Causal Horizons, Trapped Sets, and Achronal Boundaries

Horizons arise from several logically different constructions. Some are boundaries of causal access to an observer or to infinity; others are compact surfaces whose future null normals locally converge; still others bound domains of prediction. This chapter develops the common achronal-boundary geometry while keeping those definitions separate. Its central lesson is diagnostic: before assigning the word “horizon,” identify the set being bounded, the data needed to locate it, and the hypotheses connecting it to black-hole physics.

Required background. We use timelike congruences and spatial projection to describe observer families and null expansions, and hypersurface normals and tangential projectors to handle null normals and codimension-two surfaces.

Helpful background. Pseudo-Riemannian metrics and signature fix the cone convention, affine and variational geodesics supply generators and focal points, Causality and Global Structure gives the volume-wide route, and Exact Solutions of Einstein’s Equations provides the Minkowski, Schwarzschild, and cosmological checks.

Enter this chapter

The most reusable construction begins with a past set $P$, satisfying $I^-(P)=P$. Its nonempty boundary $\partial P$ is achronal, has empty edge, and is locally a Lipschitz topological hypersurface. Those properties alone do not make it null: in Minkowski spacetime $P=\{t\lt0\}$ has the spacelike boundary $t=0$. When $P=I^-(W_+)$ for a suitable future-endless observer set or ideal endpoint set, the additional endpoint conditions give a future causal horizon: its null generators continue toward the future on the horizon but may enter at past endpoints. The time reverse begins with a future set. The precise future-set and achronal-boundary results appear in Minguzzi 2019, §2.14, pp. 59–64.

That common geometry does not erase the choice of $P$. For an eternally accelerated observer in Minkowski spacetime, $P$ is the observer’s chronological past and its boundary is a Rindler horizon. For a selected asymptotically flat end, the relevant set is the causal past of future null infinity. For a compact spacelike two-surface $S$, by contrast, no causal past is required: one computes the two future null expansions $\theta_{(\ell)}$ and $\theta_{(n)}$. The first two constructions are global relative to a target; the last is local to the embedding of $S$, although finding a preferred marginal surface on an entire slice is a quasi-local problem.

This distinction prevents several false inferences. A null hypersurface need not be a causal horizon. An observer horizon need not enclose a black hole. A marginal surface need not lie on an event horizon. A coordinate singularity supplies none of these structures until an invariant causal or expansion calculation identifies one.

Six questions and their answers

1. Achronal Boundaries and Null Generators asks when the frontier of a chronological future or past is null ruled without first being assumed smooth. It states the achronal-boundary theorem with its hypotheses and separates null segments on the boundary from complete null geodesics in spacetime.
2. Future and Past Causal Horizons fixes the orientation convention: $\partial I^-(W_+)$ is a future horizon, while $\partial I^+(W_-)$ is a past horizon. The Rindler calculation shows that the same flat spacetime can give different answers for different observer histories.
3. Event Horizons as Global Boundaries defines a black-hole region only after choosing an asymptotic end and its future null infinity. A collapsing null shell then demonstrates teleology quantitatively: part of the selected event horizon lies in a locally flat region before the shell arrives.
4. Observer, Acceleration, and Cosmological Horizons compares Rindler, particle, and cosmological event horizons. The relevant integrals involve the entire past or future conformal-time interval, not merely the instantaneous Hubble radius.
5. Trapped and Marginal Surfaces defines trapped, anti-trapped, marginal, and marginally outer trapped surfaces using a stated normalization of the two null normals. It also explains which signs survive a positive boost rescaling of those normals.
6. Horizon Regularity, Caustics, and Crease Sets studies where generators enter and smoothness fails. Endpoints, multiple-generator creases, and focal caustics are related but not interchangeable, and the topology of a horizon cut can depend on the time slicing.

The order is deliberate. First establish what causal boundaries guarantee, then specify whose causal access is bounded, then compare that global information with null-expansion data. Smooth horizon geometry is used only after the nonsmooth possibilities have been identified.

Global, observer-relative, and quasi-local tests

For an asymptotically flat physical spacetime $\widetilde M$ with a chosen regular component $\mathscr I^+$ of future null infinity, the exact event-horizon construction is

$$\begin{aligned} \mathcal B(\mathscr I^+) &=\widetilde M\setminus\bigl(J^-(\mathscr I^+)\cap\widetilde M\bigr),\\ \mathcal H^+(\mathscr I^+) &=\partial_{\widetilde M}\mathcal B(\mathscr I^+). \end{aligned}$$

The completion, the component of $\mathscr I^+$, and the physical-spacetime boundary are part of the definition. If there are several ends, the answer may depend on which one is selected. If future evolution has not been specified, membership in $J^-(\mathscr I^+)$ may be undecidable from the available finite region. These qualifications are integral to the asymptotic definition reviewed in Chruściel 2002, §1.4, pp. 11–17.

An observer horizon uses the same causal logic with a different target. For a future-complete timelike history $\gamma$, $\partial I^-(\gamma)$ bounds the events that can ever signal that observer. Altering the future of $\gamma$ can remove the horizon while leaving the metric unchanged. In FLRW cosmology, a particle horizon depends on the past integral $\int_{t_i}^{t}dt'/a(t')$, whereas a cosmological event horizon depends on the future integral $\int_t^{t_f}dt'/a(t')$. Neither is identified merely by $H^{-1}$; the distinctions and convergence tests are set out in Ellis and Uzan 2015, §§4.1–4.2, pp. 8–10.

A trapped-surface test instead begins at a compact spacelike codimension-two surface. With future null normals normalized by $\ell\mathbin{\cdot}n=-2$, a future-trapped surface has $\theta_{(\ell)}\lt0$ and $\theta_{(n)}\lt0$ everywhere. A marginally outer trapped surface has $\theta_{(\ell)}=0$ after an outward direction is chosen; the additional condition $\theta_{(n)}\lt0$ selects the usual future black-hole case. This is geometric information available on $S$; it is not the definition of an event horizon. Relations to singular behavior or to an asymptotic black-hole region require the particular theorem’s causal, energy, topological, and asymptotic hypotheses, as emphasized in Senovilla 2012, §§2.1–2.2, pp. 3–6.

A reliable classification procedure

When a proposed “horizon” appears in a metric, diagram, or numerical slice, use four checks.

First, name the target or surface. Is the construction relative to $\mathscr I^+$, an observer history, a domain of dependence, or a compact two-surface? Without this datum, the term is incomplete.

Second, state the operation. A causal horizon comes from a boundary such as $\partial I^-(W_+)$. A trapped surface comes from two expansion scalars. A vanishing coordinate coefficient, a null coordinate level set, or a large redshift is evidence to investigate, not a definition.

Third, test the global dependence. Extend the chart before accepting a coordinate boundary. Change the observer’s late-time motion before calling a Rindler boundary a spacetime property. Compare two future developments that agree up to a finite time before claiming an event horizon has been locally detected.

Fourth, list the hypotheses of any implication. The causal definition alone uses neither Einstein’s equation nor an energy condition. Area monotonicity, singularity theorems, topology restrictions, and theorems placing trapped surfaces inside black-hole regions add assumptions that must remain visible. Smooth optical equations also apply directly only on regular portions; general horizons can be merely Lipschitz and differentiable almost everywhere Chruściel 2002, §1.5, pp. 17–19 and Appendix A, pp. 32–35.

Boundaries of the discussion

This chapter gives the exact event-horizon definition, the causal-boundary structure supporting it, the observer and cosmological comparisons, and the elementary trapped-surface classification. The broader taxonomy of Killing, isolated, dynamical, trapping, and event horizons—and the zeroth, first, and second laws—belongs to Black-Hole Geometry and Mechanics. Flux balances and numerical horizon tracking belong to Black-Hole Dynamics and Computational Relativity. Cauchy horizons answer a different question and are treated in Cauchy Horizons and Their Generators.

The separation is scientific rather than merely terminological. Global causal boundaries encode what can communicate with a specified future. Quasi-local expansions encode how light fronts initially change at a chosen compact surface. Stationary mechanics encodes symmetry, charges, and flux laws. In important black-hole solutions these structures can coincide, but the coincidence is a theorem with hypotheses, not a reason to merge the definitions.

Chapter diagnostic

Classify each observation as an event horizon, an observer-dependent causal horizon, a trapped or marginal surface, a crease-set feature, or insufficient information. State the decisive test.

1. In Minkowski spacetime, the null plane $T=X$ bounds the causal past of an eternally uniformly accelerated worldline.
2. A compact spacelike two-surface satisfies $\theta_{(\ell)}\lt0$ and $\theta_{(n)}\lt0$ everywhere.
3. In a fixed asymptotically flat completion, a hypersurface is $\partial_{\widetilde M}\bigl(\widetilde M\setminus J^-(\mathscr I^+)\bigr)$.
4. Two distinct future null generators enter a future horizon at the same past endpoint.
5. A metric component $g_{tt}$ vanishes on $r=r_0$ in one coordinate chart.

Solution

1. This is the future Rindler horizon of the specified accelerated observer. It is an observer-dependent causal horizon, not a black-hole event horizon: inertial observers in the same Minkowski spacetime have no such inaccessible black-hole region.
2. With the stated future-null orientation, the surface is future trapped. The expansion signs alone neither locate nor prove the existence of an event horizon; a conclusion about infinity needs additional global hypotheses.
3. This is precisely the future event horizon relative to the chosen $\mathscr I^+$, provided the displayed set and boundary are interpreted in the completion and physical spacetime as stated. It is a global construction.
4. The shared past endpoint lies in the crease set under the multiplicity convention used in this chapter. The horizon is nondifferentiable there, but the point need not be a curvature singularity or a caustic.
5. There is insufficient information. One must extend or compare coordinates and then compute a causal boundary, a null-expansion condition, or another invariant structure. A zero of one component can be purely coordinate dependent.

References

• Chruściel, Piotr T. “Black Holes.” Lecture Notes in Physics 604, 61–102 (2002). arXiv.
• Ellis, George F. R., and Jean-Philippe Uzan. “Causal Structures in Cosmology.” Comptes Rendus Physique 16, 928–947 (2015; arXiv version 2016). arXiv.
• Minguzzi, Ettore. “Lorentzian Causality Theory.” Living Reviews in Relativity 22, article 3 (2019). DOI.
• Senovilla, José M. M. “Trapped Surfaces.” International Journal of Modern Physics: Conference Series 7, 37–58 (2012; arXiv version 2011). arXiv.