Theorem 8 Consider the set of smooth, periodic initial data to Equations (16
)–(17
) satisfying
Equation (50
). There is a subset
of
with the following properties
For the sake of completeness, let us also recall the definition of C2-inextendibility.
Definition 9 Let be a connected Lorentz manifold, which is at least C2. Assume there is a
connected C2 Lorentz manifold
of the same dimension as
and an isometric embedding
such that
. Then
is said to be C2-extendible. If
is not
C2-extendible, it is said to be C2-inextendible.
It might be possible to obtain this result using different methods. In fact, let 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
-topology, it is
possible to show that
is a dense
set [77]. Due to its definition, it is clear that solutions
corresponding to initial data in
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.
http://www.livingreviews.org/lrr-2010-2 |
Living Rev. Relativity 13, (2010), 2
![]() This work is licensed under a Creative Commons License. E-mail us: |