Vacuum Einstein metrics with bidimensional Killing leaves. I-Local aspects

TL;DR

This paper explicitly describes vacuum Einstein metrics with bidimensional Killing leaves, characterizing solutions via a transcendental or linear differential equation, and identifying conditions for additional symmetries.

Contribution

It provides a detailed classification of vacuum Einstein metrics with non-Abelian bidimensional Killing fields, linking solutions to specific differential equations and symmetry properties.

Findings

Solutions parametrized by the tortoise equation or linear differential equations.

Metrics with solutions of the tortoise equation admit a 3D Lie algebra of Killing fields.

Explicit description of metrics with bidimensional Killing leaves in vacuum Einstein's equations.

Abstract

The solutions of vacuum Einstein's field equations, for the class of Riemannian metrics admitting a non Abelian bidimensional Lie algebra of Killing fields, are explicitly described. They are parametrized either by solutions of a transcendental equation (the tortoise equation), or by solutions of a linear second order differential equation in two independent variables. Metrics, corresponding to solutions of the tortoise equation, are characterized as those that admit a 3-dimensional Lie algebra of Killing fields with bidimensional leaves.