Geometric interpretation and classification of global solutions in generalized dilaton gravity

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 kown problem of removing the Schwarzschild singularity by a field redefinition.

Effective Action for Scalar Fields in Two-Dimensional Gravity

“…As a consequence one obtains the generalized dilaton model, the dilaton field coinciding with the momentum π(x) conjugate to the space component of the Lorentz connection ω 1 . The equivalence between gravity with torsion and dilaton gravity was first proved in [17]. There the transformation of variables included the Weyl transformation of the metric and therefore did not preserve global structure of solutions to the equations of motion [17,36].…”

“…The equivalence between gravity with torsion and dilaton gravity was first proved in [17]. There the transformation of variables included the Weyl transformation of the metric and therefore did not preserve global structure of solutions to the equations of motion [17,36]. Here we follow the improved procedure which does not change global properties of space-time [18,19].…”

“…Later it was realized that in fact the two-dimensional gravity models with torsion and dilaton gravity models are equivalent [17,18,19]. From a geometric point of view the dilaton field is the momentum canonically conjugate to the space component of the Lorentz connection.…”

Dilaton gravity in two dimensions

“…The most attractive feature of theories of type (2.17) is that an important subclass of them is in a one-to-one correspondence with the GDT-s (1.1). This dynamical equivalence, including the essential feature that also the global properties are exactly identical, seems to have been noticed first in [248] and used extensively in studies of the corresponding quantum theory [281,285,284].…”

Graded Poisson-sigma models and dilaton-deformed 2D supergravity algebra

“…Much of the recent progress [7,[15][16][17][18] to understand bosonic gravity theories in two dimensions also at the quantum level is based upon the equivalence [19,20] of a torsion free general dilaton theory [21][22][23][24][25] and a Hamiltonian action of the type of a Poisson-Sigma model (PSM) [26,27]. A (graded) PSM ((g)PSM) is defined by the action (1.1)…”