What is meant by globally hyperbolic spacetime? [duplicate]

Question

Can somebody explain what is meant by globally hyperbolic spacetime? What other kinds can there be?

Answer 1

Wald has a nice chapter on "Causal Structure" that lays it all out very well. He points out that there are three distinct definitions of globally hyperbolic spacetimes, but that they are precisely equivalent. And I find Wald's to be the simplest, so that's what I'll explain here. The basic idea is that there is a single surface that we might think of like an "inital-data slice" for which, if you know everything about all physical fields on that surface (and maybe an infinitesimal neighborhood, so that you can define derivatives), then you can (in principle) predict everything about the future of the universe and retrodict everything about its past.

Start out with the notion of causality. Basically, a point $p$ in a spacetime is in the "causal future" of another point $q$ if $p$ can "be influenced" by a signal sent from $q$. In GR (and mainstream physics more generally), this is basically taken to mean that $p$ can be connected to $q$ by a timelike or null curve (representing possible paths of massive or massless particles, respectively).

Next, we think of an achronal set $S$. This is basically a slice through some part of the spacetime (we'd usually think of it as being a "spacelike" slice, though it could be null), for which no two points can be connected to each other by a timelike curve. That is, if we have two points $p\in S$ and $q \in S$, then — at best — only a null signal from one can influence the other. (Of course, there are usually other points that are not on $S$ that could be timelike connected to both, but such points would be in the past or future of $S$.) The achronal set doesn't have to extend very far, but it could extend "across the entire spacetime" in some sense.

Now, we can get to the idea of a domain of dependence for an achronal set $S$. The future domain of dependence $D^+(S)$ is the collection of points in the spacetime for which every single causal curve that goes through those points will go through $S$ (assuming the causal curves are extended far enough back in time). Roughly speaking, this means that the only information that can affect anything in $D^+(S)$ is information about $S$ itself. So $D^+(S)$ represents all the points that we can predict given knowledge of $S$. Similarly, we can talk about the past domain of dependence $D^-(S)$, representing all the points that we can retrodict given knowledge of $S$. And the full domain of dependence of $S$ is just the union of all those points: \begin{equation} D(S) = D^+(S) \cup D^-(S). \end{equation} This is the whole set of points that can be either predicted or retrodicted from knowledge of $S$.

Finally, we can define a globally hyperbolic spacetime as a spacetime for which there exists (at least one) achronal set $\Sigma$ for which $D(\Sigma)$ is the entire spacetime. That is, the entire past and future of the whole spacetime can be either predicted or retrodicted just from knowledge of $\Sigma$. Such a surface $\Sigma$ is called a "Cauchy surface".

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Now, the question is what good is this definition? What other type of spacetime could there be? Wald says

There are some good reasons for believing that all physically realistic spacetimes must be globally hyperbolic.

So, I don't think there's any realistic suggestion that some relevant physical model involves non-globally hyperbolic spacetime. But its certainly possible to imagine them. One prime source of this imagined spacetimes is where there are points that are simply "missing". This is a weird idea that you might expect to see in Star Trek or something, and it shouldn't feel intuitive. But just imagine that some region of spacetime ceased to exist suddenly, and then came back into existence after a moment. When it comes back into existence, the physical fields will have some new values inside that region that may or may not have anything to do with how they were before the region blinked out of existence. That means that anything in the future domain of dependence of this region will depend on the values of the physical fields when they came back into existence. There's no reason to think you could have predicted those values from some much earlier achronal set, for example. Similarly, you couldn't retrodict the values from a later achronal set. Therefore, such a spacetime could not have a $\Sigma$ whose domain of dependence is the entire spacetime, and it's not globally hyperbolic.

Again, this isn't something that we expect to happen on physical grounds, but you can construct a mathematical model that has these features. And if we're trying to model physics with math, then we need to specify which mathematical models we find relevant to physics. That's all.

Answer 2

Short answer:

The term globally "hyperbolic" refers to "hyperbolic differential equations". The term originates from the notion that the gravitational field equations (which are a set of "hyperbolic" differential equations) can be solved "globally" on the manifold. This means that providing initial data on some "Cauchy surface" (all of "space" at a single "time") determines a solution everywhere in the spacetime. If a Cauchy surface exists, the spacetime is globally hyperbolic. A Cauchy surface can fail to exist for various reasons, for example due to a naked singularity or a topology change. Having a Cauchy surface is equivalent to various causal structure properties, which you can read about in many places.