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 T2-symmetry; see Section 2.2. That the maximal globally-hyperbolic vacuum development of T2-symmetric initial data is covered by areal coordinates is proven in [9Jump To The Next Citation Point]. 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 T2-symmetric case. In the Gowdy case, the areal time coordinate is such that the metric is conformal to the Minkowski metric in the tšœƒ-direction; see Equation (2View Equation). In the general T2-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 Einstein–Vlasov system with T3-Gowdy symmetry are contained in [4], see also [5], which treats solutions to the Einstein–Vlasov system in the general T2-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].
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