Abstract
The problem of two relativistically-moving pointlike particles of constant mass is undertaken in an arbitrary Lorentz frame using the classical Lagrangian mechanics of Stückelberg, Horwitz, and Piron. The particles are assumed to interact at events along their world lines at a common “world time,” an invariant dynamical parameter which is not in general synchronous with the particle proper time. The Lorentz-scalar interaction is assumed to be the Coulomb potential (i.e., the inverse square spacetime potential) of the spacetime event separation. The classical orbit equations are found in 1 + 1 spacetime dimensions in the hyperbolic angle coordinates for the reduced problem. The solutions to the reduced motion in these coordinates are the spacetime generalizations of the nonrelativistic Kepler solutions. and they introduce an invariant eccentricity which is a function of other known constants of the motion for the reduced problem. Solutions compatible with physical scattering are obtained by the assumption that the eccentricity is a given function of the ratio of the particle masses.
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M. A. Trump, Ph.D. dissertation. The University of of Texas at Austin, December 1997.
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Trump, M.A., Schieve, W.C. Classical Scattering in the Covariant Two-Body Coulomb Potential. Foundations of Physics 28, 1211–1236 (1998). https://doi.org/10.1023/A:1018806308530
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DOI: https://doi.org/10.1023/A:1018806308530