Does direct interparticle action imply advanced inertial forces?

  • #1
jcap
170
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In his Nobel lecture (https://www.nobelprize.org/prizes/physics/1965/feynman/lecture/) Richard Feynman states that by varying the Schwarzschild-Tetrode-Fokker direct interparticle action
$$A=-\sum_i m_i\int\big(\mathbf{\dot X_i}\cdot\mathbf{\dot X_i}\big)^{1/2}d\alpha_i+\frac{1}{2}\sum_{i\ne j}e_ie_j\iint\delta(I_{ij}^2)\ \mathbf{\dot X_i}\cdot\mathbf{\dot X_j}\ d\alpha_i\ d\alpha_j\tag{1}$$
where
$$I_{ij}^2=\big[\mathbf{X_i}(\alpha_i)-\mathbf{X_j}(\alpha_j)\big]\cdot\big[\mathbf{X_i}(\alpha_i)-\mathbf{X_j}(\alpha_j)\big]$$
one can reproduce classical electrodynamics without using the concept of the electromagnetic field.

In their paper Classical Electrodynamics in Terms of Direct Interparticle Action (https://cds.cern.ch/record/1062647/files/RevModPhys.21.425.pdf) John Wheeler and Richard Feynman showed in the section Action and Reaction on pages 429-430 that the energy-momentum transferred by retarded forces from particle ##i## to particle ##j## along a null worldline connecting them is equal and opposite to the energy-momentum transferred by advanced forces from particle ##j## back to particle ##i## along the same null worldline. Therefore they had discovered a Lorentz covariant generalization of Newton's principle of action and reaction.

Following the Feynman Lectures vol.1 ch. 28 section 28-2 (https://www.feynmanlectures.caltech.edu/I_28.html) I imagine two stationary particles with charges ##e_1## and ##e_2## separated by a large distance ##r## so that only the radiative electromagnetic forces, which decay as ##1/r##, are relevant.

Let us suppose that I apply a contact force to particle ##1## at time ##t## that gives it an acceleration ##\mathbf{a}(t)## perpendicular to the line joining the two particles.

The retarded electromagnetic force received by particle ##2## at time ##t+r/c##, having been emitted by particle ##1## at time ##t##, is given by

$$\mathbf{F^{21}}(t+r/c)=\frac{-e_1 e_2}{4\pi\epsilon_0 c^2r}\mathbf{a}(t)\tag{2}$$

For simplicity let us suppose that the mass of particle ##2## is very large so that it hardly accelerates at all and therefore does not produce a retarded force back on particle ##1##.

However if the direct interparticle action ##(1)## is a correct description of Nature then there should be an advanced reaction force back on particle ##1## at time ##t##, emitted by particle ##2## at time ##t+r/c##, given by

$$\mathbf{F^{12}}(t)=\frac{e_1 e_2}{4\pi\epsilon_0 c^2r}\mathbf{a}(t)\tag{2}$$

As this back reaction force is proportional to the acceleration then it will manifest itself as an apparent change in the inertia of particle ##1##, ##\Delta m_1##, given by

$$\Delta m_1=\frac{e_1 e_2}{4\pi\epsilon_0 c^2r}\tag{3}$$

This apparent change in inertia of particle ##1## should be detectable. Has any such effect been measured?
 
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  • #2
You appear to be mixing up two different theories: standard Maxwell electrodynamics (presented in the Feynman lectures) and the "direct interparticle action" theory (presented in the Wheeler-Feynman paper). That isn't valid.

The "retarded force" formula you give is from standard Maxwell electrodynamics. There is no "advanced force" in Maxwell electrodynamics. So you can't use formulas from Maxwell electrodynamics to predict any observable consequences from an "advanced force".

If you want to assess whether the "direct interparticle action" theory predicts a change in the inertia of a charged particle due to advanced forces, you need to use the math of that theory.
 

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