(1 + 1)-Dimensional Methods for General Relativity

Summary

Abstract

We present the (1+1)-dimensional method for studying general relativity of 4-dimensions. We first discuss the general formalism, and subsequently draw attention to the algebraically special class of space-times, following the Petrov classification. It is shown that this class of space-times can be described by the (1+1)-dimensional Yang-Mills action interacting with matter fields, with the spacial diffeomorphisms of the 2-surface as the gauge symmetry. The (Hamiltonian) constraint appears polynomial in part, whereas the non-polynomial part is a non-linear sigma model type in (1+1)-dimensions. It is also shown that the representations of w_∞-gravity appear naturally as special cases of this description, and we discuss briefly the w_∞-geometry in term of the fibre bundle.

Introduction

For past years many 2-dimensional field theories have been intensively studied as laboratories for many theoretical issues, due to great mathematical simplicities that often exist in 2-dimensional systems. Recently these 2-dimensional field theories have received considerable attention, for different reasons, in connection with general relativistic systems of 4-dimensions, such as self-dual spaces [1] and the black-hole space-times [2, 3]. These 2-dimensional formulations of self-dual spaces and blackhole space-times of allow, in principle, many 2-dimensional field theoretic methods developed in the past relevant for the description of the physics of 4-dimensions.