Geometric Interpretation and Classification of Global Solutions in Generalized Dilaton Gravity

TL;DR

This paper explores the global solutions of generalized dilaton gravity in two dimensions, revealing how torsion and curvature influence their properties and how certain solutions can be asymptotically flat or singularity-free.

Contribution

It demonstrates the equivalence between 2D gravity with torsion and generalized dilaton gravity, analyzing their global solutions and properties, including asymptotic flatness and singularity removal.

Findings

Global solutions split into sets of solutions in generalized dilaton gravity.

Solutions in torsionless models can be asymptotically flat.

The Schwarzschild singularity can be removed via field redefinition.

Abstract

Two dimensional gravity with torsion is proved to be equivalent to special types of generalized 2d dilaton gravity. E.g. in one version, the dilaton field is shown to be expressible by the extra scalar curvature, constructed for an independent Lorentz connection corresponding to a nontrivial torsion. Elimination of that dilaton field yields an equivalent torsionless theory, nonpolynomial in curvature. These theories, although locally equivalent exhibit quite different global properties of the general solution. We discuss the example of a (torsionless) dilaton theory equivalent to the $R^2 + T^2$--model. Each global solution of this model is shown to split into a set of global solutions of generalized dilaton gravity. In contrast to the theory with torsion the equivalent dilaton one exhibits solutions which are asymptotically flat in special ranges of the parameters. In the simplest case of ordinary dilaton gravity we clarify the well known problem of removing the Schwarzschild singularity by a field redefinition.