Global Hyperbolicity in Space-time Manifold

Abstract

Global hyperbolicity is the most important condition on causal structure space-time, which is
involved in problems as cosmic censorship, predictability etc. An open set O is said to be
globally hyperbolic if, i) for every pair of points x and y in O the intersection of the future of x
and the past of y has compact closure i.e., a space-time
M, g
is said to be globally hyperbolic if
the sets
J x J y
  
are compact for all
x, y M
(i.e., no naked singularity can exist in spacetime
topology), and ii) strong causality holds on O i.e., there are no closed or almost closed time
like curves contained in O. Here
J x

is causal future and
J x

is the causal past of an event
x. If a space-time is timelike or null geodesically incomplete but cannot be embedded in a larger
space-time then we say that it has a singularity. An attempt is taken here to discuss global
hyperbolicity and space-time singularity by introducing definitions, propositions and displaying
diagrams appropriately.