On the Self-force in Electrodynamics and Implications for Gravity
We consider the motion of charged point particles on Minkowski spacetime. The questions of whether the self-force is finite and whether mass renormalisation is necessary are discussed within three theories: In the standard Maxwell vacuum theory, in the no
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Abstract We consider the motion of charged point particles on Minkowski spacetime. The questions of whether the self-force is finite and whether mass renormalisation is necessary are discussed within three theories: In the standard Maxwell vacuum theory, in the non-linear Born-Infeld theory and in the higher-order BoppPodolsky theory. In a final section we comment on possible implications for the theory of the self-force in gravity.
1 Introduction The problem of the electromagnetic self-force has a long history. It began in the late 19th century when Lorentz, Abraham and others tried to formulate a classical theory of the electron. The idea was to model the electron as an extended, at least approximately spherical, charged body and to determine the equations of motion for the electron. Based on earlier results by Lorentz, Abraham succeeded in writing the equation of motion in terms of a power series with respect to the radius of the electron. If the radius tended to zero, i.e., for a point charge, an infinity occurred. The reason for this infinity is in the fact that, in the point-particle limit, the electric field strength diverges so strongly at the position of the charge that the field energy in an arbitrarily small sphere becomes infinitely large. To get rid of this infinity, it was necessary to “renormalise the mass” of the particle by assuming that it carries a negative infinite “bare mass”. After this mass renormalisation, one got a differential equation of third order for the motion of the particle which is known as the AbrahamLorentz equation. It is a non-relativistic equation in the sense that, on the basis of special relativity, it can hold only if the particle’s speed is small in comparison to the speed of light. A fully relativistic treatment of the problem had to wait until Dirac’s work [1] of 1938. The resulting equation of motion is known as the Lorentz-Dirac equation or as the Abraham-Lorentz-Dirac equation. Clearly, everyone would call it the Dirac equation except for the fact that this name was already occupied by another, even V. Perlick (B) ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany e-mail: [email protected] © Springer International Publishing Switzerland 2015 D. Puetzfeld et al. (eds.), Equations of Motion in Relativistic Gravity, Fundamental Theories of Physics 179, DOI 10.1007/978-3-319-18335-0_15
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more famous equation. Neither Lorentz nor Abraham has ever seen the (Abraham-) Lorentz-Dirac equation, because both had passed away in the 1920s. In particular in the case of Abraham it is rather clear that he would not have liked this equation because he was an ardent opponent of relativity. Therefore, it seems appropriate to omit his name and call it the Lorentz-Dirac equation. For the derivation of the Lorentz-Dirac equation, again mass renormalisation was necessary and one arrived at a third-order equation of motion. The latter fact means that, in contrast to other equations of motion, not only the position and the velocity but also the acce
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