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ASYMPTOTIC BEHAVIOR OF FUTURE-COMPLETE COSMOLOGICAL SPACE-TIMES
 

Summary: ASYMPTOTIC BEHAVIOR OF FUTURE-COMPLETE COSMOLOGICAL
SPACE-TIMES
MICHAEL T. ANDERSON
Dedicated to Vince Moncrief on his 60 th Birthday
Abstract. This work discusses the apriori possible asymptotic behavior to the future, for (vacuum)
space-times which are geodesically complete to the future, and which admit a foliation by constant
mean curvature compact Cauchy surfaces.
1. Introduction.
Let (M, g) be a CMC cosmological space-time, i.e. a space-time with a compact constant mean
curvature Cauchy surface (; g; K). The main focus will be on the vacuum case in 3+1 dimensions
although we will occasionally consider generalizations to non-negative energy conditions and higher
dimensions.
A fundamental issue in general relativity is to understand the global structure of (M, g), and
in particular the evolution of the geometry of the CMC foliation   generated by the CMC slice
: Singularities of (M, g) will generally form in nite proper time, both to the future and to the
past of . Roughly, these may correspond either to big bang or big crunch singularities of the
space-time as a whole, or to localized gravitational collapse within only parts of the space-time.
The understanding of the mechanism and structure of such singularity formation is of course a
central issue in general relativity.
Here we concentrate instead on the simpler situation where there is no singularity formation (in

  

Source: Anderson, Michael - Department of Mathematics, SUNY at Stony Brook

 

Collections: Mathematics