###### Abstract

In the light of his recent (and fully deserved) Nobel Prize, this pedagogical
paper draws attention to a fundamental tension that drove Penrose’s work on
general relativity. His 1965 singularity theorem (for which he got the prize)
does not in fact imply the existence of black holes (even if its assumptions
are met). Similarly, his versatile definition of a singular space-time does
not match the generally accepted definition of a black hole (derived from his
concept of null infinity). To overcome this, Penrose launched his cosmic
censorship conjecture(s), whose evolution we discuss. In particular, we review
both his own (mature) formulation and its later, inequivalent reformulation in
the pde literature. As a compromise, one might say that in “generic” or
“physically reasonable” space-times, weak cosmic censorship postulates the
_appearance and stability of event horizons_ , whereas strong cosmic
censorship asks for the _instability and ensuing disappearance of Cauchy
horizons_. As an _encore_ , an appendix by Erik Curiel reviews the early
history of the _definition_ of a black hole.

Singularities, black holes, and cosmic censorship:

A tribute to Roger Penrose

Klaas Landsman

Department of Mathematics, Radboud University, Nijmegen, The Netherlands
Email<EMAIL_ADDRESS>
###### Contents

  1. 1 Historical introduction
  2. 2 Definitions
    1. 2.1 Null infinity
    2. 2.2 Trapped surfaces
    3. 2.3 Singularities in space-time
  3. 3 Cosmic censorship
    1. 3.1 Cosmic censorship à la Penrose
    2. 3.2 Cosmic censorship in the initial value (pde) formulation
  4. 4 Examples
  5. 5 Epilogue: Penrose’s final state conjecture
  6. A A potted early history of “black hole” (by Erik Curiel)

_Conformal diagram (Penrose 1968, p. 208, Fig. 37): ‘The Kruskal picture with
conformal infinity represented.’ Penrose usually drew his own figures in a
professional, yet playful and characteristic way._

##  1 Historical introduction

Roger Penrose got half of the 2020 Physics Nobel Prize ‘for the discovery that
black hole formation is a robust prediction of the general theory of
relativity’. This prize was well deserved, since, jointly with Hawking and
others, Penrose has shaped our (mathematical) thinking about general
relativity (gr) and black holes since the 1960s–70s. He would also deserve the
Abel Prize for this, shared with Yvonne Choquet-Bruhat: their combination
would highlight the fact that two originally distinct traditions in the
history of mathematical gr have now converged. In the wake of the work of
Einstein (1915), these traditions may be said to have originated with Hilbert
(1917) and Weyl (1918ab), respectively, as follows.

It would be fair to say that Hilbert mainly looked at gr from the point of
view of pdes,111See Stachel (1992) for an analysis of Hilbert’s contribution,
as well as for the history of the Cauchy problem of gr up to Choquet-Bruhat,
whose contributions were reviewed by Ringström (2015) as well as by herself
(Choquet-Bruhat, 2014, 2018). whereas Weyl–once Hilbert’s PhD student in
functional analysis–had a more geometric view, combined with an emphasis on
causal structure. These different perspectives initially developed separately,
in that the causal theory did not rely on the pde theory whilst for a long
time the pde results were local in nature. Penrose contributed decisively to
the causal approach to gr, with its characteristic emphasis on the _conformal
structure_ ,222Inspired by special rather than general relativity, Robb (1914,
1936), Reichenbach (1924), Zeeman (1964), and others axiomatized causal
structure as a specific _partial order_ , as Penrose knew well.  i.e. the
equivalence class of the metric tensor $g$ under a rescaling
$g_{\mu\nu}(x)\mapsto e^{\lambda(x)}g_{\mu\nu}(x)$, with $\lambda$ an
arbitrary smooth function of space and time. Though Weyl (1918b), p. 397,
mentions the analogy with Riemann surfaces,333Riemann surfaces may
equivalently be defined as either one-dimensional complex manifolds or as two-
dimensional Riemannian manifolds _up to conformal equivalence_. Modestly, Weyl
does not cite his own decisive contribution to their theory (Weyl, 1913). This
equivalence undoubtedly also influenced Penrose’s work on gr and its
derivatives like twistor theory.  his real argument for conformal invariance
was that what he calls _Reine Infinitesimalgeometrie_ must go beyond
Riemannian geometry, which (so he thinks) suffers from the inconsistency that
parallel transport of vectors (through the metric or Levi-Civita connection, a
concept Weyl himself had co-invented) preserves their length. This makes
length of vectors an absolute quantity, which a ‘pure infinitesimal geometry’
or a theory of _general_ relativity should not tolerate. To remedy this, Weyl
introduced the idea of _gauge invariance_ , in that the laws of nature should
be invariant under the above rescaling. To this end, he introduced what we now
call a gauge field $\varphi=\varphi_{\mu}dx^{\mu}$ and a compensating
transformation
$\varphi_{\mu}(x)\mapsto\varphi_{\mu}(x)-\partial_{\mu}\lambda(x)$, and
identified $\varphi$ with the electromagnetic potential (i.e. $A$). Dancing to
the music of time, he then proposed that the pair $(g,\varphi)$ describes all
of physics. The idea of gauge symmetry has lasted and forms one of the keys to
modern high-energy physics and quantum field theory: though misplaced in the
_classical gravitational_ context in which he proposed it,444See Einstein’s
negative reaction to Weyl (1918b) in Einstein (2002), Doc. 8. See also Goenner
(2004), §4.1.3.  through the Standard Model it has ironically become a
cornerstone of _non-gravitational quantum_ physics!

The conformal structure of a Lorentzian manifold determines the light cones,
and as such Weyl was not the only author to discuss causal structure. For
example, Einstein (1918) himself wondered if gravitational waves propagate
with the speed of light, and showed this in a linear approximation; Weyl
mentions this also.555See e.g. page 251 of the English translation of the
fourth edition of _Raum - Zeit - Materie_ (Weyl, 1922). The themes of
gravitational radiation, conformal invariance, and causal order were combined
and came to a head in the work of Penrose,666This history largely remains to
be written (Dennis Lemkuhl is working on this). For now, see e.g. Thorne
(1994), Frauendiener (2000), Friedrich (2011), Wright (2013, 2014), and Ellis
(2014). Furthermore, both the written AIP interview by Lightman (1989) and the
videotaped interview by Turing’s biographer Hodges (2014) are great and
intimate portraits of Penrose.  who also received additional inspiration from
the Dutch artist M.C. Escher and the spinor theory of Dirac. In Penrose (1963,
1964, 1965, 1966, 1968, 1972) he introduced most of the global causal
techniques and topological ideas that are now central to any serious
mathematical analysis of both gr and Lorentzian geometry (O’Neill, 1983;
Minguzzi, 2019). An important exception is global hyperbolicity, which has its
roots in the work of Leray (1953) and was adapted to gr by Choquet-Bruhat
(1967) and Geroch (1970). Global hyperbolicity is the main concept through
which the causal theory meets the pde theory, but Penrose hardly worked on the
pde side.777Although he was well aware of it: the singularity theorem in
Penrose (1965) assumes the existence of a Cauchy surface.

The second piece of history one needs in order to understand Penrose’s
contributions that are relevant to his Nobel Prize, is astrophysical.
Briefly:888See Israel (1987), Luminet (1992), Thorne (1994), Melia (2009),
Sanders (2014), Curiel (2019a), and Falcke (2020) for history, and Misner,
Thorne, & Wheeler (1973), Joshi (1993, 2007), Poisson (2004), and Weinberg
(2020) for theory. relatively light stars retire as white dwarfs, in which
nuclear burning has ended and inward gravitational pressure is stopped by a
degenerate electron gas. In 1931 Chandrasekhar discovered that this works only
for masses $m\leq 1.46M_{\odot}$, where $M_{\odot}$ is the solar mass. Heavier
stars collapse into neutron stars (typically after a supernova explosion), but
also these have an upper bound on their mass, as first suggested by
Oppenheimer & Volkoff (1939); the current value is about $2.3M_{\odot}$. Stars
that are more massive cannot stop their gravitational collapse and unless they
get rid of most of their mass/energy they collapse completely.999Supermassive
black holes like Sagittarius A* and M87* are probably formed by mergers and
accretion rather than collapse. But what does this mean mathematically?

Most of the early intuition came from the Schwarzschild solution, seen as a
model of the final state of such a collapse. This solution is spherically
symmetric, and by Birkhoff’s theorem any such vacuum solution must be
Schwarzschild (or Minkowski). It has two very notable features, namely _a
curvature singularity as $r\rightarrow 0$_ and _an event horizon at $r=2m$_.
Here it should be mentioned that initially both caused great confusion, even
among the greatest scientists involved such as Einstein and Hilbert, though
Lemaître was ahead of his time.101010See Tipler, Clarke, Ellis (1980), Godart
(1992), Eisenstaedt (1993), Thorne (1994), Earman (1995, 1999), and Earman &
Eisenstaedt (1999). Our understanding of the event horizon as a one-way
membrane is usually attributed to Finkelstein (1958). Apart from their exact
locations, these two features, then, may be taken to be the defining
characteristics of a black hole, _but especially the event horizon, which is
held to be responsible for the “blackness” of the “hole”._ However, even short
of a correct technical understanding of these features, from the 1920s until
the 1950s most leading researchers in gr (including Einstein, Eddington, as
well as Landau’s school in the Soviet Union, which covered all of theoretical
physics) felt that at least the singularity was an artifact of the perfect
spherical symmetry of the solution (and likewise for the big bang as described
by the spherically symmetric Friedman/flrw solution). This negative view also
applied to the first generally relativistic collapse model (Oppenheimer &
Snyder, 1939), now seen as groundbreaking, which is spherically symmetric and
therefore terminates in the Schwarzschild solution.

The achievement usually attributed to Penrose (1965), culminating in his Nobel
Prize, is that he settled (in the positive) the question whether a more
general (i.e. non-spherically symmetric) collapse of sufficiently heavy stars
(etc.) also leads to a black hole. But if anyone understood this was _not_ the
case for the construal of a black hole as an astrophysical object with event
horizon, it was Penrose himself! He must have been the first to recognize that
his singularity theorem from 1965 did not prove the existence of black holes;
under suitable hypotheses (involving both the concentration of matter and the
causal structure of space-time) it proved merely the existence (but not even
the precise nature or location) of incomplete null geodesics. As such, this
implies neither the existence of a _curvature_ singularity (not even if the
solution is close to Schwarzschild), nor that of an event horizon. Leaving the
former aside for the moment, the latter became the topic of what is now called
the _weak cosmic censorship conjecture_ :111111 It is worth stressing that
Penrose included a genericity restriction right from the beginning, _pace_
Dafermos (2012), p. 55. The emphasis on initial data in the second formulation
does not recur in the strong version of cosmic censorship below, but it is
unavoidable in any form of weak cosmic censorship in order to exclude the
naked big bang singularity from the conjecture: the point of the second (1979)
formulation is that the singularity lies to the future of the Cauchy
hypersurface in question.

> We are thus presented with what is perhaps the most fundamental question of
> general-relativistic collapse theory, namely: does there exist a “cosmic
> censor” who forbids the appearance of naked singularities, clothing each one
> in an absolute event horizon? In one sense, a “cosmic censor” can be shown
> _not_ to exist. For it follows from a theorem of Hawking that the “big bang”
> singularity is, in principle, observable. But it is not known whether
> singularities observable from outside will ever arise in a generic
> _collapse_ which starts off from a perfectly reasonable nonsingular initial
> state. (Penrose, 1969, p. 1162)
>
> A system which evolves, according to classical general relativity with
> reasonable equations of state, from generic non-singular initial data on a
> suitable Cauchy hypersurface, does not develop any spacetime singularity
> which is visible from infinity. (Penrose, 1979, p. 618)

Following Penrose (1979) we give a precise mathematical version in §2, but in
any case it should be clear that in order to prove the existence of black
holes from suitable assumptions one needs _both_ Penrose’s singularity theorem
(which gives at least some kind of singularity) _and_ the weak cosmic
censorship hypothesis (which gives the event horizon): the latter is the
missing link between theorem and reality.

Expanding the scope of cosmic censorship, Penrose (1979), p. 619, subsequently
argued that:

> It seems to me to be comparatively unimportant whether the observer himself
> can escape to infinity. Classical general relativity is a scale-invariant
> theory, so if locally naked singularities occur on a very tiny scale, they
> should also, in principle, occur on a very large scale in which a ‘trapped’
> observer could have days or even years to ponder upon the implications of
> the uncertainties introduced by the observations of such a singularity. (…)
> Indeed, for inhabitants of recollapsing closed universes (as possibly we
> ourselves are) there is no ‘infinity’, so the question of being locally
> ‘trapped’ is one of degree rather than principle. It would seem, therefore,
> that if cosmic censorship is a principle of Nature, it should be formulated
> in such a way as to preclude such _locally_ naked singularities.

This ban is called _strong cosmic censorship_ , which as first shown by
Penrose (1979) himself, comes down to the requirement of global hyperbolicity
(see §3). However: global hyperbolicity _of which space-time?_

More generally, in Penrose’s singularity theorem as well as in his two
versions of cosmic censorship, it is ambiguous to which space-times the
theorem and the hypothesis are applied. Traditionally, in gr one typically
studied analytically extended solutions to the Einstein equations like–in the
context of black holes–the Kruskal extension of the Schwarzschild solution,
and similarly (but now not doubled but “infinitely extended”) for
Reissner–Nordström and Kerr. The pde approach to gr, on the other hand, is
based on two slogans, appealing to the fundamental theorem of Choquet-Bruhat &
Geroch (1969):121212A _Cauchy surface_ in a space-time $(M,g)$ is a subset
$\Sigma\subset M$ such that each endless timelike curve intersects $\Sigma$
exactly once. This makes $\Sigma$ a closed connected $3d$ submanifold of $M$
which can be chosen space-like and hence Riemannian. A space-time is _globally
hyperbolic_ iff it has a Cauchy surface. Non-characteristic initial values for
the Einstein equations form a triple $(\Sigma,h,K)$, i.e. a $3d$ Riemannian
manifold $(\Sigma,h)$ equipped with an additional symmetric tensor $K_{ij}$,
satisfying four constraint equations. A _Cauchy development_ of such initial
data is a triple $(M,g,i)$, where $(M,g)$ is a $4d$ space-time solving the
Einstein equations, and $i:\Sigma\rightarrow M$ is an embedding such that
$i^{*}g=h$ and $i(\Sigma)$ is a Cauchy surface in $M$ with extrinsic curvature
$K$. Hence $(M,g)$ is globally hyperbolic. Choquet-Bruhat & Geroch (1969)
showed that there exists a Cauchy development of the given $(\Sigma,h,K)$,
called the _maximal globally hyperbolic development_ (mghd), that is maximal
_as a globally hyperbolic space-time solving the Einstein equations with
Cauchy surface $i(\Sigma)$ and given $(h,K)$_. This mghd is unique up to time-
orientation-preserving isometries preserving $\Sigma$, i.e. if $(M,g,i)$ and
$(M^{\prime},g^{\prime},i^{\prime})$ qualify then there is an isometry
$\psi:M\rightarrow M^{\prime}$ such that $\psi\circ i=i^{\prime}$. See
Choquet-Bruhat (2009) and Ringström (2009) for introductions to the pde
approach, supplemented by Sbierski (2016).

  * •

All valid _assumptions_ are about initial data;

  * •

All valid _questions_ are about the maximal globally hyperbolic development
(mghd) thereof.

This clearly affects strong cosmic censorship, in that asking that a
physically reasonable space-time be globally hyperbolic is now deemed empty,
since the mghd of any inital data automatically has this property. For similar
reasons also Penrose’s version of weak cosmic censorship needs to be
reformulated. His singularity theorem does make sense for both traditional
solutions and mghd, but the causes of geodesic incompleteness is quite
different is these two cases (except for the Kruskal solution).

Written about and by a mathematician, we start in §2 with _definitions_. In §3
we trace the evolution of Penrose’s idea of cosmic censorship, which is
illustrated by three black hole examples in §4. In the last section §5, the
whole story culminates in Penrose’s amazing and influential _final state
conjecture_. The conclusion is that although arguably Penrose did not quite
achieve what the Nobel Prize committee says, he developed most of the
techniques, saw the need for singularity theorems (of which he proved the
first) as well as cosmic censorship, and, perhaps most importantly, showed the
way to others.131313 In particular, using the pde approach Christodoulou
(1991, 1999b, 2009) finally established the formation of black holes both in
spherically symmetric collapse models with (scalar field) matter and in vacuum
solutions through focusing of gravitational waves, by proving both causal
geodesic incompleteness and the existence of an event horizon. See also
follow-ups by Klainerman & Rodnianski (2012) and Klainerman, Luk, & Rodnianski
(2014), and reviews by Bieri (2018) and Dafermos (2012). For the incorporation
of more realistic matter models see e.g. Burtscher & LeFloch (2014) and
Burtscher (2020). Finally, an appendix written by Erik Curiel traces the
_definitional_ history of the concept of a black hole.

##  2 Definitions

In mathematical physics it is essential to start from _definitions_ that are
physically relevant, mathematically precise, and workable. Penrose had a
remarkable gift for this.141414Penrose was clearly very good at capturing the
general spirit of the time in mathematical concepts; this is why his ideas so
quickly became mainstream, despite the unfamiliarity of even theoretical
physicists at the time with a field like topology (see Thorne, 1994, Chapter
13, which describes Penrose’s role in the gr community). But he did so in his
own unique individual way: ‘It was important for me always, if I wanted to
work on a problem, to think I had a different angle on it from other people.
Because I wasn’t good at following where everybody else went. I wasn’t the
kind of person who could pick up the prevalent arguments and knowledge of the
time. Other people were good at that. They could suck it all out and put it
together and make advances. I was the kind of person who’d have some kind of
quirky way of looking at something on my own, which I would hide away and work
at. So it meant that I had to have some way of looking at a problem that was
my own.’ (Lightman, 1989).  For our purpose, i.e. what to make of the citation
for his Nobel Prize, Penrose contributed at least five great definitions to
gr, namely of:

  * •

_Null infinity_ , in turn implying a definition of an event horizon and hence
of a black hole;

  * •

_Trapped surfaces_ , formalizing the condition that gravity is strong enough
to focus light-rays;

  * •

_Singularities in space-time_ , which he characterized through incomplete
causal geodesics;

  * •

_Weak cosmic censorship_ , stating that space-time singularities are covered
by event horizons;

  * •

_Strong cosmic censorship_ , forbidding even nearby causal contact with space-
time singularities.

In this section we explain the first three definitions, leaving the last two
for a separate section (§3).151515 Unexplained notions may be found in the
standard gr textbooks such as Wald (1984) or Chruściel (2019). A _space-time_
$(M,g)$ is a $4d$ connected time-orientable Lorentzian manifold with time
orientation, i.e. the metric has signature $(-+++)$ and one has a way of
distinguishing past from future by selecting, at each point in a continuous
and consistent way, a _forward_ and a _backward_ light-cone. This leads to one
of Penrose’s most effective notations, namely the relation $J\subset M\times
M$, where $(x,y)\in J$, also written as $y\in J^{+}(x)$ or $x\in J^{-}(y)$ or
$x\leq y$, iff there exists a future-directed (fd) causal curve from $x$ to
$y$. For $A\subset M$ we write $J^{\pm}(A)=\cup_{x\in A}J^{\pm}(x)$.
Similarly, $I\subset M\times M$ is defined by replacing ‘causal’ by
‘timelike’; one writes $x\ll y$ iff $(x,y)\in I$.

###  2.1 Null infinity

Null infinity and the ensuing concept of a black hole are predicated on the
following concept:161616 See originally Penrose (1964), who–in the context of
gravitational waves–adds the condition that every null geodesic has two end-
points on $\partial\tilde{M}$, defining $(M,g)$ to be _asymptotically simple_.
In that case each connected component of $\partial\tilde{M}$ is diffeomorphic
to ${\mathbb{R}}\times S^{2}$, as is often the case even more generally (and
as such is sometimes included in Definition 2.1). See also Hawking & Ellis
(1973), §6.9, Geroch (1977), Wald (1984), §11.1, Penrose & Rindler (1986),
Chapter 9, Stewart (1991), Chapter 3, Frauendiener (2000), Valiente Kroon
(2016), and Chruściel (2020), §3.1. The question how Definition 2.1 relates to
asymptotic flatness as defined through conditions on the metric, either in
space-time ($4d$) or in the initial value problem ($3d$), is very subtle;
smoothness of $(\tilde{M},\tilde{g})$ implies detailed fall-off (or ‘peeling’)
properties of the Weyl tensor at infinity. See e.g. Geroch (1977), Stewart
(1991), Klainerman & Nicolò (2003), Friedrich (2004, 2018), Adamo, Newman, &
Kozameh (2012), Dafermos (2012), Chruściel & Paetz (2015), and Paetz (2014).
However, for the usual stationary black hole solutions and more generally for
stationary space-times satisfying standard energy conditions the boundary is
smooth (Chruściel et al., 2001).

###### Definition 2.1

  1. 1.

A _conformal completion_ of a space-time $(M,g)$ is a space-time
$(\tilde{M},\tilde{g})$, where $\tilde{M}$ is a manifold with boundary, with
an embedding $\iota:M\hookrightarrow\tilde{M}$ such that
$\iota(M)=\mathrm{int}(\tilde{M}):=\tilde{M}\backslash\partial\tilde{M}$, and
$\iota$ is _conformal_ in that $\iota^{*}\tilde{g}=(\iota^{*}\Omega^{2})g$ for
some smooth positive function $\Omega:\tilde{M}\rightarrow{\mathbb{R}}^{+}$
that satisfies:

$\displaystyle\Omega>0\>\>\>\mathrm{on}\>\>\>\iota(M);$
$\displaystyle\Omega=0\>\>\>\mathrm{on}\>\>\>\partial\tilde{M};$
$\displaystyle\tilde{\nabla}\Omega\neq
0\>\>\>\mathrm{on}\>\>\>\partial\tilde{M}.$ (2.1)

  2. 2.

$(M,g)$ is _asymptotically flat at null infinity_ if it has a conformal
completion $(\tilde{M},\tilde{g})$ for which:

    1. (a)

$\partial\tilde{M}={\mathcal{I}}^{+}\cup{\mathcal{I}}^{-}$, where
${\mathcal{I}}^{\pm}:=\partial\tilde{M}\cap J^{\pm}(M)$, with $J^{\pm}(M)$
computed in $\tilde{M}$;

    2. (b)

The Ricci tensor of the original metric $g$ is such that
$R_{\mu\nu}=O(\Omega^{3})$ pointwise near $\partial\tilde{M}$.

In clause 2 and in what follows we tacitly identify $M$ with $\iota(M)$. Here
are some comments on this clause.

  1. 2(a)

The boundary ${\mathcal{I}}$ (pronounced, as Penrose suggests, “scri”) is
called _null infinity_. Its components ${\mathcal{I}}^{+}$ and
${\mathcal{I}}^{-}$ are called _future null infinity_ and _past null infinity_
, respectively. The idea is that ${\mathcal{I}}^{+}$ (${\mathcal{I}}^{-}$)
consists of limit points of future (past) directed null curves along which
$r\rightarrow\infty$.

  2. 2(a)

Asking $O(\Omega^{3})$ is on the safe side (one might ask
$O(\Omega^{2+\varepsilon})$ for $1/2<\varepsilon\leq 1$), and implies that
$\Omega^{-2}R_{\mu\nu}$ extends by continuity from $\iota(M)$ to zero on
$\partial\tilde{M}$, as in $R_{\mu\nu}(r)\sim 1/r^{3}$ as
$r\rightarrow\infty$. The simplest way to satisfy this is to assume that
$(M,g)$ solves the vacuum Einstein equations $R_{\mu\nu}=0$; in the presence
of matter one equivalently asks that $T_{\mu\nu}$ be $O(\Omega^{3})$.

A crucial fact, noted (_mutatis mutandis_) without proof in Penrose (1964,
1968), is that:171717If $R_{\mu\nu}=\lambda g_{\mu\nu}$, then
$\tilde{g}(\tilde{\nabla}\Omega,\tilde{\nabla}\Omega)=-\mbox{\footnotesize$\frac{1}{3}$}\lambda$
on $\partial\tilde{M}$, so that $\tilde{\nabla}\Omega$ is timelike and hence
$\partial\tilde{M}$ is spacelike if $\lambda>0$, and _vice versa_ if
$\lambda<0$ (Penrose, 1964, Lecture II; Penrose, 1968, p. 181). See Ashtekar,
Bonga, & Kesavan (2015) and Ashtekar & Magnon (1984), respectively, for these
cases. But as the king of null geometry in gr, Penrose must have taken special
pleasure in $\lambda=0$!

###### Proposition 2.2

On the boundary $\partial\tilde{M}$ the scaling function $\Omega$ satisfies
the eikonal equation

$\tilde{g}(\tilde{\nabla}\Omega,\tilde{\nabla}\Omega)=0,$ (2.2)

so that $\partial\tilde{M}$ (more precisely: each connected component thereof)
is a null hypersurface in $\tilde{M}$.181818 Short of the very subtle
regularity issues discussed in footnote 16, the boundary ${\mathcal{I}}$ is
smooth, and points like $i^{\pm}$ and $i^{0}$, typically included in Penrose
diagrams, are _not_ part of it. However, if one is interested in _spatial_
infinity (Geroch, 1977; Ashtekar, 1980, 2015) one could extend the definition
of a conformal completion so as to include these points.

_Proof_. A simple computation, based on a conformal rescaling of the Ricci
tensor,191919It is easily verified by direct computation, and found in many
books (Valiente Kroon, 2016, §5.2.2; Chruściel, 2020, Appendix H.6) that if
$g^{\prime}=\varphi^{2}g$, then
$R^{\prime}_{\mu\nu}=R_{\mu\nu}-\varphi^{-1}(2\nabla_{\mu}\nabla_{\nu}\varphi+g_{\mu\nu}\Delta_{g}\varphi)+\varphi^{-2}(4\nabla_{\mu}\varphi\nabla_{\nu}\varphi-
g_{\mu\nu}g(\nabla\varphi,\nabla\varphi))$. Now replace $g^{\prime}\leadsto g$
and $g\leadsto\tilde{g}$, so that $\varphi=1/\Omega$. This gives
$R_{\mu\nu}=\tilde{R}_{\mu\nu}+\Omega^{-1}(2\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}\Omega+\tilde{g}_{\mu\nu}\Delta_{\tilde{g}}\,\Omega)-3\Omega^{-2}\tilde{g}(\tilde{\nabla}\Omega,\tilde{\nabla}\Omega)\tilde{g}_{\mu\nu}$,
where
$\tilde{\nabla}=\tilde{g}^{\rho\sigma}\tilde{\nabla}_{\rho}\tilde{\nabla}_{\sigma}$.
This is eq. (11.1.16) in Wald (1984), which immediately yields (2.3). shows
that

$\tilde{g}(\tilde{\nabla}\Omega,\tilde{\nabla}\Omega)=\mbox{\footnotesize$\frac{1}{12}$}(\Omega^{2}\tilde{R}-R)+\mbox{\footnotesize$\frac{1}{2}$}\Omega\,\Delta_{\tilde{g}}\,\Omega.$
(2.3)

Since $\tilde{g}$ is regular on $\partial\tilde{M}$ (where $\Omega=0$) and
$R_{\mu\nu}=O(\Omega^{3})$ gives $R=O(\Omega)$, eq. (2.2) follows.  $\Box$

Following Hawking and Penrose–we leave the tangled history to the appendix–we
may then define

$\displaystyle\mathcal{B}:=M\backslash J^{-}({\mathcal{I}}^{+});$
$\displaystyle\mathcal{W}:=M\backslash J^{+}({\mathcal{I}}^{-}),$ (2.4)

called the _black hole region_ and the _white hole region_ in $M$,
respectively; each connected component of $\mathcal{B}$, if not empty, is then
simply a _black hole_.202020 See the appendix by Erik Curiel for historical
information on this definition. See also Thorne (1994), Chapter 7. It can be
shown that $J^{\mp}({\mathcal{I}}^{\pm})$ is open.212121 Since
$I^{\mp}({\mathcal{I}}^{\pm})\cap M=J^{\mp}({\mathcal{I}}^{\pm})\cap M$, one
could have used $I$ instead of $J$ in (2.4); see Wald (1984), p. 308. The
boundaries

$\displaystyle\mathcal{H}_{E}^{+}:=\partial\mathcal{B};$
$\displaystyle\mathcal{H}_{E}^{-}:=\partial\mathcal{W},$ (2.5)

decompose into the _future_ and _past event horizons_ of each of the black and
white holes in $M$ respectively. Since the hole regions $\mathcal{B}$ and
$\mathcal{W}$ are closed, the event horizons form part of the black/white
holes.

The analysis of such space-times is greatly facilitated by Penrose’s
_conformal diagrams_ ,222222 These confirm what Penrose often says, namely
that he prefers to think in terms of pictures. Since Penrose started in
algebraic geometry as a PhD student of Hodge in Cambridge, he was undoubtedly
influenced by the theory of Riemann surfaces in finding this concept (like
Weyl, as mentioned in the historical introduction). For example, _mutatis
mutandis_ the closed Poincaré disk is a Penrose diagram of the Poincaré upper
half plane. Penrose must also have been influenced by the famous _Circle
Limit_ woodcuts by Escher (nos. I–IV, dating from 1958–1960). See also Wright
(2013, 2014).  now called _Penrose diagrams_.232323Or, sometimes,
_Penrose–Carter diagrams_. Carter himself speaks of _PC-diagrams_ , perhaps
tongue-in-cheek saying that _PC_ stands for _Projective Conformal_. See
Chruściel (2020), Chapter 6 for an axiomatic theory of such diagrams. On a
pragmatic case-by-case basis, construct and draw $\tilde{M}$, suppress two-
spheres, and use coordinates in which null geodesics are at $\pm 45\degree$,
as in Minkowski space-time (indeed the $\pm 45\degree$ idea goes back to
Minkowski himself, who also drew his own diagrams).  These became an important
tool for visualizing black holes (Carter, 1973; Hawking & Ellis, 1973). The
title page shows one of the first such diagrams, drawn by Penrose himself.

However, the above definition of a black hole, though mathematically sweet, is
not uncontroversial:

> This definition depends on the whole future behaviour of the solution; given
> the partial Cauchy surface $\mathcal{S}(\tau)$,242424A _partial Cauchy
> surface_ $\Sigma$ is an acausal edgeless subset of $M$ (Hawking & Ellis,
> 1973, p. 204; Minguzzi, 2019, p. 95). This makes $\Sigma$ a closed
> hypersurface in $M$ which, because it is edgeless, is inextendible as an
> acausal set (though not necessarily maximal in $M$ as such). A sufficient
> condition for the existence of a partial Cauchy surface is the existence of
> a time function; see Minguzzi (2019), Theorems 3.39 and 4.100. In the pde
> approach it arises when strong cosmic censorship fails, see §3.2, and a
> Cauchy surface for the mghd turns into a partial one for the extension. In
> that case $\Sigma$ acquires a non-empty _Cauchy horizon_
> $H_{C}(\Sigma)=\partial D(\Sigma)$, where $D(\Sigma)$ is the domain of
> dependence of $\Sigma$, splitting into past and future ones
> $H_{C}(\Sigma)=H_{C}^{-}(\Sigma)\cup H_{C}^{+}(\Sigma)$.  one cannot find
> where the event horizon is without solving the Cauchy problem for the whole
> future development of the surface.’ (Hawing & Ellis, 1973, p. 319)
>
> [The future event horizon] is the boundary of an interior spacetime region
> from which causal signals can never be sent to the asymptotic observers, _no
> matter how long they are prepared to wait._ The region is therefore “black”
> in an absolute sense.’ (Ashtekar & Galloway, 2005, p. 2)
>
> The idea that nothing can escape the interior of a black hole once it enters
> makes implicit reference to all future time–the thing can never escape no
> matter how long it tries. Thus, in order to know the location of the event
> horizon in spacetime, one must know the entire structure of the spacetime,
> from start to finish, so to speak, and all the way out to infinity. As a
> consequence, no local measurements one can make can ever determine the
> location of an event horizon. That feature is already objectionable to many
> physicists on philosophical grounds: one cannot operationalize an event
> horizon in any standard sense of the term. Another disturbing property of
> the event horizon, arising from its global nature, is that it is prescient.
> Where I locate the horizon today depends on what I throw in it
> tomorrow–which future-directed possible paths of particles and light rays
> can escape to infinity starting today depends on where the horizon will be
> tomorrow, and so that information must already be accounted for today.
> Physicists find this feature even more troubling. (Curiel, 2019b, p. 29)

It is amusing how differently even top gr experts (and textbook authors!)
respond to this charge:

> [The above definition of an event horizon] is probably very useless, because
> it assumes we can compute the future of real black holes, and we cannot.
> (Rovelli, quoted in Curiel, 2019b, p. 30)
>
> I have no idea why there should be any controversy of any kind about the
> definition of a black hole. There is a precise, clear definition in the
> context of asymptotically flat spacetimes (…) I don’t see this as any
> different than what occurs everywhere else in physics, where one can give
> precise definitions for idealized cases but these are not
> achievable/measurable in the real world. (Wald, _ibid._ , p. 32)

What seems at stake here is what may be called _Earman’s Principle_ :

> While idealizations are useful and, perhaps, even essential to progress in
> physics, a sound principle of interpretation would seem to be that no effect
> can be counted as a genuine physical effect if it disappears when the
> idealizations are removed. (Earman, 2004, p. 191)

Note that two kinds of idealizations are involved in the case of event
horizons of black holes:

  1. 1.

The ability to know an entire space-time $(M,g)$, either from initial data or
by direct construction;

  2. 2.

The construction of null _infinity_ in terms of which black holes and event
horizons are defined.

Rovelli’s comment seems to apply to the first point but Wald’s to the second,
in which case they would not contradict each other. The need to idealize the
idea (!) that an event horizon prevents sending signals from the singularity
to observers “far away” by taking the latter to mean “at (null) infinity”
arises because _in general_ any _finite_ distance could potentially lie within
the event horizon. For _specific space-times_ like Kruskal or Kerr, the
horizons $\mathcal{H}_{E}^{\pm}$ as defined in (2.5) can be explicitly located
in $M$ without reference to (null) infinity.252525 This is different from the
idealization in phase transitions and spontaneous symmetry breaking, where
even in exactly solvable models one needs the idealization of the
thermodynamic limit to have such effects, at least according to their official
definition. See Butterfield (2011) and Landsman (2017), Chapter 10, for the
way to deal with Earman’s principle in these cases. Even if the space-time is
not known explicitly, the event horizon (if it has one) by definition lies at
some _finite_ distance from the singularity (if it has one). Hence in locating
the horizon, the phrase “at infinity” could be replaced by “sufficiently far
away” (from the singularity), which agrees with Earman’s principle–and
therefore Wald’s stance seems valid provided it concerns the second point.

On the other hand, the second point is predicated on the first, which remains
unresolved. Thus we are entering an almost axiomatic approach to gr here,
liable to the famous charge that it has ‘the advantages of theft over honest
toil’ (Russell, 1920, p. 71). However, nothing is wrong with an axiomatic
approach as long as one can find realistic models for the axioms (or
definitions) that show that they are reasonable. This is the case in Penrose’s
approach. The fact that we cannot ‘compute the future of black holes’ does not
disqualify the event horizon as an object of nature we can prove theorems
about (whose desirability may be different for theoretical and mathematical
physicists). What _is_ worrying is the precise relationship between the black
hole “shadow” in the EHT image of M87* and the event horizon as defined by
(2.5), which we cannot possibly know _now_.262626More precisely, where it was
53.5 million years ago. An additional complication is that (ignoring the
rotation of M87* for simplicity) the edge of the disk is not the event horizon
at $r=2m$ but the photon sphere at $r=3m$, further dislocated by optical
effects so that we actually see it at $r=\sqrt{27}m$, cf. Chruściel (2019),
§3.9.6 and Event Horizon Telescope Collaboration (2019).  This raises
epistemological questions about the role of theory in observation, which will
not even be addressed here, let alone answered. See also Franklin (2017).

###  2.2 Trapped surfaces

In their excellent review of Penrose’s 1965 singularity theorem, Senovilla &
Garfinkle (2015) explain that all singularity theorems in gr share the
following three assumptions (we quote _verbatim_):

  1. (i)

a condition on the curvature;

  2. (ii)

a causality condition;

  3. (iii)

an appropriate initial and/or boundary condition.

In Penrose (1965) condition (i) states that
$R_{\mu\nu}\dot{\gamma}^{\mu}\dot{\gamma}^{\nu}\geq 0$ along all null
geodesics $\gamma$. Condition (ii) states that the space-time be globally
hyperbolic with _non-compact_ Cauchy surface; the topological assumption
reflects the idea that the theorem is supposed to apply to black holes and
hence to asymptotically flat space-times. His condition (iii) is the existence
of a closed _trapped surface_ , which is one of the most important concepts in
all of black hole (mathematical) physics.272727See initially Hawking & Ellis
(1973), Chapter 9. The study of trapped surface formation from the pde point
of view began with Schoen & Yau (1983), who gave initial values that _already
contain_ trapped surface; see also Alaee, Lesourd, & Yau (2019). Christodoulou
(1991, 1999a, 2009) first proved the evolution of asymptotically flat initial
data _into_ trapped surfaces. Later literature may be traced back from Li & Yu
(2015) and Athanasiou & Lesourd (2020). See also references in footnote 13.
Here is Penrose’s own definition:

> A _trapped_ surface [is] defined generally as a closed, spacelike two-
> surface $T^{2}$ with the property that the two systems of null geodesics
> which meet $T^{2}$ orthogonally converge locally in future directions at
> $T^{2}$. (Penrose, 1965, p. 58)

In the presence of a radial coordinate $r$ as in the Schwarzschild,
Reissner–Nordström, and Kerr solutions, this condition is equivalent to the
(metric) gradient $\nabla r$ being _timelike_ , which in the Schwarzschild
solution happens for $r<2m$, and which in the other two (subcritical) cases is
the case at least for a while after crossing the event horizon. In general,
the convergence condition can be stated in terms of the null hypersurface $C$
generated by the future directed null congruence emanating from some
(instantaneous) spacelike two-sphere $S^{2}$, so that $\partial C=S^{2}$. In
terms of a tetrad $(e_{1},e_{2},L,\underline{L})$ with $(e_{1},e_{2})$
spacelike and tangent to $C$ , $L$ null and tangent as well as orthogonal to
$C$, and $\underline{L}$ null and pointing off $C$, normalized such that
$g(e_{i},e_{j})=\delta_{ij}$, $g(e_{i},L)=g(e_{i},\underline{L})=0$ for
$i,j=1,2$, and finally $g(L,\underline{L})=-1$ and of course
$g(L,L)=g(\underline{L},\underline{L})=0$, all defined on $C$, null extrinsic
curvatures are $2\times 2$ matrices $k_{ij}=g(\nabla_{j}L(t),e_{i}(t))$ and
$\underline{k}_{ij}=g(\nabla_{j}\underline{L},e_{i})$, with traces
$\theta=\mathrm{tr}(k)$ and $\underline{\theta}=\mathrm{tr}(\underline{k})$.
Then $S^{2}$ is trapped iff $\theta<0$ and $\underline{\theta}<0$ throughout
$S^{2}$. This condition is local and there are none of the problems afflicting
null infinity (cf. §2.1).

###  2.3 Singularities in space-time

Ironically, although this is also seen as one of Penrose’s most important
contributions to gr (and was immediately recognized as such by his
contemporaries like Hawking), his definition of a singular space-time (i.e. as
being causally geodesically incomplete) has to be inferred from his proof by
contradiction of his singularity theorem in Penrose (1965, 1968), to the
effect that properties (i), (ii), and (iii) of the previous subsection exclude
the possibility that $(M,g)$ is also future null geodesically complete.282828
A geodesic $\gamma$, which we take by definition to be affinely parametrized,
is called _complete_ if it can be extended to arbitrary values of its
parameter, i.e. is defined as a map $\gamma:{\mathbb{R}}\rightarrow M$. It is
_future complete_ if it is defined as a map $\gamma:[a,\infty)\rightarrow M$
for some $a\in{\mathbb{R}}$, etc. In the Riemannian case, by the Hopf–Rinow
theorem geodesic completeness is equivalent to completeness in the topological
metric $d$ derived from the Riemannian metric $g$ as the infimum over the path
length (computed from $g$) of all curves connecting two given points. Since a
Lorentzian metric no longer defines a topological metric, this result is lost.

Hawking (1966), §6.1, much more explicitly discusses the need for a good
definition of a singular space-time, upon which he arrives at the
contrapositive: ‘We shall say that $M$ is singularity-free if and only if it
is timelike and null geodesically complete.’292929As in Hawking & Ellis
(1973), §8.1, Penrose is not mentioned here but there is generic
acknowledgement in the Preface. However, unlike Penrose (1965, 1968), Hawking,
and later Hawking & Ellis (1973), §8.1, emphatically apply this definition to
the case where $(M,g)$ is metrically inextendible, in that it cannot be
isometrically embedded as an open submanifold of a larger space-time
$(M^{\prime},g^{\prime})$, subject to certain regularity conditions on both
the manifold and the metric. Adapting a definition given by Clarke (1993), p.
10 (who however uses arbitrary curves) we may formalize this by:

###### Definition 2.3

A space-time is _singular_ if it contains an incomplete causal geodesic
$\gamma:[0,a)\rightarrow M$ such that there is no extension
$\theta:M\rightarrow M^{\prime}$ for which $\theta\circ\gamma$ is extendible.

This refinement of Penrose’s definition was originally proposed in order to
avoid trivial cases: removing any point from Minkowski space-time makes it
geodesically incomplete, but also think of Schwarzschild for $r>2m$ only. But
with hindsight, we can say it makes a big difference to impose inextendibility
also in nontrivial cases where strong cosmic censorship fails, as will be
explained in due course. We therefore follow Penrose in defining a space-time
to be singular iff it is causally geodesically incomplete, leaving it open
whether it can be extended–indeed his 1965 singularity theorem (or any later
version thereof) gives no information about metric inextendibility at all. As
we shall see, if the three conditions in Penrose’s singularity theorem hold,
the cases where the space-time in question can or cannot be extended are quite
different in so far as the nature of the incompleteness is concerned, and both
cases are equally interesting. Even apart from this, Penrose’s definition is
once again controversial; it ended a long period of confusion, but it did so
at a price, as was recognized right from the start. As Geroch
(1968),303030Further to this classical paper on singularities, see also Earman
(1995, 1996), Senovilla (1997), and Curiel (1999, 2019a).  p. 526, states:

>   1. (a)
>
> there is no widely accepted definition of a singularity in general
> relativity;
>
>   2. (b)
>
> each of the proposed definitions is subject to some inadequacy.
>
>

For example, the link between singularities and diverging curvature is lost,
although this was the original intuition from both the Schwarzschild and the
Friedman “singularities”. Furthermore, even within the confines of defining
singularities through incomplete curves, singling out (causal) geodesics
excludes some interesting space-times intuitively felt to be singular–but this
can only be detected through the incompleteness of more general curves
(Geroch, 1968, appendix). In fact, Penrose (1979) did incorporate these at a
later stage, as will be discussed in §3.1. But ultimately we side with Hawking
and Ellis:313131Reminiscent of the great slogan ‘ _A good definition should be
the hypothesis of a theorem_ ’ (attributed to J. Glimm).

> ‘Timelike geodesic completeness has an immediate physical significance in
> that it presents the possibility that there could be freely moving observers
> or particles whose histories did not exist after (or before) a finite
> interval of proper time. This would appear to be an even more objectionable
> feature than infinite curvature and so it seems appropriate to regard such a
> space as singular. (…) The advantage of taking timelike and/or null
> incompleteness as being indicative of the presence of a singularity is
> [also] that on this basis one can establish a number of theorems about their
> occurrence.’ (Hawking & Ellis, 1973, p. 258).

##  3 Cosmic censorship

In this section we review Penrose’s original versions of cosmic censorship,
followed by pde reformulations now in use. Penrose (1979) gave a precise
statement of strong cosmic censorship that seems almost forgotten, but
translated this into an equivalent characterization in terms of global
hyperbolicity that became very influential. Since only seasoned relativists
will be able to relate global hyperbolicity to the original ideas behind
cosmic censorship (as reviewed in the historical introduction), we first give
a unified formulation of both weak and strong cosmic censorship along the
lines of Penrose (1979).

###  3.1 Cosmic censorship à la Penrose

Remarkably, where Penrose (1965) defined singularities in terms of _incomplete
causal geodesics_ , Penrose (1979) switches to _endless timelike curves_. It
turns out that the change from ‘causal’ to ‘timelike’ does not matter,323232In
the light of the analysis below, this follows from Theorem (2.3) in Geroch,
Kronheimer, & Penrose (1972). but the change from _geodesics_ to _curves_ is
quite substantial.333333Penrose’s timelike curves are smooth by convention
(Penrose, 1972, pp. 2–3). Following Minguzzi (2019), we prefer to work with
_continuous causal_ curves, which behave better under limits (e.g. smooth
timelike curves typically converge uniformly, if they do, to continuous causal
curves, whereas limits of the latter, if they exist, lie in the same class).
We say that a continuous curve $c:I\rightarrow M$ is _causal_ if every point
$x=c(t)$ on the curve ($t\in I$) has a normal neigbourhood $U_{x}$ such that
the unique geodesic connecting $x$ with any later point $y\in U_{x}$ (with
$y=c(t^{\prime})$ for $t^{\prime}>t$) is causal. To analyse such curves we
introduce an auxiliary (complete) Riemannian metric $h$ on $M$ (which always
exists), with associated topological metric $d_{h}$ defined as in footnote 28,
and defining things like absolute continuity etc. A continuous curve
$c:I\rightarrow M$ is causal iff (possibly after reparametrization) it is
absolutely continuous and a.e. differentiable on $I$ with $\dot{c}$ causal.
Moreover, for $[s,u]\in I$ the Riemannian length
$L_{h}(c_{|[s,u]})=\int_{s}^{u}dt\,\sqrt{h(\dot{c}_{n}(t),\dot{c}_{n}(t))}$ is
well defined and finite. See e.g. Theorem 2.3.2 in Chruściel (2011), §2.3, and
Theorem A.1 in Candela _et al_ (2010). Since the function $u\mapsto
L_{h}(c_{|[s,u]})$ is strictly increasing and hence invertible, any continuous
causal curve $c$ may be parametrized by $h$-arc length. If an fd (i.e. future-
directed) continuous causal curve $c:[a,b)$ is parametrized by (or
proportional to) $h$-arc length, then $b=\infty$ iff $c$ is future endless
(Minguzzi, 2008, Lemmas 2.6 and 2.17).  Forbidding signaling by singularities
thus defined turns out to be equivalent to global hyperbolicity (of all of
space-time in case of strong cosmic censorship and of
$J^{-1}({\mathcal{I}}^{+})$ in the weak version), which is very neat and may
justify this change. However, had the original definition in terms of causal
_geodesics_ been used, then presumably some weaker causality condition than
global hyperbolicity would have been found.343434It is the second (‘converse’)
part of the proof of Theorem 3.2 below that does not work for causal geodesics
instead of curves, since the curve $\gamma$ constructed there is not
necessarily a geodesic. This goes back to the definition of domains of
dependence and Cauchy surfaces in terms of causal curves rather than
geodesics, and may explain Penrose’s (1979) choices.

In any case, the basic problem is to express mathematically what it means for
a signal to emanate from a singularity, since the latter is not part of space-
time. Happily, it is precisely his own definition of singularities in terms of
incomplete causal geodesics–now general causal curves–that enabled Penrose to
overcome this problem, drawing on earlier work (Geroch, Penrose, & Kronheimer,
1972), as follows.

An endless causal curve may either be complete, i.e. have infinite length, or
incomplete (finite length). In the first case it may either go off to
infinity, or hover around in a compact set, which is impossible in a strongly
causal space-time; hence Penrose makes this assumption. In the second case
(also assuming $\gamma$ is future directed for simplicity) it may either be
thought of as crashing into a singularity, or leading to the edge of an
extendible space-time. If $\gamma$ is not endless but has a future endpoint
$y$, then

$I^{-}(\gamma)=I^{-}(y).$ (3.1)

If $(M,g)$ is strongly causal, then $I^{\pm}(x)=I^{\pm}(y)$ iff $x=y$. The
idea, then, is that an _endless_ (continuous) causal curve $\gamma$
corresponds to an _ideal_ point $y$ of space-time, which is not contained in
$M$ but is still defined by $I^{-}(\gamma)$, this time without (3.1). By the
above case distinction, at least in strongly causal space-times ideal points
may be either points at infinity, or singularities, or boundary points,
respectively.353535 Geroch, Kronheimer, & Penrose (1972) and in their wake
Hawking & Ellis (1973), §6.8, show that (assuming strong causality) both real
points and ideal points of $M$ correspond to subsets $U\subset M$ that are:
(i) open, (ii), past sets, i.e. $I^{-}(U)\subset U$, and (iii) indecomposable,
in that $U\neq U_{1}\cup U_{2}$ where $U_{1}$ and $U_{2}$ have properties (i)
and (ii) and are neither empty nor equal to $U$. Such sets are called IP (for
Indecomposable Past set), and those that are not of the form $U=I^{-}(x)$ for
some $x\in M$ are called TIP’s (for Terminal IP’s); these TIP’s are
$U=I^{-}(\gamma)$ for some future-endless timelike curve $\gamma$. It would be
more natural if strong cosmic censorship merely excluded visible TIP’s coming
from incomplete curves, but Penrose (1979) gives various arguments for
including complete curves $\gamma$, too, and in any case his Theorem 3.2 below
holds only if all TIP’s are included.

Now, if $\gamma$ does have a future endpoint $y\neq x$, the crucial condition
$I^{-}(\gamma)\subset I^{-}(x)$ occurring in Definition 3.1 below–albeit in
the endless case–is evidently equivalent to $I^{-}(y)\subset I^{-}(x)$, i.e.
$y\ll x$, which states that there exists an fd _timelike_ curve or signal from
$y$ to $x$. If $\gamma$ is endless, on the other hand, there is no such point
$y$, but we may still interpret (3.3) as saying that timelike signals
emanating from the ideal point $y$ defined by $\gamma$ (such as a
singularity), or from arbitrarily nearby points, can reach $x$. This exegesis
also applies to the condition $I^{-}(\gamma)\subset J^{-}(x)$, in which case
some _causal_ curve from $y$ reaches $x$.

The following definition then captures the two notions of cosmic censorship in
Penrose (1979).363636See also Królak (1986) for a different axiomatization of
weak cosmic censorship à la Penrose, as well as useful analysis. We recall
that these definitions and the ensuing theorem presuppose that $(M,g)$ is
strongly causal.373737Strong causality is used through its implication that
$I^{\pm}(x)=I^{\pm}(y)$ iff $x=y$, without which Definition 3.1 would make
little sense, as well as through its implication of non-total imprisonment,
without which the invocation of Theorem 2.53 in Minguzzi (2019) in the proof
of Theorem 3.2 below would fail. It is also part of one of the traditional
definitions of global hyperbolicity (namely strong causality plus compactness
of causal diamonds $J^{+}(y)\cap J^{-}(x)$), but since the proof of Theorem
3.2 is based on contradicting compactness of causal diamonds, in that role it
is hardly necessary anymore, since Hounnonkpe & Minguzzi (2019) proved that a
non-compact space-time with $\dim(M)\geq 3$ is globally hyperbolic iff all
double cones are compact.

###### Definition 3.1

In both cases below, let $\gamma$ denote a future-directed future-endless
causal curve.383838There is a similar definition in terms of _past-directed_
endless causal curves, in which $I^{-}(\cdot)$ is replaced by $I^{+}(\cdot)$
throughout. As far as strong cosmic censorship is concerned this definition
turns out to be equivalent to the given one, cf. Theorem 3.2 below, whilst for
weak cosmic censorship the above definition is the appropriate one. Note that
strong cosmic censorship does not imply weak cosmic censorship since (3.2) has
$J^{-}(x)$ with $x$ possibly in ${\mathcal{I}}^{+}\subset\partial\tilde{M}$,
whilst (3.3) has $I^{-}(x)$ with $x\in M$.

  * •

A space-time $(M,g)$ that is asymptotically flat at null infinity (§2.1)
contains a _naked singularity_ if there is a curve $\gamma$ as above, and a
point $x\in J^{-}({\mathcal{I}}^{+})$ in its causal future in $\tilde{M}$, in
the sense that

$I^{-}(\gamma)\subset J^{-}(x).$ (3.2)

Penrose’s _weak cosmic censorship conjecture_ states that space-times that are
asymptotically flat at null infinity and arise from “generic” regular initial
conditions contain no naked singularities.

  * •

A space-time $(M,g)$ contains a _locally naked singularity_ if there is a
curve $\gamma$ as above, and a point $x\in M$ in its chronological future, in
the sense that

$I^{-}(\gamma)\subset I^{-}(x).$ (3.3)

Penrose’s _strong cosmic censorship conjecture_ states that “generic” [in his
own words: “physcially reasonable”] space-times do not contain locally naked
singularities.

It should be defined precisely what “generic” means, lest these conjectures
turn into a definition of genericity! Penrose did not do this, and we will
return to this point in §3.2. It is important to realize that in this
definition Penrose does not require $\gamma$ to be _incomplete_ , but merely
_endless_. Indeed the notion of (in)completeness is hard to define for non-
geodesic curves since it depends on the parametrization; if, as we do,
continuous causal curves are parametrized by arc length as measured by an
auxiliary complete Riemannian metric (see footnote 33), then the distinction
between endlessness and incompleteness cannot even be made, because any
endless curve has infinite arc length.393939Recall that affinely parametrized
geodesics are incomplete iff they are endless and have finite parameter
length. Beyond moving from causal geodesics to general causal curves, this
further generalization allows even more singularities, and has the effect of
making the notion(s) of cosmic censorship more stringent–in excluding a larger
class of naked or locally naked singularities–than Penrose’s (1965)
singularity theorem would suggest.404040The conditions (3.2) and (3.3) do make
sense for complete future endless causal curves: for example, in anti-de
Sitter space one has endless causal curves $\gamma$ and points $x$ such that
$I^{-}(\gamma)\subset J^{-}(x)$, but this space is not regarded as singular
(it is a space-time of constant negative curvature). Penrose (1979), p. 623
notes that this is impossible in space-times that are asymptotically flat at
null infinity, and indeed anti-de Sitter space has a negative cosmological
constant with timelike future null infinity.

The following theorem, of which part 2 is due to Penrose (1979) with a
slightly different proof, and part 1 is an almost trivial addition, is the
main characterization of cosmic censorship in his sense.

###### Theorem 3.2

  1. 1.

If $(M,g)$ is asymptotically flat at null infinity, then it has no naked
singularities iff the exterior region $J^{-}({\mathcal{I}}^{+})$ in
$\tilde{M}$ (which by definition includes
${\mathcal{I}}^{+}\subset\partial\tilde{M}$) is globally
hyperbolic.414141Tipler, Clarke & Ellis (1980), p. 176, made this the
_definition_ of weak cosmic censorship. It may be closer to Penrose’s (1979)
formulation to require $x\in J^{-}({\mathcal{I}}^{+})\cap J^{+}(\Sigma)$ in
the first part of Definition 3.1, where $\Sigma$ is some partial Cauchy
surface in $M$, in which case Theorem 3.2 yields global hyperbolicity of
$J^{-}({\mathcal{I}}^{+})\cap J^{+}(\Sigma)$. This is similar to the condition
${\mathcal{I}}^{+}\subset\overline{D^{+}(\Sigma)}$ making $(M,g)$ _future
asymptotically predictable_ from $\Sigma$ (Hawking & Ellis, 1973, p. 312), but
is equivalent to it only under further regularity assumptions (Królak, 1986,
Lemma 2.10). See also Wald (1984), §12.1 and Chruściel (2020), §3.5.1.

  2. 2.

In general, a space-time $(M,g)$ has no locally naked singularities iff it is
globally hyperbolic.

It should be clear intuitively that at least part 2 of the theorem is true (in
the contrapositive): if a space-time contains a locally naked singularity,
represented by $\gamma$ as in Definition 3.1, then $\gamma$ will not reach any
partial Cauchy-surface $\Sigma$ lying in the future of $x$, since it crashes
at the singularity lying in the past of $x$. Conversely, if no Cauchy surface
exists then one can construct such a curve $\gamma$. See also §4.

_Proof._ We prove the inference from a locally naked singularity to non-global
hyperbolicity by contradiction. Suppose that (3.3) holds for some $\gamma$ and
$x$ and that $(M,g)$ is globally hyperbolic. Take $y\in\gamma$ and then a
future-directed sequence $(y_{n})$ of points on $\gamma$, with $y_{0}=y$.
Because of (3.3) this sequence lies in $J^{+}(y)\cap J^{-}(x)$, which is
compact by assumption. Hence $(y_{n})$ has a limit point $z$ in $J^{+}(y)\cap
J^{-}(x)$. Now define curves $(c_{n})$ as the segments of $\gamma$ from $y$ to
$y_{n}$. By the curve limit lemma,424242One needs Theorem 2.53 in Minguzzi
(2019), of which part (i) applies: Let $(c_{n}:[0,b_{n}]\rightarrow M)$ be a
sequence of fd continuous causal curves parametrized by $h$-arc length in a
non-imprisoning space-time such that $c_{n}(0)\rightarrow x$ and
$c_{n}(b_{n})\rightarrow y\neq x$. Then there exists an fd continuous causal
curve $c:[0,b]\rightarrow M$, where $b<\infty$ as well as a subsequence of
$(c_{n})$ that converges uniformly to $c$ (including $b_{n}\rightarrow b$ at
the endpoint). Penrose (1979) gives a more complicated argument, perhaps since
the version of the curve limit lemma just cited was not available at the time,
or because he wanted to use his TIP’s (which we avoid). these curves have a
uniform limit, whose arc length (as measured by an auxiliary complete
Riemannian metric, see footnote 33) is on the one hand infinite (since
$\gamma$ is endless and hence has infinite arc length, which is approached as
the $y_{n}$ move up along $\gamma$), but on the other hand is finite, since it
must end at $z$ (and fd continuous causal curves have finite arc length iff
they have an endpoint). Hence $(M,g)$ cannot be globally hyperbolic.434343This
also works for part 1, where $x\in J^{-}({\mathcal{I}}^{+})$), eq. (3.2) also
implies $y_{n}\in J^{+}(y)\cap J^{-}(x)$. Conversely, if there are $x,y$ for
which $J^{-}(x)\cap J^{+}(y)$ is not compact, one can easily construct a
future-directed future-endless causal curve $\gamma$ such that (3.3) holds for
the given $x$ (Penrose, 1979, p. 624). Since (3.3) trivially gives
$I^{-}(\gamma)\subset J^{-}(x)$, also this implication works for part 1.
$\Box$

Especially its definition through the existence of a Cauchy surface relates
global hyperbolicity to _determinism_ , the idea being that any event in a
globally hyperbolic space-time is determined by certain initial data on a
Cauchy surface in it, at least as long as the (classical) universe is governed
by hyperbolic partial differential equations. This is clearly true for the
gravitational field itself (as long as it satisfies the Einstein equations),
and also has considerable backing for other fields.444444See e.g. Choquet-
Bruhat (2009) and Bär, Ginoux, and Pfäffle (2007), respectively, as well as
Earman (1995, 2007). This does not imply that _non_ -globally hyperbolic
space-times are necessarily _in_ deterministic: the point is rather that
signals from a (locally) naked singularity can reach an event without
ultimately coming from a Cauchy surface, so that the event is influenced by
data other than those at an initial-value surface. Thus the event in question
may still be fully determined–but it is not determined by the initial data
that were supposed to do so.454545 The closest analogue to this generally
relativistic situation occurs in _non-relativistic_ mechanics, where bodies
may disappear to infinity in finite time (Xia, 1992; Saari & Xia, 1995), and
hence, by the same (time-reversed) token, may _appear_ from nowhere in finite
time and hence influence affairs in a way unforeseeable from any Cauchy
surface. See Earman (2007), §3.6.

Conversely, the flagrant indeterminism concerning the unknown fate of someone
falling into a black hole singularity is compatible with global hyperbolicity
(as in e.g. the Schwarzschild solution). Furthermore, suppose some globally
hyperbolic space-time $(M,g)$ solving the Einstein equations is metrically
extendible (see §2.3), such that the extension $(M^{\prime},g^{\prime})$ is
either not globally hyperbolic at all, or is globally hyperbolic but not with
respect to any Cauchy surface $\Sigma$ in $M$. Then again, although all things
in $(M^{\prime},g^{\prime})$ may be determined, they are not determined by the
initial data on $\Sigma$ one expected to do so.464646Doboszewski (2017, 2019,
2020) analyzes the connection between global hyperbolicity, extendibility, and
determinism.

###  3.2 Cosmic censorship in the initial value (pde) formulation

Definition 3.1 of the cosmic censorship conjectures is inappropriate from the
point of view of the initial-value problem. Recall from §1 that in this
approach all valid questions are about the maximal globally hyperbolic
development $(M,g,\iota)$ of initial data $(\Sigma,h,K)$. Since the mghd is
always globally hyperbolic, the strong version is trivial by Theorem 3.2. For
weak cosmic censorship there is a subtle issue about which asymptotically flat
initial data lead to mghd that are asymptotically flat at null infinity (see
footnote 16), but even granting this, the real problem is that even in clear
counterexamples to the Penrosian conjecture (such as $m<0$ Schwarzschild, see
§4), the space $J^{-}({\mathcal{I}}^{+})$ computed for the mghd is globally
hyperbolic. Thus the Penrosian version, applied to the mghd, would hold
despite naked singularities!

There is no crystal-clear logical path from Penrose’s formulation of the
cosmic censorship conjectures to the current versions used in the pde
literature, but there is some continuity of ideas. First, as to the weak
version, in order to strengthen Definition 2.1 (or rather some slight
variation thereof) Geroch & Horowitz (1978) proposed that _future null
infinity ${\mathcal{I}}^{+}$ be null geodesically complete_.474747 This
condition is nontrivial to state; for example, in the metric $\tilde{\eta}$
used below even ${\mathcal{I}}^{+}$ for the standard conformal completion
$(\tilde{\mathbb{M}},\tilde{\eta})$ of Minkowski space-time
$(\mathbb{M},\eta)$ is incomplete. Completeness of curves depends on their
parametrization. Geodesics are affinely parametrized by definition (and an
affine reparametrization does not affect their (in)completeness), but a change
in $\Omega$ changes the unphysical metric $\tilde{g}$ (for given physical
metric $g$). Hence the notion of a geodesic and its (in)completeness depends
on the choice of $\Omega$. As recognized by Geroch & Horowitz (1978)
themselves, the correct approach is to use the freedom of rescaling $\Omega$
to ensure that $\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}\Omega=0$ on
${\mathcal{I}}^{+}$, and require null geodesic completeness of the null
hypersurface ${\mathcal{I}}^{+}$ in this “gauge”, in which the flow of
$\tilde{\nabla}\Omega$ is geodesic. See also Wald (1984), §11.1 or Stewart
(1991), §3.6.  This was motivated by the following example. In light-cone
coordinates, the standard conformal completion
$(\tilde{\mathbb{M}},\tilde{\eta})$ of Minkowski space-time
$(\mathbb{M},\eta)$, described for example in Penrose (1968), pp. 175–177, is
given by

$\displaystyle\hat{\mathbb{M}}$
$\displaystyle:=\\{(p,q,\theta,\varphi)\mid(p,q)\in(-\mbox{\footnotesize$\frac{1}{2}$}\pi,\mbox{\footnotesize$\frac{1}{2}$}\pi)^{2},p\geq
q,(\theta,\varphi)\in S^{2}\\}\cup{\mathcal{I}}^{+}\cup{\mathcal{I}}^{-};$
(3.4) $\displaystyle{\mathcal{I}}^{+}$
$\displaystyle=\\{(p,q,\theta,\varphi)\mid
p=\mbox{\footnotesize$\frac{1}{2}$}\pi,q\in(-\mbox{\footnotesize$\frac{1}{2}$}\pi,\mbox{\footnotesize$\frac{1}{2}$}\pi),(\theta,\varphi)\in
S^{2}\\};$ (3.5) $\displaystyle{\mathcal{I}}^{-}$
$\displaystyle=\\{(p,q,\theta,\varphi)\mid
p\in(-\mbox{\footnotesize$\frac{1}{2}$}\pi,\mbox{\footnotesize$\frac{1}{2}$}\pi),q=-\mbox{\footnotesize$\frac{1}{2}$}\pi,(\theta,\varphi)\in
S^{2}\\};$ (3.6) $\displaystyle\hat{\eta}$
$\displaystyle=-dp\,dq+\mbox{\footnotesize$\frac{1}{4}$}\sin^{2}(p-q)(d\theta^{2}+\sin^{2}\theta
d\varphi^{2});$ (3.7) $\displaystyle\Omega$ $\displaystyle=\cos p\cos q.$
(3.8)

Now on the one had, truncating ${\mathcal{I}}^{+}$ to for example
$\\{(p,q,\theta,\varphi)\mid
p=\mbox{\footnotesize$\frac{1}{2}$}\pi,q\in(-\mbox{\footnotesize$\frac{1}{2}$}\pi,0)\\}$
instead of (3.5) would still define a conformal completion of
$(\mathbb{M},\eta)$, with respect to which the future light-cone $J^{+}(0)$ is
a fake black hole $\mathcal{B}$ in $\mathbb{M}$. On the other hand, removing
$\mathcal{B}=J^{+}(0)$, the ensuing space-time $(\mathbb{M}\backslash
J^{+}(0),\eta)$ has a conformal completion (such as the one just described),
which by design is free of black holes. In both undesirable cases future null
infinity is incomplete (in the sense of footnote 47).484848This clause seems
to be an improvement over an inextendibility condition proposed by Geroch
(1977), which did not exclude cases like $(\mathbb{M}\backslash
J^{+}(0),\eta)$. However, inextendibility plus some regularity condition
enabled Geroch (1977) to prove uniqueness of conformal completions, a result
that seems to have no analogue for Definition 2.1, even in strengthened form.

Completeness of future null infinity in the above sense, then, was taken to be
the pde reformulation of weak cosmic censorship (Christodoulou, 1999a),
although it turns the Penrosian version on its head! For whereas his version
states that _outgoing_ signals from a black hole singularity are blocked by an
event horizon $\mathcal{H}^{+}_{E}$, the new version is about _incoming_
(null) signals: the further these are away from $\mathcal{H}^{+}_{E}$, the
longer it takes them to enter $\mathcal{H}^{+}_{E}$, and in the limit at null
infinity this takes infinitely long, making ${\mathcal{I}}^{+}$ complete. Yet
the pde version appears to strengthens Penrose’s: heuristically, and
contrapositively, lack of global hyperbolicity of $J^{-}({\mathcal{I}}^{+})$
gives a partial Cauchy surface $\Sigma$ a Cauchy horizon which cuts off
${\mathcal{I}}^{+}$, making it incomplete (this will perhaps be clearer from
the examples in §4).494949Christodoulou (1999a) actually reformulates the
above definition of weak cosmic censorship in such a way that the idealization
${\mathcal{I}}^{+}$ no longer occurs. Let $(\Sigma,h,K)$ be asymptotically
flat initial data for the Einstein equations (satisfying the constraints),
with mghd $(M,g,i)$. He then defines $(M,g)$ to have “complete future null
infinity” iff for any $s>0$ there exists a region $B_{0}\subset
B\subset\Sigma$ such that $\partial D^{+}(B)$, which is ruled by null
geodesics, has the property that each null geodesic starting in $\partial
J^{+}(B_{0})\cap\partial D^{+}(B)$ can be future extended beyond parameter
value $s$. Here $D^{+}(B)$ is the future domain of dependence of $B$, and each
null geodesic in question is supposed to have tangent vector $L=T-N$, where
$T$ is the fd unit normal to $\Sigma$ in $M$ and $N$ is the outward unit
normal to $\partial B$ in $\Sigma$. See also Christodoulou & Klainerman (1993)
for background on these constructions.  Thus we obtain:

###### Definition 3.3

  * •

The _weak cosmic censorship conjecture_ states that if “generic” complete
initial data have a mghd that is asymptotically flat at null infinity, then
future null infinity is complete.

  * •

The _strong cosmic censorship conjecture_ states that the mghd of “generic”
complete initial data is metrically inextendible (as a space-time in a
regularity class to be specified in detail).

For convenience, we have added the strong version of cosmic censorship used in
the pde approach, whose path from the Penrosian formulation we now try to
trace.505050I am greatly indebted to Juliusz Doboszewski for drawing my
attention to the early papers by Moncrief _et al_. First, in a paper on _weak_
cosmic censorship, Moncrief & Eardley (1981), p. 889, propose an ‘(informally
stated) global existence conjecture’:

> Every asymptotically flat initial data set with $\mathrm{tr}K=0$ may be
> evolved to arbitrarily large times (…)

adding that its proof would ‘in essence prove the [weak] cosmic censorship
conjecture for asymptotically flat space-times’. For initial data given on a
compact Cauchy surface they propose something similar, and in doing so they
opened the door to regarding cosmic censorship as a global existence problem
for the (vacuum) Einstein equations, as indeed the title of their paper
already expresses. In this spirit, Moncrief (1981), p. 88, paraphrases
Penrose’s strong version as expressed by Theorem 3.2 as

> i.e., that the maximal Cauchy development of a generic initial data set is
> inextendible.

This is made more precise by Chruściel, Isenberg, & Moncrief (1990), who open
their abstract as follows:

> The strong cosmic censorship conjecture states that ‘most’ spacetimes
> developed as solutions of Einstein’s equations from prescribed initial data
> cannot be extended outside of their maximal domains of dependence.

They later (§3) specify the word ‘most’ in terms of open and dense subsets in
the space of initial data.515151Detailed mathematical criteria for genericity
(which are suggested by pde theory and whose physical relevance is doubted
outside the pde community) may be also be found for example in Dafermos (2003)
and Luk & Oh (2019a), §3.

Chruściel (1992) introduced the notion of a _development_ of initial data
$(\Sigma,h,K)$ as a triple $(M,g,i)$, where $(M,g)$ is a $4d$ space-time
solving the vacuum Einstein equations, and $i:\Sigma\rightarrow M$ is an
embedding such that $i^{*}g=h$ and $i(\Sigma)$ has extrinsic curvature $K$;
the difference from a Cauchy development (see footnote 12) is that $i(\Sigma)$
is no longer required to be Cauchy surface in $M$, so that $(M,g)$ is not
necessarily globally hyperbolic. He calls such a development _maximal_ if
there is no extension $(M^{\prime},g^{\prime})$ that also satisfies the vacuum
Einstein equations, and proves existence of maximal developments (but not
uniqueness up to isometry, as in the globally hyperbolic case, cf. footnote
12). Applying Penrose’s strong cosmic censorship to such a maximal
development, he asks it to be globally hyperbolic. If this is the case,
then–up to isometry as usual–$(M,g)$ must coincide with the mghd of given
initial data.525252Continuing footnote 12, the set of isometry classes
$[M,g,i]$ of Cauchy developments $(M,g,i)$ of given initial data
$(\Sigma,h,K)$ is partially ordered by
$[M_{1},g_{1},i_{1}]\leq[M_{2},g_{2},i_{2}]$ provided there are
representatives $(M^{\prime}_{1},g^{\prime}_{1},i^{\prime}_{1})$ and
$(M^{\prime}_{2},g^{\prime}_{2},i^{\prime}_{2})$ and an embedding
$\psi:M^{\prime}_{1}\rightarrow M^{\prime}_{2}$ for which
$\psi^{*}g^{\prime}_{2}=g^{\prime}_{1}$ and
$\psi\circ\iota^{\prime}_{1}=\iota^{\prime}_{2}$. The mghd
$[M_{t},g_{t},i_{t}]$ is the top element of this poset (Sbierski, 2016) and so
if some maximal development $(M_{m},g_{m},i_{m})$ à la Chruściel is globally
hyperbolic then $[M_{m},g_{m},i_{m}]\leq[M_{t},g_{t},i_{t}]$. On the other
hand, since $(M_{t},g_{t},i_{t})$ is a solution and $(M_{m},g_{m},i_{m})$ is
maximal also the converse holds, so
$(M_{m},g_{m},i_{m})\cong(M_{t},g_{t},i_{t})$. Consequently, this specific
application of strong cosmic censorship à la Penrose is equivalent to asking
the mghd of given initial data to be inextendible _as a solution to the vacuum
(or any kind of) Einstein equations_.535353See Doboszewski (2017, 2019) and
Manchak (2011, 2017) for conceptual studies of the (in)extendibility of space-
times.

Adding suitable regularity conditions on the extensions,545454As pointed out
to me by Juliusz Doboszewski, Chruściel, Isenberg, & Moncrief (1990) as well
as Chruściel & Isenberg (1993) only consider smooth extensions, so that
looking at lower regularity seems a refinement postdating this early phase.
this would be a meaningful and natural pde version of strong cosmic censorship
but the version used in the pde literature is stronger: one requires metric
inextendibility of the mghd full stop, _whether or not this extension
satisfies the vacuum Einstein equations_. And although it would make sense in
general, in practice the ensuing conjecture is posed for either non-compact
$\Sigma$ with asymptotically flat initial data or compact $\Sigma$; better
safe than sorry!

The need for a restriction on the scope of the conjectures was clearly
realized and stated–albeit purely qualitatively–already by Penrose himself
(see footnote 11). Indeed, without such a restriction some of the best-known
exact black hole solutions (cf. §4) provide counterexamples to one or both of
the conjectures, as was of course well known to Penrose and his circle (for
the Penrosian version, that is). To get around this, in one of his most
prophetic insights, Penrose (1968, p. 222) suggested that because of a
blueshift instability of the Cauchy horizon under perturbations, it turns into
a curvature singularity:

> Our contention in this note is that if the initial data is generically
> perturbed then the Cauchy horizon does not survive as a non-singular
> hypersurface. It is strongly implied that instead, genuine space-time
> singularities will appear along the region which would otherwise have been
> the Cauchy horizon. (Simpson & Penrose, 1973, p. 184)

Since then, this instability has been confirmed in a large number of studies,
starting with Hiscock (1981) in the physics literature and Dafermos (2003) in
the mathematical one; recent papers include Chesler, Narayan & Curiel (2020)
and Van de Moortel (2020), respectively. The conclusion seems to be that
Cauchy horizons turn into so-called _weak null singularities_ ,555555These are
null boundaries with $C^{0}$ metric but Christoffel symbols not locally in
$L^{2}$ (Luk & Sbierski, 2016; Luk, 2017). behind which–at least for one-ended
asymptotically flat initial data–there is a strong curvature singularity at
$r=0$. See also Luk & Oh (2019ab) for the two-ended case. Unappealingly, the
sense in which strong cosmic censorship (in the pde formulation) then fails or
holds depends critically on the regularity assumptions of the
extension.565656The results below concern cosmological constant $\lambda=0$
and subextremal black holes (i.e. $e^{2}<m^{2}$ for R–N and $a^{2}<m^{2}$ for
Kerr). See Dias, Reall, & Santos (2018) for $\lambda>0$: strong cosmic
censorship seems true in pure gravity and false for the Einstein–Maxwell
system, but again this depends critically on the regularity of the extension.
For extremal Reissner–Nordström ($e^{2}=m^{2}$) at $\lambda=0$ see Gajic & Luk
(2017), suggesting failure of strong cosmic censorship, as is trivially the
case for $e^{2}>m^{2}$.

For example, for two-ended asymptotically flat data for the spherically
symmetric Einstein–Maxwell-scalar field system (to which the conjecture, so
far discussed for the vacuum case, can be extended in the obvious way), the
strong cosmic censorship conjecture _fails_ in $C^{0}$ (Dafermos & Luk,
2017),575757Their general result assumes the (widely expected) stability of
the Kerr metric under perturbations of the initial data. but it _holds_ in
$C^{0}$ with the additional requirement that the associated Christoffel
symbols are locally $L^{2}$ (Luk & Oh, 2019ab). This is not just a
technicality, since having the metric in $C^{0}$ and its Christoffel symbols
locally $L^{2}$ is a borderline regularity condition for metric extensions in
strong cosmic censorship: it is the least regular case in which the metric can
still be defined as a weak solution to Einstein’s equations (Christodoulou,
2009, p. 9; Luk, 2017, footnote 1). Indeed, a weak solution of the vacuum
Einstein equations is a metric $g$ for which for all compactly supported
$X,Y\in\mathfrak{X}(M)$,

$\int_{M}d^{4}x\,\sqrt{-\det(g(x))}R_{\mu\nu}(x)X^{\mu}(x)Y^{\nu}(x)=0.$ (3.9)

Partial integration shows that this is well defined iff the
$\Gamma_{\mu\nu}^{\rho}$ are locally $L^{2}$. This simple observation should
not be confused with the very deep result that having the _Ricci tensor_ in
$L^{2}$ is sufficient for the (vacuum) Einstein equations to be weakly
_solvable_ at least locally (Klainerman, Rodnianski, & Szeftel, 2015).
Ironically, in Definition 3.3 of strong cosmic censorship the extension is not
required to satisfy the Einstein (or indeed any other) equations! See also
Ringström (2009) for a review of the ‘cosmological’ case where the Cauchy
surface is compact, in which the strong (pde) conjecture seems to hold.

The status of weak cosmic censorship is even less clear. Christodoulou (1999b)
proves the conjecture for the spherically symmetric gravitational collapse of
a scalar field, but on the basis of genericity conditions whose relevance has
been questioned in the physics literature (Gundlach & Martin-Garcia, 2007,
§3.4). More generally, the status of weak cosmic censorship seems mixed also
in earlier heuristic formulations in terms of an event horizon; see e.g. Joshi
(1993, 2007), Królak (1999), and Ong (2020).

##  4 Examples

The relationship between the Penrosian and the pde versions of the cosmic
censorship conjectures is best understood from three key black hole examples
and their Penrose diagrams:585858Even more so than the previous sections this
one is purely pedagogical and drawn largely from Hawking & Ellis (1973), pages
158 and 160, as well as from Dafermos & Rodnianski (2008) and Dafermos (2013,
2014ab, 2017, 2019) for the pde side.

  * •

Maximally extended Schwarzschild (i.e. Kruskal) with $m>0$ (and two-sided
initial data);

  * •

Schwarzschild with $m<0$, which in so far as singularities and horizons are
concerned also looks like supercharged Reissner–Nordström ($e^{2}>m^{2}>0$),
or ultrafast rotating Kerr ($a^{2}>m^{2}>0$);

  * •

Reissner–Nordström with $0<e^{2}<m^{2}$, which qualitatively also represents
Kerr with $0<a^{2}<m^{2}$.

In the first case the solution coincides with the mghd of the corresponding
(two-ended) initial data, so the difference between the Penrosian and the pde
approach evaporates. Here is the Penrose diagram:

$r=0$$r=0$$\mathcal{H}_{E}^{+}$$\mathcal{H}_{E}^{-}$IIIV$\Sigma$$\Sigma$IIIIIIII$\mathcal{I}^{+}$$\mathcal{I}^{+}$$\mathcal{I}^{-}$$\mathcal{I}^{-}$$i^{0}$$i^{0}$$i^{+}$$i^{+}$$i^{-}$$i^{-}$

_Penrose diagram of the maximally extended Schwarzschild solution with $m>0$.
This solution coincides with the maximal Cauchy development (marked in light
grey) of a generic two-sided Cauchy surface $\Sigma$ with suitable initial
data, drawn as a horizontal blue line. Thus the Cauchy horizon
$\mathcal{H}^{\pm}_{C}$ is empty. The upper two green lines form the future
event horizon $\mathcal{H}^{+}_{E}$ of the black hole area, which is the
upside-down upper triangle (labeled region II), whereas the lower two green
lines form the past event horizon $\mathcal{H}^{-}_{E}$ of the white hole
area, i.e. the lower triangle (region IV). The right-hand diamond is region I,
the left-hand diamond is region III. Fd causal curves cannot _leave_ region II
and they cannot _enter_ region IV._

Both cosmic censorship conjectures hold in both versions (i.e. Penrose and
pde):

  * •

_Weak cosmic censorship._ Penrose: $\Sigma$ is a Cauchy surface for
$J^{-}({\mathcal{I}}^{+})$, making it globally hyperbolic.595959Alternatively:
any _incomplete_ future inextendible timelike curve $\gamma$ must crash in the
upper $r=0$ singularity. Hence $I^{-}(\gamma)$ lies partly in region II, which
is disjoint from $J^{-}({\mathcal{I}}^{+})$, so that $I^{-}(\gamma)\nsubseteq
J^{-}(x)$ for all $x\in J^{-}({\mathcal{I}}^{+})$. pde: each component of
${\mathcal{I}}^{+}$ ends at timelike infinity and hence all its null geodesics
are future complete (as confirmed by explicit parametrization and
computation).

  * •

_Strong cosmic censorship._ Penrose: Kruskal space-time is globally hyperbolic
(since the causal structure of the diagram is such that the line $\Sigma$
represents a Cauchy surface). pde: For smooth extensions Remark 5.45 on page
155 of O’Neill (1983) or Proposition 4.4.3 in Chruściel (2020) plus a detailed
study of the geodesics shows that Kruskal space-time is metrically
inextendible.606060 If for any maximally extended timelike geodesic
$\gamma:[0,b)\rightarrow M$ in $M$ there is a curvature invariant (such as $R$
or $R^{\rho\sigma\mu\nu}R_{\rho\sigma\mu\nu}$, etc.) that blows up as
$\gamma(t)\rightarrow b$, then $(M,g)$ is inextendible. See O’Neill (1983),
Chapter 13, for a study of Kruskal geodesics, proving the antecedent. Sbierski
(2018ab) proves that Kruskal space-time is inextendible even in $C^{0}$.

However, for $m<0$ Kruskal, Reissner–Nordström, and Kerr, differences arise
between the Penrosian and the pde perspectives, since in these cases the
maximal (analytic) solutions, deemed unphysical by the pde aficionados, differ
from the mghd of the pertinent initial data. In particular, although
(curvature) singularities are not part of space-time in any case, they can at
least be drawn as boundaries in the maximal solutions, where they lie behind a
Cauchy horizon. But precisely for that reason singularities are outside any
kind of scope of the corresponding mghd. Here are the Penrose diagrams:

$r=0$$\mathcal{I}^{+}$$\mathcal{I}^{-}$$\mathcal{H}_{C}^{+}$$\mathcal{H}_{C}^{-}$$i^{0}$$i^{+}$$i^{-}$$\Sigma$

$r=0$$r=0$$r=0$$r=0$$\mathcal{H}_{E}^{+}$$\mathcal{H}_{E}^{-}$$\Sigma$$\Sigma$$\mathcal{H}_{C}^{+}$$\mathcal{I}^{+}$$\mathcal{I}^{+}$$\mathcal{H}_{C}^{-}$$\mathcal{I}^{-}$$\mathcal{I}^{-}$$i^{0}$$i^{+}$$i^{+}$$i^{-}$$i^{-}$$i^{0}$$i^{0}$

__Left picture:_ Penrose diagram of $m<0$ Schwarzschild, or supercharged
Reissner–Nordström ($e^{2}>m^{2}>0$), or fast Kerr ($a^{2}>m^{2}>0$). These
solutions have a singularity at $r=0$, but unlike the $m>0$ Kruskal case it is
not shielded by an event horizon. Instead, the red lines labeled
$\mathcal{H}_{C}^{-}$ and $\mathcal{H}_{C}^{+}$ are past and future Cauchy
horizons with respect to the blue line, indicating a maximal spacelike surface
whose initial data give rise to the metrics in question and whose maximal
Cauchy development is the grey area. _

__Right picture:_ Penrose diagram of subcritical Reissner–Nordström
($0<e^{2}<m^{2}$), whose event and Cauchy horizons (despite the different
structure of the singularity) also resemble those of slowly rotating Kerr
($0<a^{2}<m^{2}$). The maximal Cauchy development of the pertinent initial
data given on the maximal spacelike hypersurface represented by the blue line
labeled $\Sigma$ is again colored in grey. It contains past and future event
horizons labeled $\mathcal{H}_{E}^{-}$ and $\mathcal{H}_{E}^{+}$, drawn in
green, but unlike the $m>0$ Schwarzschild case the singularity they are
supposed to shield cannot be reached directly from the maximal Cauchy
development, which is bounded by the various fictitious boundaries
${\mathcal{J}}^{\pm}$, $i^{\pm}$, and $i^{0}$, which lie at infinity, as well
as by the Cauchy horizons $\mathcal{H}_{C}^{\pm}$, drawn in red, which can be
reached in finite proper time.616161 This diagram can be infinitely extended
in both directions (Hawking & Ellis, 1973, pp. 158, 165): to the north,
another grey area folds inside the upper two red line segments, and similarly
to the south, _et cetera_ , but we do _not_ do so here._

Despite the different space-times they apply to, the outcomes of the Penrosian
version and the pde version of both weak and strong cosmic censorship are once
again the same, _mutatis mutandis_ :626262For $m<0$ Kruskal the initial data
are not complete in this case, so strictly speaking the cosmic censorship
conjectures do not apply here and their falsity is unimportant. Nonetheless,
they can be stated and the comparison is instructive.

  * •

$m<0$ Kruskal (etc.): For the Penrosian total space-time the difference
between weak and strong cosmic censorship fades since
$J^{-1}({\mathcal{I}}^{+})=M\cup{\mathcal{I}}^{+}$, which, like $M$ itself is
not globally hyperbolic: wherever one tries to place a partial Cauchy surface
$\Sigma$ (such as the blue line), above the surface inextendible causal curves
can be drawn that enter $i^{+}$ or ${\mathcal{I}}^{+}$ in the future and enter
the singularity at $r=0$ in the past, without crossing $\Sigma$. Similarly,
below $\Sigma$ one may draw inextendible causal curves converging to the
singularity in the future, and to $i^{-}$ or ${\mathcal{I}}^{-}$ in the past,
which once again do not cross $\Sigma$. Thus neither weak nor strong cosmic
censorship holds for this space-time.

The pde picture applies to the grey area, which is the mghd of the initial
data given on the blue line marked $\Sigma$ in the left-hand Penrose diagram.
Then weak cosmic censorship fails because future null infinity
${\mathcal{I}}^{+}$ is clearly incomplete: null geodesics terminate at the
Cauchy horizon (where they “fall off’ space-time) and hence are incomplete. On
the other hand, strong cosmic censorship fails because the grey space-time,
though globally hyperbolic (in contrast with the entire space as we have just
seen), is evidently (smoothly–even analytically) extendible, namely by the
total space. Though they do not coincide, we see that strong and weak cosmic
censorship are closely related: future incompleteness of null geodesics at
null infinity happens because the mghd is extendible.

  * •

Subcritical Reissner–Nordström ($0<e^{2}<m^{2}$): for both Penrose and the pde
people strong cosmic censorship fails whereas the weak version holds. In the
Penrosian version the total space fails to be globally hyperbolic because of
the part above the grey area (i.e. beyond the future Cauchy horizon
$\mathcal{H}_{C}^{+}$): one has past-directed inextendible causal curves that
(backwards in time) end up in the singularity and hence never cross $\Sigma$
(e.g. those crossing the upper left, NW-pointing red line from N to SW).
Morally, weak cosmic censorship holds because of the future event horizon
$\mathcal{H}_{E}^{+}$, which shields the upper $r=0$ singularity above it, but
legally this is only the case if we stop the Penrose diagram at the past
Cauchy horizon $\mathcal{H}_{C}^{-}$, as we have done in drawing the picture
(for otherwise causal curves below it may crash at the lower $r=0$ singularity
and hence never reach $\Sigma$).

The pde view is cleaner here: roughly speaking, as in the $m>0$ Kruskal or
Schwarzschild case (but unlike the $m<0$ case) future null infinity
${\mathcal{I}}^{+}$ ends at future timelike infinity $i^{+}$ and hence is
complete, so that weak cosmic censorship holds. Strong cosmic censorship, on
the other hand, fails because the mghd (marked in grey) is clearly extendible
(namely into the Penrosian space-time!).

More generally, if the strong Penrosian conjecture fails for some space-time
$(M_{P},g_{P})$, then its lack of global hyperbolicity typically occurs
because $(M_{P},g_{P})$ is an extension of the mghd $(M,g)$ of some given
initial data, whose Cauchy surface $\Sigma$ fails to be one for
$(M_{P},g_{P})$. Similarly, if $J^{-1}({\mathcal{I}}^{+})$ is not globally
hyperbolic (so that there is a naked singularity), $M_{P}$ usually comes from
extending some $(M,g)$, as above, whose Cauchy surface becomes a partial
Cauchy surface in $M_{P}$, with an associated future Cauchy horizon that cuts
off ${\mathcal{I}}^{+}\cap\tilde{M}$, causing its
incompleteness.636363However, these aren’t rigorous deductions: there are
pathological cases where strong cosmic censorship holds whilst the weak
version fails. See the Penrose diagram at the end of §2.6.2 of Dafermos &
Rodnianski (2008) for an example. As already mentioned, any counterexamples
are believed to be “non-generic”, assuming of course that the conjectures
hold!

Such reasoning, which applies to many case studies, also suggests a compromise
between the Penrosian and pde versions of cosmic censorhsip: informally one
might say that, in physically reasonable space-times, weak cosmic censorship
postulates the _appearance and stability of event horizons_ , whereas strong
cosmic censorship requires the _instability and ensuing disappearance of
Cauchy horizons_.

##  5 Epilogue: Penrose’s final state conjecture

In practice, the cosmic censorship conjectures are not put in the full
generality of either Penrose’s own version as expressed by Theorem 3.2 or of
the pde version as Definition 3.3, but are posed in the context of black holes
as they are expected to occur in the universe, i.e. as described by something
like the Kerr metric (at least outside its Cauchy horizon and more safely even
outside its event horizon). As such, on the one hand they gain focus, but on
the other hand they can be thought of as forming part of a broader conjecture
that also originated with Penrose himself and is often called the _final state
conjecture_ :646464This is sometimes stated somewhat differently, in that
‘generic asymptotically flat vacuum initial data (…) evolve to a solution
which either disperses (in which case there are no black holes) or else
eventually asymptotes to finitely many Kerr solutions (…) moving away from
each other’ (Coley, 2019, p. 78). See also the fascinating lecture by
Klainerman (2014).

> A body, or collection of bodies, collapses down to a size comparable to its
> Schwarzschild radius, after which a trapped surface can be found in the
> region surrounding the matter. Some way outside the trapped surface region
> is a surface which will ultimately be the absolute event horizon. But at
> present, this surface is still expanding somewhat. Its exact location is a
> complicated affair and it depends on how much more matter (or radiation)
> ultimately falls in. We assume only a finite amount falls in and that gic is
> true. Then the expansion of the absolute event horizon gradually slows down
> to stationarity. Ultimately the field settles down to becoming a Kerr
> solution (in the vacuum case) or a Kerr-Newman solution (if a nonzero net
> charge is trapped in the “black hole”). (Penrose, 1969, pp. 1157–1158)

Here gic refers to what Penrose (1969) called the _Generalized Israel
Conjecture_ ,656565The reference is to Israel (1967, 1968). See Israel (1987),
§7.9 and Thorne (1994), Chapter 7, for interesting history. which states that:

> if an absolute event horizon develops in an asymptotically flat space-time,
> then the solution exterior to this horizon approaches a Kerr-Newman solution
> asymptotically with time. (Penrose, 1969, pp. 1156)

In the stationary case, which is what Israel himself conjectured and proved
under fairly restrictive assumptions, this would simply say that the solution
exterior to this horizon _equals_ a Kerr-Newman solution. As such, the
conjecture is an outgrowth of what (following Wheeler) used to be called the
“no hair” property of black holes, to the effect that stationary black holes
(and eventually all black holes) are characterized by by just three
parameters, viz. mass, angular momentum, and electric charge. As such, the
final state conjecture incorporates not only weak cosmic censorship (notably
in the compromise version suggested at the end of the previous section) but
also what in the pde literature is called _Kerr stability_ , as well as _Kerr
rigidity_. The former is the conjecture that generic perturbations of the
initial data for the Kerr metric lead to a mghd that is close to the original
one (at least outside the event horizon).666666There is certain numerical
evidence for this (Zilhão et al., 2014), but mathematical results so far are
preliminary (Dafermos, Holzegel, & Rodnianski, 2019b; Giorgi, Klainerman, &
Szeftel, 2020), except for positive cosmological constant and small $a$ (Hintz
& Vasy, 2018), where the problem is solved. Even the Schwarzschild case is
still open, despite impressive progress (Klainerman & Szeftel, 2017; Dafermos,
Holzegel, & Rodnianski, 2019a).  This would generalize the remarkable theorem
on the stability of Minkowski space-time (Christodoulou & Klainerman, 1993),
which launched the modern era in pde-oriented mathematical relativity. The
latter is a more modest version of the no-hair or black hole uniqueness
theorems of Israel, Carter, Hawking, Robinson, and others,676767These theorems
are reviewed in Hawking & Ellis (1973), §9.3, Carter (1979, 1986), Heusler
(1996), Robinson (2009), Chruściel, Lopes Costa, & Heusler (2012), and
Cederbaum (2019). See also Cardoso & Gualtieri (2016) for possible tests.
where short of proving the stationary case of the above gic, which requires
unphysical analyticity assumptions, one tries to show that at least stationary
solutions to Einstein’s vacuum (electrovac) equations that are close to Kerr
(–Newman) actually coincide with the latter.686868See e.g. Alexakis, Ionescu,
& Klainerman (2014) as well as the review by Ionescu & Klainerman (2015).

In conclusion, “the discovery that black hole formation is a robust prediction
of the general theory of relativity” still lies in the future as far as
mathematical proof is concerned. Penrose’s Nobel Prize was effectively awarded
for a theorem and a conjecture, but it was fully deserved in every conceivable
way!

##  Appendix A A potted early history of “black hole” (by Erik Curiel)

In reply to a query in an early draft of this paper concerning the origin of
the definitions $\mathcal{B}:=M\backslash J^{-}({\mathcal{I}}^{+})$ of a black
hole (region) and of the event horizon as its boundary
$\mathcal{H}_{E}^{+}:=\partial\mathcal{B}$, see (2.4) and (2.5), phrased as:
‘Such definitions are routinely used in e.g. Carter (1971a) and Hawking &
Ellis (1973), §9.2, but who stated them first?’, Erik Curiel very kindly
supplied the following information.696969As in the rest of the paper, single
quotation marks below denote literal quotation whereas double ones are scare
quotes. As we see, the question still remains somewhat open. A puzzling point
is that although Penrose himself would have been the obvious person to state
these definitions mathematically, apparently he did not do so!

> Penrose (1968) defines (p. 188), an event horizon as the boundary of the
> chronological past of a timelike curve (essentially the same definition,
> including the name, as given by Rindler 1956), and notes (p. 206) that
> $r=2M$ in Schwarzschild is one. The term “black hole” does not appear in
> that essay, nor any definition remotely like ‘the complement of the causal
> past of future null infinity’. Given the magisterial depth and encyclopedic
> scope of that essay, I must conclude that the definition was not then yet
> extant. Penrose (1969) does use the term “black hole” (the first use of it I
> know in the general relativity literature, though it reportedly was used in
> the early 1960s by Dicke in discussion with a popular science writer), but
> he always encloses it in scare quotes, leading me to believe that the name
> and the general concept both were still inchoate. This is buttressed by the
> fact that he does give here (p. 1146, footnote 3) the classic definition of
> an ‘absolute event horizon’ (the boundary of the chronological past of
> future null infinity), but he does not explicitly link it to the term “black
> hole”. That is the first appearance of the classic definition I know of in
> the literature.
>
> Ruffini and Wheeler gave a series of lectures in September 1969 at the
> _Interlaken Colloquium on the Significance of Space Research for Fundamental
> Physics_ (Interlaken, Switzerland), one of which was entitled ‘Black Holes’,
> at least according to the expanded version of the lectures published as
> Ruffini and Wheeler (1971a), from which Ruffini and Wheeler (1971b) was
> adapted. This is the first use of the term I have been able to find recorded
> in a public forum in the relativity community. They explain the idea in
> informal, intuitive terms. Bardeen (1970), received 23 January 1970, has
> “black hole” in the title, the first publication I know of to do so. He
> introduces “black hole” using scare quotes and equates it with a ‘collapsed
> object’. Israel (1971), originally read at the _Gwatt Seminar on the
> Bearings of Topology upon General Relativity_ on 19 May 1970, uses “black
> hole” without defining it, but it is clear from context that he means
> something like ‘system that quickly settles down so that its exterior is
> modeled by the Kerr solution’. Christodoulou (1970), received 17 September
> 1970, uses the term without blushing, not even an informal gloss given for
> its meaning. He simply begins by talking of a ‘black hole [with] angular
> momentum’ without even citing Kerr (1963). Three papers then appear in 1971
> with “black hole” in the title, Penrose and Floyd (1971), received 16
> December 1970, Carter (1971), received 18 December 1970, and Hawking (1971),
> received 11 March 1971. They are the only other papers from 1971 I can find
> related to the topic whose authors plausibly could have proposed the classic
> definition. Penrose and Floyd (1971) uses scare-quotes around the first use
> of “black hole”; their discussion relies only on the event horizon and
> ergosphere (which they refer to as the ‘stationary limit’) defined by the
> Kerr metric, with no attempt at (or mention of the possibility of)
> generalization. Carter (1971) does not give a formal definition of “black
> hole”, but he does give an informal definition of ‘domain of outer
> communication’, and says (p. 331) that ‘ “black holes” [are] regions of
> space-time beyond the domain of outer communication.’ Note the scare-quotes.
> Hawking (1971) uses scare-quotes as well, going out of his way to assimilate
> the idea of a black hole to one more widely known (‘there are initially two
> collapsed objects or “black holes”’, p. 1345). He also comes achingly close
> to defining a black hole as a connected component of the complement of the
> causal past of future null infinity, but never quite does it. He rather says
> things like, ‘On $\Sigma_{i}$ [a spacelike hypersurface], there will be two
> separate regions, $B_{1}$ and $B_{2}$ which contain closed, trapped surfaces
> (…) Just outside $B_{1}$ and $B_{2}$ will be two two-spheres which are the
> intersection of $\dot{J}^{-}({\mathcal{I}}^{+})$ with $\Sigma_{i}$.’ The
> first explicit definition I know of “black hole” as ‘connected component of
> the complement of the causal past of future null infinity’ is in Hawking
> (1972). But I am not confident that I have found all relevant sources in the
> literature; even if I have, one cannot be confident based only on this that
> Hawking was the one who finally put all the pieces together.
>
> (Erik Curiel, private communication, January 6, 2021, reprinted with
> permission)

Acknowledgement. This paper grew out of a seminar in honour of Roger Penrose
and his Nobel Prize at Nijmegen on October 28th, 2020, at the kind invitation
of Timothy Budd. The author is grateful to Carlo Rovelli for encouragement to
publish it in _Foundations of Physics_ , as well as to Béatrice Bonga, Jeremy
Butterfield, Erik Curiel, Juliusz Doboszewski, John Earman, Dejan Gajic, and
Dennis Lemkuhl for helpful feedback on earlier drafts. I also learned a lot
form various seminars by Martin Lesourd.

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