Aspects of electrostatics in BTZ geometries

Abstract

In the present paper the electrostatic of charges in non rotating BTZ black hole and wormhole space times is studied. In particular, the self force of a point charge in the geometry is characterized analitically. The differences between the self force in both cases is a theoretical experiment for distinguishing both geometries, which otherwise are locally indistinguishable. This idea was applied before to four and higher dimensional black holes by the present and other authors. However, the particularities of the BTZ geometry makes the analysis considerable more complicated than usual electrostatic in a flat geometry, and its even harder than its four dimensional counterparts. First, the BTZ space times are not asymptotically flat but instead asymptotically AdS. In addition, the relative distance $d(r,r+1)$ between two particles located at a radius $r$ and $r+1$ in the geometry tends to zero when $r\to\infty$. This behavior, which is radically different in a flat geometry, changes the analysis of the asymptotic conditions for the electrostatic field. In addition, there are no summation formulas that allow a closed analytic expression for the self force. We find a method to calculate such force in series, and the resulting expansion is convergent to the real solution. However, we suspect that the convergence is not uniform. In other, for points that are far away from the black hole the calculation of the force requires higher order summation. These subtleties are carefully analyzed in the paper, and it is shown that they lead to severe problems when calculating the self force for asymptotic points in the geometry.