Domains of Dependence and Global Hyperbolicity

Predictability in relativity is not secured by a local light cone alone. One must know that every causal history relevant to an event crosses the proposed data set, that causal curves cannot escape through holes between fixed endpoints, and that the resulting evolution can be organized by global time. This chapter builds that chain from domains of dependence to smooth Cauchy foliations, then states exactly what the chain gives for linear hyperbolic fields—and what changes when a timelike boundary is present.

Enter this chapter

The durable question is: when does a spacelike data set control an entire spacetime rather than only a local neighborhood? The answer has three layers.

At the set-theoretic layer, a domain of dependence contains events whose every past or future causal history meets the data set. At the global-geometric layer, a Cauchy surface meets every inextendible timelike history, and compact causal diamonds make that condition equivalent to global hyperbolicity. At the analytic layer, a smooth Cauchy temporal function produces a foliation on which normally hyperbolic equations have well-posed evolution and causally supported Green operators.

The chapter does not prove the nonlinear Einstein initial-value theorem, solve the gravitational constraints, or develop timelike-boundary evolution. Those require a hyperbolic reduction, gauge and constraint propagation, or a specified boundary operator. Here the aim is to make the causal hypotheses of those analyses precise.

The six-page argument

The pages are arranged so that each removes one ambiguity left by the previous one.

1. Domains of Dependence defines $D^+(S)$ and $D^-(S)$ using every inextendible causal curve. A finite Minkowski data interval shows how its edge cuts out a causal diamond, while a deleted point shows that local flatness does not prevent a global loss of prediction.
2. Cauchy Horizons and Their Generators identifies the boundary $H^\pm(S)$ where interception by $S$ ceases. The horizon is achronal and null-ruled; its generators either begin at the edge of the data set or are inextendible within the spacetime.
3. Cauchy Surfaces upgrades a partial data set to a universal slice. It reconciles the “exactly once for every inextendible timelike curve,” “full domain of dependence,” and “empty Cauchy horizon” definitions, including the nuance that a rough achronal surface can contain a null segment.
4. Equivalent Criteria for Global Hyperbolicity proves that a Cauchy surface is equivalent, in a smooth boundaryless spacetime, to causality plus compact causal diamonds. The compactness condition prevents sequences of causal curves from disappearing through a hole or escaping to infinity between fixed endpoints.
5. Cauchy Time Functions and Metric Splitting strengthens the topological result to a smooth Cauchy temporal function and an orthogonal product metric $g=-\beta d\tau^2+h_\tau$. It also explains why this splitting is neither unique nor a stationarity theorem.
6. Why Global Hyperbolicity Matters for Field Propagation applies the geometry to normally hyperbolic operators. It states the Cauchy and support theorems, fixes the future/past support behind advanced–retarded naming conventions, and separates the boundaryless result from timelike-boundary problems.

Together these pages establish the classical equivalences summarized in Minguzzi and Sánchez 2008, §§3.11.1–3.11.4, pp. 42–50, while keeping the smooth splitting and operator theorem as distinct later inputs.

Objects that should not be conflated

A domain of dependence is relative to an achronal set. It can be a proper region even in flat spacetime. A Cauchy horizon is the boundary of that relative region, not necessarily an event horizon. A Cauchy surface is global: its domain of dependence is all of $M$. Global hyperbolicity is a property of the spacetime, equivalent to possessing such a surface under the chapter’s hypotheses.

A time function is continuous and increases on future causal curves. A temporal function is smooth with timelike gradient. Either can exist on a spacetime with a causal hole. A Cauchy temporal function adds the global requirement that every level be a Cauchy hypersurface. Its existence gives the smooth product and metric splitting of Bernal and Sánchez 2005, Theorem 1.1, pp. 44–49.

Finally, a normally hyperbolic operator is a second-order linear operator whose principal symbol is the Lorentzian metric times the fiber identity. It is an operator class, not a synonym for every relativistic field equation. Gauge systems must first be reduced, and nonlinear Einstein evolution needs arguments beyond this chapter.

Reader routes through causal evolution

Readers should be able to classify causal tangents, work with inextendible curves, and distinguish a curve from a geodesic. If that distinction is uncertain, repair it through Causal Curves and Time Orientation and Affine and Variational Geodesics. Readers mainly interested in propagation may also review PDEs and Green Functions for Gravitation.

For a causal-geometry route, read the first five pages in order. This route leads naturally to conformal boundaries, horizons, singularity theorems, and censorship. For a field-theory route, read domains of dependence, Cauchy surfaces, the equivalence theorem, the smooth splitting, and the propagation bridge; the Cauchy-horizon page can then be used to diagnose what fails in a non-global evolution. For a quick theorem check, begin with the equivalence page and return to the definitions whenever a quantifier or boundary term is unclear.

Recurring tests

Four examples recur because they isolate different logical failures.

Full Minkowski slice. The surface $t=0$ is Cauchy, diamonds are compact, and the standard wave equation has unique causal evolution.

Finite Minkowski interval. The set is achronal but has an edge. Its domain is only a diamond and its null sides form a Cauchy horizon. No curvature or black hole is involved.

Punctured Minkowski spacetime. Strong causality survives, but a diamond straddling the missing point is noncompact in the punctured topology. An inextendible curve can end at the hole, so coordinate time is temporal but not Cauchy.

Minkowski half-space. A timelike boundary admits incoming waves. Initial data alone are not unique; a boundary condition is additional mathematical input. This is the simplest model for why anti-de Sitter dynamics is not covered by the ordinary boundaryless theorem.

These cases form a useful decision procedure: test causal recurrence, then compactness of diamonds, then universal intersection with a proposed slice, and finally whether a boundary or operator-specific condition remains to be supplied.

Review the chapter

Consider the four examples above. For each one, decide whether $t=0$ is a Cauchy surface, whether the spacetime is globally hyperbolic in the boundaryless sense, and whether compactly supported initial data alone determine a normally hyperbolic field everywhere.

Solution

In full Minkowski spacetime, $t=0$ is Cauchy, the spacetime is globally hyperbolic, and the normally hyperbolic Cauchy theorem gives unique global evolution.

For a finite interval on $t=0$, the spacetime is still globally hyperbolic, but the interval is not a Cauchy surface. Its data determine only $D(S)$; outside that diamond, additional data are required. This separates a property of a set from a property of the ambient spacetime.

Punctured Minkowski spacetime is not globally hyperbolic: a diamond around the deletion is noncompact, and no surface crossing the hole can be Cauchy. A normally hyperbolic operator remains locally well behaved, but the standard global uniqueness theorem no longer applies.

For the half-space, $t=0$ is not a Cauchy surface under the ordinary boundaryless definition because causal curves can enter through $x=0$. Initial data alone are insufficient. A well-posed initial–boundary-value problem may be recovered only after an admissible boundary condition is specified.

Where to go next

The next chapter, Conformal Completion, Causal Boundaries, and Penrose Diagrams, studies how ideal endpoints make global causal relations visible without turning a diagram into a proof. Later, Cosmic Censorship and Determinism asks whether maximal globally hyperbolic developments are generically inextendible. The definitions here are the fixed vocabulary for that question: one must know which data surface, development, horizon, boundary, and regularity class are being discussed.

References

• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, Chapter 3, §§3.2–3.5, pp. 73–102, 2007.
• 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, and Miguel Sánchez, “The Causal Hierarchy of Spacetimes”, in Recent Developments in Pseudo-Riemannian Geometry, §3.11, pp. 42–51 in the arXiv version, European Mathematical Society, 2008.