1.1 Outline, basic material

Definition of the Gowdy class. Since the present article is concerned with cosmic censorship in Gowdy spacetimes, a natural starting point is to define the Gowdy class. This is the subject of Section 2. However, in order to obtain a good understanding, it is of interest to put this symmetry class into perspective. Therefore, in Section 2.1, we discuss the role of symmetry in cosmology. In particular, we mention different ways of imposing symmetry and describe the place the Gowdy spacetimes occupy in the symmetry hierarchy. In Section 2.2, we then define the Gowdy spacetimes. The essential condition is that there be a two-dimensional isometry group with two-dimensional spacelike orbits. However, Gowdy makes some additional restrictions, which we explain in Section 2.2.1. We end the section by defining an important subclass called polarized Gowdy; see Section 2.4.

The existence of foliations. After the Gowdy class has been defined, a natural first question to ask is if there are preferred foliations. For example, is there a CMC foliation, and, if so, does it cover the maximal globally-hyperbolic development (MGHD)? We address such questions in Section 3.

Formulation of the strong cosmic-censorship conjecture. In Section 4, we turn to the formulation of the strong cosmic-censorship conjecture. We shall here phrase it in terms of the initial value problem. Therefore, in Section 4.1, we define the initial value problem for Einstein’s equations. First, we give an intuitive motivation for some aspects of the formulation. We then provide a formal definition. After having phrased the problem, we mention the standard results concerning the existence of developments. The emphasis is on the existence of the MGHD. In Section 4.2, we then state the strong cosmic censorship conjecture. Two words that require a detailed definition occur in the formulation: generic and inextendible. There are several possible technical definitions of these concepts, and we provide some examples. We end the section by formulating a related conjecture concerning curvature blow up in Section 4.3 and by mentioning some pathologies that can occur in Gowdy in Section 4.4.

The BKL conjecture. The results that exist concerning strong cosmic censorship in Gowdy spacetimes have been obtained through a detailed analysis of the asymptotic behavior of solutions. One point of view that has played an important role in the analysis is the circle of ideas often referred to as the “BKL conjecture” (after Belinskii, Khalatnikov, and Lifshitz). For this reason, in Section 5, we give a brief description of these ideas as well as some recent developments. A related topic is that of asymptotic expansions, which we discuss in Section 5.2. We also describe the Fuchsian methods that can be used to prove that there are solutions with a prescribed type of asymptotic behavior.

The equations. In Section 6, we write down Einstein’s equations in terms of the components of a T3-Gowdy metric. It is important to note that the essential equations have the structure of a wave-map with hyperbolic space as a target. We describe this structure and mention some of its consequences.


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