Deformations of three-dimensional metrics

TL;DR

This paper explores three-dimensional metric deformations via tetrad transformations, classifies the deforming matrices based on scalar and vector properties, and applies the findings to spherically symmetric and Kerr space geometries.

Contribution

It provides a systematic classification of metric deformations in three dimensions using scalar and vector parameters, extending previous results and analyzing causal structures.

Findings

Deforming matrices classified by scalar $\sigma$ and vector $ extbf{s}$ properties.

Causal structures of deformed geometries analyzed.

Applications to spherical symmetry and Kerr metric sectors.

Abstract

We examine three-dimensional metric deformations based on a tetrad transformation through the action the matrices of scalar fields. We describe by this approach to deformation the results obtained by Coll et al. in [1], where it is stated that any three--dimensional metric was locally obtained as a deformation of a constant curvature metric parameterized by a 2--form.To this aim, we construct the corresponding deforming matrices and provide their classification according to the properties of the scalar $\sigma$ and of the vector $\mathbf{s}$ used in [1] to deform the initial metric. The resulting causal structure of the deformed geometries is examined, too.Finally we apply our results to a spherically symmetric three geometry and to a space sector of Kerr metric.