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.