Domain of Dependence and Cauchy Horizons

Determinism in a relativistic theory is a strictly local promise, and the light cones fix its terms: the field at an event can be swayed only by data lying in its causal past. Hand the theory a partial slice of "initial data" on a set $S$ and one urgent question follows — for which later events is the whole story already written? The answer is the domain of dependence $D(S)$ , the region every one of whose past-directed signals is forced to register on $S$ before reaching it. Introduced into hyperbolic PDE theory by Leray and geometrized for relativity by Choquet-Bruhat, Geroch, Penrose, and Hawking, it converts the analyst's "domain of determinacy" into a purely causal, coordinate-free object built from cones alone. The construction demands only that $S$ be achronal — no two of its points are joined by a timelike curve — so that $S$ can play the role of a single instant. An event $p$ lies in $D^{+}(S)$ when every past-inextendible causal curve through $p$ meets $S$ ; if even one such curve slips past $S$ to infinity or to a singularity, an unrecorded influence can reach $p$ and prediction there collapses. The boundary of the predictable region is the future Cauchy horizon $H^{+}(S)$ : an achronal null hypersurface, woven from null-geodesic generators, marking the exact frontier where data on $S$ stop determining the future. Cauchy horizons are not mere bookkeeping. When $H^{+}(S)$ refuses to retreat however you enlarge $S$ — as at the inner horizons of the Reissner-Nordström and Kerr black holes — the spacetime is announcing that determinism is fundamentally bounded there. Beyond such a horizon the maximal analytic extension opens onto timelike singularities, closed timelike curves, or fresh universes whose physics no exterior slice can fix. Whether generic perturbations amputate these horizons into singularities and rescue predictability is the content of strong cosmic censorship, which keeps this circle of ideas at the live edge of mathematical relativity.

Key takeaways

The future domain of dependence $D^{+}(S)$ of an achronal set $S$ is the set of events every past-inextendible causal curve through which meets $S$ ; physics there is fully fixed by data on $S$ .
The full domain is $D(S)=D^{+}(S)\cup D^{-}(S)$ ; if $D(S)$ is the whole spacetime then $S$ is a Cauchy surface and $(M,g)$ is globally hyperbolic.
The Cauchy horizon $H^{+}(S)=\overline{D^{+}(S)}-I^{-}\!\left[D^{+}(S)\right]$ is the achronal null boundary beyond which data on $S$ no longer determine the evolution.
$H^{+}(S)$ is generated by null geodesics that, traced to the past, either run to $\operatorname{edge}(S)$ or stay in the horizon forever; predictability fails because new influences enter from beyond.
The interior Cauchy horizons of charged and rotating black holes — $r_{-}=M-\sqrt{M^{2}-Q^{2}}$ (RN), $r_{-}=M-\sqrt{M^{2}-a^{2}}$ (Kerr) — mark the limit of deterministic evolution and are the focus of strong cosmic censorship.
Domains of dependence underpin the entire well-posed initial-value formulation of general relativity through the Choquet-Bruhat-Geroch maximal Cauchy development.
Key idea: $D^{+}(S)$ is the set of events whose entire past causal history is anchored on $S$ ; its future boundary, the Cauchy horizon $H^{+}(S)$ , is the null surface where data on $S$ cease to determine the future, and globally hyperbolic spacetimes are exactly those admitting an $S$ with $D(S)=M$ and empty Cauchy horizon.

The core insight

$D^{+}(S)$ is the set of events whose entire past causal history is anchored on $S$ ; its future boundary, the Cauchy horizon $H^{+}(S)$ , is the null surface where data on $S$ cease to determine the future. Globally hyperbolic spacetimes are precisely those for which some $S$ has $D(S)=M$ and hence empty Cauchy horizon.

Where this lives today

Domains of dependence are the working grammar of every attempt to evolve Einstein's equations. Numerical relativity — the engine behind the LIGO-Virgo-KAGRA waveform banks — computes only inside $D(\Sigma)$ : constraint-satisfying data $(\Sigma,h_{ij},K_{ij})$ are laid on a slice, and the artificial outer boundary is placed far enough out that the merging binary sits strictly inside the domain of dependence, so spurious boundary reflections cannot reach the answer before the waveform is read off. The frontier open problem is causal too: strong cosmic censorship asks whether the maximal Cauchy development of generic data is inextendible — whether the Cauchy horizons of Kerr and Reissner-Nordström are always destroyed by the blueshift instability (mass inflation) that perturbations trigger at $r_{-}$ . The story turned subtle in the 2010s when Dafermos, Luk, Cardoso, and others showed that near-extremal Reissner-Nordström-de Sitter holes can have Cauchy horizons stable enough to admit weak, finite-energy extensions — a genuine potential violation that keeps the conjecture unsettled. The same machinery polices chronology: any engineered spacetime with closed timelike curves (Gödel, the Tipler cylinder, wormhole time machines) develops a Cauchy horizon — the chronology horizon — on which Hawking's chronology-protection argument predicts the quantum stress-energy diverges, mirroring the mass-inflation catastrophe at a black hole's inner horizon.

The physical picture

Picture $S$ as one tick of a universal clock and ask, event by event, whether the future is already scripted. An event $p$ is determined precisely when you cannot trace any allowed signal — anything travelling at or below the speed of light — backwards from $p$ to infinity or to a singularity without first crossing $S$ . If every such backward path is forced through $S$ , nothing unaccounted-for can arrive, and $p$ belongs to $D^{+}(S)$ . The Cauchy horizon is the "sunset line" of this predictability: just inside it every influence is on the books; just outside there is at least one past-directed causal ray that threads past the rim of $S$ and reaches into the unknown. Because the failure is always carried by a limiting null ray, the horizon is itself a null surface, ruled by light.

Figure

Domain of dependence of a finite disk of radius R = 3 in Minkowski space c = 1: the achronal slice S is the segment t = 0 out to radius 3; D⁺(S) is the solid diamond of events whose past rays all land on S, apex at t = 3, determined out to radius R − t (radius 2 at t = 1); the null faces at radius R − t are the Cauchy horizon H⁺(S) grazing the rim edge(S) at radius 3, and the escaping ray through q slips past the rim so q is undetermined

A useful mental model comes with finite data. A flat-space spatial disk $S$ of radius $R$ determines a solid diamond above it — every event within radius $R-t$ of the centre at time $t$ — and the diamond's slanted null faces are the Cauchy horizon. Enlarge the disk and the horizon tilts outward, ceding more of spacetime to prediction; send $R\to\infty$ and the horizon vanishes, $S$ becomes a full Cauchy hyperplane, and Minkowski space is globally hyperbolic. The truly interesting case is the opposite: a horizon that will not retreat no matter how you enlarge $S$ . That is what happens inside a charged or rotating black hole, where the inner horizon at $r_{-}$ is a Cauchy horizon for any slice of the exterior, and no enlargement can push it away because new causal influences pour in from the timelike singularity and from other universe sheets of the maximal extension.

Figure

Causal diagram of a Reissner-Nordström black hole with Q = 0.8M so r± = M ± √(M² − Q²) gives event horizon r₊ = 1.6M and inner horizon r₋ = 0.4M: a Cauchy surface Σ through regions I and II has domain of dependence D(Σ) covering r greater than r₋ shaded blue, but its future boundary is the Cauchy horizon H⁺(Σ) at r = r₋, beyond which a causal curve from the timelike singularity r = 0 reaches events with no record on Σ

The core equations

The whole subject is fixed by a single membership rule: an event belongs to the future domain of dependence when there is no way for a past-directed causal curve to reach it without being caught by $S$ . Everything else — the horizon, the generator theorem, the equivalence with global hyperbolicity — is a consequence of reading this condition carefully.

D^{+}(S)=\{\,p\in M : \text{every past-inextendible causal curve through }p\text{ meets }S\,\} \qquad (1)

where: $M$ — the spacetime manifold with Lorentzian metric $g_{ab}$ ; $S$ — a closed achronal set, the candidate initial-data surface; $p$ — an event of $M$ ; a past-inextendible causal curve is one that cannot be prolonged any further into the past as a causal curve. Reading: $p$ is determined exactly when data on $S$ intercept every signal that could ever reach it — one escaping curve is enough to disqualify $p$ . Reversing the time direction gives the past domain of dependence, and their union is the total region controlled by $S$ — the maximal Cauchy development of the data.

D^{-}(S)=\{\,p\in M : \text{every future-inextendible causal curve through }p\text{ meets }S\,\},\qquad D(S)=D^{+}(S)\cup D^{-}(S) \qquad (2)

where: $D^{-}(S)$ — the region whose evolution is fixed by retrodicting the data on $S$ ; $D(S)$ — the full domain of dependence. Reading: $S$ predicts its future through $D^{+}$ and reconstructs its past through $D^{-}$ ; together they are everything $S$ can know. The future boundary of the determined region is the Cauchy horizon. It is what remains of the closure of $D^{+}(S)$ once you strip away every point that still has determined events to its own future.

H^{+}(S)=\overline{D^{+}(S)}-I^{-}\!\left[D^{+}(S)\right]=\{\,p\in\overline{D^{+}(S)} : I^{+}(p)\cap D^{+}(S)=\varnothing\,\} \qquad (3)

where: $\overline{D^{+}(S)}$ — the topological closure of $D^{+}(S)$ ; $I^{-}[A]$ — the chronological past of the set $A$ ; $I^{+}(p)$ — the chronological future of the event $p$ . Reading: a horizon point is a genuine future edge — nothing determined lies chronologically above it — and symmetrically $H^{-}(S)$ uses $I^{+}$ and $D^{-}(S)$ , with $H(S)=H^{+}(S)\cup H^{-}(S)$ . Once a Cauchy horizon exists it is never a smooth spacelike wall: it is achronal and null, ruled by null geodesic segments. This is the single most useful structural fact about horizons of prediction.

H^{+}(S)\neq\varnothing\ \Rightarrow\ H^{+}(S)\ \text{is achronal, and is generated by null geodesics with no future endpoint on it} \qquad (4)

where: a null geodesic generator is a null geodesic segment lying entirely within $H^{+}(S)$ ; traced to the past each generator either meets $\operatorname{edge}(S)$ or stays in $H^{+}(S)$ without end. Reading: the boundary of predictability is literally woven from light rays, which is why it inherits the null, achronal character of an event horizon while meaning something quite different. When the domain of dependence swallows the entire manifold, $S$ has become a Cauchy surface and the horizon disappears — and this is equivalent to global hyperbolicity, the exact condition for a well-posed deterministic evolution.

\Sigma\ \text{a Cauchy surface} \iff D(\Sigma)=M \iff (M,g)\ \text{globally hyperbolic} \qquad (5)

where: $\Sigma$ — a smooth spacelike, or at least achronal and edgeless, hypersurface; $(M,g)$ — the spacetime. Reading: the three statements are one; a spacetime predicts everything from a single slice precisely when that slice has empty Cauchy horizon. The deepest physical Cauchy horizons are the inner horizons of the charged and rotating black holes, located as the smaller root of the metric function. They coincide with no event horizon and cannot be pushed away.

r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}\quad(\text{Reissner-Nordström}),\qquad r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}\quad(\text{Kerr},\ a=J/M) \qquad (6)

where: $M$ — the black hole mass in geometrized units $G=c=1$ ; $Q$ — electric charge; $a=J/M$ — spin per unit mass; $r_{+}$ — the outer event horizon; $r_{-}$ — the inner horizon, which is the Cauchy horizon for a slice of the exterior. Reading: $r_{-}$ is a real horizon only when the discriminant is positive ( $Q<M$ or $a<M$ ); at extremality the roots merge and beyond it the singularity is naked.

Derivation AWhy a Cauchy horizon is a null surface ruled by geodesics

The goal is to prove the structural backbone of the whole subject: that whenever $H^{+}(S)$ is non-empty it is an achronal, imbedded null hypersurface threaded by null geodesic generators which, followed to the past, either terminate on $\operatorname{edge}(S)$ or continue in the horizon forever. This Penrose-Hawking result gives Cauchy horizons the same local geometry as event horizons while endowing them with a different global meaning, and it is what lets one study a horizon by shooting light rays along it. We argue in three moves — achronality, the null-hypersurface property, and the generator structure — each resting only on the definitions above.

Step 1. Suppose two points $p,q\in H^{+}(S)$ were timelike related, say $p\ll q$ . Then $q\in I^{+}(p)$ , but the definition (3) requires $I^{+}(p)\cap D^{+}(S)=\varnothing$ for a horizon point, and since $q\in\overline{D^{+}(S)}$ its own chronological neighbourhood meets $D^{+}(S)$ — a contradiction. Hence no two horizon points are timelike related: $H^{+}(S)$ is achronal. Why: achronality is exactly the statement that the horizon never "doubles back" inside anyone's cone, which is what forces its tangent planes to be null rather than timelike.

Step 2. Achronality alone buys an imbedded topological ( $C^{0}$ ) hypersurface that is nowhere timelike; it does not buy nullness, since the slice $t=0$ in Minkowski is an achronal boundary and is spacelike throughout. Nullness is specific to a Cauchy horizon: no point of $H^{+}(S)$ lies in $D^{+}(S)$ while every neighbourhood of one meets it, and the Cauchy-horizon generator theorem then puts a null geodesic of the horizon through each point, so the tangent plane touches the light cone rather than cutting inside it. So $H^{+}(S)$ is a null hypersurface. Why: a spacelike tangent plane would let the determined region reach past $p$ and a timelike one would violate achronality — only the null case survives.

Step 3. Through each $p\in H^{+}(S)$ runs a null geodesic lying entirely in $H^{+}(S)$ which, traced to the past, either reaches $\operatorname{edge}(S)$ — the rim where $S$ stops being locally spacelike — or remains in $H^{+}(S)$ with no past endpoint. The restriction is on the past alone: to the future a generator may leave the horizon at a crossover point, as every generator of the disk horizon in the next derivation does at its apex. These are the generators. Why: if a generator kinked, one could cut the corner with a timelike curve and violate achronality, so the ruling curves are unbroken null geodesics, and their only allowed past fate is the edge of the data or an eternal life on the horizon.

Go deeper · Derivation B · 2 minThe domain of dependence of a finite disk in Minkowski space

Derivation BThe domain of dependence of a finite disk in Minkowski space

The goal is to construct $D^{+}(S)$ and its Cauchy horizon explicitly for the cleanest possible example — a flat spatial disk — so that the abstract membership rule (1) becomes a solid diamond one can draw and measure, and so that the generator theorem (4) is visible as a set of light rays ending on a rim. Work in $\mathbb{R}^{1,3}$ with $c=1$ and line element $\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2}$ , and take $S=\{t=0,\ r\le R\}$ , a spacelike hence achronal disk of coordinate radius $R$ . We locate the worst-case escaping ray, impose that it still meets $S$ , and read off both the determined region and its null boundary.

Step 1. Consider a candidate event $p=(t_{0},\mathbf{x}_{0})$ with $t_{0}>0$ and study its past-inextendible causal curves. The one most likely to miss the disk is the past-directed null ray fired radially outward, since it gains radius as fast as physically allowed. It reaches the slice $t=0$ at radius $|\mathbf{x}_{0}|+t_{0}$ . Why: tilting a past causal curve toward the outgoing light cone maximizes the radius at which it crosses $t=0$ , so if even this extreme ray lands on the disk, every gentler curve does too.

Step 2. Require that this extreme ray still land inside the disk, $|\mathbf{x}_{0}|+t_{0}\le R$ , i.e. $|\mathbf{x}_{0}|\le R-t_{0}$ . Impose it for every past ray at once and this single inequality is exactly the membership condition (1) for $p\in D^{+}(S)$ . Why: the outgoing null ray is the binding constraint; once it is satisfied no past-directed causal curve can escape the disk, so the condition is both necessary and sufficient.

Step 3. The set of qualifying events is therefore $D^{+}(S)=\{(t,\mathbf{x}):0\le t\le R,\ |\mathbf{x}|\le R-t\}$ , a solid cone that shrinks linearly from the disk at $t=0$ to a single apex at $(t,\mathbf{x})=(R,\mathbf{0})$ ; reflecting in $t$ gives $D^{-}(S)$ and the full diamond $D(S)$ . Why: the determined radius $R-t$ falls by one unit of length per unit of time — the imprint of the $45^{\circ}$ light cones — so the region is a right cone of height $R$ over a disk of radius $R$ .

Step 4. The lateral face $H^{+}(S)=\{|\mathbf{x}|=R-t,\ 0\le t\le R\}$ is null and is swept out by inward-directed null geodesics that, traced to the past, all terminate on the rim $\operatorname{edge}(S)=\{t=0,\ r=R\}$ — precisely the generator theorem in action. Beyond this face, data from $r>R$ could have arrived, so prediction is impossible. Why: each generator is the past-outgoing null ray that just grazes the rim; the rim is where $S$ ends, so it is the natural past endpoint of every generator, confirming (4) in a case one can see.

Go deeper · Derivation C · 2 minThe inner horizon of Reissner-Nordström as a Cauchy horizon

Derivation CThe inner horizon of Reissner-Nordström as a Cauchy horizon

The goal is to show how the inner horizon $r_{-}$ of a non-extremal charged black hole becomes a genuine Cauchy horizon for data on a Cauchy surface of the exterior, and why the maximal analytic extension beyond it is not fixed by that data — the sharpest physical example of predictability breaking down inside general relativity. Take the non-extremal Reissner-Nordström metric with $0<Q<M$ in units $G=c=1$ , locate its two horizons, choose a slice, compute its domain of dependence, and identify where determinism ends. Numbers appear because the field admits them: the horizon radii are explicit roots.

Step 1. Start from $\mathrm{d}s^{2}=-f(r)\,\mathrm{d}t^{2}+f(r)^{-1}\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega^{2}$ with $f(r)=1-\tfrac{2M}{r}+\tfrac{Q^{2}}{r^{2}}$ , and solve $f(r)=0$ . Clearing denominators gives $r^{2}-2Mr+Q^{2}=0$ , whose roots are $r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}$ — the outer event horizon $r_{+}$ and the inner horizon $r_{-}$ , both null hypersurfaces since $g^{rr}=f$ vanishes there. Why: horizons of a static metric sit where the norm of the timelike Killing field vanishes, i.e. where $f=0$ , and there are two distinct real roots exactly when $Q<M$ .

Step 2. Choose $\Sigma$ to be a spacelike slice threading the asymptotically flat exterior $r>r_{+}$ (region I) and continuing across the event horizon into region II, $r_{-}<r<r_{+}$ . One checks $\Sigma$ is achronal and edgeless, a legitimate Cauchy surface for the region it can control. Why: the exterior together with region II is globally hyperbolic on its own, so a well-behaved slice through it is exactly the setting in which to test how far its prediction reaches.

Step 3. Compute the domain of dependence: every past-inextendible causal curve in regions I and II is forced back through $\Sigma$ , so $D(\Sigma)$ covers all of $r>r_{-}$ . But at $r=r_{-}$ this fails — new past-directed causal curves appear that never crossed $\Sigma$ , arriving instead from the timelike singularity at $r=0$ (visible to region III) and from the other asymptotic universes of the maximal extension. Why: unlike Schwarzschild's spacelike singularity, the RN singularity is timelike and sits to the future side of $r_{-}$ , so it can emit signals into the region beyond $r_{-}$ without ever registering on $\Sigma$ .

Step 4. Therefore $H^{+}(\Sigma)=\{r=r_{-}\}$ : the inner horizon is the Cauchy horizon of the exterior slice. An observer crossing it enters a region whose metric is not determined by data on $\Sigma$ , and where the analytic extension is non-unique. Strong cosmic censorship conjectures that generic infalling radiation blueshifts without bound at $r_{-}$ (mass inflation), converting the smooth Cauchy horizon into a singular boundary and restoring determinism by making the spacetime inextendible. Why: the mathematical non-uniqueness beyond $r_{-}$ is physically intolerable, so the conjecture asks whether real perturbations seal the horizon off — turning a failure of prediction into an edge of spacetime. *The horizon radii $r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}$ are the roots of $r^{2}-2Mr+Q^{2}=0$ , and the causal structure — including the timelike character of the RN singularity — is faithfully represented in the Penrose diagram of the maximal extension.*

Limiting cases

The character of $D(S)$ and $H(S)$ depends sharply on the global causal setting; the same construction yields a filled spacetime or a hard wall of unpredictability depending on what lies beyond the data.

Case	Result	Meaning
Globally hyperbolic spacetime	$D(S)=M,\ H(S)=\varnothing$	$S$ is a Cauchy surface; initial data determine everything and no horizon forms — Minkowski, maximally extended Schwarzschild, and FLRW are examples.
Finite / partial data surface	$H^{+}(S)\neq\varnothing$ , generated by null geodesics ending on $\operatorname{edge}(S)$	A bounded $S$ in Minkowski has a Cauchy horizon rising from its rim; enlarging $S$ pushes the horizon out, and $S\to$ hyperplane sends $H\to\varnothing$ .
Charged / rotating black-hole interior	$H^{+}(\Sigma)$ at $r_{-}=M-\sqrt{M^{2}-Q^{2}}$ (RN) or $r_{-}=M-\sqrt{M^{2}-a^{2}}$ (Kerr)	A Cauchy surface for the exterior fails to determine the geometry past $r_{-}$ ; the maximal extension is non-deterministic — the arena of strong cosmic censorship.
Extremal / super-extremal	$Q\to M$ or $a\to M$ : $r_{+}=r_{-}=M$ ; $Q>M$ or $a>M$ : no real root	At extremality the two horizons merge; beyond it the discriminant is negative and the singularity is naked, with no horizon of any kind.
Anti-de Sitter asymptotics	spatial infinity timelike $\Rightarrow H(S)\neq\varnothing$	AdS has a timelike conformal boundary through which information enters; no spacelike slice is a Cauchy surface without boundary conditions imposed at infinity.

Common trapA Cauchy horizon is just another name for an event horizon.

It is not. An event horizon is a global causal boundary defined relative to future null infinity — $\partial J^{-}(\mathscr{I}^{+})$ , the frontier of the events that can still signal to infinity — and it depends on the entire future of the spacetime. A Cauchy horizon $H^{+}(S)$ is the boundary of the region predictable from a chosen achronal surface $S$ ; it depends on $S$ , not on infinity. The two coincide nowhere in general: in Reissner-Nordström the event horizon sits at the outer root $r_{+}$ while the Cauchy horizon of an exterior slice sits at the distinct inner root $r_{-}<r_{+}$ . Schwarzschild sharpens the contrast — it has an event horizon but no Cauchy horizon at all, because its maximal extension is globally hyperbolic and a suitable slice determines the whole spacetime. And note the standing requirement: $S$ must be achronal. Feed the construction a set two of whose points are timelike related and $D(S)$ becomes meaningless, because the "initial data" would already constrain itself.

Worked examples

1 · The diamond over a finite disk, and its limit to a Cauchy surface (fully worked)

In Minkowski space with $c=1$ , take the achronal disk $S=\{t=0,\ r\le R\}$ with $R=3$ light-years. Find the apex of $D^{+}(S)$ , the radius still determined at $t=1$ and $t=2$ light-years, the fraction of the $t=1$ slice that is predictable, and the behaviour as $R\to\infty$ .

Step 1. Apex. The determined radius at time $t$ is $R-t$ , which reaches zero at $t=R$ . So the domain of dependence is a solid cone with its apex at $t=3$ light-years on the central worldline; above that single event nothing at all is determined by data confined to a $3$ -light-year disk.

Step 2. Determined radius on later slices. At $t=1$ : $R-t=3-1=\mathbf{2}$ light-years; at $t=2$ : $3-2=\mathbf{1}$ light-year. Each unit of elapsed time costs one unit of predictable radius — the direct fingerprint of the $45^{\circ}$ light cones, since the binding influence is a null ray from the rim travelling inward at speed $1$ .

Step 3. Predictable fraction of a slice. The infinite $t=1$ hyperplane is mostly undetermined, but comparing the determined disk of radius $2$ to the original data disk of radius $3$ , the determined area has shrunk to $(2/3)^{2}=\mathbf{0.44}$ of the disk's area — and against the whole infinite slice it is a set of measure zero. Finite data predict only a finite, shrinking diamond, never a whole later slice.

Step 4. The limit $R\to\infty$ . As the disk grows, the apex $t=R$ recedes to infinity and the null faces tilt outward until, in the limit, $S$ becomes the full hyperplane $t=0$ , $D(S)$ fills all of Minkowski space, and $H(S)\to\varnothing$ . The disk becomes a Cauchy surface exactly in the limit that its rim runs off to infinity — consistent with Minkowski being globally hyperbolic.

2 · Inner-horizon radii of charged and rotating black holes (fully worked)

Using $r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}$ (RN) and $r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$ (Kerr) in units $G=c=1$ , locate both horizons for a Reissner-Nordström hole with $Q=0.8\,M$ and a Kerr hole with $a=0.6\,M$ , identify which radius is the Cauchy horizon, and state what happens at extremality.

Step 1. Reissner-Nordström, $Q=0.8M$ . The discriminant is $\sqrt{M^{2}-Q^{2}}=M\sqrt{1-0.64}=M\sqrt{0.36}=0.6\,M$ . Hence $r_{+}=M+0.6M=\mathbf{1.6\,M}$ (event horizon) and $r_{-}=M-0.6M=\mathbf{0.4\,M}$ (inner horizon). The Cauchy horizon of an exterior slice is $r_{-}=0.4\,M$ , well inside the event horizon.

Step 2. Kerr, $a=0.6M$ . Now $\sqrt{M^{2}-a^{2}}=M\sqrt{1-0.36}=M\sqrt{0.64}=0.8\,M$ , giving $r_{+}=\mathbf{1.8\,M}$ and $r_{-}=\mathbf{0.2\,M}$ . Again $r_{-}$ is the Cauchy horizon — cross it and the geometry (which contains the ring singularity and closed timelike curves in its deep interior) is no longer fixed by exterior data.

Step 3. Extremal limit. Sending $Q\to M$ or $a\to M$ drives the discriminant to zero, so $r_{+}$ and $r_{-}$ merge at $r=M$ : the event and Cauchy horizons coincide and the surface gravity vanishes. Push past extremality ( $Q>M$ or $a>M$ ) and the roots become complex — no horizon exists and the singularity is naked, which cosmic censorship conjectures cannot arise from generic collapse.

Step 4. Reading. In every non-extremal case the Cauchy horizon is the inner root $r_{-}$ , strictly smaller than the event horizon $r_{+}$ , and it cannot be pushed away by enlarging the slice because the influences that evade $\Sigma$ come from the timelike singularity and other universe sheets, not from a too-small disk. The RN and Kerr inner horizons are the physical home of the strong cosmic censorship conjecture.

Physics of the portrait

The domain of dependence is where the initial-value problem of general relativity became rigorous geometry. Before the 1950s, "predict the metric from data on a slice" was a hope; the theorems that made it a theorem — local well-posedness, the maximal development, and the topology of global hyperbolicity — are the work of a small group, and the name attached most directly to this construction is Robert Geroch, whose 1970 paper is titled, simply, "Domain of Dependence".

Origins · 1 minRobert Geroch and how this was discovered

Robert Geroch

1942– · domain of dependence and the topology of global hyperbolicity

Robert Geroch is one of the architects of the global, causal way of thinking about general relativity that this cluster rests on. His 1970 paper "Domain of Dependence" (J. Math. Phys. 11, 437) proved the theorem that anchors the whole subject: a spacetime is globally hyperbolic if and only if it admits a Cauchy surface, and in that case it is topologically $\mathbb{R}\times\Sigma$ , foliated by a continuous one-parameter family of Cauchy surfaces $\Sigma_{t}$ . That single result turns the algebraic statement " $D(\Sigma)=M$ " into a geometric picture — spacetime as a stack of instants, each determining the next — and it is why numerical relativity can speak of evolving data "forward in time" at all. With Yvonne Choquet-Bruhat he proved in 1969 that every set of constraint-satisfying data has a unique maximal globally hyperbolic development, the precise sense in which Einstein's equations are deterministic and the exact object whose boundary a Cauchy horizon is. Geroch also introduced the "Geroch splitting" behind that foliation and sharpened what "singularity" and "predictability" mean globally. Where Penrose supplied conformal infinity and the achronal-boundary machinery, Geroch supplied the theorems that tie a Cauchy surface, its domain of dependence, and global hyperbolicity into a single equivalence — the backbone of every statement in this article.

History

1952

— Choquet-Bruhat proves local existence and uniqueness for the Einstein vacuum Cauchy problem

1953

— Leray casts the domain of determinacy of a hyperbolic system in geometric form

1969

— Choquet-Bruhat and Geroch establish the unique maximal globally hyperbolic development

1970

— Geroch proves a globally hyperbolic spacetime is foliated by Cauchy surfaces

1973

— Hawking and Ellis systematize domains of dependence and Cauchy horizons in their monograph

Going further

Texts. Wald, General Relativity (1984), Sec. 8.3 · Hawking & Ellis, The Large Scale Structure of Space-Time (1973), Sec. 6.5 and Sec. 5.5 · Penrose, Techniques of Differential Topology in Relativity (1972) · Misner, Thorne & Wheeler, Gravitation (1973), Ch. 34 · Carroll, Spacetime and Geometry (2004), Appendix A and Ch. 6 · Geroch, Domain of Dependence, J. Math. Phys. 11, 437 (1970) · Choquet-Bruhat & Geroch, Global aspects of the Cauchy problem in general relativity, Comm. Math. Phys. 14, 329 (1969). The Cauchy horizon formula $H^{+}(S)=\overline{D^{+}(S)}-I^{-}[D^{+}(S)]$ and the generator theorem are the standard results of the achronal-boundary theory. The Reissner-Nordström and Kerr horizon radii $r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}$ and $r_{\pm}=M\pm\sqrt{M^{2}-a^{2}}$ are obtained as roots of $r^{2}-2Mr+Q^{2}=0$ (RN) and $r^{2}-2Mr+a^{2}=0$ (Kerr) in units $G=c=1$ ; direct substitution gives $r_{+}=1.6M,\ r_{-}=0.4M$ at $Q=0.8M$ and $r_{+}=1.8M,\ r_{-}=0.2M$ at $a=0.6M$ . The flat-disk domain of dependence $|\mathbf{x}|\le R-t$ agrees with the wave-equation domain of determinacy. No numerical constants beyond these geometric quantities are used.

Prerequisites. Causal Structure of Spacetime · Chronological and Causal Futures and Pasts · Achronal Sets and Edges · Null Geodesics and Light Cones. Leads to. Global Hyperbolicity · Cauchy Surfaces · Causality Conditions Hierarchy · Penrose-Carter Conformal Diagrams.

Ask the tutor: Want to walk through how mass inflation is conjectured to convert the Kerr inner horizon into a singular boundary, or how Geroch's foliation theorem turns " $D(\Sigma)=M$ " into spacetime as a stack of Cauchy slices?

What is the difference between the domain of dependence D⁺(S) and the causal future J⁺(S)?

They point in opposite directions and answer different questions. The causal future $J^{+}(S)$ is everything $S$ can influence — the events reachable from $S$ by future-directed causal curves. The domain of dependence $D^{+}(S)$ is everything $S$ fully controls — the events all of whose past-directed causal curves are forced back onto $S$ . An event can easily lie in $J^{+}(S)$ without lying in $D^{+}(S)$ : it may be reachable from $S$ yet also reachable from elsewhere, so data on $S$ do not determine it. $D^{+}(S)$ is always a subset of $J^{+}(S)$ , and the gap between them — events influenced by $S$ but not determined by it — is exactly the region a Cauchy horizon seals off.

Why must the surface S be achronal?

Because a set that is not achronal cannot serve as a single instant of time. If two points of $S$ were joined by a timelike curve, one would lie to the future of the other, so prescribing "initial data" at the earlier point would already constrain what you are independently prescribing at the later one — the data would be self-overdetermined or outright inconsistent. Achronality, the condition $S\cap I^{+}(S)=\varnothing$ , guarantees that no point of $S$ can causally influence another, so the whole surface is a genuine "now" on which data can be freely specified. Only then is the forward construction of $D^{+}(S)$ as the region determined by that now well posed; hand the machinery a non-achronal set and $D(S)$ is meaningless.

How exactly does a Cauchy horizon differ from an event horizon?

An event horizon is defined by reference to infinity: it is $\partial J^{-}(\mathscr{I}^{+})$ , the boundary of the region that can still send signals to future null infinity, and deciding where it lies requires knowing the entire future of the spacetime. A Cauchy horizon $H^{+}(S)$ is defined by reference to a chosen slice $S$ : it is the boundary of the region predictable from data on $S$ . The two answer different questions — "can light escape to infinity?" versus "is the future fixed by this data?" — and generally sit at different places. Reissner-Nordström makes this vivid: the event horizon is the outer root $r_{+}$ , the Cauchy horizon of an exterior slice is the inner root $r_{-}$ , and Schwarzschild has an event horizon with no Cauchy horizon anywhere.

What is the maximal Cauchy development, and why does it matter?

Given initial data $(\Sigma,h_{ij},K_{ij})$ satisfying the Einstein constraint equations, the Choquet-Bruhat-Geroch theorem (1969) guarantees a unique, largest globally hyperbolic spacetime that evolves from those data — the maximal Cauchy development. "Unique" is the precise statement that general relativity is deterministic: two developments of the same data agree on their common globally hyperbolic region. "Maximal" means you cannot enlarge it without either violating the equations or leaving the globally hyperbolic regime. Its future boundary, where you would leave that regime, is precisely a Cauchy horizon. So the maximal development and the Cauchy horizon are two sides of one object — the horizon is the edge of the largest region physics can determine from the data.

Does the existence of a Cauchy horizon always signal a physical pathology?

Not by itself — it depends on why the horizon is there. For a finite data surface, like a bounded disk in Minkowski, the Cauchy horizon is completely benign: it simply marks the reach of limited data, and enlarging the surface pushes it away, eventually to nothing. That is unpredictability by insufficient information, not by any defect of the spacetime. The pathological case is a horizon that cannot be removed by enlarging the slice, as at a black hole's inner horizon $r_{-}$ : there the failure is intrinsic, because influences from a timelike singularity or other universes enter from beyond the maximal development. Strong cosmic censorship is the conjecture that nature avoids this second kind by making generic Cauchy horizons singular and hence unreachable.

What is strong cosmic censorship, and how does it "rescue" determinism?

Strong cosmic censorship conjectures that the maximal Cauchy development of generic initial data is inextendible — that you cannot smoothly continue the spacetime past its Cauchy horizon. Physically, the mechanism is mass inflation: perturbations falling into a charged or rotating black hole pile up and blueshift without bound as they approach the inner horizon $r_{-}$ , so the curvature diverges there and the smooth Cauchy horizon of the idealized solution is replaced, for any realistic hole, by a singular boundary. Because nothing can be extended across a singularity, there is no longer a region "beyond $r_{-}$ " whose physics the data fail to fix — the failure of prediction is amputated. The conjecture remains open, and recent near-extremal de Sitter counterexamples show the rescue is not automatic.