Global Hyperbolicity in Space-Time Manifold
Keywords:
Cauchy surface, causality, global hyperbolicity, space-time manifold, space-time singularities.Abstract
Global hyperbolicity is the most important condition on causal structure space-time, which isinvolved in problems as cosmic censorship, predictability etc. An open set O is said to beglobally 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.
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