Lineal gravity from planar gravity

TL;DR

This paper demonstrates how to derive a two-dimensional black hole action from three-dimensional Einstein gravity with a cosmological constant, using a novel dimensional reduction approach that connects different formulations of 2+1 gravity.

Contribution

It introduces a new dimensional reduction method linking 3D Einstein gravity to 2D black hole models, avoiding the need for a central charge in the algebra.

Findings

Derived 2D black hole action from 3D Einstein gravity.

Connected Chern-Simons and gauge formulations of gravity.

Revealed natural emergence of Lagrange multipliers from 3D connections.

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.