The most convenient formal definition of global
hyperbolicity is the following. In a spacetime , a subset
of
is called a Cauchy
surface if every inextendible causal (i.e., timelike or
lightlike) curve intersects
exactly once. A spacetime is
globally hyperbolic if and only if it admits a Cauchy surface. The
name globally hyperbolic refers to the fact that for hyperbolic
differential equations, like the wave equation, existence and
uniqueness of a global solution is guaranteed for initial data
given on a Cauchy surface. For details on globally hyperbolic
spacetimes see, e.g., [154
, 25
]. It was
demonstrated by Geroch [133] that every
gobally hyperbolic spacetime admits a continuous function
such that
is a Cauchy
surface for every
. A complete proof of the
fact that such a Cauchy time function can be chosen differentiable
was given much later by Bernal and Sánchez [27, 26]. The topology of a
globally hyperbolic spacetime is determined by the topology of any
of its Cauchy surfaces,
. Note, however, that the
converse is not true because
may be homeomorphic (and
even diffeomorphic) to
without
being homeomorphic to
. For instance, one
can construct a globally hyperbolic spacetime with topology
that admits a Cauchy surface which is not
homeomorphic to
[238
].
In view of applications to lensing the following
observation is crucial. If one removes a point, a worldline
(timelike curve), or a world tube (region with timelike boundary)
from an arbitrary spacetime, the resulting spacetime cannot be
globally hyperbolic. Thus, restricting to globally hyperbolic
spacetimes excludes all cases where a deflector is treated as
non-transparent by cutting its world tube from spacetime (see
Figure 24 for an example).
Note, however, that this does not mean that globally hyperbolic
spacetimes can serve as models only for transparent deflectors.
First, a globally hyperbolic spacetime may contain
“non-transparent” regions in the sense that a light ray may be
trapped in a spatially compact set. Second, the region outside the
horizon of a (Schwarzschild, Kerr,
) black hole is
globally hyperbolic.