Self-force of a static electric charge near a Schwarzschild star

When a charge is held static near a constant density spherical star, it experiences a self-force $F_{\mathrm{self}}$, which is significantly different from the force $F_{\mathrm{self}}^{\mathrm{BH}}$ it would experience when placed near a black hole of the same mass. In this paper, an expression for the self-force (as measured by a locally inertial observer) is given for an insulating Schwarzschild star, and the result is explicitly computed for the extreme density case, which has a singularity at its center. The force is found to be repulsive. A similar calculation of the self-force is also performed for a conducting star. This calculation is valid for any static, spherically conducting star, since the result is independent of the interior metric. When the charge is placed very close to the conducting star, the force is found to be attractive but when the charge is placed beyond a certain distance (2.95 M for a conducting star of radius 2.25 M), the force is found to be repulsive. When the charge is placed very far from the star (be it conducting or insulating), the charge experiences the same repulsive force it would experience when placed in the space-time of a black hole with the same mass as the star.

Figure 1

For various values of $x$, the convergence of ${\eta }^{\ell }(\mathrm{x})$ and ${\zeta }^{\ell }(\mathrm{x})$ in $\ell$ is shown. Note the difference in vertical scales.

Figure 2

The self-force on the charge is plotted as a function of its position for the two cases considered in the text: when the star is an insulator (upper curve) and when it is a conductor (lower curve).