4.2 Inhomogeneous models with symmetry

In the spatially homogeneous case the metric can be written in a form that is independent of the spatial variables and this leads to an enormous simplification. Another class of spacetimes that are highly symmetric but require the metric to be spatially dependent are those that admit a group of isometries acting on two-dimensional spacelike orbits, at least after passing to a covering manifold. The group may be two-dimensional (local U (1) × U (1) or T2 symmetry) or three-dimensional (spherical, plane, or hyperbolic symmetry). In all these cases, the quotient of spacetime by the symmetry group has the structure of a two-dimensional Lorentzian manifold Q. The orbits of the group action (or appropriate quotients in the case of a local symmetry) are called surfaces of symmetry. Thus, there is a one-to-one correspondence between surfaces of symmetry and points of Q. There is a major difference between the cases where the symmetry group is two- or three-dimensional. In the three-dimensional case no gravitational waves are admitted, in contrast to the two-dimensional case where the evolution part of the Einstein equations are non-linear wave equations.

Three types of time coordinates that have been studied in the inhomogeneous case are CMC, areal, and conformal coordinates. A CMC time coordinate t is one where each hypersurface of constant time has constant mean curvature and on each hypersurface of this kind the value of t is the mean curvature of that slice. In the case of areal coordinates, the time coordinate is a function of the area of the surfaces of symmetry, e.g., proportional to the area or proportional to the square root of the area. In the case of conformal coordinates, the metric on the quotient manifold Q is conformally flat. The CMC and the areal coordinate foliations are both geometrically-based time foliations. The advantage with a CMC approach is that the definition of a CMC hypersurface does not depend on any symmetry assumptions and it is possible that CMC foliations will exist for general spacetimes. The areal coordinate foliation, on the other hand, is adapted to the symmetry of spacetime but it has analytical advantages and detailed information about the asymptotics can be derived. The conformal coordinates have mainly served as a useful framework for the analysis to obtain geometrically-based time foliations.

4.2.1 Surface symmetric spacetimes

Let us now consider spacetimes (M, g) admitting a three-dimensional group of isometries. The topology of M is assumed to be ℝ × S1 × F, with F a compact two-dimensional manifold. The universal covering Fˆ of F induces a spacetime (Mˆ ,ˆg) by Mˆ = ℝ × S1 × ˆF and ˆg = p∗g, where ˆ p :M → M is the canonical projection. A three-dimensional group G of isometries is assumed to act on (Mˆ ,ˆg). If F = S2 and G = SO (3), then (M, g) is called spherically symmetric; if F = T 2 and G = E 2 (Euclidean group), then (M, g) is called plane symmetric; and if F has genus greater than one and the connected component of the symmetry group G of the hyperbolic plane H2 acts isometrically on Fˆ= H2, then (M, g) is said to have hyperbolic symmetry.

In the case of spherical symmetry the existence of one compact CMC hypersurface implies that the whole spacetime can be covered by a CMC time coordinate that takes all real values [151Jump To The Next Citation Point, 42]. The existence of one compact CMC hypersurface in this case was proven by Henkel [94Jump To The Next Citation Point] using the concept of prescribed mean curvature (PMC) foliation. Accordingly, this gives a complete picture in the spherically symmetric case regarding CMC foliations. In the case of areal coordinates, Rein [136Jump To The Next Citation Point] has shown, under a size restriction on the initial data, that the past of an initial hypersurface can be covered, and that the Kretschmann scalar blows up. Hence, the initial singularity for the restricted data is both a crushing and a curvature singularity. In the future direction it is shown that areal coordinates break down in finite time.

In the case of plane and hyperbolic symmetry, global existence to the past was shown by Rendall [151] in CMC time. This implies that the past singularity is a crushing singularity since the mean curvature blows up at the singularity. Also in these cases Rein showed [136Jump To The Next Citation Point] under a size restriction on the initial data, that global existence to the past in areal time and blow up of the Kretschmann scalar curvature as the singularity is approached. Hence, the singularity is both a crushing and a curvature singularity in these cases too. In both of these works the question of global existence to the future was left open. This gap was closed in [25Jump To The Next Citation Point], and global existence to the future was established in both CMC and areal time coordinates. The global existence result for CMC time is a consequence of the global existence theorem in areal coordinates, together with a theorem by Henkel [94] which shows that there exists at least one hypersurface with (negative) constant mean curvature. In addition, the past direction is analyzed in [25Jump To The Next Citation Point] using areal coordinates, and global existence is shown without a size restriction on the data. It is not concluded if the past singularity, without the smallness condition on the data, is a curvature singularity as well. The issues discussed above have also been studied in the presence of a cosmological constant, cf. [182, 184]. In particular it is shown that in the spherically-symmetric case, if Λ > 0, global existence to the future holds in areal time for some special classes of initial data, which is in contrast to the case with Λ = 0. In this context we also mention that surface symmetric spacetimes with Vlasov matter and with a Maxwell field have been investigated in [183].

An interesting question, which essentially was left open in the studies mentioned above, is whether the areal time coordinate, which is positive by definition, takes all values in the range (0,∞ ) or only in (R0, ∞ ) for some positive R0. It should here be pointed out that there is an example for vacuum spacetimes with T2 symmetry (which includes the plane symmetric case) where R0 > 0. This question was first resolved by Weaver [192Jump To The Next Citation Point] for T2 symmetric spacetimes with Vlasov matter. Her result shows that if spacetime contains Vlasov matter (f ⁄= 0) then R0 = 0. Smulevici [174Jump To The Next Citation Point] has recently shown, under the same assumption, that R0 = 0 also in the hyperbolic case. Smulevici also includes a cosmological constant Λ and shows that both the results, for plane (or T2) symmetry and hyperbolic symmetry, are valid for Λ ≥ 0.

The important question of strong cosmic censorship for surface-symmetric spacetimes has recently been investigated by neat methods by Dafermos and Rendall [67Jump To The Next Citation Point, 65Jump To The Next Citation Point]. The standard strategy to show cosmic censorship is to either show causal geodesic completeness in case there are no singularities, or to show that some curvature invariant blows up along any incomplete causal geodesic. In both cases no causal geodesic can leave the maximal Cauchy development in any extension if we assume that the extension is 2 C. In [67Jump To The Next Citation Point, 65Jump To The Next Citation Point] two alternative approaches are investigated. Both of the methods rely on the symmetries of the spacetime. The first method is independent of the matter model and exploits a rigidity property of Cauchy horizons inherited from the Killing fields. The areal time described above is defined in terms of the Killing fields and a consequence of the method by Dafermos and Rendall is that the Killing fields extend continuously to a Cauchy horizon, if one exists. Now, since global existence has been shown in areal time it follows that there cannot be an extension of the maximal hyperbolic development to the future. This method is useful for the expanding future direction. The second method is dependent on Vlasov matter and the idea is to follow the trajectory of a particle, which crosses the Cauchy horizon and shows that the conservation laws for the particle motion associated with the symmetries of the spacetime, such as the angular momentum, lead to a contradiction. In most of the cases considered in [67Jump To The Next Citation Point] there is an assumption on the initial data for the Vlasov equation, which implies that the data have non-compact support in the momentum space. It would be desirable to relax this assumption. The results of the studies [67Jump To The Next Citation Point, 65Jump To The Next Citation Point] can be summarized as follows. For plane and hyperbolic symmetry strong cosmic censorship is shown when Λ ≥ 0. The restriction that matter has non-compact support in the momentum space is here imposed except in the plane case with Λ = 0. In the spherically-symmetric case cosmic censorship is shown when Λ = 0. In the case of Λ > 0 a detailed geometric characterization of possible boundary components of spacetime is given. The difficulties to show cosmic censorship in this case are related to possible formation of extremal Schwarzschild-de-Sitter-type black holes. Cosmic censorship in the past direction is also shown for all symmetry classes, and for all values of Λ, for a special class of anti-trapped initial data.

Although the methods developed in [67, 65] provide a lot of information on the asymptotic structure of the solutions, questions on geodesic completeness and curvature blow up are not answered. In a few cases, information on these issues has been obtained. As mentioned above, blow up of the Kretschmann scalar curvature has been shown for restricted initial data [136]. In the case of hyperbolic symmetry causal future geodesic completeness has been established by Rein [140] when the initial data are small. The plane and hyperbolic symmetric cases with a positive cosmological constant are analyzed in [185Jump To The Next Citation Point]. The authors show global existence to the future in areal time, and in particular they show that the spacetimes are future geodesically complete. The positivity of the cosmological constant is crucial for the latter result. A form of the cosmic no-hair conjecture is also obtained in [185]. It is shown that the de Sitter solution acts as a model for the dynamics of the solutions by proving that the generalized Kasner exponents tend to 1∕3 as t → ∞, which in the plane case is the de Sitter solution.

4.2.2 Gowdy and T2 symmetric spacetimes

The first study of spacetimes admitting a two-dimensional isometry group was carried out by Rendall [155Jump To The Next Citation Point] in the case of local T2 symmetry. For a discussion of the possible topologies of these spacetimes we refer to the original paper. In the model case the spacetime is topologically of the form ℝ × T 3, and to simplify our discussion later on we write down the metric in areal coordinates for this type of spacetime:

g = e2(η−U )(− αdt2 + d𝜃2) + e−2Ut2[dy + Hd 𝜃 + M dt]2 2U 2 +e [dx + Ady + (G + AH )d𝜃 + (L + AM )dt] . (52 )
Here the metric coefficients η, U, α, A, H, L, and M depend on t and 𝜃 and 1 𝜃,x, y ∈ S. In [155Jump To The Next Citation Point] CMC coordinates are in fact considered rather than areal coordinates. Under the hypothesis that there exists at least one CMC hypersurface, Rendall proves for general initial data that the past of the given CMC hypersurface can be globally foliated by CMC hypersurfaces and that the mean curvature of these hypersurfaces blows up at the past singularity. The future direction was left open. The result in [155] holds for Vlasov matter and for matter described by a wave map. That the choice of matter model is important was shown in [154], where a non-global existence result for dust is given, which leads to examples of spacetimes [101] that are not covered by a CMC foliation.

There are several possible subcases to the T2 symmetric class. The plane case, where the symmetry group is three-dimensional, is one subcase and the form of the metric in areal coordinates is obtained by letting A = G = H = L = M = 0 and U = logt∕2 in Equation (52View Equation). Another subcase, which still admits only two Killing fields (and which includes plane symmetry as a special case), is Gowdy symmetry. It is obtained by letting G = H = L = M = 0 in Equation (52View Equation). In [6Jump To The Next Citation Point] Gowdy symmetric spacetimes with Vlasov matter are considered, and it is proven that the entire maximal globally hyperbolic spacetime can be foliated by constant areal time slices for general initial data. The areal coordinates are used in a direct way for showing global existence to the future, whereas the analysis for the past direction is carried out in conformal coordinates. These coordinates are not fixed to the geometry of spacetime and it is not clear that the entire past has been covered. A chain of geometrical arguments then shows that areal coordinates indeed cover the entire spacetime. The method in [6Jump To The Next Citation Point] was in turn inspired by the work [37] for vacuum spacetimes, where the idea of using conformal coordinates in the past direction was introduced. As pointed out in [25], the result by Henkel [95] guarantees the existence of one CMC hypersurface in the Gowdy case and, together with the global areal foliation in [6], it follows that Gowdy spacetimes with Vlasov matter can be globally covered by CMC hypersurfaces as well. The more general case of T2 symmetry was considered in [26], where global CMC and areal time foliations were established for general initial data. In these results, the question whether or not the areal time coordinate takes values in (0,∞ ) or in (R0, ∞ ), R0 > 0, was left open. As we pointed out in Section 4.2.1, this issue was solved by Weaver [192] for T2 symmetric spacetimes with the conclusion that R0 = 0, if the density function f is not identically zero initially. In the case of T2 symmetric spacetimes, with a positive cosmological constant, Smulevici [174] has shown global existence in areal time with the property that t ∈ (0,∞ ).

The issue of strong cosmic censorship for T2 symmetric spacetimes has been studied by Dafermos and Rendall using the methods, which were developed in the surface symmetric case described above. In [66] strong cosmic censorship is shown under the same restriction on the initial data that was imposed in the surface symmetric case, which implies that the data have non-compact support in the momentum variable. Their result has been extended to the case with a positive cosmological constant by Smulevici [173].


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