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Time-Symmetric Action-at-a-Distance Electrodynamics and the Principle of Action and Reaction for Particles at Relative Rest

Calin Galeriu

TL;DR

The paper tackles deriving the relativistic action-reaction principle (PAR) within a time-symmetric, action-at-a-distance framework where four-forces depend only on charges, rest masses, positions, and four-velocities. It builds a symmetry-based derivation by combining conformal inversion with an axis reflection in flat conformal space, demonstrating PAR for identical particles at relative rest and partially extending to different masses. Key steps include mass exchange under conformal transformations, velocity/force transformations with conformal weights, and a-reflection that maps advanced to retarded contributions. The work underscores conformal symmetry as a potential underpinning of PAR in relativistic, time-symmetric interactions, while acknowledging that a complete account for particles in arbitrary relative motion requires an additional symmetry.

Abstract

We investigate a theory of time-symmetric action-at-a-distance electrodynamic or gravitational interactions where the four-forces depend only on the electric charges, the rest masses, the position four-vectors, and the four-velocities of the two interacting particles, but not on their four-accelerations or higher derivatives. The goal is to prove that the principle of action and reaction is obeyed due to the fact that, for a given pair of corresponding infinitesimal segments along the worldlines of the two particles, the impulse of the advanced four-force and the impulse of the retarded four-force are equal in magnitude but opposite in direction. For two particles at relative rest, we derive this result as the outcome of a symmetry operation in flat conformal space. We start by assuming a positive spacetime interval between the two interacting particles, and we perform a conformal inversion (an improper inversion in a hypersphere), after which the two particles exchange their position and time coordinates and their rest masses. Then we perform an improper reflection across the axis connecting the two particles, after which the two particles recover their initial four-velocities, up to a minus sign. Two improper coordinate transformations are needed together in order to obtain a resulting positive Jacobian determinant. In the final step we go to the limit of a null spacetime interval between the endpoints of the corresponding infinitesimal segments. When the two interacting particles are at rest, or move with the same velocity, the initial and the final physical parameters that determine the four-forces are identical. Due to symmetry, the principle of action and reaction emerges. We make the conjecture that another, undetermined yet, symmetry operation must also apply, in order for the principle of action and reaction to hold even for particles moving with different velocities.

Time-Symmetric Action-at-a-Distance Electrodynamics and the Principle of Action and Reaction for Particles at Relative Rest

TL;DR

The paper tackles deriving the relativistic action-reaction principle (PAR) within a time-symmetric, action-at-a-distance framework where four-forces depend only on charges, rest masses, positions, and four-velocities. It builds a symmetry-based derivation by combining conformal inversion with an axis reflection in flat conformal space, demonstrating PAR for identical particles at relative rest and partially extending to different masses. Key steps include mass exchange under conformal transformations, velocity/force transformations with conformal weights, and a-reflection that maps advanced to retarded contributions. The work underscores conformal symmetry as a potential underpinning of PAR in relativistic, time-symmetric interactions, while acknowledging that a complete account for particles in arbitrary relative motion requires an additional symmetry.

Abstract

We investigate a theory of time-symmetric action-at-a-distance electrodynamic or gravitational interactions where the four-forces depend only on the electric charges, the rest masses, the position four-vectors, and the four-velocities of the two interacting particles, but not on their four-accelerations or higher derivatives. The goal is to prove that the principle of action and reaction is obeyed due to the fact that, for a given pair of corresponding infinitesimal segments along the worldlines of the two particles, the impulse of the advanced four-force and the impulse of the retarded four-force are equal in magnitude but opposite in direction. For two particles at relative rest, we derive this result as the outcome of a symmetry operation in flat conformal space. We start by assuming a positive spacetime interval between the two interacting particles, and we perform a conformal inversion (an improper inversion in a hypersphere), after which the two particles exchange their position and time coordinates and their rest masses. Then we perform an improper reflection across the axis connecting the two particles, after which the two particles recover their initial four-velocities, up to a minus sign. Two improper coordinate transformations are needed together in order to obtain a resulting positive Jacobian determinant. In the final step we go to the limit of a null spacetime interval between the endpoints of the corresponding infinitesimal segments. When the two interacting particles are at rest, or move with the same velocity, the initial and the final physical parameters that determine the four-forces are identical. Due to symmetry, the principle of action and reaction emerges. We make the conjecture that another, undetermined yet, symmetry operation must also apply, in order for the principle of action and reaction to hold even for particles moving with different velocities.

Paper Structure

This paper contains 8 sections, 76 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A derivation of the PAR for two identical elementary particles at relative rest
  3. A review of conformal inversion formulas
  4. A derivation of the PAR for two different elementary particles at relative rest (1/3)
  5. A review of reflection across an axis formulas
  6. A derivation of the PAR for two different elementary particles at relative rest (2/3)
  7. A derivation of the PAR for two different elementary particles at relative rest (3/3)
  8. Concluding remarks

Figures (3)

  • Figure 1: The spacetime configurations before and after the PT operation.
  • Figure 2: The reflection of vector $V$ across the axis of vector $X$.
  • Figure 3: In the limit, when the interacting particles $AS$ and $BT$ become two corresponding segments of infinitesimal length, the two parallel worldlines of the two particles at rest, together with the two parallel null lines connecting the endpoints of the two segments, will make an infinitely thin parallelogram.