Casimir effect in the curved background and the black hole in three dimensions

Abstract

We consider the quantum correction to the Lagrangean by the massless free boson in the curved background in three dimensions where one of the coordinates is periodic. The correction term is given by an expansion of the metric with respect to the derivative and the first term expresses to the usual Casimir energy. As an application, we investigate the change of the geometry in three dimensional black hole due to the quantum effect and we show that the geometry becomes like that of the Reissner-Nordstr\o m solution.