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.
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 is effective if
for all
implies
.
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 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 [70
, pp. 176–177]; see
also [17
, 18].
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 are such that there is a smooth and effective Lie
group action of
on
. 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 cosmic-censorship conjecture,
this list yields a hierarchy of classes of spacetimes in which one can study it in a simplified
setting.
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 [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
].
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 two-dimensional isometry group. This leads us to the Gowdy class of spacetimes.
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Living Rev. Relativity 13, (2010), 2
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