Self-force as probe of internal structure

TL;DR

This paper investigates how the self-force on a charge outside a massive body reveals information about the body's internal structure, using exact general relativity and polytropic models.

Contribution

It provides a detailed calculation of the scalar and electromagnetic self-force for relativistic polytropic spheres, linking the self-force to internal density distribution.

Findings

Self-force scales as 1/r^3 at leading order, depending only on charge and mass.

Next order 1/r^5 term depends on internal structure via a structure factor.

More centrally dense bodies produce smaller self-force for fixed mass and radius.

Abstract

The self-force acting on a (scalar or electric) charge held in place outside a massive body contains information about the body's composition, and can therefore be used as a probe of internal structure. We explore this theme by computing the (scalar or electromagnetic) self-force when the body is a spherical ball of perfect fluid in hydrostatic equilibrium, under the assumption that its rest-mass density and pressure are related by a polytropic equation of state. The body is strongly self-gravitating, and all computations are performed in exact general relativity. The dependence on internal structure is best revealed by expanding the self-force in powers of 1/r, with r denoting the radial position of the charge outside the body. To the leading order, the self-force scales as 1/r^3 and depends only on the square of the charge and the body's mass; the leading self-force is universal. The dependence on internal structure is seen at the next order, 1/r^5, through a structure factor that depends on the equation of state. We compute this structure factor for relativistic polytropes, and show that for a fixed mass, it increases linearly with the body's radius in the case of the scalar self-force, and quadratically with the body's radius in the case of the electromagnetic self-force. In both cases we find that for a fixed mass and radius, the self-force is smaller if the body is more centrally dense, and larger if the mass density is more uniformly distributed.