F_{ret}+F_{adv})/2 components. These fields both involve advanced solutions, but their sum does not. The first one is particularly nice in that it sometimes captures everything you might be interested in while also being a solution of the vacuum Maxwell equations.

This was a crucial point in Dirac's electron theory that you mentioned a couple of times. Since you brought that up, I guess you're thinking about the motions of point particles. The procedure I described above then breaks down. Unless extremely odd boundary conditions are chosen, the field diverges at the locations of these particles. There is therefore no way to even write down their equations of motion. I interpret this to mean that point particles are not rigorously admitted in classical physics. None have ever been observed, so this isn't much of a problem. There are no inconsistencies for any extended charge distributions.

For various reasons that were much better motivated in 1938, Dirac wanted to force point particles into the theory. This required introducing an additional axiom applying only to these objects. It stated that point particles only interact with the radiative component of their self-field. This is everywhere finite, so it allowed Dirac to derive an equation of motion for point particles (whatever that may mean). It has completely crazy properties that can be tamed only by considering the physically meaningful analog of this problem: the motion of "small" extended charge distributions.

To be fair, my personal views that classical point particles are silly is not universal. These issues are currently an active area of research in the context of general relativity. A generalization of Dirac's ideas has now resurfaced to describe the effects of gravitational radiation reaction on compact masses. This is useful for some things, but not fundamental. It can be derived entirely using retarded fields. I'm actually writing up a paper related to this right now.