Globally Hyperbolic Spacetime: What Is It?

[robousy]

What is a globally hyperbolic spacetime?

I'm reading birrel and davies 'quantum fields' in curved space and chapter 3 starts with this assumption...

Thanks in advance.

 

[Stingray]

It's basically a spacetime that admits that admits Cauchy surfaces. There's a theorem which states that all such spacetimes can be assigned continuous "time functions" t:M \rightarrow \mathbb{R} where t^{-1}(s) gives a Cauchy surface for any s. Also, if each Cauchy surface has topology \Sigma, the manifold has topology \Sigma \times \mathbb{R}. Wald's GR book goes into these things extensively.

 

[pervect]

The next logical question is "What is a Cauchy surface"?

One of the less technical definitions is that it is a space-like surface representing an "instant of time" in the universe, and has the property that the future state of the universe and the past state of the universe can both be predicted/retrodicted from the values of "conditions" on the Cauchy Surface alone. (Of course this arises from a classical, deterministic viewpoint, but then GR is a classical theory, not a quantum theory).

The more technical defintion (also in Wald, as was this less technical defintion which I paraphrased a bit) involves a lot of discusion of achronal sets and domains of dependency.

 

[robousy]

Ok thanks!

The terminology sounds a bit confusing. 'Hyperbolic' makes me think if conic sections and the like - but pretty much it has nothing to do with geometry then?

 

[robphy]

It's "hyperbolic" as in "hyperbolic partial differential equation".
Check out this article from the Living Reviews site:
http://relativity.livingreviews.org/Articles/lrr-1998-3/node2.html#SECTION00011000000000000000

 

[selfAdjoint]

Hey, it does too have to do with geometry. It's hyperbolic as in "hyperboloid of one sheet, non-euclidean geometry on", the 2_D hyperboloc space.

 

[Careful]

It is a spacetime in which
(a) no signals can come back arbitrarily close to themselves
(b) in which for any two events a,b where b is in the future of a one has a compact set of events c to the future of a and in the past of b
For more info, see Hawking and Ellis.