Flat deformation of a spacetime admitting two Killing fields

TL;DR

This paper presents a local deformation method for 4D Lorentzian metrics with two Killing vectors, transforming them into flat metrics while preserving symmetries, with applications to stationary axisymmetric spacetimes.

Contribution

It introduces a deformation law for Lorentzian metrics with two Killing vectors that yields flat metrics while maintaining the original symmetries, including special cases with quotient metrics.

Findings

Existence of a local deformation law for metrics with two Killing vectors.

Characterization of cases where the projector coincides with the quotient metric.

Application of results to stationary axisymmetric spacetimes.

Abstract

It is shown that given an analytic Lorentzian metric on a 4-manifold, $g$, which admits two Killing vector fields, then it exists a local deformation law $\eta = a g + b H$, where $H$ is a 2-dimensional projector, such that $\eta$ is flat and admits the same Killing vectors. We also characterize the particular case when the projector $H$ coincides with the quotient metric. We apply some of our results to general stationary axisymmetric spacetimes