Derivation of Newton's Third Law (?)

[nonequilibrium]

Griffiths says "unlike the first two, Newton's third law does not, in general, extend to the relativistic domain."

I don't understand, what is wrong with the following derivation?

So we take it as an experimental fact that momentum is conserved in isolated systems (for clarity, the momentum we're working with is the spatial part of the 4-vector momentum or momenergy).

Now imagine an isolated system consisting of two particles. Call $$\vec P$$ the momentum of the whole system, and $$\vec p_1$$ and $$\vec p_2$$ those of the respective particles in the system. By definition, $$\vec P = \vec p_1 + \vec p_2$$.

Differentiate both sides with respect to coordinate time, then by definition of ordinary force:
$$\frac{\mathrm d \vec P}{\mathrm d t} = \boxed{ \vec F_{\textrm{ext}}= \vec F_{12} + \vec F_{21}} = \frac{\mathrm d \vec p_1}{\mathrm d t} + \frac{\mathrm d \vec p_2}{\mathrm d t}$$.

Due to the experimental fact that $$\vec P = \textrm{constant} \Rightarrow \vec F_{\textrm{ext}} = 0$$ and thus as a result of the boxed equation: $$\vec F_{12} = - \vec F_{21}$$.

Did I make an error in the above derivation, or is it simply that Newton's 3rd law only fails for systems consisting of 3+ particles?

 

[Sam Gralla]

your capital P is not constant when electromagnetic fields are involved. EM fields carry momentum and only the total momentum, bodies + EM field, is conserved.

 

[bcrowell]

Is Griffiths referring to SR, or GR? In the context of GR, conservation of momentum is local, not global, and the way your proof breaks down is that you're assuming that vectors expressed in different local coordinate charts can be added without any ambiguity caused by the path-dependence of parallel transport.

 

[nonequilibrium]

Sam - Hm, I don't agree with your comment for two reasons:
1) Griffiths wasn't talking specifically about electrodynamics, he seemed to claim it wasn't true in general
2) we still have conservation of momentum in case we define the new momentum

Bcrowell - SR, actually. But if I understand correctly: it's not my derivation you have problems with, but the use of "conservation of momentum"? (Griffiths does state momentum is preserved, as an experimental fact)

 

[atyy]

Conservation of momentum does generalize, but Newton's third law in its "original form" ("action"="reaction" at a "time") does not. In Newtonian physics, Coulomb's law and the law of universal gravitation which implemented the third law globally were ok.

In relativistic electrodynamics, the force between two electrically charged particles is mediated by electromagenetic waves which travel with finite speed, so the two particles cannot have "equal and opposite" forces on each other at the same "time". If you want to make it "local" in spacetime, you'd have to somehow define the force of a particle on the electromagnetic field, which I don't know how to do easily.

 

[nonequilibrium]

atyy, what is wrong with my derivation (first post) in a field-free case?

 

[atyy]

atyy, what is wrong with my derivation (first post) in a field-free case?

I think your point is interesting, whether it really cannot be done.

I would say that in your derivation, you have specified that two particles at an arbitary distance apart can "know" what the other is doing instantaneously, which would imply faster than light transmission, and thus be inconsistent with special relativity.

Now, wavefunction collapse is faster than light, and there is no violation of special relativity. I believe the proper restriction is that no "classical" information travels faster than light. This needs some care to phrase correctly. Usually JesseM has know this very well, if you search for FTL stuff here.

 

[Sam Gralla]

Sam - Hm, I don't agree with your comment for two reasons:
1) Griffiths wasn't talking specifically about electrodynamics, he seemed to claim it wasn't true in general
2) we still have conservation of momentum in case we define the new momentum

Well, Newton's third law doesn't hold in a scalar field theory either, if that makes you feel better. The point is that conservation of momentum is the basic principle, not the 3rd law. If the only carriers of momentum in the problem are bodies, then the 3rd law holds, as you just proved (at least for 2 bodies). But if--like in EM or pretty much any other field theory--there are fields that can carry momentum, then the 3rd law doesn't hold. This is a very standard and well-established point, by the way.

 

[atyy]

atyy, what is wrong with my derivation (first post) in a field-free case?

If the only carriers of momentum in the problem are bodies, then the 3rd law holds, as you just proved (at least for 2 bodies).

This is interesting. So it may be possible for Newton's 3rd law to hold relativistically for 2 particles - is there a solution that is relativistically ok, apart from the case where both forces are equal and opposite and zero? If so, would there any additional restriction, eg. the forces cannot be central, since I would presume Coulomb's law is not relativistically covariant, even for 2 particles? My intuition is that generally one would run into trouble in cases where the acceleration of a particle is not parallel to its velocity, which is why I am thinking about central forces.

 

[PAllen]

This is interesting. So it may be possible for Newton's 3rd law to hold relativistically for 2 particles - is there a solution that is relativistically ok, apart from the case where both forces are equal and opposite and zero? If so, would there any additional restriction, eg. the forces cannot be central, since I would presume Coulomb's law is not relativistically covariant, even for 2 particles? My intuition is that generally one would run into trouble in cases where the acceleration of a particle is not parallel to its velocity, which is why I am thinking about central forces.

Any field can't carry change instantly (in SR or GR), so 3d law can't hold for any field in SR or GR. It can hold in SR or GR (locally) for collisions (and probably nothing else).

 

[atyy]

Any field can't carry change instantly (in SR or GR), so 3d law can't hold for any field in SR or GR. It can hold in SR or GR (locally) for collisions (and probably nothing else).

Agreed. I think what mr. vodka is asking about is the "probably nothing else". Can we change that to "certainly nothing else", and if so, what do we have to add to his derivation in the OP to demonstrate it.

 

[PAllen]

Agreed. I think what mr. vodka is asking about is the "probably nothing else". Can we change that to "certainly nothing else", and if so, what do we have to add to his derivation in the OP to demonstrate it.

In any field, given momentum change at one point, you can't instantly propagate this to a body at another point. You carry momentum change in the field, arriving at other body later. So, at no time does F12 = -F21, exactly.

 

[atyy]

In any field, given momentum change at one point, you can't instantly propagate this to a body at another point. You carry momentum change in the field, arriving at other body later. So, at no time does F12 = -F21, exactly.

How about restricting to 2 particles, and no field?

 

[PAllen]

How about restricting to 2 particles, and no field?

There can't be influence at a distance. I think that does bring it down exclusively to collisions. How else would you avoid influence at a distance (which would violate SR causality).

 

[atyy]

There can't be influence at a distance. I think that does bring it down exclusively to collisions. How else would you avoid influence at a distance (which would violate SR causality).

Heuristically, yes, but I don't see why the reasoning in the OP fails for spatially separated particles. I do think it implies the force law cannot be central, but it's not obvious that it rules out velocity dependent forces on spatially separated particles.

 

[PAllen]

Heuristically, yes, but I don't see why the reasoning in the OP fails for spatially separated particles. I do think it implies the force law cannot be central, but it's not obvious that it rules out velocity dependent forces on spatially separated particles.

Velocity dependent forces also fail. Even for the most simple motion, you would need acceleration dependent forces (because force produces acceleration). If you allow for changing acceleration, then you need higher derivative contributions to potential. I guess mathematically you can construct some simple case for which the 3d law is true for separated particles, but no physically interesting force law would work. Certainly, EM and all hypothetical gravity theories don't work (except Newton's action at a distance).

 

[atyy]

Velocity dependent forces also fail. Even for the most simple motion, you would need acceleration dependent forces (because force produces acceleration). If you allow for changing acceleration, then you need higher derivative contributions to potential. I guess mathematically you can construct some simple case for which the 3d law is true for separated particles, but no physically interesting force law would work. Certainly, EM and all hypothetical gravity theories don't work (except Newton's action at a distance).

I'd say any non-zero force would be interesting, if not physically realistic.

meopemuk has often mentioned a no-interaction theorem, which I see was quite recently discussed https://www.physicsforums.com/showthread.php?t=225026 .

 

[Sam Gralla]

The statement of Newton's third law doesn't really make sense unless the bodies are moving slow enough that retardation can be ignored. So I was assuming there was some sort of implicit slow-motion assumption here, so that action at a distance makes sense. Otherwise I'm not sure you can even ask the question about Newton's third law. (You'd have to try to compare forces at retarded times, or something.)

 

[atyy]

It looks like only zero forces are allowed even between 2 particles interacting "at a distance" without fields.

http://www.jstor.org/pss/52561
http://arxiv.org/abs/hep-th/0008163

 

[atyy]

Here's a picture of Sudarshan, who with Currie and Jordan proved the no interaction theorem. http://prizes.ictp.it/Dirac/dirac-medallists-2010