Abstract
We develop an approach to calculate the self-force on a charged particle held in place in a curved spacetime, in which the particle is attached to a massless string and the force is measured by the string's tension. The calculation is based on the Weyl class of static and axially symmetric spacetimes, and the presence of the string is manifested by a conical singularity; the tension is proportional to the angular deficit. A remarkable and appealing aspect of this approach is that the calculation of the self-force requires no renormalization of the particle's field. This is in contract with traditional methods, which incorporate a careful and elaborate subtraction of the singular part of the field. We implement the approach in a number of different situations. First, we examine the case of an electric charge in Schwarzschild spacetime, and recover the classic Smith-Will force in addition to a purely gravitational contribution to the self-force. Second, we turn to the case of electric and magnetic dipoles in Schwarzschild spacetime, and correct expressions for the self-force previously obtained in the literature. Third, we replace the electric charge by a scalar charge, and recover Wiseman's no-force result, which we generalize to a scalar dipole. And fourth, we calculate the force exerted on extended bodies such as Schwarzschild black holes and Janis-Newman-Winicour objects, which describe scalarized naked singularities.
Highlights
Smith and Will [1] calculated the self-force acting on an electric charge held in place in the Schwarzschild spacetime of a nonrotating black hole
The electric field lines emanating from the charge would be isotropically distributed around the particle, and the net force on the charge would vanish
The electric field is modified by the spacetime curvature, the field lines are no longer isotropic, and the net force no longer vanishes
Summary
Smith and Will [1] calculated the self-force acting on an electric charge held in place in the Schwarzschild spacetime of a nonrotating black hole. In addition to providing a precise operational meaning to the force and answers to these questions, a substantial advantage of the method developed in this paper is that the calculation of the force requires no regularization and no renormalization of the particle’s electric field, which is badly singular at the particle’s position. The approach, is limited by the restrictions inherent to the Weyl class of metrics: the spacetime must be static and axially symmetric, and the energy-momentum tensor must be such that ρ z This last condition is fairly accommodating, being met by electromagnetic and massless scalar fields, and by point particles.
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