# A paradox for two moving protons?

• I
• FranzDiCoccio
In summary: It is not really analyzed in depth, and it is never mentioned again. The book goes on analyzing the relativistic field transformations, both for electromagnetic and other fields, but it stops there.
FranzDiCoccio
TL;DR Summary
a textbook example considers two protons at a given distance having the same constant velocity. The analysis of their interaction in two different reference frames results in an (apparent) paradox.
This is used to highlight the inconsistency of Newtonian Mechanics and Maxwell's Electrodynamics in an introductory discussion of Special Relativity.
I'm wondering whether there's a simple way of "solving" the paradox.
Before introducing Special Relativity, a textbook highlights the inconsistency of Maxwell's Electrodynamics and Newtonian Mechanics through the standard discussion about the velocity of light in different frames of reference.
A further inconsistency discussed.

In some inertial frame of reference two protons are initially at distance ##d## and move with velocity ##v## perpendicular to the segment joining their positions.
In this frame of reference each proton generates both an electric and a magnetic field. The fields generated by each of the protons affect the other particle. Specifically, the electric and magnetic field produce an attractive and a repulsive force, respectively.

The same situation is then analyzed in a frame of reference moving at the same velocity as the (initial) velocity of the protons. There, no magnetic field is present, so only the repulsive interaction exists.

From a "classical" (non-relativistic) point of view this is a problem, because the two inertial frames of reference should be equivalent in the Galilean sense. In this case the two protons would move with different accelerations in two different inertial frames of reference.
But according to Galilean invariance their accelerations should be the same in the two frames of reference.

After discussing this, the book introduces the postulates of Special Relativity, and never goes back to this example.
In a way, this is understandable, because its discussion does not go as far as the transformations of the fields.

I was wondering whether a sufficiently simple argument exists that "solves" the paradox. I think I have something, but I am not sure I am interpreting it correctly.

The magnetic field that each proton generates is
$$\vec B = \frac{\mu_0}{4\pi} e \frac{\vec v \times \vec r}{r^3}$$
Using ##\vec v = v \hat x## and ##\vec r = d \hat y## I get
$$\vec B = \frac{\mu_0}{4\pi}\frac{e v}{d^2} \hat z$$
The force resulting from the combined electric and magnetic field should have a magnitude
$$F=\frac{1}{4\pi \varepsilon_0} \frac{e^2}{d^2} (1-\frac{v^2}{c^2})$$

Classically, in the frame of reference where the protons are at rest ##v=0## and the magnitude is
$$F'=\frac{1}{4\pi \varepsilon_0} \frac{e^2}{d^2}$$

Using the transformation of the fields between the two inertial frames I get
$$E' =\gamma \frac{1}{4\pi \varepsilon_0} \frac{e}{d^2}(1-\frac{v^2}{c^2}),\qquad B'=0$$
so that
$$F'=\gamma F$$

If I get this other discussion right, this is the way the force is actually expected to transform according to Special Relativity, which should prove that there is no real paradox here.

Does this make sense?

PeroK and Dale
FranzDiCoccio said:
Before introducing Special Relativity, a textbook highlights the inconsistency of Maxwell's Electrodynamics and Newtonian Mechanics through the standard discussion about the velocity of light in different frames of reference.
A further inconsistency discussed.
Which textbook? You will likely get better answers if you tell us this, especially if someone here is already familiar with it.

FranzDiCoccio said:
Does this make sense?

Yes. You showed that force calculations in special relativity and force calculations in Maxwell's electrodynamics do not disagree.

Or should I say : No. You only showed that force calculations in special relativity and force calculations in Maxwell's electrodynamics do not disagree. But acceleration calculations in special relativity and acceleration calculations in Maxwell's electrodynamics disagree. (Energy has inertia)

And then there is still the Newtonian mechanics that does not agree with the Maxwell's electrodynamics. Which is not a paradox, it's a disagreement.

jartsa said:
But acceleration calculations in special relativity and acceleration calculations in Maxwell's electrodynamics disagree. (Energy has inertia)
I’m not sure what you mean here, but in this particular problem (two charged particles at rest relative to one another) the accelerations are the same whether calculated using SR or classical electrodynamics. So that’s not what original poster’s textbook is saying. Without knowing what that textbook is, it’s hard to know what it is saying…. but it sounds like the same concern (“asymmetry”) that Einstein raises in the first few paragraphs of his 1905 paper. That’s not a paradox, but it is a reason to be deeply unhappy with the classical model.
And then there is still the Newtonian mechanics that does not agree with the Maxwell's electrodynamics. Which is not a paradox, it's a disagreement.
Newtonian mechanics works just fine with Maxwell’s electrodynamics. However that combination also predicts the existence of a medium (the ether) in which electromagnetic radiation propagates, and that prediction has not aged well.

FranzDiCoccio said:
After discussing this, the book introduces the postulates of Special Relativity, and never goes back to this example.
In a way, this is understandable, because its discussion does not go as far as the transformations of the fields.
This was the historical route to SR that Einstein took. See the introduction to the famous 1905 paper:

https://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf

There is, of course, a much simpler route to SR using only the invariance of light. The textbook I have only tackles an electromagnetic example in the appendix. Not least because the theory of classical EM requires significantly more advanced mathematics than an introduction to SR.

You did extremely well to get the solution to that problem!

vanhees71
Concerning the original question, it's much simpler to use a covariant description. What you have is the (fictitious) situation, where in one inertial frame of reference two protons are kept at fixed positions. The electromagnetic field in this frame is of course simply the superposition of the two electrostatic Coulomb fields and the force on one of the protons is the corresponding Coulomb force. Now you simply have to Lorentz boost both the fields and the force, described as the four-vector Minkowski force, as it occurs in the covariant form of the equation of motion,
$$m \mathrm{d}_{\tau} p^{\mu} = \frac{q}{mc} F^{\mu \nu} p_{\nu}=K^{\mu}.$$
Then it's immediately clear that there are no contradictions anymore as you'd get when using the Galilei transformation between the inertial reference frames.

Nugatory said:
Which textbook? You will likely get better answers if you tell us this, especially if someone here is already familiar with it.
Hi,
sorry, it's just unlikely anyone here is familiar with that book.
It is an Italian high school textbook, mostly adapted from Physics 9th edition by Cutnell and Johnson. A lot of the material there is the same as in the original book.
However, the Italian authors added a sort of introductory section where the example with the two protons is discussed. I was not able to find that part in the original book.

Their discussion is kind of handwaving. They also mention a simplified version of de Sitter's double star experiment in order to "support" Einstein's "speed of light postulate".

I find that argument a bit confusing, basically because it disproves a ballistic theory of light. But just a few chapter earlier the book spent quite a lot of time discussing the wave nature of light.
My first reaction was: "wait, no. Isn't light a wave? Why are you showing me that it does not behave like a particle now?"

Nugatory said:
Newtonian mechanics works just fine with Maxwell’s electrodynamics. However that combination also predicts the existence of a medium (the ether) in which electromagnetic radiation propagates, and that prediction has not aged well.
Not sure I understand.

Are you saying that assuming the existence of aether one could make sense of the disagreement in the "two proton example"? Does that mean that the different acceleration in the two frames of reference can be explained by the fact that (at least) one of them is moving wrt the "special" frame of reference where the medium is at rest?

What would the radiation be in that situation? Would that arise from the particles accelerating in the direction perpendicular to their motion because of their interaction?

PeroK said:
This was the historical route to SR that Einstein took. See the introduction to the famous 1905 paper:

https://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf

There is, of course, a much simpler route to SR using only the invariance of light. The textbook I have only tackles an electromagnetic example in the appendix. Not least because the theory of classical EM requires significantly more advanced mathematics than an introduction to SR.

You did extremely well to get the solution to that problem!
Thanks for the paper, I'll look into it.

As I mention, the textbook is aimed at high school students. The material is basically the same as in chapter 28 of Cutnell and Johnson's "Physics" but with a few additions.

Though not very advanced, my calculations would fly over most of the students' heads. A discussion of the transformation of the fields is definitely beyond the average level of complexity in the chapter.

I was thinking that maybe an example referring to something simpler and more familiar could be helpful.
The students should be familiar with Galilean relativity and the most basic Galilean transformations.

One could perhaps start with a situation where a person in a train car moving at a constant speed drops a ball. In their (inertial) frame of reference the ball falls vertically.
Someone on the station platform sees the "experiment", but from their point of view something different happens: the ball follows a parabolic trajectory.
This does not mean that the two observers disagree on the outcome of the "experiment". In fact, using Galilean transformations each of them can make sense (or predict) what the other would observe.
They also realize that the outcome comes from "the same physics".

Then one describes the "two proton experiment" pointing out the disagreement.

Next, one could explain that Einstein's theory results in a set of transformations that are analogous to Galilean transformations. These allow us to make sense of the behavior of the two protons in the two frames of reference, much in the same way we can make sense of the "dropped ball" experiment.

One could also mention that unlike Galileo's, these (Lorentz) transformations affect the temporal coordinate as well, and that Galileo's are in fact their "low speed limit".

vanhees71 said:
Concerning the original question, it's much simpler to use a covariant description. What you have is the (fictitious) situation, where in one inertial frame of reference two protons are kept at fixed positions. The electromagnetic field in this frame is of course simply the superposition of the two electrostatic Coulomb fields and the force on one of the protons is the corresponding Coulomb force. Now you simply have to Lorentz boost both the fields and the force, described as the four-vector Minkowski force, as it occurs in the covariant form of the equation of motion,
$$m \mathrm{d}_{\tau} p^{\mu} = \frac{q}{mc} F^{\mu \nu} p_{\nu}=K^{\mu}.$$
Then it's immediately clear that there are no contradictions anymore as you'd get when using the Galilei transformation between the inertial reference frames.
Whoa. This is way too advanced.
The level of the textbook is basically the same as "Physics" by Cutnell and Johnson.
The authors added a paragraph on Lorentz boost, but not with this formalism.
Most of the students are already struggling with what's in the book.

I do realize that my calculation is also too much for them, and definitely not the best way to show that there is no contradiction.
I was just exploring a way to follow up on that "example" that could be understood (even intuitively) by the students.

Sorry, but I didn't realize that this discussion is to introduce relativity to high-school students. I doubt that you can explain this in an understandable way at the high-school level. I'm also not aware that "covariant electrodynamics" is treated at this level in any high-school textbook. I'm only aware of German high-school textbooks, which usually are not very convincing, although there is at least one good one, and there they treat only the kinematic effects up to the complete derivation of the Lorentz boost in one spatial direction and then some mechanics (but this part in a quite confusing way with relativistic mass).

Can give a short description, how your book treats this at the high-school level? I don't know the textbook by Cutnell and Johnson.

bhobba and BvU
FranzDiCoccio said:
Does that mean that the different acceleration in the two frames of reference can be explained by the fact that (at least) one of them is moving wrt the "special" frame of reference where the medium is at rest?
Not quite. The idea behind cutting edge physics in the late 19th century was basically that Maxwell's equations were a low velocity-relative-to-the-ether approximation to something more complicated. The full theory was expected to be Galilean invariant, and would provide a consistent view of the acceleration of those protons, as you said.

However, as we now know, they were trying to fix the wrong theory.

bhobba, vanhees71 and PeroK
Nugatory said:
I’m not sure what you mean here, but in this particular problem (two charged particles at rest relative to one another) the accelerations are the same whether calculated using SR or classical electrodynamics. So that’s not what original poster’s textbook is saying. Without knowing what that textbook is, it’s hard to know what it is saying…. but it sounds like the same concern (“asymmetry”) that Einstein raises in the first few paragraphs of his 1905 paper. That’s not a paradox, but it is a reason to be deeply unhappy with the classical model.
Does the inertia of two protons depend upon the kinetic energy content of the two protons?

SR: Yes

Classical electrodynamics: No

OP calculated that force gets divided by gamma. Einstein calculated that inertia gets multiplied by gamma.

Maybe Einstein did not bother calculating acceleration, but OP talks about acceleration, so
combining the calculations of OP and Einstein we get: transverse acceleration transforms as inverse of gamma squared.

weirdoguy, bhobba and PeroK
jartsa said:
Classical electrodynamics: No
No, classical electrodynamics: no answer since "inertia" is not a concept that is even covered by classical electrodynamics. SR can answer "yes" because SR covers things that classical electrodynamics does not.

bhobba and vanhees71
Ibix said:
Not quite. The idea behind cutting edge physics in the late 19th century was basically that Maxwell's equations were a low velocity-relative-to-the-ether approximation to something more complicated. The full theory was expected to be Galilean invariant, and would provide a consistent view of the acceleration of those protons, as you said.

However, as we now know, they were trying to fix the wrong theory.
So I'm still not clear on how "Newtonian mechanics works just fine with Maxwell’s electrodynamics.".
This sounds more like "Newtonian mechanics was expected to work just fine with something more intricate than Maxwell's equations, but this turned out not to be the case in the end".
I never had the time to go deeper, but this is was my idea too.

An earlier edition of my book mentions a simplified version of this view. The chapter started with a simple review of the aether theory and the Michelson Morley experiment.
Now that part is not in the introductory paragraph any more. It has been moved to a sort of appendix, at the end of the chapter.
I was told that the reason would be some kind of "historical accuracy", because Einstein did not really know about M&M's experiment, or perhaps did not think of SR because of that experiment.
But my book does not seem very consistent in that respect because, after discarding the introductory paragraph about aether and M&M's experiment, it introduced a new paragraph about the de Sitter experiment. As far as I understand, de Sitter's argument is against a ballistic theory of light that emerged in contrast to SR.

In my opinion starting with aether and the M&M's experiment does make sense, especially if emission theories were devised to explain the results of that experiment, and in contrast with SR.
I guess that before M&M everyone was sure light was a wave propagating in some medium, like all waves known on Earth. I wonder what was the support for a ballistic theory of light, at that time.

vanhees71 said:
Sorry, but I didn't realize that this discussion is to introduce relativity to high-school students [...].
No worries. I would like to have time to brush on that formalism.

So my book starts with the (in my opinion) handwaving and confusing introductory part, then gives the postulates of SR, discussing (a simplified version of) the de Sitter argument in support of the invariance of c. The explicit calculation is nice, but the argument per se is also confusing, in my opinion.
Next the relativity of simultaneity, time dilation and length contraction are discussed via standard thought experiments (lights on a train and light clock).

After that, the Lorentz boosts in 1D are given (but not derived), and the book shows how they "contain" the previously discussed phenomena, as well as the Galilean low speed counterparts. Next the velocity addition formula is "dropped from the sky", although its derivation is not that hard, at this point. The Doppler effect is discussed.
The book then proceeds with space time diagrams and the invariant interval. I like space-time diagrams, and I find the discussion in the book kind of unsatisfactory.

There is a paragraph on experiments confirming SR: muon decay (which makes sense) and Hafele-Keating (which seems a bit outside of the scope of the discussion).

Finally "relativistic dynamics" is discussed: force and momentum, mass and energy, relativistic kinetic energy, energy momentum relation and a brief discussion of relativistic collisions.

Most of this stuff is taken straight from Cutnell and Johnson's book, but there are several hapazard additions. For instance, it seems to me that C&J's book does not discuss of any of the concepts between the thought experiments and relativistic dynamics, with the only exception of the velocity addition formula.

vanhees71 said:
Sorry, but I didn't realize that this discussion is to introduce relativity to high-school students. I doubt that you can explain this in an understandable way at the high-school level.
I am not aware of any HS curricula anywhere with textbooks that discuss it at that level. Certainly not in Australia. But with the background of having done calculus in year 11, which is usual in Aus (good schools here in Aus have an accelerated math stream where it is done in year 10) and concurrently doing linear algebra and multivariable calculus in year 12, then the author believes, as do I, they can study the following book in HS:
https://www.amazon.com.au/dp/0521876222/

His exact words are - he thinks they will find it a hoot.

It does everything correctly. Nothing handwavy and SR is dealt with from the POR postulate alone, leaving the constant, that is of course the speed of light, to be determined by experiments, of which many are given.

I do not know of any school that has adopted it as a textbook, but some advanced high schools in the US have adopted the following excellent book in grade 12:
https://matrixeditions.com/5thUnifiedApproach.html

Most would say it is far too advanced for HS students. But IMHO, good students, with a good teacher, can learn at that level.

Thanks
Bill

If the goal is to introduce special relativity to high school students,
I would use the ideas from
Bondi’s Relativity and Common Sense.
Although the method is called the k-calculus, it uses no calculus. With a minimal amount of algebra and some good operational definitions using radar to assign space and time coordinates, one gets to special relativity.

Standard formulas in cartesian coordinates are obtained as corollaries to Bondi’s result expressed in terms of doppler factors rather than time-dilation factors. (I would argue that it’s easier to motivate doppler factors than to motivate time-dilation factors).

Secretly, Bondi is working in the eigenspace of the lorentz boost (using light cone coordinates), where the doppler factors are the eigenvalues and lightlike vectors are the eigenvectors. In terms of rapidities, k is the exponential function of the rapidity. The time dilation factor is (1/2)(k+k^-1)=cosh(rapidity) and (k-k^-1)/(k+k^-1)=tanh(rapidity)=velocity/c.I have an insight on the Bondi k-calculus (check the link in my signature).

FranzDiCoccio said:
So I'm still not clear on how "Newtonian mechanics works just fine with Maxwell’s electrodynamics.".
This sounds more like "Newtonian mechanics was expected to work just fine with something more intricate than Maxwell's equations, but this turned out not to be the case in the end".
I never had the time to go deeper, but this is was my idea too.

An earlier edition of my book mentions a simplified version of this view. The chapter started with a simple review of the aether theory and the Michelson Morley experiment.
Now that part is not in the introductory paragraph any more. It has been moved to a sort of appendix, at the end of the chapter.
I was told that the reason would be some kind of "historical accuracy", because Einstein did not really know about M&M's experiment, or perhaps did not think of SR because of that experiment.
But my book does not seem very consistent in that respect because, after discarding the introductory paragraph about aether and M&M's experiment, it introduced a new paragraph about the de Sitter experiment. As far as I understand, de Sitter's argument is against a ballistic theory of light that emerged in contrast to SR.

In my opinion starting with aether and the M&M's experiment does make sense, especially if emission theories were devised to explain the results of that experiment, and in contrast with SR.
I guess that before M&M everyone was sure light was a wave propagating in some medium, like all waves known on Earth. I wonder what was the support for a ballistic theory of light, at that time.
When I'm learning something, I'm generally quite ruthless about ignoring what I consider to be extraneous material. If I were learning complex numbers, for example, I wouldn't get tangled up in their history and why 16th century mathematicians were suspicious of them and got into esoteric debates about them. Nor would I concurrently look at quaternions or other alternatives to and generalisations of complex numbers. I would focus on mastering the theory of complex numbers as taught and used in modern mathematics and physics.

Likewise, the modern basis of SR can be presented remarkably simply, without getting drawn into the historical compexity of its development, the alternative aether theory or the mathematically more complicated relationship with Maxwell's equations.

These ideas are worth mentioning in passing, of course, but it gets confusing if you extend your purview beyond the fundamentals of the theory itself - at a first reading.

Once you have mastered the basics of SR, then you can move on. If we are talking high-school level, then you need a serious textbook aimed at those with high-school mathematics and physics knowledge. The ones I like, because they focus on teaching the fundamentals as directly as possible, are Morin and Helliwell:

https://scholar.harvard.edu/files/david-morin/files/relativity_chap_1.pdf

bhobba and Dale
FranzDiCoccio said:
So I'm still not clear on how "Newtonian mechanics works just fine with Maxwell’s electrodynamics.".
I would say @Nugatory slightly over reached with that statement. If you have Galilean relativity and Maxwell's equations and you're reasonably sure Galileo is right, your problem is that Maxwell appears to be indistinguishable from reality. So you need to fix Maxwell. Since he predicts electromagnetic waves, proposing a medium for electromagnetism in which we are nearly at rest is an obvious way to try to preserve Galilean relativity. It imposes a frame that's important for electromagnetism in the same way the rest frame of the ocean is important for shipping, and doesn't break Galilean relativity.

However, I'm not aware of any successful extension of Maxwell's equations that is fully consistent with Newtonian physics plus an ether.

FranzDiCoccio said:
So I'm still not clear on how "Newtonian mechanics works just fine with Maxwell’s electrodynamics.".
It doesn't work "just fine". One example are the endless debates about "hidden momentum" or phenomena like the "homopolar generator". Nevertheless often the non-relativistic approximation for the "mechanical part" of electrodynamics works well, but one must keep in mind that it's an approximation, which is not entirely consistent with electromagnetic phenomena to avoid the corresponding confusion.
FranzDiCoccio said:
This sounds more like "Newtonian mechanics was expected to work just fine with something more intricate than Maxwell's equations, but this turned out not to be the case in the end".
I never had the time to go deeper, but this is was my idea too.

An earlier edition of my book mentions a simplified version of this view. The chapter started with a simple review of the aether theory and the Michelson Morley experiment.
Now that part is not in the introductory paragraph any more. It has been moved to a sort of appendix, at the end of the chapter.
I was told that the reason would be some kind of "historical accuracy", because Einstein did not really know about M&M's experiment, or perhaps did not think of SR because of that experiment.
Einstein's approach was much more general than this one experimental result, although it was very important as an experimental tool in its accuracy beyond the linear order in ##\beta##. Einstein's really big achievement of the famous paper of 1905 was the emphasis of symmetry principles as a way to think about physical theories. It turned out as the paradigm leading to the great success of 20th-century physics, including Einstein's masterpiece, General Relativity but also quantum theory and particularly quantum-field theory and high-energy particle physics with the development of the Standard Model.
FranzDiCoccio said:
But my book does not seem very consistent in that respect because, after discarding the introductory paragraph about aether and M&M's experiment, it introduced a new paragraph about the de Sitter experiment. As far as I understand, de Sitter's argument is against a ballistic theory of light that emerged in contrast to SR.

In my opinion starting with aether and the M&M's experiment does make sense, especially if emission theories were devised to explain the results of that experiment, and in contrast with SR.
I guess that before M&M everyone was sure light was a wave propagating in some medium, like all waves known on Earth. I wonder what was the support for a ballistic theory of light, at that time.
I don't see too much merit in discussing outdated theories instead of concentrating on the modern ones. This should indeed be put in an extra section or appendix of a textbook, because it might be useful to understand the modern point of view better when knowing about some of the historical background, but it's distracting from the consistent formulation of the modern theory.

bhobba, robphy and PeroK
PeroK said:
the modern basis of SR can be presented remarkably simply, without getting drawn into the historical compexity of its development
I agree with this. My textbook focused on the history and the thought experiments. I think it hindered my learning more than it helped.

bhobba, robphy, vanhees71 and 1 other person
Ibix said:
However, I'm not aware of any successful extension of Maxwell's equations that is fully consistent with Newtonian physics plus an ether.
Even if we limit ourselves to frames at rest relative to the ether?

Nugatory said:
Even if we limit ourselves to frames at rest relative to the ether?
I see what you mean. You can always assert that you're at rest in the ether frame. But you don't really have a transformation law for EM in that case, do you?

vanhees71 said:
I don't see too much merit in discussing outdated theories instead of concentrating on the modern ones. This should indeed be put in an extra section or appendix of a textbook, because it might be useful to understand the modern point of view better when knowing about some of the historical background, but it's distracting from the consistent formulation of the modern theory.

Dale said:
I agree with this. My textbook focused on the history and the thought experiments. I think it hindered my learning more than it helped.
It seems to me that intro physics texts follow the physics-timeline of how relativity developed (ether, michelson-morley, "effects", "paradoxes", etc...) because it's a story of "how new physics was developed".

Unfortunately, it seems to me that the students (and maybe faculty) never really get over the various stumbling blocks because the more-streamlined spacetime viewpoint is never really developed,
...probably because to the typical physics teacher "it's too mathematical"
(in the spirit of Einstein's first reaction to Minkowski's spacetime approach "Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore.” - Einstein (as reported by Sommerfeld).
Of course, we know that Einstein later embraced the spacetime approach to develop general relativity.)

Recently, I've been taking this approach:
I use the term "position-vs-time graph" first, then later call it a "spacetime diagram".
Then, I say (in the spirit of Steve Jobs) Minkowski spacetime diagrams seem confusing
because "You're using it (a PHY101 position-vs-time graph) wrong"...
then show what an accurate position-vs-time graph would really say
on everyday physics scales (e.g. v=30 m/s).

vanhees71, bhobba and Dale
robphy said:
it's a story of "how new physics was developed".
What is strange to me about that is that the same textbook did not grab on to that story approach for any other topic.

vanhees71 and robphy
FranzDiCoccio said:
So I'm still not clear on how "Newtonian mechanics works just fine with Maxwell’s electrodynamics.".
This sounds more like "Newtonian mechanics was expected to work just fine with something more intricate than Maxwell's equations, but this turned out not to be the case in the end".

Ibix said:
I would say @Nugatory slightly over reached with that statement. If you have Galilean relativity and Maxwell's equations and you're reasonably sure Galileo is right, your problem is that Maxwell appears to be indistinguishable from reality. So you need to fix Maxwell. Since he predicts electromagnetic waves, proposing a medium for electromagnetism in which we are nearly at rest is an obvious way to try to preserve Galilean relativity. It imposes a frame that's important for electromagnetism in the same way the rest frame of the ocean is important for shipping, and doesn't break Galilean relativity.

However, I'm not aware of any successful extension of Maxwell's equations that is fully consistent with Newtonian physics plus an ether.

Here's an interesting article that motivated my interest in a spacetime geometric formulation of Galilean relativity:

http://dx.doi.org/10.1119/1.12239
"If Maxwell had worked between Ampère and Faraday: An historical fable with a pedagogical moral" by Jammer and Stachel ( American Journal of Physics 48, 5 (1980) )
...which discusses how you can re-order the history of EM to see a Galilean-invariant theory along the way... then be forced into a Lorentz-invariant one. The idea is based on the paper "Galilean Electromagnetism" by Le Bellac and Levy-Leblond ( (B 14, 217). Nuovo Cimento , 1973 ). (see https://en.wikipedia.org/wiki/Galilean_electromagnetism )
[not to be confused with "Galilean Electrodynamics"]
If one drops the Faraday induction term from Maxwell’s equations, they become exactly Galilei invariant. This suggests that if Maxwell had worked between Ampère and Faraday, he could have developed this Galilei‐invariant electromagnetic theory so that Faraday’s discovery would have confronted physicists with the dilemma: give up the Galileian relativity principle for electromagnetism (ether hypothesis), or modify it (special relativity). This suggests a new pedagogical approach to electromagnetic theory, in which the displacement current and the Galileian relativity principle are introduced before the induction term is discussed.

vanhees71 and Ibix
Dale said:
What is strange to me about that is that the same textbook did not grab on to that story approach for any other topic.
It seems the intro-textbook development of quantum mechanics also goes by a physics-timeline...
Rayleigh-Jeans to Planck, and Bohr to Schrodinger, etc...

Some have argued that we should move away from that storyline to present quantum mechanics in a more streamlined way. (e.g., Edwin Taylor's "Rescuing Quantum Mechanics from Atomic Physics"
https://www.eftaylor.com/pub/RescuingQM.pdf )

vanhees71 and Dale
PeroK said:
The ones I like, because they focus on teaching the fundamentals as directly as possible, are Morin and Helliwell:

I gave Morins Classical Mechanics textbook, including relativity. I forgot he did one specifically for relativity which is also excellent. I guess which you use depends on your goals. The point is good students at HS can handle good material at this level and get the basics understood properly.

Thanks
Bill

vanhees71 and Dale
bhobba said:
I gave Morins Classical Mechanics textbook, including relativity. I forgot he did one specifically for relativity which is also excellent. I guess which you use depends on your goals. The point is good students at HS can handle good material at this level and get the basics understood properly.

Thanks
Bill
That's what I think. Morin and Helliwell would be good for high school students.

bhobba and vanhees71
How much class time is being devoted to special relativity? Is it a week or two? Or is it a whole semester?
(If it's just a week or so, I repeat my suggestion of Bondi's Relativity and Common Sense.)
My \$0.02.

vanhees71

## 1. What is the "A paradox for two moving protons"?

The paradox for two moving protons is a thought experiment that explores the concept of relativity in physics. It involves two protons moving towards each other at high speeds and the paradox arises when considering the effects of time dilation and length contraction on the protons' measurements.

## 2. How does time dilation play a role in this paradox?

Time dilation is a phenomenon predicted by Einstein's theory of relativity, which states that time moves slower for objects moving at high speeds. In the paradox for two moving protons, time dilation causes the protons to perceive time differently, leading to a paradoxical situation where both protons experience different amounts of time passing.

## 3. What is the significance of length contraction in this paradox?

Length contraction is another consequence of Einstein's theory of relativity, which states that an object's length appears shorter when it is moving at high speeds. In the paradox for two moving protons, length contraction causes the protons to measure different lengths for the same object, leading to a paradoxical situation where both protons observe different lengths.

## 4. How does this paradox challenge our understanding of relativity?

The paradox for two moving protons challenges our understanding of relativity by highlighting the counterintuitive nature of the theory. It forces us to question our assumptions about time and space and how they are affected by motion. It also demonstrates the need for a deeper understanding of the concept of relativity and its implications in physics.

## 5. Is this paradox applicable to real-world situations?

While the paradox for two moving protons is a thought experiment, it is based on the principles of relativity, which have been proven to be accurate through various experiments and observations. Therefore, it can be applied to real-world situations involving objects moving at high speeds, such as particles in accelerators or spacecraft traveling at near-light speeds.

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