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 three-dimensional compact manifold Σ on which a Riemannian metric, a symmetric covariant two-tensor 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 [70Jump To The Next Citation Point, pp. 176–177]; see also [17Jump To The Next Citation Point18].

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 three-dimensional 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 [17Jump To The Next Citation Point] contains a list. Readers interested in the underlying mathematics are referred to [59Jump To The Next Citation Point]. Given a specific topic of interest, such as, the strong cosmic-censorship 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 locally-homogeneous 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 [87Jump To The Next Citation Point]. 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 [67Jump To The Next Citation Point].

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, [72734546] and references cited therein for partial results concerning the asymptotics and [44Jump To The Next Citation Point] 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 two-dimensional isometry group. This leads us to the Gowdy class of spacetimes.


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