Lineal gravity from planar gravity

Abstract

We show how to obtain the two-dimensional black hole action by dimensional reduction of the three-dimensional Einstein action with a non-zero cosmological constant. Starting from the Chern-Simons formulation of 2+1 gravity, we obtain the 1+1 dimensional gauge formulation given by Verlinde. Remarkably, the proposed reduction shares the relevant features of the formulation of Cangemi and Jackiw, without the need for a central charge in the algebra. We show how the Lagrange multipliersin these formulations appear naturally as the remnants of the three dimensional connection associated to symmetries that have been lostin the dimensional reduction. The proposed dimensional reduction involves a shift in the three dimensional connection whose effect is to make the length of the extra dimension infinite.