Future and Past Causal Horizons

A causal horizon is not specified by a null surface alone. One must first name the region—or observer history—whose causal access is being bounded. A suitable future-endless observer history gives a boundary of its past with generators that continue toward the future; the time-dual construction gives a past horizon. An arbitrary past set has only an achronal boundary and need not have a null boundary at all. The endpoint assumptions explain the extra null ruling without invoking Einstein’s equation, an energy condition, or stationarity.

Required background. We use timelike congruences and spatial projection to specify observer families, and hypersurface normals and tangential projectors to interpret smooth pieces of a null frontier.

Helpful background. Pseudo-Riemannian metrics and signature fix future and past cones, affine and variational geodesics supply the generator language, Causality and Global Structure orients the larger argument, and Exact Solutions of Einstein’s Equations supplies Rindler and de Sitter test cases.

Horizons relative to causal access

Let $W_+$ be a future-inextendible timelike worldline or a specified family of such worldlines. When ideal endpoints are used instead, $W_+$ must be understood as a subset of a stated completion. Its chronological past

$$P(W_+)=I^-(W_+)$$

is the set of physical events that can send a timelike signal to some member of $W_+$. It is a past set, so its nonempty boundary

$$\mathcal H^+[W_+]=\partial I^-(W_+)$$

is an achronal boundary. We call it a future causal horizon when it is ruled by future-directed null generators with no future endpoints on the boundary; the generators may have past endpoints where they enter. The plus sign records this generator orientation, not affine completeness.

Time reversal begins with a past-inextendible timelike worldline or family $W_-$. Its future set has boundary

$$\mathcal H^-[W_-]=\partial I^+(W_-),$$

which is a past causal horizon when it is ruled by past-directed null generators with no past endpoints on the boundary; future endpoints where generators leave are allowed.

Using $J^\pm$ instead of $I^\pm$ is common when the target is conformal infinity. The boundary construction must then be read in the completed spacetime and accompanied by the causal regularity needed to identify the same frontier. Chronological pasts are open, so $\partial I^\pm$ makes the achronal-boundary structure immediate. The relevant future-set identities and generator alternatives are proved in Minguzzi 2019, §2.14, pp. 59–64.

Generator direction and endpoint behavior

Every nonconstant causal curve segment contained in an achronal boundary is a null geodesic segment. A future horizon strengthens this fact into a ruling by future-inextendible generators. Suppose two distinct generators meet, and one also extends to the past of the meeting point on the horizon. Following that past segment and then the other generator’s future segment makes a corner that can be rounded into a timelike curve between horizon points. Achronality forbids this. Distinct generators can therefore meet only at their common past endpoints, where neither supplies such a past segment. This is why a crease is allowed but interior branching is not; see Minguzzi 2019, §2.14, Proposition 2.81.

Past endpoints are different. Several generators may enter a future horizon at one point, or the number of generators through nearby points may change. The horizon can fail to be differentiable there while remaining a closed Lipschitz topological hypersurface. The time-dual statement holds for future endpoints of past horizons. Chruściel’s mathematical definition and regularity discussion make this orientation explicit in Chruściel 2002, §1.5, pp. 17–19 and Appendix A, pp. 32–35.

The endpoint rule is not a claim of affine completeness in the whole spacetime. A generator can be future-inextendible within the horizon yet reach a spacetime boundary at finite affine parameter. Conversely, a geodesically complete null curve can fail to be a generator after it leaves the achronal boundary. “No future endpoint on $\mathcal H^+$” is a causal statement about membership in the frontier, not a conclusion about curvature or singularity.

Observer horizons in flat and curved spacetimes

In two-dimensional Minkowski spacetime, consider the eternally uniformly accelerated worldline

$$T(\tau)=a^{-1}\sinh(a\tau), \qquad X(\tau)=a^{-1}\cosh(a\tau),$$

with $a>0$ and $-\infty<\tau<\infty$. Direct use of the null coordinates $u=T-X$ and $v=T+X$ gives

$$I^-(\gamma)=\{u<0\}, \qquad \mathcal H^+[\gamma]=\{u=0\}.$$

The future horizon is the null line $T=X$. The time-dual past horizon is $T=-X$. Both depend on maintaining the accelerated history for unlimited proper time. An inertial worldline complete in both time directions has $I^-(\gamma)=I^+(\gamma)=M$, so it has neither observer horizon even though the ambient spacetime is identical. This comparison isolates the dependence on $W_\pm$, not on curvature. The worldline calculation and its global qualification are also presented in Ellis and Uzan 2016, §2.2, pp. 4–5.

In de Sitter spacetime, a complete timelike observer has both future and past cosmological horizons. Other observers have different static patches, related by de Sitter isometries. The horizon is therefore invariant once the worldline is specified, but it is not a single observer-independent subset assigned to the whole spacetime. By contrast, a black-hole event horizon is defined relative to a chosen asymptotic end and its future null infinity, as developed on Event Horizons as Global Boundaries.

What the adjective “causal” does not guarantee

A Cauchy horizon, discussed in Cauchy Horizons and Their Generators, bounds a domain of dependence rather than an observer’s causal past. It fits the broad null-horizon geometry but answers a predictability question. A trapping horizon or a marginally trapped tube is instead defined from null expansions of codimension-two leaves. It can be spacelike, timelike, or null and need not be an achronal boundary. Those local and quasi-local conditions are introduced on Trapped and Marginal Surfaces.

Nor does a coordinate boundary automatically become a causal horizon. Rindler coordinates happen to end on the boundary of causal access for the chosen eternal observers; a poorly chosen chart can end on an entirely artificial surface. One must compute $I^-(W_+)$ or $I^+(W_-)$ in the extended spacetime. For a finite timelike segment from $a$ to $b$, its chronological past is $I^-(b)$; in Minkowski spacetime this boundary is a past light cone whose generators end at $b$. The finite causal diamond is instead $J^+(a)\cap J^-(b)$. Neither construction has the future-endless generator property of a future horizon. The target’s full future history is essential.

Exercises

Let $\gamma$ be the uniformly accelerated worldline above. Show directly in null coordinates that an event $q=(u_q,v_q)$ can chronologically influence some point of $\gamma$ if and only if $u_q<0$. Explain why the answer would change if the acceleration stopped at a finite proper time.

Solution

Along the worldline,

$$u_\gamma(\tau)=-a^{-1}e^{-a\tau}<0, \qquad v_\gamma(\tau)=a^{-1}e^{a\tau}>0.$$

In two-dimensional Minkowski spacetime, $q\ll\gamma(\tau)$ precisely when $u_q<u_\gamma(\tau)$ and $v_q<v_\gamma(\tau)$. If $u_q<0$, choose $\tau$ large enough that $u_\gamma(\tau)>u_q$ and $v_\gamma(\tau)>v_q$; hence $q\in I^-(\gamma)$. If $u_q\geq0$, the first inequality is impossible. Thus $I^-(\gamma)=\{u<0\}$ and its boundary is $u=0$. If acceleration stops at finite $\tau$, the later worldline can cross the null asymptote, allowing signals from part or all of $u\geq0$ to arrive; the horizon depended on the complete future history.

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.