Abstract
A classical action which describes the motion of a system of small-point massive charged particles including the existence of the electromagnetic and gravitational self-forces, Maxwell equations and Einstein field equations is presented. The action possesses the particularity of being a functional of the variables zi (τi), the trajectory of the i-particle, Aα (x), the electromagnetic 4-potential, and gαβ(x), the metric tensor. It is also considered that the metric tensor gαβ (x) and the potential Aα (x) are not functions of the trajectory of each particle when the variations with respect to the trajectories of the particles are done. That is, the action is complete. The electromagnetic and the gravitational self-forces are analyzed.
Highlights
Classical Electrodynamics and General Relativity represent two of the most important theories developed during the last century
Even if for many authors the Lorentz-Dirac equation [1] represents the best description of the motion of a charged point particle in Special Relativity, different proposals have appeared in order to avoid the unphysical predictions of such equation as the runaway solutions and the preaccelerations [2]
DeWitt and Brehme [9], or Hobbs [10], among others. It highlights the work done by Poisson et al [17], using techniques developed by Detweiler [19], in order to deduce the expressions for the self-electromagnetic force for a point massive charge with a fixed metric tensor and for the self-gravitational effect for a non-charged massive point particle in a vacuum background spacetime
Summary
Classical Electrodynamics and General Relativity represent two of the most important theories developed during the last century. As we have noted above, the different techniques developed to obtain an expression for the self-forces avoid the divergences by using renormalization processes as Dirac [1] did or by considering regular Greens functions as Detweiler did [17] [19], among others In both cases, an average of a regular field (not divergent) over a small surface is accomplished in order to calculate the self-force; in this order of ideas the concept of small-point massive charged particle is consistent with the technique cited above. Summarizing, the difference with the standard treatments consists of including the parallel propagator in the particle and interaction actions in order to obtain an action for a system of charged small-point particles which functionality does not possesses inconsistencies and the variations of each degree of freedom, zi , gαβ ( x) and Aα ( x) , lead to the equations of motion for each particle, the Einstein field equations and the Maxwell equations in curved space.
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