The notion of determinism is tied with the ability to predict i.e. having given sufficient information at some ‘instance’, we can predict what the in-formation will be at a later instance and also retrodict what the inin-formation was at an earlier instance which lead to the information ‘now’. These intuitive ideas from a Newtonian view of the dynamics are formulated in a relativistic view, in terms of domains of dependence of a suitable submanifold. As we will see, the ‘now’ surface (submanifold) can be generalized to an achronal submanifold (which can have light-like portions which are absent in the New-tonian causality).
Let S be a closed, achronal set.
Definition 6.12 (Edge Set) p ∈ S is an edge point of S if for every neigh-borhood up, ∃ q, r ∈ up such that q ∈ I+(p), r ∈ I−(p) and there is a time-like curve λ from r to q which does not intersect S.
The set of all edge points of S is called the edge of S, denoted edge(S).
Note that S is only a set, no smoothness properties are implied. The x-axis of usual Minkowski space-time is clearly closed, achronal and has each of its points as its edge point since we can always skirt around by a time-like curve, hence in this case edge(S) = S. The x-y plane in three-dimensional Minkowski space-time however has no edge points - edge(S) = ∅. In fact we have the theorem,
Theorem 6.10 If a non-empty, closed, achronal set S has no edge points, then S is a three-dimensional, embedded, C0 submanifold of M .
Such an edge-less S is called a slice.
For a generic closed, achronal set S, we define its domains of dependences.
Definition 6.13 (Future/Past Domains of Dependence)
D+(S) := {p ∈ M/ Every, past in-extendible causal curve through p intersects S } ;
D−(S) := {p ∈ M/ Every, future in-extendible causal curve through p intersects S } ;
D±(S) are called future/past domain of dependence of S.
Clearly, S ⊂ D±(S) ⊂ J±(S). However, since S is achronal, D+(S) ∩ I−(S) =
∅ = D−(S) ∩ I+(S). The qualifier every is important since it implies that the future domain of dependence precisely consists of only those events whose causes have been registered on S and have no other causes un-registered on S.
Likewise, the past domain of dependence consists of only those ‘causes’ whose
‘effects’ have to be registered on S. Therefore, D(S) := D+(S) ∪ D−(S), the Domain of Dependence of S, is the set of events at which all physical properties should be completely determined by the properties at events on S.
Evidently for a space-time supporting predictability, we would like existence of an achronal set whose domain of dependence is the full space-time! This leads to the central definition of this section:
Definition 6.14 (Cauchy Surface and Global Hyperbolicity)
If S is an achronal, closed subset of M such that D(S) = M , then S is called a Cauchy Surface while the space-time is said to be Globally Hyperbolic.
It follows immediately that,
Theorem 6.11 A Cauchy surface is edge-less i.e. a slice which is of course an embedded three-dimensional, C0 submanifold.
The proof is simple. If edge(S) 6= ∅, then ∃ p ∈ S such that every neighborhood upcontaining q ∈ I+(p), r ∈ I−(p) and a time-like curve λ connecting the two without intersecting S. Hence q, r do not belong to the domain of dependence of S. But this contradicts global hyperbolicity, hence S must be edge-less.
To consider the converse, let M be a globally hyperbolic space-time so that it can admit a Cauchy surface. Consider a three-dimensional, achronal, closed, edge-less submanifold, S ⊂ M . Under what conditions can such an S be a Cauchy surface?
Theorem 6.12 S is a Cauchy surface iff every, in-extendible null geodesic intersects S and enters I±(S).
Edge-less, achronal, closed submanifolds which are not intersected by all in-extendible null geodesics are called Partial Cauchy Surfaces. The domain of dependence of a partial Cauchy surface, while clearly not all of the space-time, is by itself a globally hyperbolic space-time with the surface being its Cauchy surface.
When the domain of dependence of a closed, achronal set S does not coincide with the space-time, we have the notion of a Cauchy Horizon. To define it, Let us note some properties of the domains of dependences for a generic closed, achronal set S. Neither the D±(S) not their closures D±(S) coincide with the full space-time. Then the following are true:
Theorem 6.13 (Properties of Domains of Dependence)
The Space-Time Arena 107 1. An event p ∈ D+(S) iff every past in-extendible time-like curve from p
intersects S;
2. It follows that,
Int[D+(S)] = I−[D+(S)] ∩ I+(S);
Int[D(S)] = I−[D+(S)] ∩ I+[D−(S)];
3. Define,
H±(S) := D±(S) − I∓[D±(S)] (Future/Past Cauchy Horizons) ,
H(S) := H+(S) ∪ H−(S) (Cauchy Horizon) ;
4. Every p ∈ H+(S) lies on a null geodesic λ contained within H+(S); it is either past in-extendible or has an end point on S;
5. Cauchy horizon is the boundary of the domain of dependence: H(S) = D(S). As a corollary, it follows that for a connected space-time, S is a˙ Cauchy surface iff its Cauchy Horizon is empty, H(S) = ∅;
6. If Σ is a Cauchy surface, every in-extendible, causal curve, λ intersects Σ, I+(Σ) and I−(Σ).
Here are some simple examples. In two-dimensional Minkowski space-time, the boundary of the ‘future’ light cone (i.e. the two 45 degree lines emanating into the future from some point, including that point), is a closed, achronal edge-less submanifold. Its future domain of dependence is the full future light cone; its past domain of dependence is just the vertex of the light cone. This set is not a Cauchy surface for the Minkowski space-time - there are several null geodesics which do not intersect the light cone. Including the past light cone does not help either. The full light cone is also not a Cauchy surface.
Now we note some of the main properties of globally hyperbolic space-times. These properties refer to the absence of causal pathologies as well as implications for the topology of the space-time itself. There are additional implications related to properties of spaces of curves. These are needed in the proofs of singularity theorems and will be discussed there.
Theorem 6.14 (Well Behaviour of Causality) A globally hyperbolic space-time satisfies the chronology condition and is strongly causal. Further-more, it is stably causal.
The first assertion is easy to see. If the chronology condition is violated, there there exist a closed time-like curve. If it intersects the Cauchy surface, it violates the achronality of the Cauchy surface and if it does not intersect, then it violates global hyperbolicity. If strong causality is violated, then there exists an event p such that every up⊃ u0p which is visited more than once at least by one causal curve. The previous logic applies again to this curve. The proof of the second assertion is by construction [17].
Theorem 6.15 (Topological Implications:)
If Σ, Σ0 are two Cauchy surfaces in (M, g), then they are homeomorphic;
A globally hyperbolic space-time admits a global time function such that its level sets are Cauchy surfaces;
Thus M can be foliated by Cauchy surfaces and topologically, M ∼ R × Σ, topology of Σ being arbitrary.
This concludes our discussion of determinism and global hyperbolicity. We will see their role in the singularity theorems as well as in the initial value formulations.