LINEAL GRAVITY FROM PLANAR GRAVITY

The 2D ''black hole'' action is obtained by a nonstandard dimensional reduction of pure 3D gravity with a nonzero 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. The Lagrange multipliers in these formulations appear naturally as remnants of the three-dimensional theory. The proposed dimensional reduction involves a shift in the three-dimensional connection whose effect is to make the physical length of the extra dimension infinite.