Gravitational fields with a non Abelian bidimensional Lie algebra of symmetries

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. PACS numbers: 04.20.-q, 04.20.Gz, 04.20.Jb In the last years a great deal of attention has been devoted to the detection of gravitational waves. However, all the experimental devices, interferometers or resonant antennas, are constructed coherently with results obtained from the non covariant linearized Einstein field equations, in close analogy with that is normally done in Maxwell theory of electromagnetic fields. Starting from the seventy’s, however, new powerful mathematical methods have been invented to deal with nonlinear evolution equations and their exact solutions. One of this methods, namely a suitable generalization of the Inverse Scattering Transform, allowed to solve reduced vacuum Einstein field equations and to obtain solitary waves solutions [3] (see [14] and references therein). This paper is the first in a series devoted to the study of gravitational fields g admitting a Lie algebra G of Killing fields. The case of a non Abelian bidimensional Killing Lie algebra has been only partially studied. Here, this case will be completely analyzed within the following general problem which, as we will see, emerges naturally. I. the distribution D, generated by the vector fields of G, is bidimensional. II. the distribution D⊥ orthogonal to D, is integrable and transversal to D. According to whether dimG is 2 or 3, two qualitatively different cases occur. A bidimensional G, is either Abelian (A2) or non-Abelian (G2). A metric g satisfying I and II, with G = A2 or G2, will be called G -integrable . The study of A2-integrable Einstein metrics goes back to Einstein and Rosen [5], Rosen [11], Kompaneyets [8], Geroch [6], Belinsky and Khalatnikov [2]. The greater rigidity of G2-integrable metrics, for which some partial results can be found in [7, 1, 4], allows an exhaustive analysis. It will be shown that they are parameterized by solutions of a linear second order differential equation on the plane which, in its turn, depends linearly on the choice of a j-harmonic