I have recently fallen down a rabbit hole in physics trying to resolve some paradoxes that I stumbled upon. In spending a lot of time and mental effort trying to understand what is going on, I realized that although I have spent my life studying and teaching physics, I did not fully understand some very fundamental aspects of space and motion and the way that the laws of physics operate. That is part of the fascination that physics provides, that it can always surprise you, leading you to learn new things.
In an occasional series of posts, I will share with readers my journey through this maze, trying to make things as clear as possible to the non-physicists out there. This will not be easy because an important prerequisite to explaining something to someone else is for you to understand it first. I cannot claim to understand completely what I am going to be writing about, for which I apologize in advance. But it is well known among teachers that it is in trying to explain something to someone else that one starts more deeply understanding what one is trying to say. Like many teachers I have used students as sounding boards for tentative ideas.
And so it is with this issue. This series of posts will largely consist of me trying to work my way through some paradoxes and puzzles to a satisfactory understanding and I hope readers will find journeying with me at least partly enlightening and will also provide them an opportunity to contribute their insights. At times I may seem overly pedantic but that is because there are a lot of subtleties involved and I am trying to pick my way gingerly through them as carefully as possible so as not to go astray, though I cannot guarantee that I will succeed in avoiding all the pitfalls.
So where to start? Perhaps it is best to start with something that almost all of us are familiar with from the time of Galileo and that is that any two objects dropped from the same height will fall at the same rate and hit the ground at the same time. Of course, when we do this under everyday conditions on Earth, this will not hold true because air resistance can affect the rate of fall of each differently. But if the experiment can be done in a vacuum, the result is expected to hold. The famous experiment done on the moon by dropping a feather and hammer seemed to support this and there have been similar experiments done on Earth in high-vacuum chambers. (See this post for videos of the two experiments.)
So let us take as a given what I will call Postulate #1, that if we can eliminate all other forces such as friction, all objects that are dropped from the same height in a gravitational field will fall at the same rate and hit the ground at the same time.
Another well-known result in physics is one that I will call Postulate #2, that an accelerating charge will radiate energy. This is a consequence of Maxwell’s equations (ME), which are foundational laws of physics. The laws expressed in ME are so fundamental that they played a key role in the transition from Newtonian physics to special relativity. The radiation of energy by accelerating charges is the basis of the electromagnetic phenomena that undergirds so much of the technology that we take for granted in our lives today. The textbook Classical Electrodynamics by J. D. Jackson is a staple of graduate physics curricula in the US and is an exhaustive and authoritative treatment of electrodynamic phenomena. In such esteem is the book held that saying, “Jackson says …” is usually enough to end an argument. In his second edition, he begins chapter 14 that has the title Radiation By Moving Charges by stating flatly, “It is well known that accelerated charges emit electromagnetic radiation”. The statement is not accompanied by any qualifications or conditions or caveats whatsoever. It helped convince me (and doubtless countless other physicists) that accelerating charges always radiate energy, period.
But if that is the case then these two Postulates that each seem to be incontrovertibly true and have a wealth of evidence supporting them lead immediately to the paradox that I came across while idly thinking. (I confess to being such a nerd that these are the kinds of things I idly think about.) The paradox is this: If we drop an electric charge and a neutral particle from a height in a gravitational field, say the ceiling of a building, Postulate #1 says that they should both hit the ground at the same time. But since the electric charge is accelerating, then by Postulate #2, it must radiate energy and this loss of energy will slow it down and it will hit the ground after the neutral particle does.
It seems like both Postulates cannot hold simultaneously. So is Postulate #1 not applicable when it comes to charged particles? Alternatively, does Postulate #2 break down so that an electric charge does not radiate when its acceleration is due to it falling in a gravitational field? Neither solution seemed palatable.
This was the seeming paradox that I encountered and that started my investigations. While the answer should be an empirical fact that should be conclusively decided by experiment, the actual experiment is not easy to do. Dropping two large items in a vacuum chamber is hard enough. Trying to do it with one charged object and one neutral one would be very difficult and, as far as I am aware, has not been done. So we are left with trying to resolve this issue theoretically.
I was of course not surprised to learn that such a fundamental issue has been recognized for a long time and has been examined by many eminent physicists. What did surprise me was that there seems to be no simple theoretical resolution to the paradox, with more than one conclusion arrived at, and that even among the experts who agree on a conclusion, they sometimes do not agree on the reasons. What also surprised me was that there is no widespread discussion of this question within the larger physics community. I would have thought that it would have been discussed at least in graduate physics curricula as a topic of interest since it involves so many fundamental questions but none of my professors did so and none of the textbooks that I used mentioned it either. The issues raised all involve very basic concepts involving the laws of physics that I once thought I understood fairly well but had clearly missed many subtleties.
So let us begin the exploration of those subtleties. I hope you enjoy going on this ride with me.
(To be continued.)
Why would the loss of energy slow it down? Einstein showed that a body emitting EM radiation experiences a drop in rest mass. A change in mass would not affect gravitational acceleration.
Postulate #1 you already stated does NOT hold when an object falls through air resistance.
I think it also does not hold for going through an electromagnetic field, in the same way.
So, to properly formulate the paradox, we need to specify if there is a relevant EM field present.
Relevant means big enough to detect its effects, which means it depends on the sensitivity of one’s detectors. If the paradox is meant as a theoretical question, not experimental, then any effect is enough, even if it is never detected.
So I think the other key is that Postulate #2 is based on real-world experience, in which all experiments have been done near physical objects that affect EM fields. So Postulate #2 needs to be defined more precisely to specify what interference conditions are intended or excluded.
Once all the above is done, I bet it will come to the question: is a particle moving past NOTHING actually moving within an EM field. That might be asking: can an EM field (say in deep space) be sustained by nothing being nearby?
To me, that sounds like an experimental test of quantum vacuum fluctuation. Or, the definition of nothing. As in discussions of the Big Bang, can the temporary existence of virtual particles that “aren’t” there be enough to impose EM resistance on a moving charge? My guess is yes, but at an experimentally currently undetectable level.
Is that a yes or a no?
Yeah, what I wrote in #1 doesn’t work for an electron or other elementary particle.
Also;
A charged particle at rest in a lab on the surface of the Earth should, by the equivalence principle, be indistinguishable from a charged particle being accelerated in an inertial frame, which would emit radiation. We don’t observe such radiation. Then again, the observers in this case are also in the accelerated frame of reference. So, would an observer in free fall detect radiation coming from the charged particle? Possibly related to the Unruh effect?
Rob Grigjanis @1: That was my first thought, too…energy isn’t exactly the same thing as speed; so loss of energy isn’t necessarily loss of speed.
I’ll bet Mano already thought of that long before we did. 😎 I look forward to hearing the answer.
OK, Rohrlich solved this in 1963. The point is that Maxwell’s equations only hold in an inertial frame of reference. An observer standing on the surface of the Earth is not in an inertial frame.
https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.205.7583&rep=rep1&type=pdf
Note: (C,) An acceleration field is locally equivalent to a gravitational field.
@Bruce:
One of the issues is that once you’re dealing with an accelerating electric charge, there is of course no such thing as the space being entirely void of magnetic effects, because the charge will generate its own magnetic field by its movement.
There’s actually a Wikipedia page on Paradox of radiation of charged particles in a gravitational field. According to the last section on that, effectively the charge is radiating, but there is an effective event horizon for any co-accelerating observer, and the radiation from the charge supported in the gravitational field effectively slips into the narrow regions of space-time that aren’t viewable to anybody else in the same situation. It has a citation to a 2006 paper which goes through the math.
This… okay, that’s weird, but it’s exactly the sort of thing you’d expect from Relativity, in that it’s a way that things happening in a shielded black box will look the same as long as the few things getting through the box are the same: in this case, specifically, if the box and all contents are in freefall, that must be indistinguishable to those inside from all contents motionless in zero gravity.
Doesn’t the fact of the particle being charged invalidate postulate #1? There must be some electrostatic attraction or repulsion acting on the particle, so gravity is not the only force at play.
And if the radiation is travelling at the speed of light (and why wouldn’t it?) then there probably is some relativistic effect going on which means it would never reach an observer outside its forward cone.
bluerizlagirl @7: If there are no other charged particles around, where would the attraction or repulsion come from?
On further reflection, my reply at #8 is too simplistic. There’s a lot I’ve forgotten…
If a charged particle is accelerated by an external force, it radiates, and so loses energy and momentum, which in turn affects its motion. That’s called ‘radiative reaction’. In many cases, this can be ignored, but ignoring it is always an approximation. Like most of physics.
Jackson doesn’t bring it up until the last chapter of his book. And he says “a completely satisfactory treatment of the reactive effects of radiation does not exist”. I’m not sure that’s changed since 1975 (the date of the second edition).
@Rob Grigjanis:
Apparently it has since then; as I noted above, Wikipedia mentions a 2006 paper by Camila de Almeida and Alberto Saa. It looks like the paper that originally solved the acceleration issue was a 1980 paper by David G. Boulware , “Radiation from a Uniformly Accelerated Charge”.
jenorafeuer @10: I’ve seen the paper, but they treat only uniform acceleration (hence the use of Rindler coordinates), and don’t even address radiative reaction until the concluding remarks;
From Jackson Ch 17, we can get an idea of why radiative reaction corrections can often be ignored in classical (i.e. non-quantum) physics.
If an electron undergoes constant linear acceleration from rest for T seconds*, the ratio of the energy it radiates to the final kinetic energy it would have without radiating is
(6.26e−24)/T
*Assuming T isn’t long enough for the electron to reach relativistic speeds.
Since the 1960s there have been quite a few discussions of this problem, including the ones by Boulware and Camila de Almeida & Alberto Saa and others. I have read them and there is a lot to unpack and reconcile. They are very technical and I am struggling to make sense of all of them.
Rob @#12,
It is true that the radiation losses are small. But as long as it is non-zero, we have to deal with the idea that it could reduce the kinetic energy of the falling charged particle and thus should slow it down, even if by an imperceptible amount.
Mano @13: Yeah, but unless I missed something in the stuff I read (quite possible!), we’re left with the problem of an observer falling with the charged and neutral particles; observer and particles are in an inertial frame. The observer sees a stationary (non-radiating) charged particle next to a neutral particle, so the two should remain stationary for the duration of the fall, i.e. hit the ground at the same time.
General remark: Theories with massless particles always seem to have various kinds of weirdness when you dig a bit.
It’s worth mentioning that not everyone thought an earthbound observer will see a freely falling charged particle radiate. In the Rohrlich paper I referenced above, he mentions a paper by Nathan Rosen (of EPR Paradox fame) which uses the same field transformations to conclude that the lab observer sees no radiation. The difference arises from different choice of Coulomb fields; Rohrlich uses retarded fields.
I couldn’t find the Rosen paper online, but perhaps you can access it for free, Mano;
N. ROSEN, Ann. Phys. (N. Y.) 17, 269 (1962).
Rob @#15,
I have a pdf of Rosen’s paper. If you would like to see it, let me know the best way to get it to you.
Mano, yes, I’d like to see it. I’ll send you an email.
I am not a physicist but I do have a blog where I discuss school text books etc. dealing with school-level physics. In particular I have series of blog posts:
https://agrumpyoldphysicstechnician.wordpress.com/2020/04/25/electrical-machinery-more-about-electrical-machinery/
where I deal with the topic without reference to electrical charges, sub-atomic or otherwise.
I have had a few thoughts about Mano’s post:
If an accelerating charge always produces electromagnetic radiation, this implies it does not matter if the velocity of the charge is increasing or decreasing or even if it is stationary. Is this correct?
Putting on my electronics engineer’s hat; how do I accelerate an electron? I would subject it to an electric field. This would seem to make it very complicated to separate out the effect of the electron accelerating from the effect of it accelerating in an electric field. Quite beyond my pay grade.
Finally, with my electrical engineers hat (and this is where my blog comes into it); when am I even concerned about accelerating electrons? The point I make strongly in my blog is that both current and magnetic flux travel in complete circuits. As permanent magnets are difficult to describe quantitatively, I replace them in my explanations with a current flowing in a conductor of zero resistance; in the simplest case this is just a single turn. The voltage that was applied to establish the current may be considered to be long since removed. Now, my understanding is that the current remains flowing indefinitely and the total flux through the loop can never be changed, also, there are no differences in electrical potential between any points. Energy was put into the system when the current was first established and there seem to be no way that it can leave. However, the electrons are moving in a circle and have an acceleration towards the centre; therefore, the system should be losing energy by electromagnetic radiation and the current should eventually cease. Where am I wrong? One clue is that the same dilemma arose when considering electrons orbiting around nuclei; are we going to be considering Quantum Mechanics along with General Relativity?
alanuk @18:
If a charge is accelerating relative to an inertial frame of reference, it radiates, according to an observer in the inertial frame.
The question in the OP is: does an observer in an accelerated frame (like someone standing on the surface of the Earth) see radiation from a freely falling charge? The freely falling charge is in an inertial frame, and an observer falling with it will see no radiation.
Yes, but the loss of energy per second is really tiny. You could estimate it from the Larmor formula (in cgs units);
P = (2/3)q²a²/c³
Here, q is the total charge, and a is the acceleration. If you can assign a unique speed v to the electrons, the acceleration would be v²/r, where r is the radius of the circle. I don’t think this is a ‘dilemma’ requiring quantum mechanics to resolve it. Just a prediction of classical electrodynamics.
Mano @13:
No kidding. I’ve done some digging, and I had no idea there was such controversy about the motion of charged particles in a gravitational field. I took a quick scan through my copy of Gravitation (1973) by Misner, Thorne, and Wheeler, and they don’t seem to even mention the problem.
A lot of the papers I read refer to a paper by Dirac, Classical theory of radiating electrons (1938). You can read it here;
https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1938.0124
As with anything I’ve read by Dirac, it is beautifully written, and probably a good place to start on the subject.
Really interesting problem Mano. Thanks.
If experts disagree over whether a charged particle undergoing uniform acceleration radiates, then what chance do I have? I found the following interesting reading, but I have nothing to add when Barut says “Yea” and Feynman says “Nay”.
https://www.mathpages.com/home/kmath528/kmath528.htm
file thirteen @22: The weird thing about Feynman’s argument is that he starts with an expression (the equation you cite) which is derived from the expression for the radiation from an accelerating particle. The most simple derivation involves cyclic motion, but since Feynman’s day there have been derivations for linear acceleration.
He’s right that if acceleration is constant, there is no radiation reaction, and thus it would seem no radiation. But that’s a flawed argument. If you start with constant acceleration, the particle radiates, and the reaction to the radiation is a change in acceleration.
Anyway, that seems irrelevant to the experiment in the OP. The charged particle in that case is undergoing free fall; it’s in an inertial frame, in which Maxwell’s equations are valid. They say that since the particle is stationary (or at most undergoing constant velocity), there is no radiation, and thus no departure from uniform motion. It is the observer who is in an accelerated frame. And if the real motion of a particle depends on the acceleration of the observer, we are left with an absurdity.
#23:
Actually, I think Dirac derived it in 1938 for arbitrary motion of the charged particle.
Rob Grigjanis @23:
Thanks Rob, that’s a really helpful explanation.
To clarify, the motion observed can be different to the real motion though, correct? How about radiation -- can a particle appear to radiate for one observer but not another?
file thirteen @25: Suppose you’re sitting at rest in an inertial frame (feeling no forces acting on you). There’s a charged particle sitting stationary in front of you, and an uncharged particle (alo stationary) to its right. There are also no EM fields other than that of the charged particle.
If an observer is accelerating to your right, they would see the particles as accelerating to the left. If, in their frame, they see the charged particle radiate, they might conclude that the kinetic energy of the charged particle is being reduced relative to that of the uncharged particle (although both kinetic energies would be increasing). That would mean they see the particles move closer together, eventually colliding.
By the same reasoning, an observer accelerating to your left would see the particles move further apart.
The particles can’t be doing both. Indeed, there is no reason for the stationary observer (you) to think the particles would move relative to each other at all.
That’s the dodgy question.
Thanks again Rob. So, extending the example you gave, if the observers to my left and right are accelerating at different rates, they see the two particles accelerate at different rates relative to themselves, and to myself in the same frame as the (stationary within this frame) particles, the acceleration appears to be zero . Now according to Classical Electrodynamics, accelerating charged particles emit radiation. But the acceleration varies depending on the observer, and may be zero! So is the dodgy answer to my dodgy question that it must be the case? What did I miss?
Whoops, didn’t mean to be ambiguous there -- I should have written something like “whereas to myself” rather “and”
file thirteen @27:
Right. According to Maxwell’s equations, which are valid in an inertial frame. For an accelerated observer (one who sees free objects accelerate), Maxwell’s equations don’t apply exactly in the simple form you usually see.
So here’s a question which I think anticipates Mano’s exposition; if you’re standing on the surface of the Earth watching a charged particle accelerate in free fall, which one of you is really accelerating? If you have an accelerometer handy, it will tell you that you are accelerating upwards at 9.8 m/s².
The question “which one of you is really accelerating” confuses me. I don’t know what “really” means in that context. I know that acceleration, velocity, distance and time are locally consistent, but measure differently relative to the frame of reference of whatever is doing the measuring. Why does there have to be a “real” value? If I imagine a single particle in an otherwise empty universe, those terms don’t make any sense to me.
So taking it a step further, are reference frames only relative and searching for an “inertial” frame like trying to find whether an object has an absolute, non-relative, velocity?
It’s not a trick question. The accelerometer is giving you a ‘real’ value.
Oh… please ignore comment #30, it’s nonsense. My inner self is chiding me now for speculating that inertia could be imaginary.
Thanks again Rob.