Two Dimensional Gravity as a modified Yang-Mills Theory

TL;DR

This paper explores a modified Yang-Mills approach to gravity in various dimensions, focusing on two-dimensional models that connect to known theories and reveal solutions with or without torsion.

Contribution

It develops a Yang-Mills based gravity model that unifies features of general relativity and gauge theories, including a Hamiltonian analysis and a generalized Birkhoff theorem in 2D.

Findings

Solutions with torsion exhibit horizons and singularities.

Unique torsion-free solution describes constant curvature space.

Hamiltonian analysis confirms the model's consistency.

Abstract

We study a deSitter/Anti-deSitter/Poincare Yang-Mills theory of gravity in d-space-time dimensions in an attempt to retain the best features of both general relativity and Yang-Mills theory: quadratic curvature, dimensionless coupling and background independence. We derive the equations of motion for Lie algebra valued scalars and show that in the geometric optics limit they traverse geodesics with respect to the Lorentzian geometry determined by the frame fields. Mixing between components appears to next to leading order in the WKB approximation. We then restrict to two space-time dimensions for simplicity, in which case the theory reduces to the well known Katanaev-Volovich model. We complete the Hamiltonian analysis of the vacuum theory and use it to prove a generalized Birkhoff theorem. There are two classes of solutions: with torsion and without torsion. The former are parametrized by two constants of motion, have event horizons for certain ranges of the parameters and a curvature singularity. The latter yield a unique solution, up to diffeomorphisms, that describes a space constant curvature .