TY - JOUR

T1 - On the self-force in Bopp-Podolsky electrodynamics

AU - Gratus, Jonathan

AU - Perlick, Volker

AU - Tucker, Robin W.

N1 - Date of Acceptance: 23/07/2015 30 pages, 6 figures; minor reformulations, some figures changed, additional explanations added

PY - 2015/10/7

Y1 - 2015/10/7

N2 - In the classical vacuum Maxwell-Lorentz theory the self-force of a charged point particle is infinite. This makes classical mass renormalization necessary and, in the special relativistic domain, leads to the Abraham-Lorentz-Dirac equation of motion possessing unphysical run-away and pre-acceleration solutions. In this paper we investigate whether the higher-order modification of classical vacuum electrodynamics suggested by Bopp, Lande, Thomas and Podolsky in the 1940s, can provide a solution to this problem. Since the theory is linear, Green-function techniques enable one to write the field of a charged point particle on Minkowski spacetime as an integral over the particle's history. By introducing the notion of timelike worldlines that are "bounded away from the backward light-cone" we are able to prescribe criteria for the convergence of such integrals. We also exhibit a timelike worldline yielding singular fields on a lightlike hyperplane in spacetime. In this case the field is mildly singular at the event where the particle crosses the hyperplane. Even in the case when the Bopp-Podolsky field is bounded, it exhibits a directional discontinuity as one approaches the point particle. We describe a procedure for assigning a value to the field on the particle worldline which enables one to define a finite Lorentz self-force. This is explicitly derived leading to an integro-differential equation for the motion of the particle in an external electromagnetic field. We conclude that any worldline solutions to this equation belonging to the categories discussed in the paper have continuous 4-velocities.

AB - In the classical vacuum Maxwell-Lorentz theory the self-force of a charged point particle is infinite. This makes classical mass renormalization necessary and, in the special relativistic domain, leads to the Abraham-Lorentz-Dirac equation of motion possessing unphysical run-away and pre-acceleration solutions. In this paper we investigate whether the higher-order modification of classical vacuum electrodynamics suggested by Bopp, Lande, Thomas and Podolsky in the 1940s, can provide a solution to this problem. Since the theory is linear, Green-function techniques enable one to write the field of a charged point particle on Minkowski spacetime as an integral over the particle's history. By introducing the notion of timelike worldlines that are "bounded away from the backward light-cone" we are able to prescribe criteria for the convergence of such integrals. We also exhibit a timelike worldline yielding singular fields on a lightlike hyperplane in spacetime. In this case the field is mildly singular at the event where the particle crosses the hyperplane. Even in the case when the Bopp-Podolsky field is bounded, it exhibits a directional discontinuity as one approaches the point particle. We describe a procedure for assigning a value to the field on the particle worldline which enables one to define a finite Lorentz self-force. This is explicitly derived leading to an integro-differential equation for the motion of the particle in an external electromagnetic field. We conclude that any worldline solutions to this equation belonging to the categories discussed in the paper have continuous 4-velocities.

KW - gr-qc

KW - math-ph

KW - math.MP

U2 - 10.1088/1751-8113/48/43/435401

DO - 10.1088/1751-8113/48/43/435401

M3 - Journal article

VL - 48

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 43

M1 - 435401

ER -