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I am been studying differential geometry and spacetime and I keep coming across the term globally hyperbolic. I am having a hard time coming up with an intuitive understanding of this idea. What is an intuitive meaning of globally hyperbolic.

My background is mostly mathematics at a higher undergraduate level.

Qmechanic
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j.d. allen
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  • Every point has the local shape of a saddle point? For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R? That kind of thing? The sums of angles in triangles are smaller than 180 degrees, too, aren't they? Maybe that's the most obvious rule that one could use if one ever had to test if one was living in a haunted house with hyperbolic space room. Dr. Who might have one of those in his Tardis. – CuriousOne Jan 15 '16 at 22:33
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    I think Wikipedia -- https://en.wikipedia.org/wiki/Globally_hyperbolic_manifold -- has a great definition for globally hyperbolic spaces. I like the third definition as it is the clearest in my opinion - namely that it is a manifold that has a Cauchy surface $\Sigma$. What this says is that globally hyperbolic manifolds are those for which one can construct well defined initial value problems - i.e. by providing the appropriate information on $\Sigma$, the theory can be solved on all of the manifold. – Prahar Jan 15 '16 at 23:03
  • @Prahar: Wow... that's at the same time cool and completely untestable... but it is absolutely cool. – CuriousOne Jan 15 '16 at 23:16
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    @CuriousOne - completely untestable? This is a definition here - what is there to test? Maybe I didn't get your point. – Prahar Jan 15 '16 at 23:25
  • @Prahar: How are you going to create a set of stimuli in every point on the Cauchy surface to test that the initial value problem is fulfilled? That's the problem with global criteria... they are enormously powerful statements, but impossible to realize experimentally. Local conditions like the sum of angles can be tested locally. An interesting question arises... so is Galilean spacetime globally hyperbolic by this definition or is it degenerate? Is the Cauchy condition fulfilled? – CuriousOne Jan 16 '16 at 00:22
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    @CuriousOne But the question is about "globally hyperbolic spacetimes", i.e. you are testing/defining a global property of the manifold. Obviously, there are no local experiments can tell you about such global properties. Any test of global hyperboliticity will require infinite number of local experiments. You can test the local shape of points as you suggested, but you'd still need to do it for every single point in the manifold to conclude that it is globally hyperbolic. So of course, global hyperbolicity is completely untestable. – Prahar Jan 16 '16 at 00:25
  • @Prahar: Exactly. We usually have to assume that what holds locally also holds globally, especially in cosmology, which is somewhat covered by those observations that we can make. – CuriousOne Jan 16 '16 at 00:29

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Globally hyperbolic refers to the fact that hyperbolic equations always have locally a well defined Cauchy problem, that is, a unique development given initial conditions. Which means that, given a matter field at a time $t_1$, there exists a unique solution of that field at a time $t_2$. It is boosted up to globally hyperbolic if that property holds globally.

Globally hyperbolic isn't directly related to this notion (you can still have hyperbolic equations for which this property does not hold up globally in a globally hyperbolic spacetime), but the two notions are close. A globally hyperbolic is, formally, a spacetime that is both causal and such that the intersection of the past and future of two different points is compact, which roughly translates to having no naked singularities (a point removed from spacetime might cause that set to fail to be compact).

Alternatively, global hyperbolicity can be defined by the existence of a Cauchy surface - an achronal (cannot be linked by any causal curve) spacelike hypersurface such that every causal curve crosses it exactly once. No more (no curves going to the same time more than once) and no less (no curve appearing out of nowhere).

Breaks in causality and naked singularities may cause a lack of existence or uniqueness of solutions for matter fields - corresponding either to matter going round in a circle, so to speak, or coming out of nowhere or disappearing. Globally hyperbolic spacetimes benefit from a theorem due to Hawking stating that, for such a spacetime with a reasonable enough matter field (basically if it already behaves nicely in flat space), any hyperbolic equation is globally hyperbolic.

Slereah
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