The violation of Newton's third Law

[reterty]

Let us consider two point charges: one of which moves (in the simplest case) rectilinearly with a constant speed and the other is at rest. The electric field of the first charge is renormalized due to the effects of retardation and generation of a solenoidal electric field (see Am. J. Phys. 62, 79 (1994); doi: 10.1119/1.17716), while the field of the second one is the usual static conservative electric field. As a result of this, the electric forces of interaction of these charges are not equal in magnitude. Thus, we are dealing with a violation of Newton's third law. Could someone please help with resolving this paradox.

 

[etotheipi]

Maybe this is besides the point, but how can one charge stay at rest and the other move at a constant speed if they're exerting forces on each other?

 

[reterty]

Let there be another external fields that cause the legitimacy of this consideration

 

[Ibix]

Without thinking about this too hard (it's nearly midnight here), you should calculate the Poynting vector of the EM field. You'll find that it carries away momentum, which should equal what's missing from your particles. If you want your particles to move with steady velocity you'll need to include the external field in your calculations.

A naive application of the third law doesn't work here because it neglects the momentum of the field.

 

[atyy]

Newton's third law fails in electromagnetism, as there is no instantaneous action at a distance, since interactions propagate at the speed of light. Rather the third law must be generalized to momentum conservation, which holds in electromagnetism.

 

[Dale]

Newton’s 3rd law does fail in electromagnetism, but is this an example? Since both charges are inertial I think the field momentum is constant and the forces are equal in this specific example.

I believe that the mistake is failing to account for the retarded time. Yes the force is altered, but it is altered in a way that makes it so that the force from the moving charge at the retarded time on the stationary charge (at the current time) is equal and opposite the force from the stationary charge on the moving charge (at the current time).

 

[reterty]

Even if we take into account the momentum of the electromagnetic field, the momentum of the center of mass of this system of charges will still change with time.

 

[Vanadium 50]

I'd like to see some equations here.
Equations > Assertions
Drat. That was an inequality.

 

[Dale]

Even if we take into account the momentum of the electromagnetic field, the momentum of the center of mass of this system of charges will still change with time.

Well, it does have an external force acting on it.

 

[vanhees71]

A fully selfconsistent solution of a system consisting of two point charges and the electromagnetic field is, afaik, not found yet. It's utmost complicated, particularly for classical point charges.

The natural description of charges and the electromagnetic field is a continuum mechanical one and there momentum conservation holds exactly and in a local way. Newton's 3rd Law in its form for point particles can only work in action-at-a-distance models for interaction, and this violates the spacetime description of relativistic physics. There cannot be causal signals propagating faster than the speed of light.

 

[Dale]

Since both charges are inertial I think the field momentum is constant and the forces are equal in this specific example.

This is incorrect. I worked out the Lienard Wiechert potentials for one point charge stationary at the origin and the other point charge moving at $\vec v = (0.6, 0, 0)$ with $r(0)=(0,1,0)$. For units where c=1, q=1, and $\epsilon_0$=1 I calculated at t=0 that the force on the moving charge is $(0,1,0)$ and the force on the stationary charge is $(0,-1.25,0)$. I didn't test other configurations.

 

[vanhees71]

Sure, that's easy, but it's not solving the problem of the motion of the two particles. The total field energy an momentum are divergent for point charges even in this simplified case of static (or boosted static) fields. As mentioned already above, I don't think that there is a complete self-consistent solution for this problem within classical electrodynamics. The best we have is the Landau-Lifshitz equation for the motion of a single point charge in an external field, including the radiation reaction in an approximate way.

 

[Dale]

Sure, that's easy, but it's not solving the problem of the motion of the two particles.

That isn't an issue here since the motion is completely specified in the problem. However the specified motion does not fix the more basic problem that you mentioned that the field energy and momentum diverge for point particles. Personally, I consider this as a "point particle pathology" rather than a paradox.

What surprises me personally, though, is that the momentum is not constant. Since there is no acceleration there should be no radiation and so I had implicitly thought that there should be constant momentum. But since the mechanical forces are not equal the field momentum must be changing also.

 

[vanhees71]

Of course, the conclusion is that there are no classical point particles in nature.

I don't understand what you mean by the problem with momentum conservation. Momentum is just diverging. So you cannot define the momentum. But that can be easily repaired by taking a model of an extended charge. You can just solve the static problem for a charged sphere (or spherical shell) and then do a Lorentz transformation to a reference frame, where the charge moves uniformly. Why do you think the total momentum isn't conserved for this model?

If you take your two charges, one at rest, one moving uniformly, of course you must then neglect their interaction to get momentum conservation right. So you neglect acceleration due to this interaction too. Otherwise you'd have to make a model with extended charges interacting, and this is a very tough problem. I don't think that this has been solved to a satisfactory degree.

For a single extended Born rigid charged particle in an external field, which is a quite artificial model too, but it's at least consistent, there are the nice papers by Medina:

https://arxiv.org/abs/physics/0508031
https://arxiv.org/abs/hep-th/0702078

I'm not aware of something similar for elastic bodies, which would be very interesting and more realistic.

The entire issue becomes even more challenging if you try to include spin.

Of course, the entire issue also exists of course in GR, when it comes to a self-consistent description of interacting black holes in motion.

 

[Dale]

I don't understand what you mean by the problem with momentum conservation. ... Why do you think the total momentum isn't conserved for this model?

I didn’t say that there was a problem with momentum conservation. Please don’t distort what I said. I said that the field part of the momentum is not constant, which surprised me. Of course total momentum is conserved.

There is a net external force on the system so the momentum of the system is not constant. The motion of the particles has constant momentum, so to conserve total momentum (fields + particles + environment) the momentum of the fields must be changing. That surprised me.

 

[vanhees71]

Then I still don't get what precise calculation you are talking about. Which external force is acting on the system and why is then momentum conserved? That's a contradiction.

Let's do the calculation. In the rest frame of a little charged sphere the mechanical four-momentum is $p_{\text{mech}}^{\mu}=(m c^2,0,0,0)$ and the momentum of the em. field is $p_{\text{em}}=\lambda Q/a,0,0,0)$, where $a$ is the radius of the sphere and $\lambda$ some constant, depending on the charge distribution, which is not so important for the discussion. In the frame where the charge is moving with four-velocity $u^{\mu}=\gamma(1,0,0,\vec{\beta})$ one simply has
$p_{\text{mech}}^{\mu}=m c^2 u^{\mu}$ and $p_{\text{em}}=\lambda Q/a u^{\mu}$.

All this of course gets much more complicated, if the particle is accelerated, i.e., under the influence of an external force, because then you have to consider radiation reaction.

 

[Vanadium 50]

The thread has taken a turn in an overcomplicated direction, I think.

We have two particles, one constrained to be at rest and one constrained to move with constant velocity. The particles exert forces on each other. The action-reaction pairs are the force applied to particle A and its constraining force, and the force applied to particle B and its constraining force. The forces of An on B and B on A are not action-reaction pairs.

Now, one can make some statements about those force pairs (using those equations I talked about earlier), but from the point of Newton's Third Law, we are done.

 

[vanhees71]

But this is a contradiction and cannot give a valid result. Under the assumptions made on neither of the two particles acts a force. On the other hand you say there's a net force on each particles. Also in the 3rd law the action and reaction force never act on the same particle!

 

[jbriggs444]

The particles exert forces on each other.

If this were so, the force of A on B and B on A would be a third law pair. However, the action-at-a-distance paradigm for electrostatic repulsion does not work in general. Instead, one can model the interaction of each particle with the field. In which case we have two third law pairs and a field that can carry momentum away.

 

[Vanadium 50]

The point is that the problem has constraining forces (you can think of them as rails). The action-reaction pairs are between the particles and their constraints.

 

[vanhees71]

Sure, then the 3rd law holds, because you have only local ("contact") forces.

 

[Dale]

Which external force is acting on the system and why is then momentum conserved? That's a contradiction.

It is not a contradiction. There are three things that can contain momentum in this problem: the environment, the fields, and the particles. The environment exerts a force on the particles exchanging momentum between the environment and the particles, and the particles exchange momentum with the fields. Since the momentum of the particles is constant by the problem setup, that means that the momentum exchanged from the environment to the particles must go to the fields. That must happen in the same amount as the forces so that total momentum (environment + particles + fields) is conserved.

 

[vanhees71]

Then it's by definition not an external force, but an interaction with the environment, but that's semantics only again.

 

[Dale]

Then it's by definition not an external force, but an interaction with the environment, but that's semantics only again.

Maybe next time just consider that as a possibility before assuming that other experts are making actual contradictions.

 

[vanhees71]

I'm sorry, I've learned obviously other definitions than you, and I didn't say that you make a contradiction. In all textbooks I know, an external force means, I have an open system, and then of course momentum is not conserved. I didn't imply that you are wrong. It was just a misunderstanding by using different definitions. It's clarified now.

 

[etotheipi]

The best we have is the Landau-Lifshitz equation for the motion of a single point charge in an external field, including the radiation reaction in an approximate way.

Hey Vanhees, do you know of any references where this equation for a radiating particle is covered/derived? Maybe it's in Landau's lectures, but if so I'll have to wait a little while to borrow it!

 

[vanhees71]

The best treatment with a very elegant use of a very clever gauge transformation can be found in Landau and Lifshitz vol. 2.

 

[etotheipi]

The best treatment with a very elegant use of a very clever gauge transformation can be found in Landau and Lifshitz vol. 2.

Awesome, I'll try and get hold of it next week. Thanks!

 

[cmb]

Let us consider two point charges: one of which moves (in the simplest case) rectilinearly with a constant speed and the other is at rest. The electric field of the first charge is renormalized due to the effects of retardation and generation of a solenoidal electric field (see Am. J. Phys. 62, 79 (1994); doi: 10.1119/1.17716), while the field of the second one is the usual static conservative electric field. As a result of this, the electric forces of interaction of these charges are not equal in magnitude. Thus, we are dealing with a violation of Newton's third law. Could someone please help with resolving this paradox.

My understanding is that electric fields and magnetic fields are frame dependent. What looks like an electric field to one observer can look like a magnetic field to another observer in a different inertial frame. Two particles traveling past an observer will appear to interact by magnetic forces (motions of particles creating magnetic fields that interact) while an observer traveling with them will only see electric fields, no magnetic fields.

So electromagnetism is very much a relativistic effect and I would have no surprise to find scenarios where Newton's laws fail to describe this.