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The Newton's law of universal gravitation is described in terms of a force, which is produced by an action at a distance. It also can be described using the concept of a field, and that would be an equivalent formulation.

Now, classical electrodynamics is formulated in terms of fields in a more complicated way that Newton's gravitation. My question is: Is there an equivalent formulation of classical electrodynamics in terms of action at a distance that is completely equivalent to the formulation in terms of fields (Maxwell's equations)? I would tempted to think that it is not possible, since in Maxwell's equations you may have electromagnetic waves without the presence of charges, and I don't know how to do it only in terms of forces. However, I'm not entirely sure, that's why I'm asking.

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    $\begingroup$ You might find this helps you sort this out: "Sources, Potentials and Fields in Lorenz and Coulomb Gauge: Cancellation of Instantaneous Interactions for Moving Point Charges": arxiv.org/abs/1110.6210 "As is well known, the scalar potential shows an action-at-a-distance behavior in Coulomb gauge. The conundrum generated by the instantaneous interaction has intrigued physicists for a long time." $\endgroup$ – Alfred Centauri Dec 15 '13 at 15:46
  • $\begingroup$ Also, see "Causality, the Coulomb field, and Newton's law of gravitation": siba.unipv.it/fisica/articoli/A/… $\endgroup$ – Alfred Centauri Dec 15 '13 at 15:51
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Is there an equivalent formulation of classical electrodynamics in terms of action at a distance that is completely equivalent to the formulation in terms of fields (Maxwell's equations)?

Yes, but only if special boundary conditions on the fields are assumed. For example, if the fields are purely retarded (wiki: Retarded and Advanced Potentials), one can solve for them in terms of the trajectories of particles. The forces acting on the particles are then functions of their past positions, velocities and accelerations. This seems to be the most natural choice.

Feynman and Wheeler considered another possibility: that the fields are half retarded + half advanced. Then it is again possible to eliminate fields and write down the equations of motion.

But in both cases, the fields due to particles are still very useful auxiliary quantities.

J. A. Wheeler, R. P. Feynman, Classical Electrodynamics in Terms of Direct Interparticle Interaction, Rev. Mod. Phys., 21, 3, (1949), p. 425-433.

http://dx.doi.org/10.1103/RevModPhys.21.425

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I strongly suggest You research Einstein's brilliant insight in "The Principal of Equivalence". You will begin to see that what we think of as a "force" or hypothetical "graviton" as a carrier of force is not valid. An accelerating field= gravity. No "pull". No force. Seems hard to grasp, because it is. 'might try and get "Gravity Revealed" by Ronald Hall--an obscure self-published book.

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