Definition 2.1 Let be a spacetime containing an asymptotically-flat end, or a
-asymptotically-flat end
, and let
be a stationary Killing vector field on
. We will say
that
is
-regular if
is complete, if the domain of outer communications
is
globally hyperbolic, and if
contains a spacelike, connected, acausal hypersurface
, the
closure
of which is a topological manifold with boundary, consisting of the union of a compact set and
of a finite number of asymptotic ends, such that the boundary
is a topological manifold
satisfying
The “” of the name is due to the
appearing in (2.23
).
Some comments about the definition are in order. First, one requires completeness of the orbits of the
stationary Killing vector to have an action of on
by isometries. Next, global hyperbolicity of the
domain of outer communications is used to guarantee its simple connectedness, to make sure that the area
theorem holds, and to avoid causality violations as well as certain kinds of naked singularities in
. Further, the existence of a well-behaved spacelike hypersurface is a prerequisite to
any elliptic PDEs analysis, as is extensively needed for the problem at hand. The existence of
compact cross-sections of the future event horizon prevents singularities on the future part of the
boundary of the domain of outer communications, and eventually guarantees the smoothness of
that boundary. The requirement Eq. (2.23
) might appear somewhat unnatural, as there are
perfectly well-behaved hypersurfaces in, e.g., the Schwarzschild spacetime, which do not satisfy this
condition, but there arise various technical difficulties without this condition. Needless to say,
all those conditions are satisfied by the Kerr–Newman and the Majumdar–Papapetrou (MP)
solutions.
http://www.livingreviews.org/lrr-2012-7 |
Living Rev. Relativity 15, (2012), 7
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