Causality and Global Structure

Global relativity asks questions that no local curvature calculation can settle. Can one event influence another? Does data on one hypersurface determine an entire spacetime? What exactly is an event horizon when its location depends on the future? What does a singularity theorem prove when a singular point is not part of the manifold? Which asymptotic conditions make energy, radiation, topology, and censorship statements meaningful? This volume builds the definitions and proof structures needed to answer those questions without confusing a diagram, coordinate chart, or model result with a global theorem.

Helpful background. Lorentzian Metrics and Causal Cones supplies the pointwise cone field; Geodesics and Jacobi Fields supplies freely falling curves, affine parameters, and focusing. The site uses signature $(-,+,+,+)$, keeps $G$ explicit, and follows the global curvature and orientation conventions throughout.

The global questions

Start with one familiar experiment: emit a flash at an event $p$ in Minkowski spacetime. Events strictly inside its future light cone can receive a slower-than-light signal from $p$; events on the cone can receive the flash; events outside cannot receive either. Curvature changes the cone field, but global relativity also asks what happens when rays travel far, meet an obstruction, or approach an endpoint missing from the spacetime. Local cones alone do not decide those questions.

The notation encountered throughout the volume records different tests:

Object	Question it answers
$I^+(p)$	Can a future-directed timelike curve from $p$ reach this event?
$J^+(p)$	Can a future-directed causal curve from $p$ reach it, allowing a light signal and including $p$ itself?
$D^+(S)$	Does every past-inextendible causal curve through this event meet the initial set $S$?
$\mathscr I^+$	Where do escaping future-directed null rays end in a specified conformal completion?
$\mathcal H^+$	Where is the boundary of the region that cannot signal the chosen future infinity?

The change from some curve in $J^+$ to every curve in $D^+$ is central. A signal may reach an event from $S$ even though other influences enter from outside $S$. Causal accessibility and prediction from initial data therefore require different definitions.

A Lorentzian metric classifies each tangent vector locally. The first global step integrates those cones into chronological and causal relations $I^\pm$ and $J^\pm$. Their topology, closure, and continuity diagnose progressively stronger levels of causal regularity. Global hyperbolicity adds compact causal diamonds or, equivalently under the standard smooth boundaryless hypotheses, a Cauchy surface. The causal hierarchy and equivalence results are developed in Minguzzi 2019, §§2–4.5. This condition connects causal order to deterministic propagation.

Infinity must then be defined rather than pictured. A conformal completion preserves light cones while moving ideal endpoints to a finite boundary. The boundary can be null, spacelike, or timelike; the distinction separates asymptotically flat, de Sitter, and anti-de Sitter evolution. A causal completion can attach endpoints using chronological sets when smooth conformal geometry fails. Penrose diagrams summarize selected consequences but suppress affine length, curvature, angular topology, differentiability, and dynamics.

Horizons and singularities make the global dependence physical. An event horizon is defined relative to a specified future null infinity,

$$\mathcal B=M\setminus J^-(\mathscr I^+), \qquad \mathcal H^+=\partial\mathcal B,$$

so its location can depend on the entire future. A trapped surface is instead diagnosed quasi-locally by null expansions. A singularity theorem concludes incomplete causal geodesics under focusing, causal, and initial-geometry hypotheses; it does not automatically prove curvature blowup, an event horizon, or inextendibility in any regularity class.

Causality, predictability, and boundaries

The first route through the volume is cumulative. Causal Relations and the Causal Ladder defines admissible curves, chronological sets, achronal sets, and the hierarchy from chronology through causal simplicity. It uses punctures, identifications, and restricted domains to show why each implication is one way unless a converse is separately proved.

Domains of Dependence and Global Hyperbolicity turns causal order into predictability. Domains of dependence and Cauchy horizons are defined from inextendible causal curves; compact diamonds, Cauchy surfaces, temporal functions, and metric splitting are then connected by precise equivalence theorems. The chapter closes by explaining why normally hyperbolic fields possess unique causal Green operators in globally hyperbolic spacetimes and why timelike boundaries require extra data.

Conformal Completion, Causal Boundaries, and Penrose Diagrams constructs ideal boundaries and tests their normals. Minkowski compactification distinguishes $\mathscr I^\pm$, $i^\pm$, and $i^0$; de Sitter and AdS examples show how changing $\Lambda$ changes boundary causal character. A reproducible diagram workflow makes coordinate extension and domain inequalities prior to boundary labels.

Causal Horizons, Trapped Sets, and Achronal Boundaries develops the null-generator structure common to many boundaries while keeping their definitions distinct. Event, observer, acceleration, cosmological, and Cauchy horizons are classified by the causal set whose boundary is taken. Trapped and marginal surfaces use null expansions and need not coincide with a global horizon. Stationary mechanics and detailed horizon taxonomy continue in Black-Hole Geometry and Mechanics.

Incompleteness and singularity theorems

Completeness, Extendibility, and Singularities separates four questions: finite affine range, extension of the metric, curvature behavior, and representation of an absent endpoint. Flat punctures show incomplete curves without curvature; weak null singularities show why $C^0$ and $C^2$ extension questions differ; abstract-boundary classes show why an ideal endpoint is not a physical event.

Classical Singularity Theorems builds the focusing-to-incompleteness argument. Raychaudhuri inequalities produce conjugate or focal points. Global causal and topological arguments then turn the loss of maximizing or boundary-generating behavior into contradiction under an assumed completeness condition. The Penrose, Hawking, and Hawking–Penrose theorems differ in causal direction, initial geometry, convergence, genericity, and causality assumptions; each conclusion is stated no more strongly than its proof.

The recurrent diagnostic is to remove one hypothesis and identify which step fails. Positive cosmological constant can defeat timelike convergence; compact Cauchy topology removes the topological contradiction in one Penrose proof; lack of a generic condition permits complete special solutions. Low-regularity extensions require replacement tools and are not inferred from the smooth theorem.

Infinity, symmetry, and charges

Asymptotically Flat Spacetimes distinguishes null and spatial definitions, asymptotic simplicity and black-hole-compatible variants, smooth peeling and polyhomogeneous expansions, and the assumptions required to match through $i^0$. “Flat at large radius” is never enough: topology, derivative falloffs, parity, conformal regularity, generator completeness, and matter decay decide which theorem or charge is available.

Asymptotic Symmetries and Gravitational Charges begins from a phase space and takes allowed transformations modulo proper gauge. ADM energy–momentum, angular momentum, and center of mass require spatial falloffs and parity. At null infinity the BMS group enlarges translations to smooth functions on the sphere. The Bondi constraint gives mass and supermomentum balance; supertranslations explain vacuum-frame and angular-momentum ambiguities; memory and soft relations are connected only under matched boundary and scattering assumptions. General charge construction continues in Formulations of General Relativity, while fluxes, detector memory, and waveforms continue in Gravitational Radiation and Waveforms.

The cosmological-constant chapters expose a useful contrast. Asymptotically de Sitter Spacetimes has spacelike infinity. Global and observer patches differ; asymptotic data resemble data on a final conformal slice; imposing conformal flatness at infinity can reduce the asymptotic group but removes half the radiative data and zeroes gravitational flux in the restricted framework.

Asymptotically Anti-de Sitter Spacetimes has timelike infinity. Bulk causal diamonds are noncompact and initial data alone do not determine evolution. Boundary conditions select flux, self-adjoint dynamics, symmetries, and charges. Fefferman–Graham expansion separates boundary conformal geometry from state data. Reflection, resonance, and trapping motivate—but do not settle—model-specific stability questions.

Topology and determinism

Spacetime Topology and Topological Censorship explains why global hyperbolicity fixes the diffeomorphism type of Cauchy surfaces and why topology change must sacrifice a standard causal, smoothness, or compactness assumption. Covering-space arguments convert nontrivial causal homotopy into a curve in a suitable cover; focusing under averaged null energy then yields topological censorship. Horizon-section theorems give positive-Yamabe restrictions in higher dimensions, not a universal spherical-topology claim.

Cosmic Censorship and Determinism begins from the unique maximal globally hyperbolic development and asks what lies beyond it. Weak censorship concerns visibility from future null infinity; strong censorship concerns inextendibility and deterministic uniqueness. Neither is one conjecture until matter model, asymptotics, initial-data topology, genericity, symmetry, and extension regularity are fixed.

Censorship evidence is formulation-dependent. Results in spherical Einstein–scalar collapse, charged or rotating interiors, de Sitter backgrounds, linear test fields, and specific function spaces are kept in their stated domains. A proof or counterexample for one formulation is not promoted to a universal verdict. Current assessments are stated on the specialist censorship pages with explicit evidence cutoffs.

A working method

For any global claim, proceed in this order:

1. Specify the spacetime manifold, metric regularity, time orientation, dimension, and boundary status.
2. Name the curve family, causal set, asymptotic end, or phase space that defines the object.
3. State causal, convergence/energy, topology, completeness, falloff, and boundary hypotheses before the conclusion.
4. Identify which part is a definition, which follows from geometry, which uses the field equations, and which remains model-dependent evidence.
5. Check Minkowski space and one discriminating puncture, quotient, horizon, or cosmological example.
6. Identify which converse fails, what regularity is required, and which questions need further geometric or dynamical analysis.

This method is deliberately conservative. Global results are strongest when their scope is explicit: a theorem with exact hypotheses transfers; a memorable slogan does not.

Choose a route

For a first encounter, begin with Chronological and Causal Futures, Domains of Dependence, and Constructing Penrose Diagrams. Work their examples before attempting the singularity proofs. The chapter overviews then explain how each new definition supports the next argument; the following routes select a longer sequence by purpose.

• For a global-relativity core, read chapters 1–6, then topology and censorship.
• For black-hole causal structure, read chapters 1–5, 7–8, 11–12, then continue to Black-Hole Geometry and Mechanics.
• For radiation and asymptotic charges, read chapters 2–3 and 7–10, then continue to Gravitational Radiation and Waveforms.
• For PDE and determinism, read chapters 1–2, 5–6, 10, and 12, then continue to Mathematical Relativity.

References

• Compère, Geoffrey, and Adrien Fiorucci, Advanced Lectures on General Relativity, §§1.2–1.5 and 3.1–3.3, pp. 10–32 and 69–84, Springer, 2019.
• Minguzzi, Ettore, “Lorentzian Causality Theory”, Living Reviews in Relativity 22, §§2–6, article 3, 2019.
• Steinbauer, Roland, “The Singularity Theorems of General Relativity and Their Low Regularity Extensions”, Jahresbericht der Deutschen Mathematiker-Vereinigung 125, §§2–4, pp. 8–40, 2023.