Chronology, causality, strong causality, global hyperbolicity and time functions

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with the causal curves and the relations $\le$, $\ll$, $I^{\pm }$, $J^{\pm }$ of Causal curves, chronological and causal reachability, and inextendibility. All curves below are curves in that sense.

(a) Chronology and causality. $(M,g,N)$ is chronological when there is no $p\in M$ with $p\ll p$, equivalently when there is no nonconstant future timelike curve $\gamma :[a,b]\to M$ with $\gamma (a)=\gamma (b)$. It is causal when there is no nonconstant future causal curve $\gamma :[a,b]\to M$ with $\gamma (a)=\gamma (b)$; a nonconstant future causal curve with equal endpoints is called a closed causal curve, and a closed future timelike curve is called a closed timelike curve.

(b) Strong causality. $(M,g,N)$ is strongly causal when for every $p\in M$ and every open set $U\subseteq M$ with $p\in U$ there is an open set $V$ with $p\in V\subseteq U$ such that for every future causal curve $\gamma :[a,b]\to M$ and all $a\le s<t\le b$ with $\gamma (s)\in V$ and $\gamma (t)\in V$ one has $\gamma ([s,t])\subseteq V$. Thus no causal curve can leave $V$ and later re-enter it; a set $V$ with this property for all causal curves is called causally convex for causal curves that meet it twice.

(c) Global hyperbolicity. $(M,g,N)$ is globally hyperbolic when it is strongly causal and for all $p,q\in M$ the causal diamond $J^{+}(p)\cap J^{-}(q)$ is compact in the subspace topology (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

(d) Time functions. A time function on $(M,g,N)$ is a function $u:M\to \mathbb{R}$ which is continuous from the manifold topology to the usual topology of $\mathbb{R}$: for every $p\in M$ and every open neighborhood $W$ of $u(p)$ there is an open neighborhood $V$ of $p$ with $u(V)\subseteq W$ (Continuity of a map of topological spaces at a point and globally). It also satisfies $u(\gamma (a))<u(\gamma (b))$ for every future causal curve $\gamma :[a,b]\to M$ with $a<b$ that is not constant on $[a,b]$.

(e) Temporal functions. A temporal function on $(M,g,N)$ is a smooth function $u:M\to \mathbb{R}$ ($C^r$ and smooth maps between smooth manifolds) whose gradient is past-directed timelike: writing $du$ for its differential (The differential of a smooth map) and $\mathrm{\nabla }u$ for the vector field with $g(\mathrm{\nabla }u,X)=du(X)$ for every smooth vector field $X$, constructed from the inverse metric of The inverse metric, musical isomorphisms and index raising, one has $g(\mathrm{\nabla }u,\mathrm{\nabla }u)<0\text{and}du(N)=g(\mathrm{\nabla }u,N)>0.$ The first inequality says that the gradient is nowhere zero and timelike; the second fixes it as past-directed for the chosen time orientation. No level set of a temporal function is asserted here to be a hypersurface or to be spacelike, and no relation between temporal functions and compactness is asserted; those claims are proved separately where they are used.

(f) Scope and conventions. These are mathematical conditions on the Lorentzian model. Different sources define global hyperbolicity by equivalent but differently phrased conditions; this page fixes the formulation of (c) and proves the implications it uses. In particular no equivalence between global hyperbolicity and the existence of a Cauchy hypersurface is asserted by this definition, and the physical reading of strong causality or global hyperbolicity as predictability of a classical model is adopted only in the framework postulates.