Equivalent Criteria for Global Hyperbolicity

Global hyperbolicity is the condition under which causal curves cannot loop locally or disappear globally between fixed endpoints. Its most useful formulations look very different: one controls compact causal diamonds, another demands a universal Cauchy surface, and another supplies a global time. Their equivalence explains why a topological compactness statement is exactly what turns the local light-cone field into a usable framework for prediction.

Required background. Geodesics and Jacobi Fields supplies limit geodesics, inextendibility, and conjugate-point language. Helpful background. Lorentzian Metrics and Causal Cones fixes causal character; Causality and Global Structure supplies the hierarchy; Partial Differential Equations and Green Functions motivates the resulting Cauchy problem.

The global-hyperbolicity theorem

Assume that $(M,g)$ is a connected, smooth, time-oriented spacetime without boundary. The following conditions are equivalent:

1. $M$ is strongly causal, and every causal diamond $$J^+(p)\cap J^-(q)$$ is compact for all $p,q\in M$.
2. $M$ is causal, and every causal diamond is compact.
3. $M$ possesses a Cauchy hypersurface: every inextendible timelike curve meets it exactly once.
4. $M$ possesses a continuous Cauchy time function, strictly increasing along future causal curves, whose level sets are Cauchy hypersurfaces.
5. $M$ possesses a smooth Cauchy temporal function $\tau$ with past-directed timelike gradient, and hence smooth spacelike Cauchy level sets.

Conditions 1 and 2 show that strong causality need not be added separately once causality and compact diamonds are known: the latter pair already implies it in the smooth theory. Conditions 3 and 4 are the classical Geroch equivalence, stated in Minguzzi and Sánchez 2008, §§3.11.1–3.11.2, pp. 42–46. Condition 5 is the smoothing and splitting result of Bernal and Sánchez 2005, Theorem 1.1, pp. 44–49.

Some authors take condition 1 as the definition, others condition 2, and still others the existence of a Cauchy surface. The theorem makes those choices equivalent under the stated hypotheses. It does not license switching formulations without checking metric regularity, a manifold boundary, or nonstandard causality definitions.

Why compact diamonds matter

The diamond $J^+(p)\cap J^-(q)$ contains every event that can occur after $p$ and before $q$. Compactness has two components: the diamond is closed in $M$, and no sequence in it can escape every compact subset. Both are global restrictions. Coordinate boundedness is not enough, and a set with a missing limit point is not compact even when it looks bounded in a chart.

Compactness powers the limit-curve argument. Given a sequence of causal curves from $p$ to $q$, select points and parametrizations so that a subsequence converges on compact portions. Because the entire sequence remains in a compact diamond, the limit cannot vanish through a hole or run off to infinity between its fixed endpoints. One obtains a causal limit curve. This makes $J^+(p)$ closed, supplies maximizing causal curves when the Lorentzian distance is finite, and prevents the loss of causal information illustrated by punctured spacetimes.

Causality is still indispensable. Compactness alone does not prevent a curve from returning to its starting point. Conversely, strong causality alone is local: it prevents almost-closed behavior near each event but does not prevent causal curves between $p$ and $q$ from escaping through a missing boundary. Global hyperbolicity combines the two logically independent controls.

From diamonds to a Cauchy surface

The proof architecture begins with causal compactness. Causality plus compact diamonds implies closedness of $J^\pm(p)$ and strong causality. Choose an admissible finite measure $\mu$ that is positive on nonempty open sets and gives zero measure to every $\partial I^\pm(p)$; a smooth positive density with finite integral suffices. The last requirement excludes atoms that could produce jumps as a light cone crosses them Minguzzi and Sánchez 2008, §3.7.1, pp. 30–31. Define volume functions

$$t^-(p)=\mu(I^-(p)), \qquad t^+(p)=\mu(I^+(p)).$$

In a globally hyperbolic spacetime these functions have the continuity and strict monotonicity needed to form, for example,

$$t(p)=\log\!\left(\frac{t^-(p)}{t^+(p)}\right).$$

Along every inextendible future-directed causal curve, $t$ tends to $-\infty$ toward the past and $+\infty$ toward the future. Each level $t=c$ is therefore crossed, and strict increase prevents repeated timelike crossing. The levels are Cauchy hypersurfaces. The key contradiction is that a curve on which one of the volume limits failed would be forced into a compact diamond; strong causality forbids an inextendible causal curve from being totally imprisoned there.

Geroch’s construction first gives a continuous time function and topological splitting. Smoothing is a separate theorem, not an automatic differentiation of $t$. Bernal and Sánchez construct a smooth temporal function whose gradient remains timelike and whose levels remain Cauchy. The resulting metric splitting is developed carefully on Cauchy Time Functions and Metric Splitting.

From a Cauchy surface to compact diamonds

Suppose instead that $S$ is a Cauchy hypersurface. Its universal timelike-curve intersection property implies causality and then strong causality. If $S$ has null portions, this step needs the Cauchy-surface theorem; one cannot simply assume unique intersection with every causal curve, which was not the definition. To prove diamond compactness, assume a sequence in $J^+(p)\cap J^-(q)$ has no convergent subsequence. Join its points to $p$ and $q$ by causal curves and apply the limit-curve theorem. The escaping sequence would generate an inextendible causal curve incompatible with interception by $S$. The full proof controls the possible splitting into past- and future-inextendible limits and the null portions of $S$ Minguzzi and Sánchez 2008, Theorem 3.75, pp. 44–46. This is the compactness argument’s structure, rather than a replacement for those steps.

This direction shows why a Cauchy surface is more than an attractive foliation slice. Its universal intersection property supplies exactly the compactness needed to retain limiting causal histories. It follows that the definitions on Domains of Dependence and Cauchy Surfaces are not merely analogous descriptions of predictability; under the theorem hypotheses, they characterize the same class of spacetimes.

Counterexamples that separate the hypotheses

Three examples diagnose the theorem’s limits.

A causal hole. Remove the origin from Minkowski spacetime and choose $p$ to its past and $q$ to its future. The spacetime remains strongly causal, but the diamond contains a sequence approaching the missing origin with no limit in $M$. It is not compact, and a would-be time slice is not Cauchy.

A timelike boundary. The open Minkowski half-plane $x>0$ is causally simple, yet a diamond large enough to reach the excluded line has sequences with $x\to0$ and no limit in the spacetime. Strong causal behavior does not replace boundary data.

A causal loop. A flat Lorentzian two-torus with a periodic timelike coordinate is compact, and in the totally vicious case its diamonds can equal all of $M$, hence are compact. But it contains closed timelike curves and has no Cauchy surface. The compact-diamond condition without causality is not sufficient.

At lower metric regularity, familiar push-up and limit-curve statements can change form, so the smooth equivalence must not be quoted as a regularity-free result. With timelike boundaries, one needs boundary-adapted definitions and initial–boundary-value theorems. Neither modification is included in the theorem above.

Exercise

For each of the following, identify the first global-hyperbolicity criterion that fails: ordinary Minkowski spacetime, Minkowski spacetime with one point removed, and the flat Lorentzian two-torus with periodic timelike direction.

Solution

Ordinary Minkowski spacetime is causal and every diamond is a closed, bounded Euclidean subset, hence compact; its constant-time planes are Cauchy surfaces.

Punctured Minkowski spacetime is still strongly causal, but a diamond straddling the deleted point is not compact in the punctured topology. A sequence approaching the missing point has no convergent subsequence in $M$.

On the Lorentzian torus, compactness of the manifold makes closed diamonds compact, but periodic timelike curves violate causality. Thus the causal condition fails even though compactness by itself is harmless. The three cases show that global hyperbolicity requires both causal exclusion and global compactness.

References

• Bernal, Antonio N., and Miguel Sánchez, “Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes”, Communications in Mathematical Physics 257, Theorem 1.1, pp. 44–49, 2005.
• Minguzzi, Ettore, “Lorentzian Causality Theory”, Living Reviews in Relativity 22, article 3, §§3–4, 2019.
• Minguzzi, Ettore, and Miguel Sánchez, “The Causal Hierarchy of Spacetimes”, in Recent Developments in Pseudo-Riemannian Geometry, §§3.11.1–3.11.2, pp. 42–46 in the arXiv version, European Mathematical Society, 2008.