Cosmic Censorship for Gowdy Spacetimes
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Abstract
Due to the complexity of Einstein’s equations, it is often natural to study a question of interest in the framework of a restricted class of solutions. One way to impose a restriction is to consider solutions satisfying a given symmetry condition. There are many possible choices, but the present article is concerned with one particular choice, which we shall refer to as Gowdy symmetry. We begin by explaining the origin and meaning of this symmetry type, which has been used as a simplifying assumption in various contexts, some of which we shall mention. Nevertheless, the subject of interest here is strong cosmic censorship. Consequently, after having described what the Gowdy class of spacetimes is, we describe, as seen from the perspective of a mathematician, what is meant by strong cosmic censorship. The existing results on cosmic censorship are based on a detailed analysis of the asymptotic behavior of solutions. This analysis is in part motivated by conjectures, such as the BKL conjecture, which we shall therefore briefly describe. However, the emphasis of the article is on the mathematical analysis of the asymptotics, due to its central importance in the proof and in the hope that it might be of relevance more generally. The article ends with a description of the results that have been obtained concerning strong cosmic censorship in the class of Gowdy spacetimes.
Keywords
Hyperbolic Space Lorentz Manifold Kinetic Energy Density Constant Mean Curvature Cauchy Hypersurface1 Introduction and Outline
Gowdy spacetimes have been used as a toy model in the context of, e.g., gravitational waves, quantum gravity, numerical relativity and mathematical cosmology. However, here we shall only be concerned with the question of strong cosmic censorship. In other words, we are interested in a mathematical problem. Nevertheless, numerical relativity has played an important role in the development of the subject, and as a consequence, we shall mention some of the key numerical observations.
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 twodimensional isometry group with twodimensional 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 globallyhyperbolic development (MGHD)? We address such questions in Section 3.
Formulation of the strong cosmiccensorship conjecture. In Section 4, we turn to the formulation of the strong cosmiccensorship 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 T^{3}Gowdy metric. It is important to note that the essential equations have the structure of a wavemap with hyperbolic space as a target. We describe this structure and mention some of its consequences.
1.2 Outline, the asymptotics in the direction towards the singularity
Asymptotic behavior in the direction of the singularity in the polarized case. The two Gowdy cases in which results concerning strong cosmic censorship exist are the polarized case and the general T^{3}case. In Section 7, we focus on the polarized case. In particular, on finding asymptotic expansions of the metric components in the direction towards the singularity. One function appearing in the expansions has a special importance. From the wavemap point of view, it has a natural interpretation as the rate at which the solution tends to the boundary of hyperbolic space; see Section 7.4. As a consequence, it is referred to as the asymptotic velocity. Beyond having a natural interpretation, the asymptotic velocity has an additional important property. In fact, it can be used as a criterion for curvature blow up along causal curves going into the singularity; see Section 7.5.
Existence of solutions with specified asymptotics, Fuchsian methods. Due to the central importance of the asymptotic expansions in the polarized case, it is of interest to obtain expansions in the general T^{3}case. One way to proceed is to try to construct solutions with prescribed asymptotics. This is the subject of Section 8. Again, it is possible to define the concept of an asymptotic velocity. It has the same geometric interpretation and importance as in the polarized case. The results on existence of expansions depend on a restriction of the asymptotic velocity. We describe the results and motivate the restriction.
Spikes. The numerical studies indicate that for most spatial points, the asymptotic expansions presented in Section 8 constitute a good description of the asymptotics. However, they also indicate that there are spatial points where the behavior is very different. Due to the visual impression of plots of the solutions in the neighborhood of the exceptional points, the corresponding features have been referred to as “spikes”. In Section 9, we describe analytic constructions of solutions with spikes.
Existence of an asymptotic velocity in the general T^{ 3 }Gowdy case. The analysis in the polarized case and the construction of solutions with prescribed asymptotics indicate the importance of the asymptotic velocity. Consequently, it is of interest to prove that the asymptotic velocity exists in general. This is the subject of Section 10. We also demonstrate that the asymptotic velocity can be viewed as a twodimensional object in the disc model. Finally, we illustrate that it can be used as a criterion for the existence of expansions.
Definition of the generic set in the general T^{ 3 }Gowdy case. As a preparation for the formulation of the theorem verifying that strong cosmic censorship holds in T^{3}Gowdy, we define the generic set of initial data in Section 11.
1.3 Outline, the expanding direction
The asymptotic behavior in the expanding direction of polarized Gowdy. Only in the case of T^{3}topology is there an expanding direction. Consequently, it is only necessary to discuss the general T^{3}case. However, there are some results of interest, which are only known in the polarized case. Consequently, we devote Section 12 to a discussion of it. It is of particular interest to note that the spatial variation of solutions dies out in the sense that the difference between the solution and its average converges to zero. On the other hand, with respect to other measures, the solution does not tend to spatial homogeneity.
The asymptotic behavior in the expanding direction of general T^{ 3 }Gowdy. In the general case, less detailed information is available. However, a clear picture of the asymptotics exists and is described in Section 13. The first step of the analysis consists of proving that a naturally defined energy converges to zero at a specific rate. This leads to the conclusion that the distance from the solution to its average converges to zero. In order to analyze the asymptotics of solutions, it is convenient to note that there are conserved quantities. When viewed in the right way, these conserved quantities can be reinterpreted as ODEs for the averages, and this leads to detailed information concerning their asymptotics. Finally, results concerning the decay of the sup norm of derivatives is derived. Such estimates are useful in order to prove future causal geodesic completeness.
1.4 Strong cosmic censorship
Strong cosmic censorship. Finally, in Section 14, we phrase the existing results on strong cosmic censorship in the class of Gowdy spacetimes. So far, the results are restricted to the polarized case and the general T^{3}case.
2 Gowdy Spacetimes
In order to put the Gowdy class of spacetimes into context, it is natural to start by discussing the role of symmetry in general relativity. We shall not discuss it here in all generality, but will restrict our attention to the fourdimensional cosmological case.
2.1 Symmetry in cosmology
By a cosmological spacetime, we mean one that is foliated by compact spacelike hypersurfaces. Moreover, we shall, most of the time, tacitly assume the spacelike hypersurfaces to be Cauchy hypersurfaces; see Definition 2. Let us mention a few different ways of imposing symmetry conditions.
2.1.1 Symmetry via the Lie algebra
In the physics literature, it is quite common to phrase the demands in terms of the Lie algebra of Killing vector fields on the spacetime under consideration.
2.1.2 Symmetry via Lie group actions on the spacetime
Another possibility is to demand that there be a Lie group acting smoothly and effectively by isometries on the spacetime. Recall that a Lie group action G × M → M is effective if gp = p for all p ∈ M implies g = e.
2.1.3 Symmetry via the initial value formulation
A third option is provided by the formulation of Einstein’s equations of general relativity as an initial value problem. We shall give a more complete presentation of the initial value formulation in Section 4.1. However, let us briefly recall the main ingredients here. In the cosmological case, the initial data consist of a threedimensional compact manifold Σ on which a Riemannian metric, a symmetric covariant twotensor and suitable matter fields are specified. Assuming the matter model to be of an appropriate type, there is a unique MGHD of the initial data; see Theorem 2. One way to impose symmetries is to demand that there be a Lie group acting smoothly and effectively by isometries on the initial data. In order for this perspective to be of any interest, such a Lie group action should give rise to a smooth effective Lie group action, acting by isometries, on the MGHD. That this is the case can be seen by the argument presented in [70, pp. 176–177]; see also [17, 18].
2.1.4 Cosmological symmetry hierarchy
In the study of the initial value problem, it is of interest to analyze what combinations of compact Lie groups and compact threedimensional manifolds Σ are such that there is a smooth and effective Lie group action of G on Σ. It turns out that there are quite a limited number of possibilities. The introduction of [17] contains a list. Readers interested in the underlying mathematics are referred to [59]. Given a specific topic of interest, such as, the strong cosmiccensorship conjecture, this list yields a hierarchy of classes of spacetimes in which one can study it in a simplified setting.
2.1.5 Limitations, different perspectives
It should be noted that requiring the existence of an effective Lie group action of the type described above excludes large classes of cosmological spacetimes that are, in some respects, of a high degree of symmetry. Most spatially locallyhomogeneous cosmological models are excluded. A more natural perspective to take would perhaps be to demand that there be an appropriate Lie group action on the universal covering space. Yet another perspective is provided by [87]. The central assumption of [87] is the existence of two commuting local Killing vectors, and a larger class of spatial topologies is thereby permitted, see also [67].
2.1.6 Present status, hierarchy
In the case of cosmology, the assumption of spatial local homogeneity is a natural starting point. However, in this setting, the issue of strong cosmic censorship is quite well understood. Note that this claim rests on our particular definition of a cosmological spacetime. In fact, most Bianchi class B solutions are excluded by the condition that they should admit spatially compact quotients. On the other hand, it should be pointed out that there are many fundamental problems that have not been sorted out even in the spatially homogeneous setting; the detailed asymptotics of Bianchi IX and the question of whether particle horizons form in Bianchi VIII and IX or not are but two examples, see, for example, [72, 73, 45, 46] and references cited therein for partial results concerning the asymptotics and [44] for a discussion of the issue of particle horizons.
When proceeding beyond spatial homogeneity, the natural next step is to consider the case of a twodimensional isometry group. This leads us to the Gowdy class of spacetimes.
2.2 Definition of the Gowdy class
Spacetimes admitting twodimensional isometry groups had been studied prior to [39]. However, the considerations had mainly been limited to the stationary axisymmetric case. To the best of our knowledge, Gowdy was the first one to systematically analyze the consequences of imposing the existence of a twodimensional isometry group with spacelike orbits [39, 38, 37]. As a consequence, a subclass of this family now bears his name. The main objective of the work presented in [39] is to write down a convenient global form for metrics admitting this type of symmetry. However, in the course of the discussion, Gowdy introduces assumptions that exclude a large family of solutions admitting a twodimensional isometry group. Let us introduce the terminology necessary for describing the discarded class.
2.2.1 Twist constants, twosurface orthogonality
2.2.2 Essential characterizing conditions

be a globallyhyperbolic cosmological spacetime,

admit an effective action by isometries by a twodimensional compact Lie group with spacelike orbits,

be such that the twist constants vanish.
2.2.3 Technical definition
Even though the above list gives the central assumptions, there are some subtleties that have been sorted out in [16]. Thus, the formally inclined are recommended to use the assumptions of [16, Theorem 4.2, p. 117] and of [16, Theorem 6.1, p. 128–129] as a definition of Gowdy initial data. The Gowdy class of solutions is then defined as the MGHDs of Gowdy initial data.
2.3 Coordinate systems
In [16], special coordinate systems are constructed on part of the MGHD. Let us describe the different cases.
2.3.1 Coordinate systems, T^{3}Gowdy
2.3.2 Working definition, T^{3}Gowdy
2.3.3 Coordinate system, S^{3} and S^{2} × S^{1}
2.4 The polarized subcase
3 Foliations
The components of a Gowdy metric, see Equations (2) and (3), are not explicit functions of the coordinates. However, imposing Einstein’s equations leads to a system of nonlinear wave equations for the components. Consequently, it is useful to analyze the asymptotic behavior of the solutions to this system in order to be able to draw conclusions concerning the global geometry of the corresponding spacetimes. One natural first question to ask is if there are any preferred global foliations. Is there, e.g., a constant mean curvature (CMC) foliation?
3.1 CMC foliations
Note that CMC foliations are unique in cosmological vacuum spacetimes. In the case of vacuum T^{3}Gowdy, there is a CMC foliation exhausting the interval (−∞, 0) [49]. However, in the case of S^{2} × S^{1} and S^{3} topology, the only general statements concerning foliations are, as far as we are aware, the ones given in [16]. It is natural to conjecture that if there is a CMC Cauchy hypersurface in the S^{2} × S^{1} or S^{3} case, there is a CMC foliation exhausting the interval (−∞, ∞) [66]. However, to the best of our knowledge there are no results to this effect.
3.2 Areal foliation
In the case of T^{3}Gowdy spacetimes, there is another natural foliation; considering Equation (2), it is clear that the area of the symmetry orbits are proportional to the time coordinate t. Consequently, such a time coordinate is referred to as an areal time coordinate. It is natural to ask if the areal time coordinate exhausts the interval (0, ∞). That the answer is yes in the case of vacuum T^{3}Gowdy was demonstrated by Moncrief [58]. Furthermore, he verified that the foliation covers the entire MGHD. However, since the starting point of the argument in [58] is a constantt hypersurface, it is of interest to note that the results of [16] yield the same conclusions starting with a general Cauchy hypersurface.
3.3 Existence of foliations, related symmetry classes
Let us, for the sake of completeness, mention some results concerning spacetimes satisfying related symmetry conditions, in particular T^{2}symmetry; see Section 2.2. That the maximal globallyhyperbolic vacuum development of T^{2}symmetric initial data is covered by areal coordinates is proven in [9]. The result states that the area of the symmetry orbits exhausts (c, ∞) for some c ≥ 0; whether c = 0 or not is left open. However, this question has been addressed and resolved in [51] and [91], see also [85]. In the context of areal coordinates, there is a fundamental difference between the Gowdy case and the general T^{2}symmetric case. In the Gowdy case, the areal time coordinate is such that the metric is conformal to the Minkowski metric in the tθdirection; see Equation (2). In the general T^{2}symmetric case, this property is lost if one insists on an areal time coordinate [9]. Results on the existence of areal coordinates covering the MGHD in the case of solutions to the EinsteinVlasov system with T^{3}Gowdy symmetry are contained in [4], see also [5], which treats solutions to the EinsteinVlasov system in the general T^{2}symmetric case (the latter paper contains results concerning both areal and CMC foliations). Existence of a CMC foliation under the assumption of the existence of two local Killing vectors was demonstrated in [67], a paper, which generalizes, among other things, the results of [49].
3.4 Prescribed mean curvature
The above results concerning the existence of CMC foliations are based on the assumption of the existence of one CMC hypersurface. There are ways to circumvent this condition. By proving the existence of a suitable prescribed mean curvature (PMC) foliation, it is sometimes possible to construct barriers that imply the existence of one CMC hypersurface; see Section 3.6 of [69] and references cited therein for more details.
4 Strong Cosmic Censorship
The idea of cosmic censorship goes back to the work of Roger Penrose; see [62] (reprinted in [64]) and [63]. It comes in two forms: weak and strong. The weak cosmiccensorship conjecture is concerned with isolated systems and essentially states that, generically, singularities should not be visible to an observer at infinity; see [90] for a more precise and extensive discussion. The strong cosmic censorship conjecture is a statement concerning the deterministic nature of the general theory of relativity. This is the form we are interested in here, and we shall phrase it in terms of initial data. Consequently, we need to formulate the initial value (or Cauchy) problem for Einstein’s equations.
4.1 The initial value problem
How does one formulate an initial value problem for Einstein’s equations? What should the initial data be? Is there uniqueness in any reasonable sense? These questions can be formulated in the presence of various matter fields, but let us, for the sake of simplicity, restrict our attention to the vacuum case here.
4.1.1 The vacuum equations
4.1.2 Formulation, intuition
If the second term in Equation (7) were not there, Equation (5) would, in local coordinates, be a nonlinear wave equation, and it would be straightforward to formulate an initial value problem. Furthermore, given a solution to Equation (5), it is possible to choose local coordinates such that Γ_{ μ } vanishes and Equation (5) takes the form of a nonlinear wave equation. On the other hand, if Equation (5) were a nonlinear wave equation when expressed with respect to arbitrary coordinates, we would obtain uniqueness for the coordinate expression of the metric. This statement is incompatible with diffeomorphism invariance. Thus, even though Equation (5) in some respects can be viewed as a hyperbolic differential equation, the geometric aspect of the equation must not be forgotten.
Due to the above observations, it seems natural to expect the right PDE problem to formulate for Einstein’s equations to be the initial value problem. Furthermore, it seems clear that this problem should be given a geometric formulation. Naively, one would expect it to be necessary to specify the metric and the first time derivative of the metric at the initial hypersurface. However, these quantities are not geometric. The induced metric and second fundamental form are, on the other hand, geometric quantities and they contain part of the information one would naively expect to need. Furthermore, they, in the end, turn out to constitute sufficient information. The question arises of what should be required of the initial hypersurface? Since we wish to avoid issues of consistency, we shall require the hypersurface to be such that it has no causal tangent vectors. In other words, we require that it be spacelike (this is, strictly speaking, not necessary; there are formulations in the null case as well; see, e.g., [65]).
4.1.3 Formulation, formal definition
Definition 1 Initial data for Einstein’s vacuum equations consist of a threedimensional manifold Σ, a Riemannian metric ρ and a covariant symmetric twotensor κ on Σ, both assumed to be smooth and to satisfy Equations (8) – (9) . Given initial data, the initial value problem is that of finding a fourdimensional manifold M with a Lorentz metric g such that Equation (5) is satisfied, and an embedding i: Σ → M such that i*g = ρ and that if k is the second fundamental form of i(Σ), then i*k = κ. Such a Lorentz manifold (M, g) is called a development of the data. Furthermore, if i(Σ) is a Cauchy hypersurface in (M, g), then (M, g) is referred to as a globallyhyperbolic development of the initial data. In both cases, the existence of an embedding i is tacit.
Since the concepts Cauchy hypersurface and globally hyperbolic are referred to above, and will be of some importance below, let us recall how they are defined.
Definition 2 Let (M, g) be a Lorentz manifold. A subset Σ of M is said to be a Cauchy hypersurface if it is intersected exactly once by every inextendible timelike curve. A Lorentz manifold that admits a Cauchy hypersurface is said to be globally hyperbolic.
Remark. Two basic examples of Cauchy hypersurfaces are the t = const. hypersurfaces in Minkowski space and the hypersurfaces of spatial homogeneity in RobertsonWalker spacetimes. The reader interested in the basic properties of globallyhyperbolic Lorentz manifolds and Cauchy hypersurfaces is referred to [60], see also [82] and references cited therein. Cauchy hypersurfaces need neither be smooth nor spacelike, but we shall tacitly assume them to be both. The reason the concept of a Cauchy hypersurface is of such central importance is that it is the natural type of surface on which to specify initial data.
4.1.4 Existence of a development
We are now in a position to ask: given initial data to Einstein’s vacuum equations, is there a development? The answer to this question is yes, due to the seminal work of ChoquetBruhat [34] (a presentation in book form is also available in, e.g., [82]):
Theorem 1 Given initial data (Σ, ρ, κ) to Einstein’s vacuum equations, there is a globallyhyperbolic development.
Clearly, this is a fundamental result. In particular, this result is what justifies the terminology “initial data to Einstein’s vacuum equations” as specified in Definition 1. On the other hand, the issue of uniqueness is not addressed. Given initial data, there are infinitely many distinct globallyhyperbolic developments. In order to obtain uniqueness, it is consequently necessary to require some form of maximality.
4.1.5 Existence of a maximal globallyhyperbolic development
The central concept in the study of uniqueness is that of an MGHD:
Definition 3 Given initial data to Einstein’s vacuum equations (5) , a MGHD of the data is a globally hyperbolic development (M, g), with embedding i: Σ → M, such that if (M′, g′) is any other globally hyperbolic development of the same data, with embedding i′: Σ → M′, then there is a map ψ: M′ → M, which is a diffeomorphism onto its image, such that ψ*g = g′ and ψ ∘ i′ = i.
Note that this definition differs from the standard notion of maximality used in set theory. The standard notion would lead to the definition of a MGHD as a globally hyperbolic development, which cannot be extended (note that this notion of maximality would not a priori rule out the possibility of two maximal elements, neither of which can be embedded into the other, as opposed to Definition 3).
Theorem 2 Given initial data to Equation (5) , there is an MGHD of the data, which is unique up to isometry.
Remark. Uniqueness of a development (M, g) up to isometry is defined as follows: if (M′, g′) is another MGHD, then there is a diffeomorphism ψ: M → M′ such that ψ*g′ = g and ψ ∘ i = i′, where i and i′ are the embeddings of Σ into M and M′ respectively.
Theorem 2 is due to the work of ChoquetBruhat and Geroch; see [13] for the original paper and [82] for a recent presentation. The proof relies, in part, on the local theory and on an argument using what is often referred to as Zorn’s lemma. This leads to the existence of an MGHD in the set theory sense of the word. However, it does not lead to the existence of an MGHD in the sense of Definition 3. In fact, the important part of the result is the uniqueness of the MGHD (in the set theory sense of the word). This requires an additional argument.
Due to Theorem 2, the initial value formulation of Einstein’s equations is meaningful. However, the MGHD might be extendible. In fact, it turns out that there are initial data such that this is the case. If the extensions were unique in their turn, this would not be a serious problem, but it turns out that there are MGHDs with inequivalent maximal extensions [20] (see also [82]). The reason these examples are unfortunate is that they demonstrate that Einstein’s general theory of relativity is not deterministic; given initial data, there is not necessarily a unique corresponding universe. Nevertheless, the examples of this pathological behavior are very special, and there is thus reason to hope that for generic initial data, the MGHD is inextendible. These speculations naturally lead us to the strong cosmiccensorship conjecture.
4.2 Strong cosmic censorship
By the strong cosmiccensorship conjecture, we mean the following statement:
Conjecture 1 For generic asymptoticallyflat and for generic spatiallycompact initial data for Einstein’s vacuum equations, the MGHD is inextendible.
Remark. We are only interested here in initial data specified on compact manifolds, a case to which the conjecture applies. Readers interested in the asymptoticallyflat case are referred to, e.g., [90] and references cited therein.
In this form, the statement is due to Chruściel; see [17, Section 1.3], based on ideas due to Eardley and Moncrief [32]. It is of course also possible to make the same (or similar) statements in the presence of matter. However, there are some matter models, which exhibit pathologies, and we do not wish to discuss such issues here. The formulation of Conjecture 1 is rather vague; the words “generic” and “inextendible” occur without having been clearly defined. The reason for this is partly that there is no a priori preferred definition of these concepts. Let us discuss them separately.
4.2.1 Genericity

a set whose complement has measure zero (with respect to the Lebesgue measure, say),

a set that is open and dense,

a dense G_{ δ } set, i.e., a countable intersection of open sets, which is dense.
Other possibilities are conceivable; see, e.g., [14]. Regardless of the choice of definition, one requirement appears to be quite clear: if a set is generic, then the complement should not be generic. In the case of infinitedimensional dynamic systems, the case we are interested in here, the measure of the theoretic notion of genericity is not so natural. Consequently, we shall here, unless otherwise stated, take generic to mean open and dense. Nevertheless, such a definition is still not precise; it requires the prior definition of a topology on the set of initial data.
4.2.2 Inextendibility

it is extendible as a Lorentz manifold, or

it is extendible as a Lorentz manifold solving Einstein’s equations.
Here, we shall say that a solution is extendible if it is extendible as a Lorentz manifold. In other words, we shall not require the extension to be a solution to Einstein’s equations. Furthermore, we shall use the differentiability class C^{2}, since we wish the differentiability class to be strong enough that curvature is still defined.
4.3 Curvature blow up
The strong cosmiccensorship conjecture is of fundamental importance since it expresses the expectation that Einstein’s general theory of relativity is a deterministic theory (with some nongeneric exceptions). On the other hand, it is of a somewhat philosophical nature. However, it is strongly connected to a statement concerning the behavior of gravitational fields close to the singularity. Here, the existence of a singularity is equated with the existence of an incomplete causal geodesic, the motivation being the work of Hawking and Penrose resulting in the singularity theorems; see [61, 41, 43] and [42, 89, 60]. Even though the commonplace existence of singularities is established by the singularity theorems, their nature remains unclear; do, for example, the gravitational fields become arbitrarily strong in the vicinity of a singularity? It is natural to state the following conjecture:
Conjecture 2 For generic asymptoticallyflat and generic spatiallycompact initial data for Einstein’s vacuum equations, curvature blows up in the incomplete directions of causal geodesics in the MGHD.
Since it is not clear how to address Conjectures 1 and 2 in all generality, the work that has been carried out so far has been concerned with the analogous questions phrased in the context of special classes of spacetimes. Here, we shall be concerned with these questions phrased in the Gowdy class.
4.4 Pathological examples in the case of Gowdy
However, there are more sophisticated examples of pathologies. In [20], the authors demonstrate that given any positive integer n, there is a polarized Gowdy vacuum spacetime with at least n inequivalent maximal extensions.
5 BKL, Fuchsian Methods and Asymptotic Expansions
The results on strong cosmic censorship in Gowdy vacuum spacetimes cover the polarized subcase as well as the general T^{3}Gowdy case. However, to the best of our knowledge, there are no results concerning strong cosmic censorship in the general S^{3} and S^{2} × S^{1} cases. The method of proof, in all the situations in which results exist, consists of a detailed analysis of the asymptotics of solutions. As a consequence, we shall devote most of this review to a description of the analysis of the asymptotic behavior. Note that it might be possible to prove strong cosmic censorship without analyzing the asymptotics in detail. In fact, there are proofs under related symmetry assumptions, which are not based on a detailed analysis of the asymptotics [25, 26, 27, 84].
The existence of asymptotic expansions in the direction towards the singularity has played a central role in proving strong cosmic censorship for T^{3} and polarized vacuum Gowdy spacetimes. In the latter case, to take one example, there is a computation of asymptotic expansions due to Isenberg and Moncrief [50]. This computation was then used in [21] in the proof of strong cosmic censorship in the polarized case. In the general T^{3}case, there is a large literature on asymptotic expansions, which we shall return to in Section 8; the starting point being the work of Grubišić and Moncrief [40]. It is worth noting that both in the case of [40] and [50], the ideas of Belinskii, Khalatnikov, and Lifshitz [55, 6, 7] (henceforth BKL) played an important role. As a consequence, we wish to give a brief description of the BKL perspective as well as of some related proposals.
5.1 The BKL picture

given a spacetime (M, g) such that all pastdirected timelike geodesics are incomplete, there is a spacelike hypersurface Σ such that the past of S is diffeomorphic to (0, t_{0}] × Σ for some t_{0} > 0, where the first coordinate in the division (0, t_{0}] × Σ defines a time function,

as t → 0+, different spatial points do not causally influence each other, i.e., if p, q ∈ Σ and p ≠ q, then, for t small enough, the past of (t, p) does not intersect the past of (t, q),

the matter content is of negligible importance for the dynamics as t → 0+,

“time derivatives” (or “kinetic terms”) dominate “spatial derivatives” (or “spatial curvature coupling terms”) as t → 0+,

for a fixed p ∈ Σ, the behavior of the solution along (t, p) is well approximated by a spatiallyhomogeneous vacuum solution, in particular by an oscillatory solution (Bianchi types VIII, IX and VI_{−1/9}).
There are some caveats. First, the statements concern generic spacetimes; there are exceptions; see TaubNUT. Second, the matter content might be important for special classes of matter models. For instance, in the case of a stiff fluid or a scalar field, the matter should play a dominant role. The statement that solutions should exhibit oscillatory behavior also depends on the matter model; stiff fluids and scalar fields are expected to suppress it. Furthermore, symmetry might prevent the appearance of oscillations.
The first statement on the list above can be ensured under general circumstances; a combination of Hawking’s theorem [60, Theorem 55A, p. 431], an energy condition and the existence of a Cauchy hypersurface satisfying suitable assumptions concerning the mean curvature will do. The statement of causal disconnectedness is, however, unsatisfactory in that it depends not only on the foliation, but also on a choice of diffeomorphism. It would be preferable to have a geometric condition, which is even independent of the foliation. However, to our knowledge there is no such definition; see the introduction of [44] for a further discussion. The last three statements are clearly very vague.
The general framework has been developed significantly since the work of BKL; see [28, 30, 47, 88] and references cited therein. We shall not describe these developments in detail, but let us mention that there are at least two somewhat different approaches. In the Hamiltonian approach, taken in [28, 30], billiards describe the asymptotic behavior (see also the work of Misner and Chitré; see [57, pp. 805–816] and references cited therein). In the dynamic systems approach, described in [47, 88], the solution is approximated asymptotically by a family of solutions to ordinary differential equations. In [3], the Gowdy spacetimes have been considered from this perspective.
Even though the two perspectives have differences, they have an essential assumption in common: causal disconnectedness in the direction toward the singularity. Furthermore, in both cases, the oscillatory spatiallyhomogeneous vacuum solution is of central importance. It is of interest to note that these two aspects are potentially contradictory; Misner’s original motivation in studying the Mixmaster Universe (Bianchi IX) [56] was the desire to demonstrate that there is no causal disconnectedness in the direction towards the singularity. In order for the pictures suggested in [28, 30] and [47, 88] to be consistent, causal disconnectedness should hold in the oscillatory spatiallyhomogeneous vacuum solutions. It is far from clear that this is the case. The reader interested in a discussion of the status of this question is referred to the introduction of [44].
Finally, let us mention that a different formulation of the BKL picture is given in [19, Conjecture 6.10, p. 58]; see also the references cited therein.
5.2 Asymptotic expansions, Fuchsian methods
It is interesting to note that, in spite of the fact that the BKL picture and related proposals emphasize the importance of oscillatory behavior, the greatest successes of the BKL point of view have been obtained in the nonoscillatory setting. The main reason is that in the absence of oscillations, it is sometimes possible to characterize the asymptotic behavior in terms of asymptotic data (of course, the lack of results in the presence of oscillations is largely due to the difficulty in analyzing the asymptotic behavior in that case). In certain situations, this characterization is strong enough that the asymptotic data are in onetoone correspondence with the solutions. If that is the case, the asymptotic data can be considered to be “initial data at the singularity”. The results in the nonoscillatory case come in different forms.
5.2.1 From solutions to asymptotics
The type of result, that is of greatest immediate interest is the one that, given a solution to the Einstein equations, provides asymptotic expansions. The means by which this is achieved vary. One method is to devise a simplified system of equations, such that the solution to the Einstein equations converges to a solution to the simplified system. In the spirit of the BKL picture, the simplified system is often obtained by omitting some (if not all) spatial derivatives. One example of a successful application is given by the analysis of Isenberg and Moncrief [50] in the polarized Gowdy case (which in some respects follows the ideas of [31]). In [50], a simplified system consisting of the Velocity Term Dominated (VTD) equations are introduced and solved, and the authors prove that solutions to the Einstein equations converge to solutions to the VTD system. Furthermore, a geometric definition for what the authors call Asymptotically VelocityTerm Dominated near the Singularity (or AVTDS) is given; see [50, pp. 88–89]. We give a brief description of the analysis of [50] in Section 7.
5.2.2 From asymptotics to solutions

express Einstein’s equations with respect to a suitable gauge (in the case of T^{3}Gowdy for instance, the areal time coordinate has turned out to be a good candidate; see [54, 68], and in the cases without symmetries concerning which results have been obtained, a Gaussian time coordinate has proven useful [2, 29]),

identify the leadingorder asymptotic behavior, where the leadingorder terms preferably should correspond to as many free functions as are required to specify regular initial data (in the case of T^{3}Gowdy, formal expansions had been suggested in [40] prior to [54] and in [2, 29], the expansions were obtained by considering “Velocity Term Dominated” systems associated with the full system of Einstein’s equations),

express the unknowns in terms of the leadingorder terms plus a remainder, and write down the equations in terms of the remainder (this equation should be of Fuchsian form),

apply the Fuchsian theory.
The standard Fuchsian theory is applicable in the real analytic setting. As a consequence, most of the results assume real analytic “data at the singularity” and lead to the conclusion that there are real analytic solutions with the corresponding asymptotic behavior. Clearly, the procedure is not always applicable. In particular, it is not expected to be applicable in the presence of oscillations.
5.2.3 Overview of results
Let us mention some of the results that have been obtained using Fuchsian methods. In [54], Fuchsian methods were applied to the T^{3}Gowdy case in the real analytic setting. The assumption of real analyticity was later relaxed to smoothness [68]. See also [86] for a similar analysis in the S^{2} × S^{1} and S^{3} cases (though there are some problems related to the symmetry axes in that case, and as a consequence, the results are less complete). An analysis of the polarized T^{2}symmetric spacetimes in the real analytic setting was carried out in [48]. In all the examples mentioned so far, the symmetry caused the suppression of oscillations. However, matter can also have the same effect. This is illustrated by [2], which consists of a study of the Einstein equations coupled to either a scalar field or a stiff fluid. The results are in the real analytic setting and associate a solution to asymptotic initial data. Finally, in [29] large classes of matter models are considered in various dimensions with similar results.
6 Equations
In our presentation, we shall focus on T^{3}Gowdy. Therefore, we will only write down here the equations in that case. The reader interested in the equations for polarized S^{2} × S^{1} or S^{3}Gowdy is referred to [50, (12a)–(12c), p. 92].
6.1 Expanding direction
6.2 The direction towards the singularity
6.3 Wavemap structure
6.3.1 Representations of hyperbolic space
6.4 Conserved quantities, kinetic energy density
Another important consequence of the wavemap structure is the fact that isometries of hyperbolic space map solutions to solutions.
7 Singularity, Polarized Case
The proof of strong cosmic censorship, in the polarized as well as in the T^{3}Gowdy case, proceeds via Conjecture 2. In other words, it consists of a proof of the fact that, generically, the curvature is unbounded in the incomplete directions of causal geodesics. In the polarized case with S^{3} and S^{2} × S^{1} topology, the causal geodesics can be proven to be incomplete both to the future and to the past [21, p. 1673]. Thus, in those cases it is only necessary to analyze the singularities. In the case of T^{3}Gowdy, there is an expanding direction, and it is necessary to prove that causal geodesics are complete in that direction. In general, it is thus necessary to analyze the behavior in the direction towards the singularity and the behavior in the expanding direction. Since the methods involved are very different, we shall consider the two cases separately. Furthermore, since the analysis in the polarized and general cases are quite different, we shall begin by describing the analysis in the direction towards the singularity in polarized Gowdy.
7.1 Equations, polarized T^{3}Gowdy
7.2 Associated Velocity Term Dominated system
7.3 Asymptotics of the solution to the polarized T^{3}Gowdy equations
7.4 Curvature blow up, polarized T^{3}case

\({v^2}({\theta _0}) \neq 1\),

\({v^2}({\theta _0}) = 1\) but \({\partial _\theta}\nu ({\theta _0}) \neq 0\), or

\({\nu ^2}({\theta _0}) = 1,\,{\partial _\theta}\nu ({\theta _0}) = 0\) but \(\partial _\theta ^2\nu ({\theta _0}) \neq 0\),
7.5 Asymptotic velocity, polarized T^{3}case
7.6 S^{2} × S^{1} and S^{3} cases
The S^{3} and S^{2} × S^{1} cases are also treated in [50]. The results and the analysis are similar but more technical due to the presence of the axes. Consequently, we refer the interested reader to [50] for the details.
8 Asymptotic Expansions Using Fuchsian Methods, General T^{3}Case
The analysis in the polarized case, which we outlined above, illustrates the importance of being able to compute asymptotic expansions. Thus, in analyzing the general case it is natural to begin by trying to carry out a similar computation. As we observed in Section 5.2, there are two different ways to proceed. One is to derive expansions given the solution. The other is to prove the existence of solutions with specified asymptotics. In the present section, we are concerned with the latter perspective.
8.1 Geometric interpretation of v_{ a }
8.2 Restriction on the velocity
Note that Equation (37) is, disregarding the difference in notation, identical to the first equation in (35). However, in the present setting, there is a restriction on v_{ a }, which did not appear in the polarized case. The essential part of this restriction is the inequality 0 < v_{ a }(θ) < 1; the sign of v_{ a } is not of central importance. The restriction v_{ a } < 1 is due to the fact that there is a potential inconsistency with Equation (16) if v_{ a } ≥ 1; assuming the derivatives of u and w with respect to time and space to tend to zero, all the terms in Equation (16) tend to zero except, possibly, \({e^{2P  2\tau}}Q_\theta ^2\), which is roughly \({e^{2({\upsilon _a}(\theta)  1)\tau}}q_\theta ^2(\theta)\). In fact, if v_{ a }(θ) ≥ 1 and q_{ θ }(θ) ≠ 0, the expansions are clearly inconsistent. Furthermore, considering Equation (16), the term \({e^{2P  2\tau}}Q_\theta ^2\) would in that case seem to have the effect of causing P_{ τ } to decrease. For a further discussion, see [10].
8.3 Geodesic loop
Consider a solution with asymptotics of the form of Equations (37)–(38) and v_{ a }(θ) > 0. Then Q(τ, θ) converges and P(τ, θ) tends to infinity as τ → ∞. In other words, for a fixed θ, the solution roughly speaking goes to the boundary along a geodesic in hyperbolic space; see Equation (21). Since P and Q, for a fixed τ, define a loop in hyperbolic space, the solution is asymptotically approximated by a “loop of geodesics”.
8.4 Existence of expansions using Fuchsian methods, T^{3}case
In [54], Kichenassamy and Rendall proved the existence of expansions in the real analytic setting. In other words, they proved that, given real analytic v_{ a }, ϕ, q, ψ with 0 < v_{ a } < 1, there is a unique solution to Equations (16)–(17) with expansions of the form of Equations (37)–(38). Note that the number of functions that are freely specifiable in the expansions coincides with the number of functions that need to be specified in order to obtain a unique solution to the initial value problem corresponding to Equations (16)–(17). For reasons mentioned in Section 8.2, the expansions suffer from a potential consistency problem in the case of v_{ a } ≥ 1 and q_{ θ } ≠ 0. However, in [54] it was proven that if q is constant, the condition on v_{ a } can be relaxed to v_{ a } > 0. The regularity condition of [54] was relaxed to smoothness in [68].
8.5 Existence of expansions using Fuchsian methods, S^{2} × S^{1} and S^{3} cases
There are results concerning the existence of expansions in the S^{2} S^{1} and S^{3} cases as well [86]. However, the analysis is complicated by the presence of the axes; regularity conditions lead to the requirement that the velocity has to be either −1 or 3 there. As a consequence, the result is different, since such values of the velocity are inconsistent with expansions containing the full number of free functions. We refer the reader interested in more details to [86].
9 Spikes
Numerical studies of solutions to Equations (16)–(17) indicate that for most spatial points, behavior similar to that described by the asymptotic expansions (37)–(38) occurs [11, 10]. However, the studies also indicate that there are exceptional spatial points at which the behavior is different. Due to the appearance of the solutions in the neighborhood of the exceptional points, the corresponding features have been referred to as “spiky features” or “spikes”. Their existence would seem to necessitate an understanding of the “spikes” on an analytical level in order to be able to describe the asymptotics of general T^{3}Gowdy solutions. An important step in this direction was achieved by demonstrating the existence of a large class of solutions to Equations (16)–(17) with spikes [71]. In order to be able to describe these solutions, we need to introduce some transformations taking solutions to solutions.
9.1 Inversion
9.2 Gowdy to Ernst transformation
9.3 False spikes
Due to the fact that the limit of P_{1τ} has a discontinuity at θ_{0}, the point θ_{0} is called a spike for the solution (Q_{1}, P_{1}).
9.3.1 True spikes
9.4 High velocity spikes
It is possible to iterate the procedure leading to a true spike. This leads to spikes with an arbitrary high velocity; see [71, p. 2972]. In [36], numerical investigations of spikes of this type were carried out.
10 Asymptotic Velocity, General T^{3}Gowdy
The results we described in Section 8 and 9 consisted of constructions of solutions with certain types of asymptotics. However, considering the formulation of the strong cosmiccensorship conjecture, it is of interest to obtain conclusions given assumptions, which are phrased in terms of the initial data. The first question to ask is if there is a condition on initial data, which ensures the existence of asymptotic expansions. In [76], such a condition was established. Being phrased in terms of L^{2}based energies, the condition is rather technical. However, it does have the advantage of applying to higher dimensional analogues of the equations (as opposed to much of the analysis to be described below). Other, less technical, conditions were later established [12, 74]. Even though these results are of interest, they still only describe a part of the dynamics (as the construction of spikes, see Section 9, demonstrates). The question then arises concerning how to proceed. Considering the analysis in the polarized case, the asymptotic expansions (37)–(38) and the spikes, it is clear that the velocity v plays a central role. Thus, it is natural to try to prove that it is possible to make sense of the concept of a velocity under more general circumstances.
10.1 Existence of an asymptotic velocity
That it is possible to define an asymptotic velocity in all generality was demonstrated in [79] (see also [12] for related results).
If there are expansions of the form (37)–(38), we have seen that v_{ a } can be computed according to Equation (39). As a consequence, it is of interest to ask if the limit on the righthand side of this equation always exists. Due to [79, Corollary 6.9, p. 1009], the answer is yes. As a consequence, we are naturally led to the following definition.
10.2 Relevance of the asymptotic velocity to the issue of curvature blow up
Due to the definition, it is clear that v_{∞} is a geometric object from the wavemap perspective. Furthermore, it is possible to prove that if v_{∞}(θ_{0}) ≠ 1, then the curvature blows up along any causal curve ending at θ_{0}; see the proof of [79, Proposition 1.19, p. 989].
Note that the Gowdy metric corresponding to P(τ, θ) = τ, Q = 0 and λ(τ, θ) = τ is the flat Kasner metric. Consequently, the curvature tensor is in that case identically zero. For this solution, v_{∞} = 1. In particular, if v_{∞}(θ_{0}) = 1, the Kretschmann scalar need not necessarily be unbounded along a causal curve ending at θ_{0}.
10.3 Interpretation of the asymptotic velocity as a rate of convergence to the boundary in hyperbolic space
10.4 Two dimensional version of the asymptotic velocity
10.5 Dominant contribution to the asymptotic velocity
It is important to note that, not only does the kinetic energy density converge pointwise, but in the limit, the only term contributing is \(P_\tau ^2\). In fact, the following result holds (see [79, Proposition 1.3, p. 983]):
Proposition 1 Consider a solution to Equations (16) – (17) and let θ_{0} ∈ S^{1}. Then, either P_{ τ }(τ, θ_{0}) converges to v_{∞}(θ_{0}) or to −v_{∞}(θ_{0}). If P_{ τ }(τ, θ_{0}) → −v_{∞}(θ_{0}), then (Q_{1}, P_{1}) = Inv(Q, P) has the property that P_{1τ}(τ,θ_{0}) → v_{∞}(θ_{0}). Furthermore, if v_{ ∞ }(θ_{0}) > 0, then Q_{1}(τ, θ_{0}) converges to 0.
Similar to what we have already seen for false spikes, see Section 9.3, we see that by applying an inversion we can always obtain a nonnegative limit for P_{ τ }.
10.6 Value of the asymptotic velocity as a criterion for the existence of expansions
Not only is the asymptotic velocity a geometric quantity (from the wavemap perspective), not only can it be used as an indicator for curvature blow up, it can also be used as a criterion to determine whether asymptotic expansions exist or not. There are many results of this form, see, e.g., [12, 76, 83]. However, we shall only describe some of them, beginning with [79, Proposition 1.5, p. 984] (note that this result was essentially obtained in a previous paper [74]):
It is worth noting that the above proposition proves that if 0 < v_{ ∞ }(θ_{0}) < 1, then v_{ ∞ } is smooth in the neighborhood of θ_{0}. In other words, knowledge concerning v_{∞} at one point can sometimes yield conclusions in the neighborhood of that point; see [79, Remark 1.6, p. 985].
11 The Generic Set, General T^{3}Gowdy
There are general results concerning the asymptotic behavior in the direction of the singularity for T^{3}Gowdy spacetimes; there is, e.g., an open and dense subset ℇ of the circle such that there are smooth expansions on ℇ; see [12, Theorem 1.3, p. 1018] and [79, Proposition 1.9, p. 985]. However, the results of [54, 68, 71] show that the asymptotic behavior of solutions is in general very complicated; there are, e.g., solutions with an infinite number of true spikes. On the other hand, much of the complicated behavior can be expected to be unstable, i.e., nongeneric. Thus, since the strong cosmiccensorship conjecture is only a statement concerning generic solutions, it is natural to try to find a set of solutions whose asymptotics are generic but less complicated than those of general solutions. The purpose of the present section is to define one generic set of initial data. Since the concepts nondegenerate true and false spikes play a central role, let us begin by defining them.
11.1 Nondegenerate true spikes
The definition of a nondegenerate true spike proceeds by running the construction of a true spike backwards. In other words, we start with a solution (Q, P) such that 1 < v_{ ∞ }(θ_{0}) < 2 and such that P_{ τ }(τ, θ_{0}) → v_{ ∞ }(θ_{0}). Letting \(({Q_1},{P_1}) = {\rm{G}}{{\rm{E}}_{{q_0},{\tau _0},{\theta _0}}}(Q,P)\), we see, by Equation (41), that P_{1τ}(τ, θ_{0}) → 1 − v_{ ∞ }(θ_{0}) < 0. Let (Q_{2}, P_{2}) = Inv(Q_{1}, P_{1}). Then, due to Proposition 1, P_{2τ}(τ, θ_{0}) → v_{ ∞ }(θ_{0}) − 1 and Q_{2}(τ, θ_{0}) → 0. Finally, Proposition 2 applies to (Q_{2}, P_{2}) so that there are smooth expansions in the neighborhood I of θ_{0}. In particular, there is a smooth function q_{2} such that Q_{2} converges to q_{2} with respect to any C^{ k }norm. Moreover, it is important to note that q_{2}(θ_{0}) = 0. We are naturally led to [79, Definition 1.12, p. 987]:
The choice of q_{0}, τ_{0}, θ_{0} is unimportant. Note that nondegenerate true spikes have punctured neighborhoods with normal expansions.
11.2 Nondegenerate false spikes
Let us recall the definition of a nondegenerate false spike, [79, Definition 1.11, p. 986]:
Note that nondegenerate false spikes have punctured neighborhoods with normal expansions.
11.3 The generic set, definition
We are now in a position to give the definition of what, in the end, will turn out to be the generic set, [79, Definition 1.14, p. 988]:
11.4 Verification of genericity, openness
If it is possible to prove that \({\mathcal G}\) is open and dense, it is justified to call it a generic set. A first step in this direction is given by [79, Proposition 1.15, p. 988]:
Proposition 3 \({{\mathcal G}_{l,m}}\) is open in the C^{2} × C^{1}topology on initial data and \({{\mathcal G}_{l,m,c}}\) is open in the C^{2} × C^{1}topology on the subset of initial data satisfying Equation (49) .
It is of interest to note that the topology can be weakened somewhat if the only information of interest concerning the asymptotics is that the asymptotic velocity is different from 1. In fact, [79, Proposition 1.16, p. 988] states:
Proposition 4 Given \(z \in {{\mathcal G}_{l,m}}\), there is an open neighborhood of the initial data for z in the C^{1} × C^{0} topology such that for each corresponding solution ẑ, v_{∞}[ẑ](θ) ∈ (0, 1) ∪ (1, 2) for all θ ∈ S^{1}.
Recall that an asymptotic velocity different from 1 implies curvature blow up.
11.5 Verification of genericity, density
Finally, [83, Theorem 2, p. 1190] yields the conclusion that \({\mathcal G}\) is dense:
Theorem 3 \({\mathcal G}\) and \({{\mathcal G}_c}\) are dense in \({{\mathcal S}_p}\) and \({{\mathcal S}_{p,c}}\), respectively, with respect to the C^{∞}topology on initial data.
Here \({{\mathcal S}_p}\) and \({{\mathcal S}_{p,c}}\) are defined in [83, Definition 1, p. 1188]:
12 Expanding Direction, Polarized Gowdy
Since there is only an expanding direction in the case when spatial topology is T^{3}, the current section is restricted to considerations of that case. For our purposes, it would be sufficient to describe the results in the general case, since that would yield all the information needed concerning the polarized case as well. However, for historical reasons, and since results have been obtained in the polarized case that are as yet unavailable in the general case, we shall describe both.
Even though it is of no immediate relevance to the question of strong cosmic censorship, let us note that there is a general picture concerning the future asymptotic behavior of vacuum solutions to Einstein’s equations; see [1, 33] and references cited therein. In particular, a strong connection between the future asymptotics and spatial topology is conjectured to exist. In [80] it was confirmed that vacuum T^{3}Gowdy fits into the general picture.
12.1 Asymptotic behavior
12.2 Comparison with spatially homogeneous solutions
13 Expanding Direction, The General Case
13.1 Energy decay
13.2 Proof of decay of the energy
13.2.1 Toy model
13.2.2 Polarized case
13.2.3 General case
It is of interest to note that the results concerning the decay rate can be generalized to a larger class of spacetimes [81].
13.3 Asymptotic ODE behavior
13.3.1 Conserved quantities
In practice, it is often convenient to apply an isometry to a solution so that the conserved quantities become as simple as possible. This is achieved in [75, Lemma 8.2, p. 681]:
Lemma 1 Consider a solution to Equations (11) – (12) . If A^{2} + 4BC > 0, there is an isometry such that if A_{1}, B_{1} and C_{1} are the constants of the transformed solution, then \({A_1} =  \sqrt {{A^2} + 4BC}\) and B_{1} = C_{1} = 0. If A^{2} +4BC = 0, there is an isometry such that the constants of the transformed solution are A_{1} = B_{1} =0 and C_{1} = 4π or C_{1} = 0.
Analyzing the asymptotic behavior of the transformed solution and then transforming back is often more convenient than analyzing the original solution.
13.3.2 Interpreting the conserved quantities as ODEs for the averages
Returning to the question of the asymptotics, we wish to interpret the conserved quantities as ODEs for 〈P〉 and 〈Q〉. Due to [75, Lemma 8.1, p. 680], we have the following result:
Naively estimating the integral on the righthand side of Equation (61) leads to the conclusion that it is bounded but no more. Consequently, it seems unreasonable to think of this term as an error term. On the other hand, integrating with respect to time might lead to an improvement. In fact, since 〈Q_{ t }〉 decays very quickly, see Equation (64), replacing Q_{ t } with 〈Q_{ t }〉 in Equation (61) leads to a term, which tends to zero. Consequently, we can replace Q_{ t } by Q_{ t } − 〈Qt〉 in (61) with a small error. Integrating Equation (61) and using such ideas leads to [75, Lemma 8.9, p. 685]:
Clearly, Lemma 3 yields important information concerning the asymptotics. Is it possible to apply similar ideas to Equations (62) and (63)? It turns out to be necessary to combine both equations in order to obtain a single equation for 〈Q〉. The problem is the last term in Equation (62) and the second to last term in Equation (63). However, combining partial integrations, Taylor expansions in the last term in Equation (62) with various estimates, such as Equation (64), leads to [75, Lemma 8.8, p. 684]:
In the case of B = 0 it is convenient to apply this result with f = t^{−α/2}. This leads to [75, Proposition 8.11, p. 685]:
In case of B = 0, we consequently have detailed information concerning the asymptotic behavior of 〈Q〉 as well. Furthermore, as was observed earlier, given a solution with the property that A^{2} + 4BC ≥ 0, there is an isometry of hyperbolic space such that the transformed solution is such that the corresponding B equals zero. Thus, the only case that remains to be analyzed is A^{2} + 4BC < 0,. This case is more complicated (but more interesting). We shall therefore omit a description of the analysis.
13.4 Geometric interpretation of the asymptotics
The analysis of the asymptotics in the various cases leads to the following conclusion [75, Theorem 1.2, p. 661]:

If all the constants A, B and C are zero, Γ is a point.

If A^{2} + 4BC = 0, but the constants are not all zero, Γ is either a horocycle (i.e., a circle touching the boundary) or a curve y = constant.

If A^{2} + 4BC > 0, Γ is either a circle intersecting the boundary transversally or a straight line intersecting the boundary transversally.

If A^{2} + 4BC < 0, Γ is a circle inside the upper half plane.
13.5 Concluding remarks
It is of interest to note that in the case of A^{2} + 4BC < 0, the spatial variation dies out and the solution behaves like a solution to an ODE asymptotically. On the other hand, the ODE of which it is approximately a solution is not the ODE, which is obtained by dropping the spatial derivatives in the original equation.
13.6 Geodesic completeness
The analysis described above concerned only the functions and their averages. In particular, no estimates for the derivatives in the sup norm were derived. However, in order to prove future causal geodesic completeness, it is of interest to have such estimates. According to [75, Proposition 1.8, p. 665]:
14 Strong Cosmic Censorship in Gowdy Spacetimes
The results concerning strong cosmic censorship that exist concern the polarized case as well as the general T^{3}case. Since the methods in the two cases are different, it is natural to divide the exposition accordingly.
14.1 The polarized case
As has already been mentioned, the proof of strong cosmic censorship in the polarized Gowdy case proceeds via Conjecture 2. In other words, via a proof of the fact that for generic initial data, the Kretschmann scalar is unbounded in the incomplete directions of causal geodesics. Future causal geodesic completeness in the polarized T^{3}Gowdy case was announced in [21] and proven in [52, Corollary 21, p. 190]. Thus the main problem is that of proving that the curvature blows up at the singularities. This is achieved in two steps in [21]. First, the existence of a diffeomorphism between asymptotic data (i.e., v and ϕ in Equation (35)) and ordinary initial data is demonstrated; see [21, p. 1675]. Second, using the observations made concerning curvature blow up in Section 7.4, it can be shown that there is an open and dense subset of the set of asymptotic data such that the curvature of the corresponding solutions blows up everywhere on the singularity. Let us state the result as given in [21, p. 1673]:
Theorem 7 (Strong cosmic censorship for polarized Gowdy spacetimes) Let Σ^{3} = T^{3}, S^{3}, or S^{2} × S^{1}, and let \({\mathcal P}({\Sigma ^3})\) he the space of initial data for the polarized Gowdy spacetimes (with C^{∞} topology). There exists an open dense subset \(\hat {\mathcal P}({\Sigma ^3}) \subset {\mathcal P}({\Sigma ^3})\) such that the maximal development of any set of data in \(\hat {\mathcal P}({\Sigma ^3})\) is inextendible.
14.2 General T^{3}case
In the general T^{3}case, the result is an immediate consequence of Proposition 3, Theorem 3 and the fact that solutions are future causallygeodesically complete. Let us quote the exact statement [83, Corollary 1, p. 1190–1191]:

\({{\mathcal G}_{i,c}}\) is open with respect to the C^{1} × C^{0}topology on \({{\mathcal S}_{i,p,c}}\),

\({{\mathcal G}_{i,c}}\) is dense with respect to the C^{∞}topology on \({{\mathcal S}_{i,p,c}}\),

every spacetime corresponding to initial data in \({{\mathcal G}_{i,c}}\) has the property that in one time direction, it is causally geodesically complete, and in the opposite time direction, the Kretschmann scalar R_{ αβγδ }R^{ αβγδ } is unbounded along every inextendible causal curve,

for every spacetime corresponding to initial data in \({{\mathcal G}_{i,c}}\), the MGHD is C^{2}inextendible.
Definition 9 Let (M, g) be a connected Lorentz manifold, which is at least C^{2}. Assume there is a connected C^{2} Lorentz manifold \((\hat M,\hat g)\) of the same dimension as M and an isometric embedding \(i:M \rightarrow \hat M\) such that \(i:(M) \neq \hat M\). Then M is said to be C^{2}extendible. If (M, g) is not C^{2}extendible, it is said to be C^{2}inextendible.
It might be possible to obtain this result using different methods. In fact, let \({\mathcal G}\) be the set of initial data such that the corresponding solutions have an asymptotic velocity, which is different from one on a dense subset of the singularity. Endowing the initial data with the C^{∞}topology, it is possible to show that \({\mathcal G}\) is a dense G_{ δ } set [77]. Due to its definition, it is clear that solutions corresponding to initial data in \({\mathcal G}\) have the property that the curvature blows up on a dense subset of the singularity. It would be natural to expect the corresponding solutions to be inextendible, but providing a proof is nontrivial. Important steps in the direction of proving this statement were taken in [22]. However, to the best of our knowledge, there is, as yet, no result to this effect.
Notes
Acknowledgments
Thanks are due to Piotr Chruściel and Alan Rendall for comments on an earlier version of this review. The author acknowledges the support of the Goran Gustafsson Foundation and the Swedish Research Council. This review article was in part written during a stay by the author at the Max Planck Institute for Gravitational Physics as a recipient of a Friedrich Wilhelm Bessel Research Award granted by the Alexander von Humboldt Stiftung. The author is a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation.
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