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Summary: REMARKS ON EVOLUTION OF SPACE-TIMES IN 3+1 AND 4+1
DIMENSIONS
MICHAEL T. ANDERSON
Abstract. A large class of vacuum space-times is constructed in dimension 4+1 from hyperboloidal
initial data sets which are not small perturbations of empty space data. These space-times are future
geodesically complete, smooth up to their future null in nity I + , and extend as vacuum space-times
through their Cauchy horizon. Dimensional reduction gives non-vacuum space-times with the same
properties in 3+1 dimensions.
PACS numbers: 04.20.Ex, 04.20.Ha
x1. Introduction
Let (; g) be a complete Riemannian Einstein n-manifold with negative scalar curvature, nor-
malized so that
Ric g = (n 1)g: (1.1)
It is well-known that the Lorentzian cone over , i.e. the metric
g = d 2 + 2 g (1.2)
on M = R + is a vacuum solution to the Einstein equations in n + 1 = (1; n) dimensions. In
the case of 3+1 dimensions, all Einstein metrics g on a 3-manifold are of constant curvature and
hence the 4-metric g is
at; in fact (M, g) is then just (a quotient of) the interior of the future
light cone of a point in empty Minkowski space. In dimensions higher than 3, solutions of (1.1) are
usually not of constant curvature and hence the space-times (M, g) are typically not
at.
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