(Previous posts in this series: Part 1, Part 2, Part 3, Part 4, Part 5, Part 6, Part 7, Part 8, Part 9, Part 10, Part 11, Part 12, Part 13, Part 14)
This series started by asking a simple question, whether a charged particle and a neutral particle would fall at the same rate when dropped from the same height and reach the ground at the same time. You would think that it would have a simple answer. But no. After a fairly long journey, we arrived at the conclusion that they would. But in the process, the series had to address a whole host of related issues along the way. While many of those were seemingly resolved, there are some fundamental questions that remain murky.
We saw in part #13 that the mass of a point charge like an electron is not a simple thing, because an electric charge has an associated electric field that itself has energy and thus should be thought of as contributing to the mass, except that the field energy density goes to infinity at zero distances, which is of course awkward for point-like charges. By looking at the radiation reaction force created by an accelerating charge, we learned about something called the acceleration energy Q that increases with the speed of a charge, and the energy radiated by a linearly accelerating charge comes from this source.
Another interesting question is how and why the radiation is emitted. In general, when something emits energy, we can identify the process by which it happened. For example, if I rub two objects together, heat is generated and we say that the heat came about as a result of the force of friction between the two objects. If we want to go deeper, we can look at the source of friction forces as being due to the electromagnetic interactions between the atoms and molecules of the two objects that come into close contact with each other. The ultimate source of the energy radiated is the internal energy of the person doing the rubbing, who is converting the sources of energy stored in their bodies to move the muscles that rub the two objects together and overcomes the friction force. That is why we get tired. Energy transformations are a fundamental feature of life, going on all the time.
For an accelerating charge to generate any form of energy, we might assume that it must be doing some work against a force. But in the case of the radiation emitted by an electron falling freely in a vacuum in a uniform gravitational field, what could be that force? After all, the electron is not in contact with anything external to it. While we have something called the radiation reaction force 𝜞𝜇 force that arises due to the radiation it emits (hence its name), from whence does this force arise? There does not seem to be a clear answer to this in the literature. At least I could not find any.
One thing that might play a role is the electric field of the electron itself. We saw with the radiation problem that it was necessary to include the energy of the electron’s own field in explaining the energy balance. It may also be necessary to explain how the radiation energy is generated. Is the electron facing a friction-like resistance from its own field and it is in overcoming this resistance that radiation energy is generated? The physical meaning of the acceleration (or Schott) energy Q and how it is being depleted while keeping the mass of the electron unchanged even as it radiates is another question to be explored.
In traditional electromagnetism, we know how to use Maxwells Equations to calculate the electric and magnetic fields generated by any distribution of charges and currents. With those fields, we can calculate the force on any charge due to those fields.This raised an immediate problem in that it was not clear if the force on a charge due to the fields should include the field created by the charge itself. But that field is infinite at the location of a point charge. To avoid such infinities, a rule was imposed by fiat that in calculating the force on a charge, we not include the field due to that same charge, only that due to all the other charges. That has worked well so far (at least at the level of accuracy we are working in) and enabled us to avoid the awkward problem of infinities. But it clearly breaks down when we look closely at some processes.
Part of the problem may be because of the concept of energy itself. The word ‘energy’ is widely used by everyone and it has become reified in that it is used as if it is some kind of tangible thing. We speak of ‘using’ energy, of ‘producing’ energy, of ‘storing’ energy, of energy moving from place to place, and so on. It is not surprising that energy is thus viewed as consisting of some kind of tangible entity that exists in things and can scurry around. The original caloric theory of heat transfer saw heat as some kind of tangible entity that flowed from hotter to colder bodies. In later experiments it was found that rotating a paddle wheel in water made the water hotter and that was explained by saying that the source of heat was the mechanical energy used by the external agent to overcome the friction posed to the paddles by the water. Hence the idea that mechanical energy and heat energy were interchangeable was born. Over time, other forms of transformations were required to explain results.
Now we view energy as an inferred quantity, not a thing that can be directly detected. So when we say that an object has kinetic energy, what we really measure is its mass m and speed v and then infer that it has energy given by ½ mv2. When an object gets hotter, we infer from the mass of the object, its specific heat, and the change in its temperature the amount of energy it gained. And so on. We cannot measure the energy that exists in something. All we can do is calculate the change in its energy by taking measurements of changes in observable quantities. Energy can be viewed as a book keeping device, so that the total amount of it in a closed system remains unchanged, even as it shifts between internal, kinetic, electric, magnetic, chemical, etc. But that original image born in caloric theory of energy as a thing has stuck even as the caloric theory was abandoned and the concept of energy became more generalized.
A similar problem arises with the photon. By introducing the idea of a corpuscle of energy, we get drawn into thinking of it as an entity that can exist independently of a source or a detector. But is it so? In reality the photon, like energy, could be just an inferred quantity arrived at by looking at what happens at the source and the detector and reconciling the two results. So the question as to where the photon is if a detector does not detect any radiation is a poorly formulated one.
So in the case of an electron that is accelerating in a uniform gravitational field and radiating, what are the energy transformations that are taking place? What makes Q increase? What makes the charge radiate? What is the role of the self-field? It is clear that there remain many mysteries when it comes to understanding the radiation and self-energy of point charges.
Since the general theory of relativity is a classical theory, we have been able to patch together with the classical theory of electromagnetism a theory of radiation in a gravitational field. It involved treating the electron as a point particle with a surrounding field. But clearly that is not the whole story. We need to treat the electron as a quantum field as is done in relativistic quantum field theory but then we are confronted with how to transform relativistic quantum fields in a gravitational field and we do not as yet have a unified theory of quantum fields and general relativity that would enable us to to do that.
The nature of the unsolved conceptual problems created by self-fields are highlighted towards the end of the widely used graduate text Classical Electrodynamics by J. D. Jackson (Second edition, p. 781):
[A] completely satisfactory treatment of. the reactive effects of radiation does not exist. The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of the elementary particle. Although partial solutions, workable within limited areas, can be given, the basic problem remains unsolved. One might hope that the transition from classical to quantum-mechanical treatments would remove the difficulties. While there is still hope that this may eventually occur, the present quantum-mechanical discussions are beset with even more elaborate troubles than the classical ones. It is one of the triumphs of comparatively recent years (~ 1948–1950) that the concepts of Lorentz covariance and gauge invariance were exploited sufficiently cleverly to circumvent these difficulties in quantum electrodynamics and so allow the calculation of very small radiative effects to extremely high precision, in full agreement with experiment. From a fundamental point of view, however, the difficulties remain.
These unanswered questions illustrate that there are fascinating questions in physics to be explored all around us. We tend to think that the frontiers of physics, the big unsolved problems, lie in things like black holes, dark matter, dark energy, and other phenomena that get a lot of media attention. It is good to remember that some parts of the frontier lie much closer to home, right under our noses, that we discover simply by looking closely at what we have come to view as mundane and asking deeper questions about what is going on.
That ends this series. I am not sure how many readers have stuck with it to the end but I hope that even those who encountered parts of it that were difficult and were tempted to give up found at least some aspects of it helpful and even enlightening.
Hi Mano -- Long time reader/first time commenter. I am a former student of yours from PHYS121 and 122. I have been reading your blog for many years and just wanted to say I have found this series absolutely fascinating. Really great work and hopefully you’ll find other physics topics worthy of such a series in the future!
Hi MattF,
It is always nice to hear from former students and I am glad that you enjoyed the series!
I have been a long time lurker of the Freethought Blogs. I particularly enjoyed this series of posts.
It seems to me that the problem of the infinities associated with a charged point particle could be avoided by treating the charged particle as a Kerr-Newman black hole. This would mean a lot of work, but the fact that radiation from coalescing black holes can be computed to fit LIGO gravitational wave data suggest that this task would not be impossible.
A lot of food for thought in this series. More later, but a couple of what might seem minor, even tangential, points; in the classical theory, what do we mean by radiation, and what do we mean by detecting radiation?
The first question has a (in principle) precise answer; that part of the em energy flux from a source which drops off at distance r from the source as 1/r².
The energy flux is given by the Poynting vector, which is a vector cross product of the electric and magnetic fields. In general, it has parts which drop off as 1/r³ and 1/r⁴, as well as 1/r². So, classically, a single ‘measurement’ (however it might be done) of the energy flux tells us nothing about radiation, unless we’re at a large enough distance from the source. Otherwise, three (or more) measurements at different times might allow one to separate out the radiation part, if they’re accurate enough.
These points are not about the deeper issues raised in the series (I’ll have more to say on those), and one could argue they are irrelevant, but they are things which I always took for granted until now.
Great series, Mano. Thanks.
Yes, great series, and I’ve trying to follow along. Although the last few I merely skimmed because of pressures from work over the past couple weeks. I’ll go back to them later to read them in more depth when I have time.
From #3, Rob Grigjanis,
That’s a question I had early in this series, although not phrased nearly as well. I’m not a physicist, but I’m not certain this question is irrelevant. Maybe there is a good answer and I just don’t know it.
Bernie @3: The Kerr-Newman solution is for a rotating, charged mass. A charged, non-rotating spherically symmetric mass might be more appropriate, for which the solution is the Reissner–Nordström metric. This was in fact considered in the context of this problem by Nathan Rosen in 1962. I couldn’t find a link to the paper, but Mano sent me a copy.
Reference: Annals of Physics: 17: 260-275 (1962)
FWIW, Rosen concluded that a charged particle accelerating in a uniform homogeneous gravitational field does not radiate, although other authors have questioned his reasoning. See footnote 11 at the bottom of pages 182 and 183 here;
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.205.7583&rep=rep1&type=pdf (by F. Rohrlich)
As you were going through the interesting list of issues, including being a point charge and worry about
internal energy, and quantum effects, I got to wondering about something without those “complications”.
What if you took a 100lb ball and charged it up with as much charge as it could reliably hold? Would it fall at the same rate as when uncharged? And I guess we’d also have to set it up to be far enough from anything else so that there were not any induced charges in any other body (or at least make them small compared to anything else).
My apologies if you already addressed this.
Further to the Kerr-Newman solution;
To me, the great thing about this series of posts is that it exposes the doubt inherent in any scientific endeavour. Experts disagree! From what I’ve learned, there’s no doubt that an accelerating charged particle radiates, whether the acceleration is cyclic or linear. Feynman disagreed. My impression is that he was, misguidedly, desperately trying to defend the equivalence principle. On the other hand, Feynman was much smarter than me. So what’s the right answer? Fucked if I know, but I’d put (some) money on Feynman being wrong in this instance.
Rob @ 6: Thanks for the references and information.
Thank you for this series Mano.
Often, it’s the little things that count.
Talking about “radiation paradoxes”, here is another one.
.
1- A thought experiment proposed by Feynman. Suppose there is only 1 excited atom in the universe. Question: will the atom emit a photon? According to field theory (local), the atom will emit. According to action-at-a-distance (global) the atom will not emit for the atom has no potential receiver to emit to. Which view is correct?
.
2- Global versus Local. Maxwell’s EM equations written in terms of electric and magnetic fields are local. Maxwell’s EM equations written in terms of scalar and vector potentials are local for scalar potential and global for vector potential. Think Aharonov-Bohm effect. Does gravitation have a vector potential component? Can one afford to neglect the global view? Think Mach’s principle.
friedfish2718 @12:
The equations are all local. See here;
https://en.wikipedia.org/wiki/Maxwell%27s_equations#Alternative_formulations
Yes, the Aharonov-Bohm effect is a non-local effect in quantum mechanics. But the fields (or potentials) are local.
friedfish2718 @12:
That is directly relevant to the current topic. Feynman and Wheeler came up with a theory in which the future can affect the present, as well as vice versa, and in which for emission to occur, there have to be absorbers (if I got it right). They conclude that a charged particle can’t interact with its own field (contrary to the arguments for radiation from an accelerating charge). I think Feynman eventually backed off that notion.
phys.org has an article about radiation reaction
https://phys.org/news/2022-01-physicist-century-problem-reaction.html
Maxwell–Lorentz without self-interactions:
conservation of energy and momentum
by Jonathan Gratus
abstract:
Since a classical charged point particle radiates energy and momentum it is
argued that there must be a radiation reaction force. Here we present an action
for the Maxwell–Lorentz without self-interactions model, where each particle
only responds to the fields of the other charged particles. The corresponding stress–energy tensor automatically conserves energy and momentum in
Minkowski and other appropriate spacetimes. Hence there is no need for any
radiation reaction.
Rob @#15,
Good timing! Thanks for the information. I will take a look at the paper and see what it says.
Rob Curtis @15: Thanks!
The article can be seen here; https://arxiv.org/pdf/2108.08644.pdf
One thing that leaps out is that the model violates the weak energy condition (see part 4). The weak energy condition says that the local mass-energy density is non-negative for any inertial observer. Gratus says this is not a problem for the classical theory, but his argument looks suspect to me. I can elaborate if you like.
Further to #17: Gratus considers a system of N point charges. So, if N=2, we have two point charges, with electric fields E₁ and E₂. In the usual model, the electric fields contribute to the energy density as
(E₁ + E₂)² = E₁² + E₂² + 2E₁·E₂
Gratus’ program amounts to ignoring the squared terms above (the ‘self-energy’ terms), so that the energy density is proportional to
E₁·E₂
So if, say, the fields point in opposite directions, this is negative.
More problematic, I think, is the Poynting vector, which is the flux of em energy. In the usual model, this is
(E₁ + E₂) x (B₁ + B₂)
where the Bs are the magnetic fields.
For Gratus (who doesn’t mention the Poynting vector), it would have to be
(E₁ x B₂) + (E₂ x B₁)
In this scheme, if the two charges are sufficiently far apart, this expression would be vanishingly small. So, regardless of the motion of either of the particles, they would emit no radiation. That can’t be right (note that Gratus’ point is that, while there is the usual radiation, there is no radiation reaction force). Maybe I’ve missed something, but I’m not seeing it.
Thanks so much for this series Mano! I’m not familiar with four-vector notation, but that didn’t prevent me from enjoying the insight that I gained into this subtle issue.