# Low regularity extensions beyond Cauchy horizons

M. Lesourd and E. Minguzzi Black Hole Initiative, Harvard University,
Cambridge, MA 02138. E-mail<EMAIL_ADDRESS>degli Studi di
Firenze, Dipartimento di Matematica e Informatica ”U. Dini”, Via S. Marta 3,
I-50139 Firenze, Italia. E-mail<EMAIL_ADDRESS>
###### Abstract

We prove that if in a $C^{0}$ spacetime a complete partial Cauchy hypersurface
has a non-empty Cauchy horizon, then the horizon is caused by the presence of
almost closed causal curves behind it or by the influence of points at
infinity. This statement is related to strong cosmic censorship and a
conjecture of Wald. In this light, Wald’s conjecture can be formulated as a
PDE problem about the location of Cauchy horizons inside black hole interiors.

##  1 Introduction

Penrose introduced strong cosmic censorship in the seminal paper [24]. In an
elegant article appearing in the same volume, Geroch and Horowitz [11]
elaborated on the differences between the strong and weak formulations of
cosmic censorship, and sketched possible approaches one could take to prove
such statements. In the context of Lorentzian geometry, they formulated a
conjecture which they deemed to be in the spirit of strong cosmic censorship.
This conjecture was picked up and sharpened by Wald in Chapter 12 of his
landmark book [29].

> Wald’s Strong Cosmic Censorship. Let $(S,h,K)$ be an initial data set for
> Einstein’s equation, with $(S,h)$ a complete Riemannian manifold and with
> the Einstein-matter equations in which matter is described by a quasi-
> linear, diagonal, second order hyperbolic system, and where the stress-
> energy tensor satisfies the dominant energy condition. Then, if the maximal
> Cauchy development of this initial data is extendible, for each $p\in
> H^{+}(S)$ in any extension, either strong causality is violated at $p$ or
> $\overline{J^{-}(p)\cap S}$ is non-compact.111In Wald’s book $J^{-}(p)$ is
> replaced by $I^{-}(p)$, but the two formulations are easily shown to be
> equivalent.

Wald’s conjecture seeks to pinpoint why Cauchy horizons form. That is, they
are associated with one of two things: the formation of almost closed causal
curves, or the possibility that some point on the horizon be influenced by
spatial infinity. With regards to the basic examples, in Taub-NUT it is the
former, and in Reissner-Nordstrom it is the latter. As for the Kerr-Newman
family (with $a\neq 0$), both occur.
With time, the weak and strong conjectures have been translated into PDE
problems about the Cauchy development of generic initial data.

> Modern Strong Cosmic Censorship. Maximal Cauchy developments of generic
> initial data are inextendible within some regularity class.

Remarkable recent work on the subject [3, 6, 15, 16, 18, 19, 28] shows that
the SCC is highly sensitive to the setting in which one poses the problem, and
the specific definitions of generic and regular at play. The SCC is now more
of a theme of plausible statements (as opposed to a fixed universal
statement), each more or less natural relative to the particular setting, and
each having varying chances of being true.
In spite of the intricacies that the modern picture has unveiled, the
simplicity and clarity of Wald’s conjecture, viewed as a structure theorem for
Cauchy horizons, grants it an enduring significance. Moreover, with the
current emphasis on low regularity extensions, a modern version of Wald’s
conjecture ought to permit for $C^{0}$ extensions.
Before going further, we note that Wald’s conjecture cannot hold for positive
cosmological constant. A counterexample is provided by the non-extremal
Reissner-Nordström de Sitter spacetime. This is a vacuum solution of
Einstein’s equation with $\Lambda>0$ which is strongly causal but for which we
can find a spacelike partial Cauchy hypersurface such that for some $p\in
H^{+}(S)$, $\overline{J^{-}(p)\cap S}$ is compact (cf. Fig. 1).

Figure 1: The Carter-Penrose diagram of the maximal analytic extension of the
non-extremal Reissner-Nordström de Sitter spacetime. The partial Cauchy
hypersurface $S$ admits a point $p\in H^{+}(S)$ such that
$\overline{J^{-}(p)\cap J^{+}(S)}$ is non-compact while
$\overline{J^{-}(p)\cap S}$ is compact. The double line is a singularity.

######  Remark 1.1.

In [7, Thm. 2.1] Etesi stated a weaker version in which
$\overline{J^{-}(p)\cap S}$ is replaced by $J^{-}(p)\cap S$. The stronger
version is clearly Wald’s, since it is easier to prove that a set is non-
compact if it can be non-closed. However, consider the next example.

Let $(M,g)$ be the 1+1 Minkowski spacetime of coordinates $(t,x)$, $g=-{\rm
d}t^{2}+{\rm d}x^{2}$ with the timelike lines $x=-1$ and $x=+1$ identified and
the point $r=(2,0)$ removed. For $S=t^{-1}(0)$ and $p=(2.5,0.5)$ we have $p\in
H^{+}(S)$, $J^{-}(p)\cap S$ is compact but neither $J^{-}(p)\cap J^{+}(S)$ nor
its closure are compact. The statement which Etesi claims to prove is thus
false. The argument fails at [7, Lemma 2.3]: it is not possible to infer the
non-compactness of $J^{-}(p)\cap S$ from that of $J^{-}(p)\cap J^{+}(S)$ or
its closure by just using causality theory.

In recent years there have been numerous studies under a $C^{0}$ assumption.
Fathi and Siconolfi studied the existence of time functions under such
regularity[8], and Chruściel and Grant [4] presented the first investigation
of causality theory under the $C^{0}$ assumption. Sbierski [26, 27] studied
causality further and proved the $C^{0}$ inextendibility of Schwarzschild
spacetime. Sämann [25] studied stable causality and global hyperbolicity and
their characterizations. Galloway and Ling [9, 17] proved the $C^{0}$
inextendibility of AdS spacetime and showed extendibility through the Big Bang
of Milne-like hyperbolic FLRW spacetimes. Chruściel and Klinger obtained a
$C^{0}$-inextendibility criterion for expanding singularities [5]. Galloway,
Ling and Sbierski [10] proved that in globally hyperbolic spacetimes timelike
completeness guarantees $C^{0}$ inextendibility, a result improved by Minguzzi
and Suhr [23] who showed that global hyperbolicity could in fact be dropped.
Bernard and Suhr [1, 2] and Minguzzi [20, 22] obtained several results on
regularity of time functions and Cauchy hypersurfaces under even weaker
conditions. Grant, Kunzinger, Sämann and Steinbauer clarified properties and
pathologies of the chronological future under low regularity [12].

In our study, we shall make repeated use of the causality theory developed by
the second author in [20]. The regularity permitted in [20] is based on the
framework of closed and proper cone structures, and the results there are
valid for Lorentz-Finsler theory under upper semi-continuity conditions on the
cone distribution and Finsler fundamental function. Spacetimes endowed with
$C^{0}$ metrics are special cases of proper cone structures, and so the
results on Cauchy horizons developed in that work apply in our case. In fact,
it is likely that the results of the present work could hold for the more
general proper cone structures of [20], but owing to being motivated by PDEs,
we do not see the need to consider lower than $C^{0}$ regularity.

Although Wald’s formulation involves conditions on the initial data (which
guarantees the existence of a unique, up to isometry, maximal Cauchy
development), one can isolate the purely Lorentzian geometric component of the
statement. In particular, what we prove is the following

######  Theorem 1.2.

Let $(M,g)$ be a $C^{0}$ spacetime and let $S$ be a complete Cauchy $C^{1}$
spacelike hypersurface for $M$. Suppose that $(M,g)$ has a $C^{0}$ extension
$(M^{\prime},g^{\prime})$ in which $S$ is still acausal and that
$H:=H^{+}(S,M^{\prime})\neq\emptyset$. Let $p$ be a point in $H$. Then there
is a past inextendible geodesic generator $\gamma$ of the horizon with future
endpoint $p$ that either

  * (a)

it is past imprisoned in a compact subset of the horizon, and so accumulates
in the past direction on a compact region where strong causality (actually
distinction) is violated,

  * (b)

it escapes every compact set in the past direction and $\overline{J^{-}(p)\cap
J^{+}(S)}$ is non-compact.

Note that although (a) implies that strong causality is violated somewhere on
the horizon, this need not occur at $p$. Rather, $p$ is connected to the
imprisoning region by way of a null achronal generator.

As for (b), note that strong causality violation can occur without (a)
happening at all. For example, every point beyond the Cauchy horizon of
maximally extended Kerr is traversed by a closed timelike curve. This means
that there cannot be arbitrarily small causally convex neighborhoods around
$p\in H^{+}(S)$. Thus strong causality violation at $p$ can occur in (b)
without the imprisonment in (a) occuring.

Note also that we allow for the original spacetime $(M,g)$ to be $C^{0}$,
whereas Wald’s formulation presumes that $(M,g)$ is at least $C^{2}$. Our
reason is simply because we can prove the result in such regularity, and that
we find it convenient to use the same regularity for the original and extended
spacetimes. In this way we need only to prove222The spacetime previously
denoted $M^{\prime}$ is now denoted $M$, that previously denoted $M$ is now
$\textrm{Int}D(S)$.

######  Theorem 1.3.

Let $(M,g)$ be a $C^{0}$ spacetime and let $S$ be a complete acausal $C^{1}$
spacelike hypersurface. Let $p\in H:=H^{+}(S)$. Then there is a past
inextendible geodesic generator $\gamma$ of the horizon with future endpoint
$p$ that either

  * (a)

it is past imprisoned in a compact subset of the horizon, and so accumulates
in the past direction on a compact region where strong causality (actually
distinction) is violated,

  * (b)

it escapes every compact set in the past direction and $\overline{J^{-}(p)\cap
J^{+}(S)}$ is non-compact.

The proof of this statement, which largely follows from the theory developed
in [20], is given in the next section. This statement is of course consistent
with the examples discussed above, and it seems the best version one can
obtain in the context of low regularity causality theory.
Of course, some of the terms need to be clarified. For instance, on $S$ the
metric $g$ induces a $C^{0}$ Riemannian metric $h$. By complete we mean that
the closed balls with respect to the associated distance are compact (we do
not have at our disposal the Hopf-Rinow theorem in such low regularity). The
notion of geodesic generator will also be clarified in the next section.
Finally, in the spirit of Wald’s original conjecture, we will also give
necessary and sufficient conditions for $\overline{J^{-}(p)\cap S}$ to be
noncompact. We do this in the context of spacetimes to which we can associate
a concept of null infinity $\mathcal{J}^{+}$. This assumption is inspired by
Dafermos-Luk’s recent groundbreaking study of strong cosmic censorship in
vacuum. Their paper deals with strong cosmic censorship as a problem about the
stability of Cauchy horizons in a given class of black hole spacetimes, i.e.,
spacetimes already possessing a future complete $\mathcal{J}^{+}$ along with
an event horizon $\partial I^{-}(\mathcal{J}^{+})\neq\emptyset$. This allows
them to sidestep the issue of global existence in the exterior region (i.e.,
weak cosmic censorship guaranteeing a future complete $\mathcal{J}^{+}$). In
that setting we show the following, where the precise definitions involving
$\mathcal{J}^{+}$ are given below.

######  Proposition 1.4.

Let $(M,g)$ be a $C^{2}$ spacetime admitting a conformal embedding into the
unphysical spacetime $(\tilde{M},\tilde{g})$ with null boundary
$\mathcal{J}^{+}\subset\tilde{M}$ such that
$\mathcal{J}^{+}\subset\overline{D^{+}(S)}$ where the closure is taken in the
unphysical spacetime $(\tilde{M},\tilde{g})$. Let $(M^{\prime},g^{\prime})$ be
a $C^{0}$ extension of $(M,g)$ with $H^{+}(S,M^{\prime})\neq\emptyset$ and
$p\in H^{+}(S,M^{\prime})$. Then $\overline{J^{-}(p)\cap S}$ is noncompact if
and only if $J^{-}(p)$ intersects every neighborhood of $\mathcal{J}^{+}$.

Thus, the character of $\overline{J^{-}(p)\cap S}$ is really about the
location of Cauchy horizons (forming dynamically within black holes) relative
to $\mathcal{J}^{+}$.
If, for example, we start with initial data $S$ that differs from a Cauchy
surface of the maximal Schwarzschild solution only within a compact subset of
the black hole interior, then, even if a Cauchy horizon forms in the
development of $S$, $\overline{J^{-}(p)\cap S}$ remains compact by domain of
dependence.
In view of this, Wald’s conjecture can be given a modern reformulation as
follows.

> Cauchy Horizon Conjecture. Let $(S,g)$ be a complete Riemannian manifold,
> and let $(S,g,k)$ be generic initial data for the Einstein system, possibly
> with non-zero cosmological constant $\Lambda$, and possibly with matter
> sources (in which case we may assume various admissibility conditions on
> $T_{ab}$). Suppose that the maximal Cauchy development $D(S)=M$ produces an
> event horizon $\partial I^{-}(\mathcal{J}^{+})\neq\emptyset$. If $M$ admits
> a $C^{0}$ extension $M^{\prime}\supset M$ and a Cauchy horizon
> $H^{+}(S,M^{\prime})\neq\emptyset$, then
>
> $\overline{J^{-}(H^{+}(S,M^{\prime}))\cap\partial
> I^{-}(\mathcal{J}^{+})}\>\>\textnormal{is noncompact}$

This conjecture can be studied by setting up characteristic initial value
problems in which one of the initial data hypersurfaces is an event horizon.
This is exactly what is done in [6] (though in [6] they study two-ended data
meaning that both hypersurfaces are event horizons), and note that the $C^{0}$
extensions constructed in [6] do indeed satisfy the above statement.
Finally, we note that since it is in principle possible that portions of
$H^{+}(S)$ differ in the regularity of permitted extensions, the $C^{0}$
qualifier widens the scope of the conjecture, and that the above statement
does not distinguish between $\Lambda<,>,=0$, though indeed its validity may
depend on $\Lambda$.

##  2 Preliminaries and proof

Let us introduce some notations and conventions. The spacetime $(M,g)$ is a
smooth connected time-oriented Lorentzian manifold (without boundary) of
dimension $n+1$ (working with $C^{1}$ manifolds would have the same generality
as every $C^{1}$ manifold has a unique smooth compatible structure
(Whitney)[14, Thm. 2.9]). The signature of $g$ is $(-,+,\cdots,+)$. The
spacetime is said to be $C^{k}$ if $g$ is $C^{k}$, $k\geq 0$. The symbol of
inclusion is reflexive, $X\subset X$. The interior $\textrm{Int}X$ of a set
$X\subset M$ might be denoted $\mathring{X}$ for shortness. The reader is
referred to [20, 22] for the $C^{0}$ causality theory and to [21] for all the
other conventions adopted without mention in this work.
Two events are chronologically related if there is a piecewise $C^{1}$
timelike curve connecting them. They are causally related if they coincide or
there is a continuous causal curve connecting them. The chronological and
causal relations are denoted $I$ and $J$, respectively. We refer to [26, 20]
for these concepts. It must be recalled that for $C^{0}$ spacetimes we have
$I\subset\mathring{J}$, and the equality holds for $g$ locally Lipschitz [4,
20].

A set is achronal if no pair of points of it belongs to $I$, and acausal if no
pair of points of it belongs to $J\backslash\Delta$ where
$\Delta=\\{(x,x):x\in M\\}\subset M\times M$ is the diagonal. Lightlike
geodesics are defined as in [20, Def. 2.6] as continuous causal curves for
which locally no pair of points of it belong to $\mathring{J}$ (this coincides
with local achronality only under a locally Lipschitz assumption on $g$).

Sets of the form $J^{\pm}(p)\backslash\overline{I^{\pm}(p)}$ have been termed
causal bubbles [4]. It is now known that they do not exist iff the Kronheimer
and Penrose’s causal space condition (push up property) $I\circ J\cup J\circ
I\subset I$ holds [20, Thm. 2.8] (see also [12, Thm. 2.12]), which is the
case, for instance, under a locally Lipschitz condition on $g$ [4, 20].

In the $C^{0}$ theory the notion of edge of an achronal hypersurface $S$ does
not differ from that of the $C^{2}$ theory [13, p. 202] [20, Sec. 2.18]. The
proof that $\bar{S}\backslash S\subset\textrm{edge}(S)\subset\bar{S}$, cf.
[20, Prop. 2.132] passes to the $C^{0}$ theory as it uses just the
chronological relation and its openness.

An acausal edgeless (and hence closed) set is a partial Cauchy hypersurface. A
Cauchy hypersurface is a partial Cauchy hypersurface intersected by every
inextendible continuous causal curve.

######  Definition 2.1.

Let $(M,g)$ be a $C^{0}$ spacetime. The future domain of dependence or future
Cauchy development $D^{+}(S)$ of a closed and achronal subset $S\subset M$,
consists of those $p\in M$ such that every past inextendible continuous causal
curve passing through $p$ intersect $S$. The future Cauchy horizon is

$H^{+}(S)=\overline{D^{+}(S)}\backslash I^{-}(D^{+}(S)).$

We shall need the following result proved in [20, Thm. 2.32] for more general
“proper cone structures”. The fact that a $C^{0}$ spacetime is a proper cone
structure is proved in [20, Prop. 2.4] (and similarly under a locally
Lipschitz assumption).

######  Theorem 2.2.

(Cauchy horizons are generated by lightlike geodesics)
Let $(M,g)$ be a $C^{0}$ spacetime and let $S$ be a closed and acausal
hypersurface (possibly with edge). The set $H^{+}(S)\backslash S$ is an
achronal locally Lipschitz topological hypersurface and every $p\in H^{+}(S)$
is the future endpoint of a lightlike geodesic $\gamma$ contained in
$H^{+}(S)$, either past inextendible or starting from some point in $S$. No
two points $p,q\in H^{+}(S)\backslash S$ can be such that
$(p,q)\in\mathring{J}$, thus every continuous causal curve contained in
$H^{+}(S)\backslash S$ is a lightlike geodesic.

Notice that this result does not state that the generators cannot intersect
other generators in their interior (as we have in the regular theory).

Now, the main strategy is to study the fate of this generator $\gamma$ in the
past direction. The first step is to show that it cannot reach the edge of $S$
as $S$ has no edge due to the completeness condition. This fact is non-trivial
as $S$ might be “lightlike” at the edge and more generally because, though $S$
can be extended as an achronal hypersurface (hence, locally, a Lipschitz
graph), it could be the case that no extension is $C^{1}$, so that no sensible
notion of space metric could make sense over the extension.333It should be
mentioned that in low regularity there are results on the extension of partial
Cauchy hypersurfaces to Cauchy hypersurfaces in globally hyperbolic spacetimes
[22, Thm. 2.7]. These theorems cannot be used as they require that the
hypersurface be extendible at least in a neighborhood of the edge while
preserving causality and regularity properties.

######  Lemma 2.3.

Let $S$ be a complete acausal $C^{1}$ spacelike hypersurface on a $C^{0}$
spacetime. Then $\textrm{edge}(S)=\emptyset$, thus $S$ is closed.

###### Proof.

Assume, by contradiction, that there is $p\in\textrm{edge}(S)$. Let
$\\{x^{a}\\}$ be local coordinates such that $g(p)=-({\rm
d}x^{0})^{2}+\sum_{i=1}^{n}({\rm d}x^{i})^{2}$, $\partial_{0}$ is future
directed timelike, $x^{a}(p)=0$ for every $a=0,1,\cdots,n$. Observe that $S$
cannot accumulate on any other point of the axis $x^{i}=0$ because $S$ is
acausal hence achronal.

The bilinear form $\eta:=b[-a({\rm d}x^{0})^{2}+\sum_{i=1}^{n}({\rm
d}x^{i})^{2}]$ evaluated at $p$ is such that for $a<1$ its light cone is
narrower than that of $g(p)$, moreover, for $b>1$ the hyperboloid
$\\{v:\eta(v,v)=1\\}$ does not intersect the corresponding unit hyperboloid
for $g(p)$. Let, for instance, $b=a^{-1}=2$. These properties extend by
continuity to a sufficiently small coordinate cylindrical neighborhood $C$ of
$p$. Let $\pi:C\to D$ be the projection on the disk $D=C\cap\\{x^{0}=0\\}$,
and let the disk have diameter $\delta>0$ in the Euclidan coordinate metric.
The properties imply that if $v\in TC$ is a $g$-spacelike vector then it is a
$\eta$-spacelike vector on $C$ and
$\sqrt{g(v,v)}\leq\sqrt{\eta(v,v)}\leq|v^{\perp}|$ where
$v^{\perp}=\pi_{*}(v)$ is the projection of $v$ on the space orthogonal to
$\partial_{0}$ and $|\cdot|$ is the coordinate Euclidean space norm. Let
$q,r\in S\cap C$. If there is curve in $S\cap C$ connecting $q$ to $r$ then
this curve has a length smaller than that of its projection to $D$.

Let $A$ be a maximal achronal set containing $S$. Without loss of generality
(take a smaller coordinate cylinder $C$ if necessary) we can assume that the
set $A\cap C$ is a graph over $D$ (because $A$ is an achronal boundary, so the
result follows from [20, Thm. 2.19]). Let $q,r\in S\cap C$, and let
$\bar{q}=\pi(q)$, $\bar{r}=\pi(r)$. The segment $\bar{\sigma}$ connecting
$\bar{q}$ to $\bar{r}$ can be lifted to a curve $\sigma$ in $A$ connecting $q$
to $r$. If this curve is not entirely contained in $S$ then there is a maximal
segment $\tau$ with starting point $q$ entirely contained in $S$ and
converging to some point in $\bar{S}\backslash S$ such that the projection
$\bar{\tau}$ has length less than that of $\bar{\sigma}$ and hence of
$\delta$. However, this implies that $\tau$ has length less than $\delta$
which gives a contradiction with the completeness of $S$, as on $S$ the closed
ball of radius $\delta$ centered at $q$ would be non-compact. We conclude that
for every $q,r\in S\cap C$ the lift $\sigma$ is actually contained in $S$
which proves that $\rho(q,r)\leq\delta$ where $\rho$ is the distance of
$(S,h)$. However, completeness of $S$ implies that $\rho(q,r)\to\infty$ for
$r\to p$, which is not the case as the bound $\delta$ is independent of $r$,
that is, it is uniform over $S\cap C$. The contradiction proves that
$\textrm{edge}(S)=\emptyset$. ∎

We are ready to prove Theorem 1.3 and hence Theorem 1.2.

###### Proof of Theorem 1.3.

Since $S$ is a complete acausal $C^{1}$ spacelike hypersurface, by Lemma 2.3,
it is actually closed and $\textrm{edge}(S)=\emptyset$. This means that $S$ is
a topological hypersurface and so, by [20, Thm. 2.33] the generators of
$H^{+}(S)$ do not reach $S$ in the past direction. Also from [20, Thm. 2.34]
the set $D^{+}(S)\backslash S$ is open and $H^{+}(S)\cap S=\emptyset$. By
Theorem 2.2 for every $p\in H^{+}(S)$ there is a lightlike geodesic $\gamma$
with future endpoint $p$ which is past inextendible and contained in
$H^{+}(S)$. If $J^{-}(p)\cap J^{+}(S)$ has compact closure, then the same is
true for the set $J^{-}(p)\cap H\subset J^{-}(p)\cap J^{+}(S)$, then $\gamma$
is past imprisoned in a compact set contained in the closed set $H$, and by
[20, Thm. 2.22] it accumulates on a compact subset of $H$ over which future
and past distinction (and hence strong causality) are violated. ∎

###### Proof of Proposition 1.4.

For convenience, write

$C\equiv\overline{J^{-}(p)\cap S}$ (1)

By assumption, $S$ has an end $S_{\textnormal{end}}\subset S$, which we define
to be that part of $S$ contained in a neighborhood of $\mathcal{J}^{+}$.
First consider $\Leftarrow$. If $J^{-}(p)$ intersects every neighborhood of
$\mathcal{J}^{+}$, then there is a sequence of points $\\{p_{i}\\}\in
J^{-}(p)\cap S_{\textnormal{end}}$ which diverges to infinity in
$S_{\textnormal{end}}$; that is, $p_{i}$ eventually escapes every compact set
of $S$.

For the converse, we show that if there is a neighborhood of $\mathcal{J}^{+}$
that does not intersect $J^{-}(p)$, then $C$ is compact. If such a
neighborhood exists, then we can define a closed subset of $S$, $C^{\prime}$,
of the form $C^{\prime}=S\backslash S_{\textnormal{end}}$ such that $C\subset
C^{\prime}$. Both $C$ and $C^{\prime}$ are closed, so we need only show that
$C^{\prime}$ is compact. This follows from the fact that $S$ is composed of a
compact core and a union of ends. ∎

## Acknowledgments

M. Lesourd thanks the John Templeton and Gordon and Betty Moore Foundations
for their support of Harvard’s Black Hole Initiative. E. Minguzzi thanks GNFM
of INDAM for partial support.

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