Gravitational fields with a non Abelian bidimensional Lie algebra of symmetries

TL;DR

This paper explicitly describes vacuum gravitational fields with a non-Abelian bidimensional Lie algebra of symmetries, providing parameterizations via solutions to a transcendental or linear differential equation, and constructs global solutions from local ones.

Contribution

It introduces explicit descriptions and parameterizations of such gravitational fields, including global solutions, expanding understanding of symmetry-invariant vacuum solutions.

Findings

Explicit solutions parameterized by the tortoise equation or linear differential equations.

Construction of global gravitational fields from local solutions.

Identification of invariance under a 3-dimensional Lie algebra with bidimensional leaves.

Abstract

Vacuum gravitational fields invariant for a bidimensional non Abelian Lie algebra of Killing fields, are explicitly described. They are parameterized either by solutions of a transcendental equation (the tortoise equation) or by solutions of a linear second order differential equation on the plane. Gravitational fields determined via the tortoise equation, are invariant for a 3-dimensional Lie algebra of Killing fields with bidimensional leaves. Global gravitational fields out of local ones are also constructed.