Smooth Gowdy-symmetric generalized Taub–NUT solutions

Abstract

Abstract. We study a class of S 3 -Gowdy vacuum models with a regular past Cauchy horizon which we call smooth Gowdy symmetric generalized Taub-NUT solutions. In particular, we prove existence of such solutions by formulating a singular initial value problem with asymptotic data on the past Cauchy horizon. We prove that also a future Cauchy horizon exists for generic asymptotic data, and derive an explicit expression for the metric on the future Cauchy horizon in terms of the asymptotic data on the past horizon. This complements earlier results about S 1 × S 2 -Gowdy models.

Smooth Gowdy-symmetric generalised Taub–NUT solutions in Einstein–Maxwell theory

“…We start from the coordinates (t, θ, ρ 1 , ρ 2 ) as used in [5] with line element ds 2 = e M (−dt 2 + dθ 2 ) + R 0 sin 2 t e u (dρ 1 + Qdρ 2 ) 2 + sin 2 θ e −u dρ 2 2 , (1) where u, Q and M are functions of t and θ alone, and R 0 is a positive constant. The angles ρ 1 , ρ 2 take on values in the regions…”

“…Note that the boundaries θ = 0, π of the Gowdy square are the symmetry axes A 1 and A 2 , and the boundaries t = 0, π correspond to the past and future Cauchy horizons H p and H f , respectively. The construction of the coordinates and the requirement that hypersurfaces t = constant have 3-sphere topology imply the following axes boundary conditions 1 [5],…”

“…Existence of a large class of (generally) inhomogeneous cosmological solutions with U (1) isometry group, the generalised Taub-NUT spacetimes, was shown by Moncrief [19]. The starting point for this paper is a related class of solutions, which was introduced in [5]: the smooth Gowdy-symmetric generalised Taub-NUT (SGGTN) solutions. Like Moncrief's spacetimes, the SGGTN solutions are vacuum solutions with spatial three-sphere topology S 3 .…”

“…However, they require less regularity (smoothness instead of analyticity), but they contain more symmetries (Gowdy symmetry with two spacelike Killing vectors). Existence of these models was shown in [5,12], and several families of exact solutions in this class were constructed with methods from soliton theory in [4,12,13]. For a related application of soliton methods to Gowdy-symmetric cosmological models with spatial topology S 2 ×S 1 , we refer to [2], and a general overview of strong cosmic censorship in the class of Gowdy spacetimes can be found in [29].…”

“…The aim of this paper is to include a nontrivial additional field by generalising the class of SGGTN solutions from vacuum to electrovacuum. In particular, we extend the soliton methods used in [5], which allows us to study global properties of SGGTN solutions with a Maxwell field present. Firstly, however, we summarise some key features of the vacuum case.…”