Explaining the "Causation Problem" of Abraham-Lorentz Force

[neworder1]

Can somebody explain the "causation problem" related to the Abraham-Lorentz force - i.e. why the character of the force ([tex]a = c\frac{da}{dt}[/tex]) implies that an external force applied in the future affects the particle's motion now? I read about it in Griffiths and elsewhere, but I'm not convinced.

By the way, doesn't the form of the force ([tex]a[/tex] proportional to [tex]\frac{da}{dt}[/tex]) violate the Newton's principle, i.e. that acceleration is a function only of position, velocity and time?

 

[Demystifier]

I do not see it as a real problem, as the AL force is not a fundamental force. Instead, it is an effective approximative description of forces which, at the fundamental level, are causal and consistent with the Newton's principle. For me, this problem only says that such an approximation is not really a good one.

 

[olgranpappy]

By the way, doesn't the form of the force ([tex]a[/tex] proportional to [tex]\frac{da}{dt}[/tex]) violate the Newton's principle, i.e. that acceleration is a function only of position, velocity and time?

The form of the force (i.e. proportional to "jerk") is chosen such that the work done will amount to that energy radiated by the accelerating charge.

Can you tell us what you already know about the problem of causality in the case of an accelerating particle? Cheers.

P.S. this is a little off-topic, but Peierls wrote an interesting piece in his book "surprises in theoretical physics" about the problem of the standard formula for power radiated (proportional to acceleration squared) and general relativity (acceleration being equivalent to grav field). That is to say, there are many problems with the seemingly ad hoc proceedure of tacking on a force proportional to the time derivative of the accleration beyond just the problem of causality.

 

[Meir Achuz]

The use of that force is just wrong. It was derived for a periodic system, and should not be applied to anything else.

 

[samalkhaiat]

Can somebody explain the "causation problem" related to the Abraham-Lorentz force - i.e. why the character of the force ([tex]a = c\frac{da}{dt}[/tex]) implies that an external force applied in the future affects the particle's motion now? I read about it in Griffiths and elsewhere, but I'm not convinced.

Read post #10 & 14 in

https://www.physicsforums.com/showthread.php?t=100343


regards

sam

 

[Demystifier]

The use of that force is just wrong. It was derived for a periodic system, and should not be applied to anything else.

I have seen also different derivations. But of course, other derivations also have certain restrictions of validity.

 

[Brad Barker]

wow. coincidentally, i was just reading about this from griffiths last night!

i must be a little rusty, but i don't understand how a radiation reaction can exist with no net force on a charged particle.

as far as i understand, a charged particle will radiate when accelerated, and then the radiation reaction will dampen this acceleration. but the initial acceleration needs to be provided by an external force, no?

in classical EM, a charged particle won't spontaneously radiate, right?

 

[olgranpappy]

Consider a simple harmonic oscillator consisting of an *uncharged* point mass attached to a spring. If I start the thing going then in the absence of friction it will keep on going forever. And (except when it momentarily at its equilibrium position) it is always accelerating.

Now, let the point mass be *charged.* It's an accelerating point charge so it will radiate and I have a formula that tells me how to calculate the power radiated when given the acceleration... but it's not obvious that I can apply it given the previous acceleration of the uncharged SHO since, because of the radiation, the SHO must loose energy.

Thus I have a damped oscillator, and the damping should be included in the equations of motion leading to a different expression for the acceleration...

But that leads to a different expression for the power radiated and that different power radiated really should have been taken into account in the equations of motion...

And that leads to a different expression for the acceleration...

But that leads to a different expression for the power radiated...

 

[cesiumfrog]

olgran', are you trying to imply those equations are intractable??

The use of that force is just wrong.

this problem only says that such an approximation [the AL force] is not really a good one.

These two statements seem to contradict what is presented by Griffiths. Obviously he isn't infallible, but then he is the author of some of the most widely used texts for electrodynamics, particle physics and quantum mechanics, so wouldn't it be appropriate to give some detailed justification for this disagreement?

 

[olgranpappy]

olgran', are you trying to imply those equations are intractable??

no it's not intractable, you basically just introduce a damping term and hope that the damping term is small. I.e. your equation of motion is just something like:
[tex]
m\frac{d^2x}{dt^2}=-kx+\frac{2e^2}{3c^3}\frac{d^3x}{dt^3}
[/tex]
which, for the case of approximately harmonic motion, is just the same as having a term on the RHS that depends on the velocity (i.e., three time derivatives is just proportional to one time derivative). The solution looks like:
[tex]
x \approx x_0e^{-i\omega t}e^{-\gamma t}
[/tex]
where [tex]\gamma=\frac{2\omega^2e^2}{3mc^3}[/tex].

 

[Demystifier]

These two statements seem to contradict what is presented by Griffiths. Obviously he isn't infallible, but then he is the author of some of the most widely used texts for electrodynamics, particle physics and quantum mechanics, so wouldn't it be appropriate to give some detailed justification for this disagreement?

Does Griffiths say that the AL force is exact? I don't believe that he does.

 

[neworder1]

Griffiths' derivation assumes only that the motion is nonrelativistic (i.e. we can use the simple formula for Larmor's power) and cyclic (AL force is the radiation force averaged over a cycle), so, modulo relativity, it seems rather exact.

Also - is there anything strange in the particle's acceleration increasing exponentially, as one of the AL force equation's solutions permits? Suppose that the area of the EM field is infinite - the particle's energy would increase without limit, but, then, it can constantly draw energy from the EM field, doesn't it? Do we reject the "exponentially increasing acceleration" solution because it violates the energy conservation law or because it is unphysical (i.e. we don't observe things like this)?

And one more thing - is the EM radiation a theoretical necessity needed to avoid violation of the energy conservation law via self-acceleration? If I understand things correctly, a moving particle can accelerate using its own EM field - in time t0 the particle generates a field, and this field affects the particle movement in later time t1 (since the force affecting the particle is calculated in retarded time, so it depends on the motion's parameters a while earlier). Would it lead to infinite increase in particle's energy if there wasn't any radiation?

 

[JK423]

Griffiths' derivation assumes only that the motion is nonrelativistic (i.e. we can use the simple formula for Larmor's power) and cyclic (AL force is the radiation force averaged over a cycle), so, modulo relativity, it seems rather exact.

Also - is there anything strange in the particle's acceleration increasing exponentially, as one of the AL force equation's solutions permits? Suppose that the area of the EM field is infinite - the particle's energy would increase without limit, but, then, it can constantly draw energy from the EM field, doesn't it? Do we reject the "exponentially increasing acceleration" solution because it violates the energy conservation law or because it is unphysical (i.e. we don't observe things like this)?

Ive been studying Electrodynamics by Griffiths. And i have reached this exact point at the moment, where it's shown that the particle's acceleration increases exponentially due to self-force even when no external force is exerted on it.
I disagree with the derivation of such a conclusion:
First of all, the acceleration of the particle 'a' is initially created by an external force. Let's suppose that 'a' is constant in time so the 'jerk' is zero which means that the self-force is zero. When the external force suddenly stops, then 'a' tends to reduce: a->0
That means that da/dt is nonzero and negative! It must be negative because 'a' is decreasing. That means, that the actual equation is
-(da/dt)~a
and from this one, we can see that the particle's acceleration decreases with time reaching a=0, constant velocity. All this seem rational to me.
However, Griffiths supposes that da/dt>0.. Why? Just because of this assumption we derive the exponentially increasing acceleration!
Any ideas?