Time‐Symmetric Action‐at‐a‐Distance Electrodynamics and the Structure of Space‐Time

Abstract

This paper begins with a critical analysis of the concept of 'material point particle'. We argue that this concept is incompatible with the force laws of action-at-a-distance electrodynamics, and we suggest that the trajectory of a particle (the world-line) should be looked upon as a string in static equilibrium. By complementing our model with a time symmetric interaction law, we are led to a straightforward derivation of the principles underlying the electromagnetic interaction between two uniformly moving charged particles. An extension of our theory to the case of particles in arbitrary motion shows that we have to modify Maxwell's equations of the microscopic electromagnetic field, in order to accommodate a field-strength tensor which is no longer antisymmetric. It is further argued that the radiation reaction force gives another example of a 4-force which is not orthogonal to the 4-velocity, thus leading to a variable rest mass. The variation of the rest mass is then discussed from the point of view of the equivalence principle of general relativity, and we end up with a theoretical derivation of the homogenous Maxwell's equations.

Electric charge in hyperbolic motion: the early history

“…[3] This result will hold true even after a Lorentz transformation. The same time symmetric geometrical description applies to the case of a source charge in uniform motion [4]. When the source charge moves with constant velocity, or with constant acceleration, our geometrical description is identical to the generally accepted causal theory.…”

“…In this most general case, we have to write m ds instead of m ds in the expression of the action. We have also pointed out that, in the case of motion with variable acceleration, the four-force obtained from our model [4] is not always orthogonal to the four-velocity. A small deviation from orthogonality could indeed exist, and our geometrical recipe in this case amounts to supplementing the antisymmetric electromagnetic field tensor F ik with a small tensor G ik .…”

“…A small deviation from orthogonality could indeed exist, and our geometrical recipe in this case amounts to supplementing the antisymmetric electromagnetic field tensor F ik with a small tensor G ik . We expect the contribution of G ik to average out to zero, and we were able to prove this conjecture in a very simple situation [4]. In any case, by requiring that the rest mass of an electron at a given point in spacetime does not depend on its history, and by using Stoke's theorem, we have obtained the two homogeneous Maxwell equations [11], which in itself is a remarkable result.…”

“…In a similar context [4], when looking at the interaction between two electrically charged particles in uniform motion, we have noticed that the four-force shows an explicit dependence on the velocity of the test particle, and an implicit dependence on the velocity of the source particle (through the chosen reference frame). However, "from a geometrical point of view, a point in Minkowski space is just a fixed point -it does not have a velocity!"…”

“…However, "from a geometrical point of view, a point in Minkowski space is just a fixed point -it does not have a velocity!" [4] The conclusion was that, from a geometrical point of view, the electrodynamic four-force, the relevant Minkowski vector describing the interaction, is incompatible with the material point particle model.…”