Premium
Existence of an asymptotic velocity and implications for the asymptotic behavior in the direction of the singularity in T 3 ‐Gowdy
Author(s) -
Ringström Hans
Publication year - 2006
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20105
Subject(s) - singularity , mathematics , geodesic , mathematical analysis , limit (mathematics) , cosmic censorship hypothesis
This is the first of two papers that together prove strong cosmic censorship in T 3 ‐Gowdy space‐times. In the end, we prove that there is a set of initial data, open with respect to the C 2 × C 1 topology and dense with respect to the C ∞ topology, such that the corresponding space‐times have the following properties: Given an inextendible causal geodesic, one direction is complete and the other is incomplete; the Kretschmann scalar, i.e., the Riemann tensor contracted with itself, blows up in the incomplete direction. In fact, it is possible to give a very detailed description of the asymptotic behavior in the direction of the singularity for the generic solutions. In this paper, we shall, however, focus on the concept of asymptotic velocity. Under the symmetry assumptions made here, Einstein's equations reduce to a wave map equation with a constraint. The target of the wave map is the hyperbolic plane. There is a natural concept of kinetic and potential energy density; perhaps the most important result of this paper is that the limit of the potential energy as one lets time tend to the singularity for a fixed spatial point is 0 and that the limit exists for the kinetic energy. We define the asymptotic velocity v ∞ to be the nonnegative square root of the limit of the kinetic energy density. The asymptotic velocity has some very important properties. In particular, curvature blowup and the existence of smooth expansions of the solutions close to the singularity can be characterized by the behavior of v ∞ . It also has properties such that if 0 < v ∞ (θ 0 ) < 1, then v ∞ is smooth in a neighborhood of θ 0 . Furthermore, if v ∞ (θ 0 ) > 1 and v ∞ is continuous at θ 0 , then v ∞ is smooth in a neighborhood of θ 0 . Finally, we show that the map from initial data to the asymptotic velocity is continuous under certain circumstances and that what will in the end constitute the generic set of solutions is an open set with respect to C 2 × C 1 topology on initial data. © 2005 Wiley Periodicals, Inc.