The mathematical analysis of black holes

in general relativity
Mihalis Dafermos

Abstract. The mathematical analysis of black holes in general relativity has been the fo-
cus of considerable activity in the past decade from the perspective of the theory of partial
differential equations. Much of this work is motivated by the problem of understanding
the two celebrated cosmic censorship conjectures in a neighbourhood of the Schwarzschild
and Kerr solutions. Recent progress on the behaviour of linear waves on black hole exte-
riors as well as on the full non-linear vacuum dynamics in the black hole interior puts us
at the threshold of a complete understanding of the stabilityand instabilityproperties
of these solutions. This talk will survey some of these developments.

Mathematics Subject Classification (2010). Primary 83C57; Secondary 83C75.

Keywords. Einstein equations, general relativity, black holes, cosmic censorship.

1. Introduction
There is perhaps no other object in all of mathematical physics as fascinating as
the black holes of Einsteins general relativity.
The notion as such is simpler than the mystique surrounding it may suggest!
Loosely speaking, the black hole region B of a Lorentzian 4-manifold (M, g) is the
complement of the causal past of a certain distinguished ideal boundary at infinity,
denoted I + and known as future null infinity; in symbols

B = M \ J (I + ). (1)

In the context of general relativity, where our physical spacetime continuum is

modelled by such a manifold M, this ideal boundary at infinity I + corresponds
to far-away observers in the radiation zone of an isolated self-gravitating system
such as a collapsing star. Thus, the black hole region B is the set of those spacetime
events which cannot send signals to distant observers like us.
It is remarkable that the simplest non-trivial spacetimes (M, g) solving the
Einstein equations in vacuum
Ric(g) = 0, (2)
The author is grateful to G. Holzegel, J. Luk, I. Rodnianski and Y. Shlapentokh-Rothman

for comments on this manuscript and to D. Christodoulou for many inspiring discussions over
the years.
2 Mihalis Dafermos

the celebrated Schwarzschild and Kerr solutions, indeed contain non-empty black
hole regions B != . Moreover, both these spacetimes fail to be future causally
geodesically complete, i.e. in physical language, there exist freely falling observers
who live for only finite proper time. The two properties are closely related in
the above examples as all such finitely-living observers must necessarily enter the
black hole region B. Far-away observers in these examples, on the other hand, live
forever; the asymptotic boundary future null infinity I + is itself complete.
In the early years of the subject, the black hole property was widely misunder-
stood and the incompleteness of the above spacetimes was considered a pathology
that would surely go away after perturbation. The latter expectation was shat-
tered by Penroses celebrated incompleteness theorem [68] which implies in partic-
ular that the incompleteness of Schwarzschild and Kerr is in fact a stable feature
when viewed in the context of dynamics. We have now come to understand the
presence of black holes not at all as a pathology but rather as a blessing, shielding
the effects of incompleteness from distant observers, allowing in particular for a
complete future null infinity I + . This motivated Penrose to formulate an ambi-
tious conjecture known as weak cosmic censorship which states that for generic
initial data for the Einstein vacuum equations (2), future null infinity I + is indeed
complete. In the language of partial differential equations, this can be thought of
as a form of global existence still compatible with Penroses theorem.
A positive resolution of the above conjecture would be very satisfying but would
still not resolve all conceptual issues raised by the Schwarzschild and Kerr solutions.
For it is reasonable to expect that our physical theory should explain the fate not
just of far-away observers but of all observers, including those who choose to
enter black hole regions B. In the exact Schwarzschild case, such observers are
destroyed by infinite tidal forces, while in the exact Kerr case, they cross a Cauchy
horizon to live another day in a region of spacetime which is no longer determined
by initial data. The former scenario is an omenous prediction indeedbut one
we have come to terms with. It is the latter which is in some sense even more
troubling, as it represents a failure of the notion of prediction itself. This motivates
yet another ambitious conjecture, strong cosmic censorship, also originally due to
Penrose, which says that for generic initial data for (2), the part of spacetime
uniquely determined by data is inextendible. In the language of partial differential
equations, this conjecture can be thought of as a statement of global uniqueness.
For this conjecture to be true, the geometry of the interior region of Kerr black
holes would in particular have to be unstable.
Despite the ubiquity of black holes in our current astrophysical world-picture,
the above conjectureseven when restricted to a neighbourhood of the explicit so-
lutions Schwarzschild and Kerrare not mathematically understood. More specif-
ically, we can ask the following stability and instability questions concerning the
Schwarzschild and Kerr family:
1. Are the exteriors to the black hole regions B in Schwarzschild
and Kerr stable under the evolution of (2) to perturbation of data? In
particular, is the completeness of null infinity I + a stable property?
2. What happens to observers who enter the interior of the black
Black holes in general relativity 3

hole region B of such perturbations of Kerr? Are the smooth Cauchy

horizons of Kerr unstable?

If our optimistic expectations on these questions are in fact not realised by the
theory, then this may fundamentally change our understanding of general relativity
and perhaps also our belief in it!
The global analysis of solutions to the Einstein vacuum equations (2) without
symmetry was largely initiated in the monumental proof [23] of the non-linear sta-
bility of Minkowski space by Christodoulou and Klainerman in 1993. As with the
stability of Minkowski space, Question 1. would be a statement of global existence
and stability, but now concerning a highly non-trivial geometry. Question 2., on
the other hand, not only concerns a non-trivial geometry but appears to concern
a regime where solutions may become unstable and in fact singular (at least, if
strong cosmic censorship is indeed true!); the prospect of proving anything about
such a regime seemed until recently quite remote. A number of rapid develop-
ments in the last few years, however, concerning linear wave equations on black
hole backgrounds as well as the analysis of the fully non-linear Einstein equations
in singularbut controlledregimes have brought a complete resolution of Ques-
tions 1. and 2. much closer. The purpose of this talk is to survey some of these
developments. In particular, we will describe the following results, which reflect
the state of the art concerning our understanding of Questions 1 and 2 above, and
had themselves been the subject of a number of open conjectures.

1. Linear scalar waves on Schwarzschild and Kerr backgrounds re-

main bounded in the black hole exterior and in fact decay polynomially.
Schwarzschild is in fact linearly stable in full linearised gravity.

2. For a spherically symmetric toy model, Cauchy horizons are

globally stable from the point of view of the metric in L , but unstable
at the level of derivatives of the metric, as the Christoffel symbols in
any regular frame become singular. For the full vacuum equations (2)
without symmetry, then, given the stability of the exterior, the above
stability statement for the Kerr Cauchy horizon again holds.

We see in particular that the final part of 2. means that the precise understand-
ing of Questions 1. and 2. is in fact coupled. Note that the result 2. is in fact at
odds with the strongest formulations of Question 2 above and this has significant
and slightly troublingimplications as to what versions of strong cosmic censorship
are indeed true. This could indicate that some of the conceptual puzzles of general
relativity are here to stay!

2. Schwarzschild and Kerr

We begin by reviewing the Schwarzschild and Kerr families.
4 Mihalis Dafermos

2.1. The Schwarzschild metric. The Schwarzschild family (M, gM ) rep-

resents the simplest non-trivial explicit family of solutions to the Einstein vacuum
equations (2). These solutions were discovered already in December 1915 [75], the
month following Einsteins final formulation of general relativity [43]. The metrics
are static and spherically symmetric and can be written in local coordinates as

gM = (1 2M/r)dt2 + (1 2M/r)1 dr2 + r2 (d2 + sin2 d2 ). (3)

Here, M is a parameter which can be identified with mass. We shall only consider
the case M > 0. Note that the case M = 0 reduces to the flat Minkowski space,
which is trivially a solution of (2).
In discussing the Schwarzschild solution, we have not yet settled on the ambient
manifold M on which (3) should live! Historically, this was indeed only understood
later, since the correct differentiable structure of the ambient manifold is not so
immediately apparent from the form (3). If we pass, however, to new coordinates
(cf. Lemaitre [57]) (t , r, , ) where

t = t + 2M log(r 2M ),

we see that the metric expression (3) can be rewritten

(1 2M/r)(dt )2 + (4M/r)drdt + (1 + 2M/r)dr2 + r2 (d2 + sin2 d2 ). (4)

! to be precisely
This suggests that we may define our underlying manifold M
! = (, ) (0, ) S2
M (5)

with coordinates t , r, , , on which gM defined by (4) manifestly yields a smooth

metric. Let us for now consider (M, ! gM ) as our spacetime.
.
One easily sees from the form of the metric (4) that the region B = {r 2M }
has the property that future directed causal curves emanating from B must stay
in B (i.e. J + (B) = B), in particular, they cannot reach large values of r. It turns
out that with a suitable definition of the asymptotic boundary future null infinity
I + , B corresponds also to the black hole region defined in (1), and I + is moreover
complete.1 The boundary H+ = {r = 2M } of B in the spacetime M is known as
the event horizon. Note that the static Killing field t of (3) extends to a Killing
field t on M which is in fact spacelike in the region {r < 2M } and null on H+ .
In contrast to the case of Minkowski space M = 0 where the above metric (4)
extends from (5) to R3+1 by adding r = 0 to the manifold, in the case M > 0, the
metric becomes singular as r 0 is approached. In fact, {r = 0} can be attached
as a spacelike singular boundary to which all future-incomplete causal geodesics
approach. This shows that the manifold M ! is future-inextendible as a suitably
regular Lorentzian manifold. It is not, however, past -inextendible. It turns out
that one can define an even larger ambient manifold M (by suitably pasting M ! to
a copy of itself) so as for (4) above to extend to a spherically symmetric solution of
1 This means that if we define a null retarded time coordinate u such that r = 1 asymp-
u
totically at I + , then I + is covered by the u-range (, ).
Black holes in general relativity 5

(2) which is now indeed also past-inextendible. This gives the so-called maximally
extended Schwarzschild solution (M, g). See [78, 56]. In what follows, it is this
(M, g) that we shall definitively refer to as the Schwarzschild manifold.
Note that this new manifold (M, g) does not admit r as a global coordinate, but
can be covered by a global system of double null coordinates (U, V ) whose range
can be normalised to the following shaded bounded subregion Q of the plane R1+1 :

I
+
H+
R

The metric takes the form

2 (U, V )dU dV + r2 (U, V )(d2 + sin2 d2 )

where and r can be described implicity. The above depiction is known as a

CarterPenrose diagram of (M, g), and gives a concrete realisation of both future
null infinity I + (as an open constant U -segment of the boundary of Q in the
ambient R1+1 ) and the singular {r = 0} past and future boundaries.
Note that the above manifold is globally hyperbolic with a Cauchy hypersurface
(possessing two asymptotically flat ends). That is to say, all inextendible causal
curves intersect exactly once. When we discuss dynamics in Section 3, this
property will allow us to view Schwarzschild (M, g) as the maximal vacuum Cauchy
development of data on .

2.2. The Kerr metrics. The Schwarzschild family sits as the 1-parameter
a = 0 subfamily of a larger, 2-parameter family (M, gM,a ), discovered in 1963 by
Kerr [52]. The parameter a can be identified with rotation. The latter metrics are
less symmetric when a != 0they are only stationary and axisymmetricand are
given explicitly in local coordinates by the expression

" # 2 2
gM,a = 2
dt a sin2 d + dr2 + 2 d2 (6)

2 " #2
sin
+ a dt (r2 + a2 )d
2
where
2 = r2 + a2 cos2 , = r2 2M r + a2 .
We will only consider the case of parameter values 0 |a| < M , M > 0, where
= (r r+ )(r r ) for r+ > r > 0. The case |a| = M is special and is known
as the extremal case.
Again, by introducing t = t (t, r) but now also a change = (, r), the
metric can be rewritten in analogy to (4) so as to make it regular at r = r+ , which
will again correspond to the event horizon H+ of a black hole B. An additional
6 Mihalis Dafermos

transformation can now make the metric regular at r = r and allows a further
extension into r < r . The set r = r will correspond to a so-called Cauchy
horizon CH+ separating a globally hyperbolic region from part of the spacetime
which is no longer determined by Cauchy data. Our convention will be to not
include the latter extensions into our ambient manifold M, which will, however,
as in Schwarzschild, be doubled by appropriately pasting two r > r regions.
For us, the Kerr spacetime (M, gM,a ) will thus again be globally hyperbolic with
a two-ended asymptotically flat Cauchy hypersurface as in the Schwarzschild
case, and, in the language of Section 3, will again be the maximal vacuum Cauchy
development of data on . See

I
+
H+
R

It is, however, precisely the existence of these further extensions to r < r which
leads to the question of strong cosmic censorship.
The Kerr solutions are truly remarkable objects with a myriad of interesting
geometric properties beyond the mere fact of the presence of a black hole region
B, for instance, their having a non-trivial ergoregion E to be discussed in Sec-
tion 4.2.1. Even the very existence in closed form of the family is remarkable,
since simply imposing the symmetries manifest in the above expression (6) is by
dimensional considerations clearly insufficient to ensure that the Einstein equations
(2) should admit closed-form solutions. It turns out that the metrics (6) enjoy sev-
eral hidden symmetries. For instance, they possess an additional non-trivial
Killing tensor and they are moreover algebraically special. It is in fact through the
latter property that they were originally discovered [52].

2.3. Uniqueness. A natural question that arises is whether there are other
stationary solutions of (2) containing black holes B besides the Kerr family gM,a .
If we impose in addition that our solutions be axisymmetric then indeed, the
Kerr family represents the unique family of black hole solutions (with a connected
horizon). See [11, 72] for the original treatments and also [24].
The expectation that the Kerr solutions are unique even without imposing
axisymmetry stems from a pretty rigidity argument due to Hawking [47]. Under
certain assumptions, including the real analyticity of the metric, he showed that
stationary black holes are necessarily also axisymmetric, and thus, the above result
applies to infer uniqueness.
The assumption of real analyticity is physically unmotivated, however, and
Black holes in general relativity 7

leaves open the possibility that there may yet still be other smooth (but non-
analytic) black hole solutions of (2). An important partial result has recently
been proven in [1], where it is shown (generalising Hawkings rigidity argument
using methods of unique continuation) that the Kerr family is indeed unique in
the smooth class provided one restricts to stationary spacetimes suitably near the
Kerr family. In particular, this means that the Kerr family is at the very least
isolated in the family of all stationary solutions.
In view of this latter fact, it indeed makes sense to focus on the Kerr family, in
particular, to entertain the question of its asymptotic stability. Before turning
to this, however, we must first make some general comments about dynamics for
the Einstein equations (2).

3. Dynamics of the Cauchy problem

One of the early triumphs of the theory of partial differential equations applied
to general relativity was the proof that the Einstein equations (2) indeed give rise
to an unambiguous notion of dynamics. In the language of partial differential
equations, this corresponds to the well-posedness of the Cauchy problem for (2),
proven by Choquet-Bruhat [13] and Choquet-BruhatGeroch [14].
We will state the foundational well-posedness statement as Theorem 3.1 of
Section 3.1 below. We will then proceed in Sections 3.2 and 3.3 to illustrate
global aspects of the problem of dynamics with the statement of the stability of
Minkowski space and with the formulation of the cosmic censorship conjectures,
already mentioned in the introduction. This will prepare us for our study of the
dynamics of black holes in Sections 4 and 5.

3.1. Well-posedness. Before formulating the well-posedness theorem, we

must first understand what constitutes an initial state. In view of the fact that
the Einstein equations (2) are second order, one expects to prescribe initially a
triple (3 , g, K), where (3 , g) is a Riemannian 3-manifold and K is an auxil-
iary symmetric 2-tensor to represent the second fundamental form. We say that
a Lorentzian 4-manifold (M, g) is a vacuum Cauchy development of (3 , g, K) if
(M, g) solves (2) and there exists an embedding i : M such that i() is
a Cauchy hypersurface2 in M and g and K are indeed the induced metric and
second fundamental form of the embedding.
The classical Gauss and Codazzi equations of submanifold geometry immedi-
ately imply the following necessary conditions on (3 , g, K) for the existence of
such an embedding:
R + (trK)2 |K|2g = 0, divK d trK = 0. (7)
We will thus call a triple (3 , g, K) satisfying (7) a vacuum initial data set. In her
seminal [13], Choquet-Bruhat proved that for regular (3 , g, K), the conditions (7)
2 In particular, developments are globally hyperbolic in the sense described at the end of

Section 2.1. Global hyperbolicity is essential for the solution to be uniquely determined by data.
8 Mihalis Dafermos

are also sufficient for the existence of a development and for a local uniqueness
statement. In the langauge of partial differential equations, this is the analogue of
local well posedness.
We are all familiar from the theory of ordinary differential equations that local
existence and uniqueness immediately yields the existence of a unique maximal
solution x : (T , T+ ), where T < T+ +. In general relativity,
maximalising Choquet-Bruhats local statement is non-trivial as there is not a
common ambient structure on which all solutions are defined so as for them to be
readily compared. Such a maximalisation was obtained in
Theorem 3.1 (Choquet-BruhatGeroch [14]). Let (3 , g, K) be a smooth vacuum
initial data set. Then there exists a unique smooth vacuum Cauchy development
!, $
(M, g) with the property that if (M g) is any other vacuum Cauchy development,
! $
then there exists an isometric embedding i : (M, g) (M, g) commuting with the
embeddings of .
The above object (M, g) is known as the maximal vacuum Cauchy development.
It is indicative of the trickiness of the maximalisation procedure that the original
proof [14] of the above theorem appealed in fact to Zorns lemma to infer the
existence of (M, g). This made the theorem appear non-constructive, a most
unappealing state of affairs in view of its centrality for the theory. A constructive
proof has recently been given by Sbierski [73].
For convenience, we have stated Theorem 3.1 in the smooth category, even
though it follows from a more primitive result expressed in Sobolev spaces H s
of finite regularity. In the original proofs, this requisite H s space was high and
did not admit a natural geometric interpretation. In a monumental series of pa-
pers (see [54]) surveyed in another contribution to these proceedings [79], this
regularity has been lowered to g H 2 , which can in turn be related to natural
geometric assumptions concerning curvature and other quantities.

3.2. Global existence and stability of Minkowski space. With

the notion of dynamics well defined, we now turn to the prototype global existence
and stability statement, the monumental stability of Minkowski space [23].
The result states that small perturbations of trivial initial data 1. lead to
geodesically complete maximal vacuum Cauchy developments, with a complete
future null infinity I + and no black holes, 2. remain globally close to Minkowski
space and in fact, 3. settle back down asymptotically to Minkowski space:
Theorem 3.2 (Stability of Minkowski space, Christodoulou and Klainerman [23]).
Let (3 , g, K) be a smooth vacuum initial data set satisfying a global smallness
assumption, i.e. suitably close to trivial initial data. Then the maximal vacuum
Cauchy development (M, g) satisfies the following:
1. (M, g) is geodesically complete and moreover, one can attach a boundary I +
which is itself complete, and M = J (I + ).3
3 Note that the statement M = J (I + ) represents the fact that these perturbed spacetimes

do not contain a non-trivial black hole region B.

Black holes in general relativity 9

2. (M, g) remains globally close to Minkowski space,

3. (M, g) asymptotically settles down to Minkowski space (at a suitably fast
rate).
In the language of partial differential equationss, the geodesic completeness of
statement 1. can be thought of as a geometric formulation of global existence.
Statement 2. then corresponds to orbital stability while statement 3. corresponds
to asymptotic stability. Due to the supercriticality of the Einstein equations, the
only known mechanism for showing long-time control of a solution is by exploiting
its dispersive properties, which here arise due to the radiation of waves to null infin-
ity I + . As a result, the more primitive statements 1. and 2. can only be obtained
in the proof by using strong decay rates to flat space, i.e. the full quantitative
version of 3. Thus, the proof of all statements above is strongly coupled.
The original proof of this theorem has been surveyed in a previous preceedings
volume [19] for this conference series. Let us only briefly mention here the central
role played by obtaining (in a bootstrap setting) decay of weighted energy quanti-
ties associated to the Riemann curvature tensor expressed in a null frame (which
satisfies the Bianchi equations) and then coupling these with elliptic and trans-
port estimates for the structure equations satisfied by the connection coefficients,
schematically

/ = + , / +
/ = D (8)
where denotes a generic connection coefficient and denotes a generic curvature
component. The problem is especially difficult precisely because the rate of decay
of waves to null infinity I + is borderline in 3 + 1 dimensions. Thus, stability is
not true for the generic equation of the degree of nonlinearity of (2), but requires
identifying special, null-type4 structure in (8). We will return to some of these
aspects of the proof when we discuss black holes.

3.3. Penroses incompleteness theorem and the cosmic cen-

sorship conjectures. The explicit examples of Schwarzschild and Kerr in-
dicate that the geodesic completeness of Theorem 3.2 cannot hold for general
asymptotically flat data if the global smallness assumption is dropped. In the
early years of the subject, one could entertain the hope that this was an artifice
of the high degree of symmetry of these special solutions. As mentioned already
in the introduction, this was falsified by the following corollary to Penroses 1965
incompleteness theorem:
Theorem 3.3 (Corollary of Penroses incompleteness thoerem [68]). Let (3 , g, K)
be a smooth vacuum data set sufficiently close to the data corresponding to Schwarz-
schild or Kerr. Then the maximal vacuum Cauchy development (M, g) is future
causally geodesically incomplete.
As noted already in the introduction, in the specific examples of Schwarzschild
and Kerr, the above incompleteness is hidden in black hole regions. That is to
4 In contrast, the classical null condition [53] does not hold when the Einstein equations (2)

are written in harmonic gauge. See, however, the remarkable proof in [58].
10 Mihalis Dafermos

say, all finitely-living observers must cross H+ into the region B. In particular,
this allows for the asymptotic boundary I + to still be complete, cf. the second
part of statement 1. of Theorem 3.2. This property is appealing because it means
that if one is only interested in far-away observers, one need not further ponder the
significance of incompleteness as the theory gives predictions for all time at I + .
This motivates the following conjecture, originally formulated by Penrose, which,
if true, would promote this feature to a generic property of solutions to (2):

Conjecture 3.4 (Weak cosmic censorship). For generic asymptotically flat vac-
uum initial data sets, the maximal vacuum Cauchy devlopment (M, g) possesses a
complete null infinity I + .5

In the language of partial differential equations, this conjecture can be thought

of as the version of global existence which is still compatible with Theorem 3.3.
While the above conjecture would indeed explain the possibility of far-away
observation for all time, it does not do away with the puzzles opened up by the
geodesic incompleteness of Theorem 3.3 from the point of view of fundamental
theory. As remarked already, it is reasonable to expect that our theory gives pre-
dictions for all observers, not just far-away ones. The examples of Schwarzschild
and Kerr tell us that the incompleteness of Theorem 3.3 may have very different
origin. The Schwarzschild manifold (M, g) is inextendible in a very strong sense:
incomplete geodesics approach what can be thought of as a spacelike singularity
corresponding to r = 0, and not only do these observers witness infinite curvature
but they are torn apart by infinite tidal forces:

r=0
CH
H+

I+
+
H+

I+

Kerr, on the other hand, terminates in what can be viewed as a smooth Cauchy
horizon CH+ , across which the solution is smoothly extendible to a larger spacetime
(the lighter shaded region) which is no longer however uniquely determined from
.6 In the latter case, we see that the maximal Cauchy development is maximal not
because it is inextendible as a smooth solution of (2) but because such extensions
necessarily fail to be globally hyperbolic and thus cannot be viewed as Cauchy
developments.
5 This particular formulation is due to Christodoulou [18], who in particular, gives a precise

general meaning for possessing a complete null infinity. Note also that this conjecture was origi-
nally stated without the assumption of generic. The necessity of genericity is to be expected in
view of the existence of the spherically symmetric examples [16, 17].
6 Recall that our conventions on the definition of the ambient Schwarzschild (M, g ) and Kerr
M
manifolds (M, gM,a ) in Sections 2.1 and 2.2 are precisely so they be the maximal vacuum Cauchy
developments of initial data (, g, K).
Black holes in general relativity 11

As explained in the introduction, we have largely come to terms with the former
possibility exhibited by Schwarzschild. It gives the theory closure as all observers
are accounted for: They either live forever or are destroyed by infinite tidal forces7 .
The implications of the existence of Cauchy horizons, however, as in the Kerr case,
would be quite problematic, for it restricts the ability of classical general relativity
to predict the fate of macroscopic objects.
The above unattractive feature of Kerr motivated Penrose to formulate his
celebrated strong8 cosmic censorship conjecture:

Conjecture 3.5 (Strong cosmic censorship). For generic asymptotically flat vac-
uum data sets, the maximal vacuum Cauchy development (M, g) is inextendible as
a suitably regular Lorentzian manifold.

The above conjecture can be thought colloquially as saying that Generically,

the future is determined by initial data since the notion of inextendibility captures
the idea that there is not a bigger spacetime where the maximal Cauchy develop-
ment embeds, and which would thus not be uniquely determined by Cauchy data.
It can thus be considered, in the language of partial differential equations, to be a
statement of global uniqueness.
Here the necessity of requiring genericity in the formulation of Conjecture 3.5 is
clear from the start. The Kerr solutions do not satisfy the required inextendibility
property. Thus, for the above conjecture to be true, this feature of Kerr must
be unstable. It is not just wishful thinking that leads to Conjecture 3.5! See
Section 5.1.
Finally, let us remark already that the question of how suitably regular should
be defined in the formulation of Conjecture 3.5 is a subtle one, as will become
apparent in view of Section 5.2 below.

4. The stability of the black hole exterior

To make progress on the general understanding of the theory, and in particular, the
cosmic censorship conjectures of Section 3.3, we begin by looking at dynamics of (2)
in a neighbourhood of the Kerr family. With the language of the Cauchy problem
developed above, we may now turn to discuss what is one of the central open
questions in classical general relativitythe non-linear stability of the Kerr family
in its exterior region. This represents not only a fundamental test of weak cosmic
censorship but a milestone result in itself with important implications for our
current working assumption of the ubiquity of objects described by Kerr metrics
in our observable universe.

4.1. The conjecture. We begin with a more precise formulation of the con-
jecture, taken from [29]:
7 Speculation on what happens to their quantum ashes is beyond the scope of both classical

general relativity and this article.

8 We note that this conjecture is neither stronger nor weaker than Conjecture 3.4. See [18].
12 Mihalis Dafermos

Conjecture 4.1 (Nonlinear stability of the Kerr family). For all vacuum ini-
tial data sets (, g, K) sufficiently near data corresponding to a subextremal
(|a0 | < M0 ) Kerr metric ga0 ,M0 , the maximal vacuum Cauchy development space-
time (M, g) satisfies:
1. (M, g) possesses a complete null infinity I + whose past J (I + ) is bounded
in the future by a smooth affine complete event horizon H+ M,
2. (M, g) stays globally close to ga0 ,M0 in J (I + ),
3. (M, g) asymptotically settles down in J (I + ) to a nearby subextremal mem-
ber of the Kerr family ga,M with parameters a a0 and M M0 .
We have explicitly excluded the extremal case |a| = M from the conjecture for
reasons to be discussed in Section 4.2.5. In particular, the smallness assumption
on data will depend on the distance of the initial parameters a0 , M0 to extremality.
One can compare the above with our formulation of Theorem 3.2. Statement
1. above contains the statement of weak cosmic censorship restricted to a neigh-
bourhood of Schwarzschild. As explained in Section 3.3, in the language of partial
differential equations, this is the analogue of global existence still compatible
with Theorem 3.3. Statement 2. can be thought to represent orbital stability,
whereas statement 3 represents asymptotic stability. As in our discussion of the
proof of the stability of Minkowski space, all these questions are coupled; it is only
by identifying and exploiting the dispersive mechanism (i.e. a quantitative version
of 3.) that one can show the completeness of null infinity I + and orbital stability.
In particular, it is essential to identify the final parameters a and M .
Like any non-linear stability result, the first step in attacking the above conjec-
ture is to linearise the equations (2) around the Schwarzschild and Kerr solutions.
The resulting system of equations is of considerable complexity; we will indeed
turn to this in Section 4.3 below. But first, let us discuss what can be thought of
a poor mans linearisation, namely the study of the linear scalar wave equation

!g = 0 (9)

on a fixed Schwarzschild and Kerr background.

4.2. A poor mans stability result: !g = 0 on Kerr. The

study of (9) in the Schwarzschild case goes back to the classic paper of Regge
and Wheeler [71] which considered the formal analysis of fixed modes. The first
definitive result about actual solutions of (9) is due to Kay and Wald [51] and gives
that solutions of !g = 0 on Schwarzschild arising from regular localised initial
data remain uniformly bounded in the exterior, up to and including H+ .
The last decade has seen a resurgence in interest in this problem so as to
prove not just boundedness but decay and to handle not just Schwarzschild but
the general subextremal Kerr case. Many researchers have contributed to this
understanding [32, 5, 33, 7, 44, 36, 81, 2] which progressed from the Schwarzschild
case a = 0 to the very slowly rotating case |a| , M and finally to the general
subextremal case |a| < M . This programme has culminated in the following result:
Black holes in general relativity 13

Theorem 4.2 (Poor mans linear stability of Kerr [39, 41]). For Kerr exte-
rior backgrounds in the full subextremal range |a| < M , general solutions of (9)
arising from regular localised data remain bounded and decay at a sufficiently fast
polynomial rate through a hyperboloidal foliation of spacetime.

See also [8, 34, 42, 50] for analysis of the wave equation on (Schwarzschild)
Kerr-(anti) de Sitter backgrounds.
A complete survey of the proof of Theorem 4.2 is beyond the scope of this
article, but it is worth discussing briefly the salient geometric properties of the
Schwarzschild and Kerr families which enter into the analysis.

4.2.1. The conserved energy and superradiance. The existence of conserved

energy identities is often crucial for boundedness results. Recall that to every
Killing field X , by Noethers theorem, there is a corresponding conserved 1-
form associated to solutions of (9) formed by contracting X with the energy-
momentum tensor T [] = 21 g . If the Killing field is causal,
then the flux terms on suitably oriented spacelike or null hypersurfaces are non-
negative definite. Let us examine this in the context of our problem.
We first consider the Schwarzschild case a = 0. As explained in Section 2.1,
the static Killing field t is then timelike in the black hole exterior, becoming null
at the horizon H+ . The associated energy identity applied in a region R
H+

I
+

R
0

indeed gives nonnegative definite flux terms, and thus yields a useful conservation
law for solutions of (9)but barely! After obtaining higher order estimates via
further commutations of (9) by Killing fields and applying the usual Sobolev esti-
mates, this is sufficient to estimate and its derivatives pointwise away from the
horizon. Since this energy is degenerate where t becomes null, it is, however, in-
sufficient to obtain uniform pointwise control of the solution and its derivatives up
to and including H+ . The original boundedness proof of Kay and Wald [51] over-
came this problem in a clever manner, but using very fragile structure associated
to the exact Schwarzschild metric.
In the Kerr case, for all non-zero values a != 0, things become much worse.
For there is now a region E in the black hole exterior where the stationary Killing
field t is spacelike! This is known as the ergoregion. As a result, the energy
flux corresponding to t is no-longer non-negative definite and thus does not yield
even a degenerate global boundedness in the exterior. This is the phenomenon of
superradiance; there is in particular no a priori bound on the flux of radiation to
null infinity I + .
Before understanding how this problem is overcome, we must first discuss two
other phenomena, the celebrated red-shift effect and the difficulty caused by the
presence of trapped null geodesics.
14 Mihalis Dafermos

4.2.2. The redshift. The red-shift effect was first discussed in a paper of Oppenheimer
Snyder [64]. One considers two observers A and B as depicted:

H+
I+
B

The more adventurous observer A falls in the black hole whereas observer B for
all time stays outside. Considering a signal emited by A at a constant frequency
according to her watch, in the geometric optics approximation, the frequency of
the signal as measured by observer B goes to zero as Bs proper time goes to
infinityi.e. it is shifted infinitely to the red in the electromagnetic spectrum.
For general sub-extremal black holes, there is a localised version of this effect
at the horizon H+ :

H+
I+
B

If both observers A and B fall into the black hole and are connected by time
translation A = B where is the Lie flow of the Killing field t , then the
frequency measured by B is shifted to the red by a factor exponential in .
It turns out that the above geometric optics argument can be captured by the
coercivity properties of a physical space energy identity near H+ , corresponding to
a well-chosen transversal vector field N to H+ . Such a vector field was introduced
in [33] and the construction was generalised in the Epilogue of [38] to arbitrary
Killing horizons with positive surface gravity > 0.9 The good coercivity proper-
ties do not hold globally however, and thus to obtain a useful estimate one must
combine the energy identity of N with additional information.
In the Schwarzschild case |a| = 0, it is precisely the conserved energy estimate
discussed in Section 4.2.1 with which one can combine the above red-shift estimate
to obtain finally the uniform boundedness of the non-degenerate N -energy. One
can moreover further commute (9) with N preserving the red-shift property at the
horizon [37, 38] to again obtain a higher order N -energy estimate, from which then
pointwise boundedness follows using standard Sobolev inequalities. This gives a
simpler and more robust understanding of Kay and Walds original [51]. See [38].
In the Kerr case a != 0, however, in view of the absense of any global a priori
energy estimate, it turns out that in order to apply the N identity, one needs some
understanding of dispersion. Thus, the problems of boundedness and decay are
9 Note that the above positivity property breaks down in the extremal case |a| = M as this is

characterized precisely by = 0. See Section 4.2.5 below.

Black holes in general relativity 15

coupled. For the latter, however, it would seem that we have to understand a certain
high-frequency obstruction to decay caused by so-called trapped null geodesics.
4.2.3. Trapped null geodesics. Again, we begin with the Schwarzschild case.
It is well known (cf. [47]) that the hypersurface r = 3M is generated by null
geodesics which neither cross the horizon H+ nor escape to null infinity I + . They
are the precise analogue of trapped rays in the classical obstacle problem. In the
context of the latter, the presence of a single such ray is sufficient to falsify certain
quantitative decay bounds [70]. A similar result holds in the general Lorentzian
setting [74]. Weaker decay bounds can still hold, however, if the dynamics of
geodesic flow around trapping is good, that is to say, the trapped null geodesics
are themselves dynamically unstable in the context of geodesic flow.
It turns out that Schwarzschild geometry indeed exhibits good trapping.
The programme of capturing this by local integrated energy decay estimates with
degeneration was initiated by [5]. See [33, 7, 35]. From these and the red-shift
identity of Section 4.2.2, the full decay statement of Theorem 4.2 in the a = 0 case
can now be inferred directly by a black box method [36]. See also [80].
The Schwarzschild results [33, 7, 35] exploited the fact that not only is the
structure of trapping good from the point of view of geodesic flow in phase
space, but it is localised at the codimensional-1 hypersurface r = 3M of physical
space. The latter feature is broken in Kerr for all a != 0. Nonetheless, in the case
|a| , M , analogues of local integrated energy decay could still be shown using
either Carters separability [38, 40], complete integrability of geodesic flow [81],
or, commuting the wave equation with the non-trivial Killing tensor [2]. Each of
these methods effectively frequency localises the degeneration of trapping and uses
the hidden symmetries of Kerr discussed in Section 2.2; implicitly, these proofs all
show that when viewed in phase space, the structure of trapping remains good.10
The above [38, 40, 81, 2] all use the assumption |a| , M in a second essential way,
so as to treat superradiance as a small parameter; in particular, this allows one
to couple integrated local energy decay with the red-shift identity of Section 4.2.2
and obtain, simultaneously, both boundedness and decay.
Although the problems of boundedness and decay are indeed coupled, a more
careful examination shows that one need not understand trapping in order to ob-
tain boundedness. Our earlier result [37] had in fact showed that, exploiting the
property that superradiance is governed by a small parameter and the ergorergion
lies well within the region of coercivitiy properties of the red-shift identity, one
could prove boundedness using dispersion only for the superradiant part of the
solution, which is itself not trapped. This in fact allowed one to infer boundedness
for (9) on suitable metrics only assumed C 1 close to Schwarzschild, for which one
cannot appeal to structural stability of geodesic flow.
It turns out that it is the above insight which holds the key to the general
|a| < M case. Remarkably, one can show that, for the entire subextremal range, not
only is trapping always good, but the superradiant part is never trapped. The latter

10 Note that the latter fact can also be inferred from structural stability properties of geodesic

flow. See [84].

16 Mihalis Dafermos

is particularly suprising since when viewed in physical space, there do exist trapped
null geodesics in the ergorergion for a close to M . The above remarks are sufficient
to construct frequency localised vector field multipliers yielding integrated local
energy decay in the high frequency regime. See the original treatment in [39].

4.2.4. Finite frequency obstructions. There is one final new difficulty that
appears in the general |a| < M case: excluding the possibility of finite frequency
exponentially growing superradiant modes or resonances.
The absense of the former was proven in a remarkable paper of Whiting [83].
Whitings methods were very recently extended to exclude resonances on the axis
by Shlapentokh-Rothman in [76]. These proofs depend heavily on the algebraic
symmetry properties of the resulting radial o.d.e. associated to Carters separation
of (9)yet another miracle of the Kerr geometry! Using a continuity argument in
a, it is sufficient in fact to appeal to the result [76] on the real axis. This is the
final element of the proof of Theorem 4.2. See [41] for the full details.

4.2.5. The extremal case and the Aretakis instability. Let us finally note
that the precise form (see [41]) of Theorem 4.2 does not in fact hold without quali-
fication for the extremal case |a| = M . This is related precisely to the degeneration
of the red-shift of Section 4.2.2.

Theorem 4.3 (Aretakis [3, 4]). For extremal Kerr |a| = M , for generic solutions
of , translation invariant transversal derivatives on the horizon fail to decay, and
higher-order such derivatives grow polynomially.

Decay results for axisymmetric solutions of (9) in the case of |a| = M have been
obtained in [4], but the non-axisymmetric case is still open and may be subject
to additional instabilities. It is on account of Theorem 4.3 that we have excluded
|a| = M from Conjecture 4.1. The nonlinear dynamics around extremality promise
many interesting features! See [63].

4.3. The full linear stability of Schwarzschild. We have motivated

our study of (9) as a poor mans linearisation of (2). Let us turn now to the
actual linearisation of (2) around black hole backgrounds, that is to say, the true
problem of linear stability.
Very recently, with G. Holzegel and I. Rodnianski, we have obtained the full
analog of Theorem 4.2 for the linearised Einstein equations around Schwarzschild.

Theorem 4.4 (Full linear stability of Schwarzschild [30]). Solutions for the lineari-
sation of the Einstein equations around Schwarzschild arising from regular admis-
sible data remain bounded in the exterior and decay (with respect to a hyperboloidal
foliation) to a linearised Kerr solution.

The additional difficulties of the above thorem with respect to the scalar wave
equation (9) lie in the highly non-trivial structure of the resulting coupled system
equations. As in the non-linear stability of Minkowski space, a fruitful way of
capturing this structure is with respect to the structure equations and Bianchi
Black holes in general relativity 17

equations captured by a null frame. Linearising (8), we schematically obtain

/ (1) = (1) (0) + (1) ,

/ (1) = D
/ (1) + (1) (0) + (0) (1) , (10)

where (1) , (1) now denote linearised spin coefficients and curvature components,
respectively, and (0) , (0) now denote background terms. Note that in the case of
Minkowski space, (0) = 0 and thus the equations for (1) decouple from those for
(1) and admit a coercive energy estimate via contracting the Bel-Robinson tensor
with t [22]. Already in the Schwarzschild case, however, (0) != 0 and the two
sets of equations in (10) are coupled. A fundamental difficulty is the absense of an
obvious coercive energy identity for the full system (10), or even just the Bianchi
part. Thus, even obtaining a degenerate boundedness statement, cf. Section 4.2.1,
is now non-trivial.
Our approach expresses (10) with respect to a suitably normalised null frame
associated to a double null foliation. We then introduce a novel quantity, defined
explicitly as
% & 3 % &
/ $2 D
P =D / $1 (1) , (1) + 0 (tr)0 (1) (1)
4

together with a dual quantity P . Here (1) , (1) denote particular linearised com-
ponents of the Riemann tensor, (1) and (1) denote the linearised shears of the
$ $
foliation, 0 and tr0 are Schwarzschild background terms and D / 2 and D
/ 1 denote
the first order angular differential operators of [23].
The quantity P decouples from (10) and satisfies the ReggeWheeler equation

/ 4 (r5 P )) (1 2M r1 )(r
/ 3 ( / 5 P ) + (4r2 6M r3 )(1 2M r1 )(r5 P ) = 0
(11)
Like (9), the above equation does indeed admit a conserved coercive energy esti-
mate. The first part of our proof obtains a complete understanding of P , which is
a relatively easy generalisation of Theorem 4.2 restricted to a = 0;

Proposition 4.5. Solutions P of (11) arising from regular localised data satisfy
boundedness and integrated local energy decay (non-degenerate at the horizon and
with good weights at infinity, cf. [36]) and decay polynomially with respect to a
hyperboloidal foliation.

See also [6]. Given Proposition 4.5, one can then exploit a hierarchial struc-
ture in (10) to estimate, one by one, all other quantitites, schematically denoted
(1) , (1) , by integration as transport equations in L2 . From integrated local en-
ergy decay and boundedness for P , one obtains integrated local energy decay and
boundedness for each quantity, after a suitable linearised Kerr solution is sub-
tracted. It is essential here that one uses the full strength of Proposition 4.5 with
respect to the non-degeneration at the horizon and the good weights at infinity.
It is interesting to compare our approach to the formal mode analysis of the
physics literature (see [12]). There one attempts to recover everything from the
linearised curvature components (1) and (1) , which also decouple and satisfy the
18 Mihalis Dafermos

so-called BardeenPress equation11 . In contrast to (11), however, this equation

does not admit an obvious coercive conserved energy, but it can nonetheless be
shown that it does not admit growing modes. From this one can in principle
formally recover control of other quantities for fixed modes [12]. This approach,
however, fails to yield an estimate beyond fixed modes, precisely because of the
absense of a mode-independent energy estimate for BardeenPress. Note that
when viewed in frequency space, our P can be related to (1) by the transformation
theory of Chandrasekhar [12].
We reiterate finally that in the above argument, obtaining even boundedness
for the full system (10) required the dispersive part of Proposition 4.5. Thus
we see that, even at the linear level, there does not appear to be a pure orbital
stability result; just as in the non-linear theory, boundedness is coupled to showing
quantitative decay.
4.4. The road to Conjecture 4.1. Before turning in Section 5 to the
black hole interior, let us revisit our fully nonlinear problem of Conjecture 4.1.
The issue of using decay rates as in Theorem 4.2 in a nonlinear setting satisfying
a null condition has been addressed in a scalar problem by Luk [59]. See also [49].
As we described in Section 4.1, to prove Conjecture 4.1, one must identify (and
linearise around) the asymptotic parameters to which the solution will asymptote
and for every open set of initial data, these parameters will generically have a != 0.
It follows that until the analogue of Theorem 4.4 has been obtained for Kerr, at
the very least for the very slowly rotating regime |a| , M , then one expects that
there is no open set in the moduli space of initial data which can be handled.
It is worth mentioning, however, that there is a restricted version of Con-
jecture 4.1 which can in principle be studied using only the Schwarzschild linear
stability result. If axisymmetry is imposed on the initial data and one moreover im-
poses that the initial angular momentum vanishes, then, since angular momentum
does not radiate to null infinity under the assumption of axisymmetry, one expects
that the solution should approach a Schwarzschild black hole and thus should be
amenable to study using only Theorem 4.4. This is the content of ongoing work.
We mention finally that under spherical symmetry, one can formulate an analo-
gous problem to that of Conjecture 4.1 concerning the Einsteinscalar field system
(see [15]) or the EinsteinMaxwellscalar field system (to be discussed in the next
section).12 The analogue of Conjecture 4.1 is then proven in [15, 26, 32]. The
above problem retains few of the difficulties described in Section 4.2in particular,
it does not exhibit superradiance or trapping. Moreover, on the nonlinear side,
it is interesting to note that spherical symmetry breaks the supercriticality of the
Einstein equations, so in particular, allows 1., 2. and 3. to be proven separately.
Nonetheless, the above models have been especially important as a source for in-
tuition on the stability and instability properties of black hole interiors. We turn
to this now.

11 In the Kerr case, this generalises to the Teukolsky equation. See [12].
12 Recall that in view of Birkhoffs theorem [47], the only spherically symmetric vacuum solu-
tions are Schwarzschild.
Black holes in general relativity 19

5. The black hole interior and singularities

We now turn to the interior of Kerr black holes and strong cosmic censorship.

5.1. The blue-shift instability. In Section 3.3, we motivated Penroses

strong cosmic censorship by little other than wishful thinkingthe possibility of
Cauchy horizons is so problematic that we hope that generically they cannot form.
There is indeed, however, a heuristic argument that suggests that at least the Kerr
Cauchy horizon may be unstable.
The argument, due to Penrose [67], goes as follows. Let A and B be again
two observers, where B now enters the black hole whereas A remains for all time
outside. If A sends a signal to B, then the frequency measured by B becomes
infinitely high as Bs proper time approaches his Cauchy horizon-crossing time.
CH
+

B i+
+

A
H

I
+

i0

That is to say, the signal is infinitely shifted to the blue.

As with the red-shift effect discussed in Section 4.2.2, this effect should be
reflected in the behaviour of waves, but now as an instability. This was in fact
studied numerically in [77] for the related case of the scalar wave equation (9) on
ReissnerNordstrom background.13 In view of the role of (9) as a poor-mans
linearisation of (2), the above heuristic arguments were the first indication that
the smooth Cauchy-horizon behaviour of Kerr could be unstable.14
A general result due to Sbierski [74] shows that the geometric optics argument
is sufficient to falsify a quantitative energy boundedness result analogous to the
precise statement of Theorem 4.2 in the exterior. Suprisingly, however, it turns
out that the blue-shift instability is not strong enough for to blow up in L .

Theorem 5.1 (Franzen [45]). Solutions of the wave equation (9) as in Theo-
rem 4.2 remain pointwise bounded || C on sub-extremal Kerr for a != 0 (or
ReissnerNordstrom Q != 0) in the black hole interior, up to and including CH+ .

This result, whose proof uses as an input the result of Theorem 4.2 restricted
to H+ , can be thought of as the first indication that rough stability results hold all
the way to CH+ . To explore this, however, let us first turn to certain spherically
symmetric toy models.

5.2. Spherically symmetric toy-models. With the Schwarzschild case

as the only example to go by, Penrose had originally speculated [67] that the
13 ReissnerNordstrom (M, g
M,Q ) is a spherically symmetric family of solutions to the Einstein
Maxwell equations and for Q $= 0 has a Cauchy horizon similar to Kerr.
14 For an another manifestation of the blue-shift instability when solving the Einstein equations

backwards in the exterior, see [29].

20 Mihalis Dafermos

blue-shift instability in the fully non-linear setting would give rise to a spacelike
singularity15 .
The simplest toy model with a true wave-like degree of freedom where this can
be studied is the EinsteinMaxwell16 real scalar field system
1 . 1 1 1
R g R = 8T = 8( (F F g F F )+ g )
2 4 4 2
(12)
F = 0, [ F] = 0, !g = 0, (13)
under spherical symmetry. It turns out that for this toy model, Penroses expec-
tation does not hold as stated: At least a part of the boundary of the maximal
development is a null Cauchy horizon through which the metric is at least contin-
uously extendible:

Theorem 5.2 (C 0 -stability of a piece of the Cauchy horizon, [25, 27]). For all
two-ended asymptotically flat spherically symmetric initial data for (12)(13) with
non-vanishing charge, the maximal development can be extended through a non-
empty Cauchy horizon CH+

r=0
CH
+

H+ I+

as a spacetime with C 0 metric.

The above theorem depends in fact also on joint work with Rodnianski [32]
on the exterior region (cf. the end of Section 4.4) which obtains upper polynomial
bounds for the decay of on H+ . Heuristic and numerical [46, 10] work suggests a
precise asymptotic tail, in particular, polynomial lower bounds on H+ . With this
as an assumption, one can obtain the following

Theorem 5.3 (Weak null singularities, [27]). For spherically symmetric initial
data as above where a pointwise lower bound on v is assumed to hold asymptot-
ically along the event horizon H+ that forms, then the above Cauchy horizon CH+
is singular: The Hawking mass (thus the curvature) diverges and, moreover, the
extension of Theorem 5.2 fails to have locally square integrable Christoffel symbols.

The above two theorems confirmed a scenario which had been suggested on the
basis of previous arguments of Hiscock [48], IsraelPoisson [69] and Ori [65] as well
as numerical studies of the above system [9, 10]. In view of the blow up of the
15 In fact, one still often sees an alternative formulation of Conjecture 3.5 as the statement that

Generically, singularities are spacelike.

16 The pure scalar field model, whose study was pioneeered by Christodoulou [15], does not

admit Cauchy horizons emanating from i+ . The system (12)(13) is the simplest generalisation
that does, in view of the fact that it admits ReissnerNordstrom as an explicit solution.
Black holes in general relativity 21

Hawking mass, the phenomenon was dubbed mass inflation. The type of singular
boundary exhibited by the above theorem, where the Christoffel symbols fail to
be square integrable but the metric continuously extends, is known as a weak null
singularity.
The above results apply to general solutions, not just small perturbations of
ReissnerNordstrom. In the stability context, it turns out that the r = 0 piece is
absent, and the entire bifurcate Cauchy horizon is globally stable:

Theorem 5.4 (Global stability of the ReissnerNordstrom Cauchy horizon [28]).

For small, spherically symmetric perturbations of ReissnerNordstrom, the maxi-
mal development is extendible beyond a bifurcate null horizon as a manifold with
continuous metric. The CarterPenrose diagramme is as in the ReissnerNordstrom
case. In particular, there is no spacelike part of the singularity.

Note that the above is precisely the result that one obtains by naively extrap-
olating Theorem 5.1 to the fully non-linear theory, identifying with the metric.

Corollary 5.5 (Bifurcate weak null singularities, [28]). Under the assumptions of
Theorem 5.4 and the additional asssumption of Theorem 5.3 on both event hori-
zons, the Cauchy horizons CH+ represent bifurcate weak null singularities and the
extensions fail to have locally square integrable Christoffel symbols.

The ultimate spherically symmetric toy model is that of the Einstein-Maxwell

charged scalar field system, that is when the scalar field is complex-valued and
carries charge and is directly coupled with the Maxwell field through this charge,
besides the gravitational coupling through the Einstein equations (as in (12)). In
his Cambridge Ph.D. thesis [55], J. Kommemi has shown an analogue of Theo-
rem 5.2 for this model, given an a priori decay assumption on the horizon.

5.3. Beyond toy models: Einstein vacuum equations without

symmetry. Whereas the above work [32, 27, 55] more or less definitively re-
solves the issue of the appearance of weak null singularities in spherically symmetric
toy models, one could still hold out hope that the vacuum Einstein equations (2) do
not allow for the formation of such singularities but favour spacelike singularities
as in the Schwarzschild case. In contrast to the spherically symmetric toy world,
for the Einstein vacuum equations without symmetry there is really no numerical
work available on this problem and very little heuristics (see however [66]).

5.3.1. Luks vacuum weak null singularities. The first order of business is
thus to construct examples of local patches of vacuum spacetime with a weak null
singular boundary. This has recently been accomplished in a breakthrough paper
of J. Luk [60], based in part on his previous work with Rodnianski [61, 62] on
impulsive gravitational waves.
Luks spacetimes have no symmetries and are constructed by solving a char-
acteristic initial value problem with characteristic data of a prescribed singular
behaviour. The problem reduces to showing existence in a rectangular domain as
well as propagation of the singular behaviour. This is given in:
22 Mihalis Dafermos

Theorem 5.6 (Luk [60]). Consider characteristic initial data for the Einstein
vacuum equations on a bifurcate null hypersurface C C whose spherical sections
are parameterised by affine u [0, u )) and u [0, u )), resepectively, and where
the outgoing shear (and sufficient angular derivatives) satisfies
|| | log(u u)|p |u u|1 . (14)
Then the maximal development can be covered by a double null foliation terminating
in a null boundary u = u

u
=
u

C

C
through which the metric is continuously extendible. The singular behaviour (14)
propagates, making this boundary a weak null singularity.
Moreover, in analogy with the LukRodnianski theory of two interacting im-
pulsive gravitational waves [62], Luk obtained
Theorem 5.7 (Luk [60]). Consider again characteristic data as above but such
that both outgoing shears and (and sufficient angular derivatives) satisfy

|| | log(u u)|p |u u|1 , || | log(u u)|p |u u|1 , (15)

and moreover, the data satisfies an appropriate smallness condition. Then the
maximal development can be covered by a double null foliation which terminates
in a bifurcate null hypersurface {u } [0, u ] [0, u ] {u } through which the
metric is continuously extendible. Relations (15) propagate, making the boundary
of spacetime a bifurcate weak null singularity.
Note that in LukRodnianski theory [61, 62], (14) is replaced by the assump-
tion that is discontinuous but bounded. Thus, it was possible in [61, 62] to
interpret the Einstein equations beyond these null hypersurfaces, which interact
simply passing through each other, leaving in their wake a regular spacetime. Here,
however, the boundaries are much more singular ( is not in any Lp for p > 1),
and thus, the solution cannot be interpreted beyond them, even as a weak solution
of (2).17
In the short space of this article, it is impossible to give an overview of the
proofs of the above theorems. As in several of the results we have discussed, the
proof expresses (8) with respect to a null frame attached to a double null foliation,
and moreover, relies on a renormalisation of this system which removes the most
singular components (extending ideas from [61, 62]). This does not completely
regularise the system, however, and a fundamental role is played by a hierarchy
of largeness/smallness which is preserved in evolution by special null structure of
(8). These ideas are in turn related to the seminal work of Christodoulou [20] on
the dynamic formation of trapped surfaces, surveyed in another article in these
proceedings [21], and his short pulse method.
17 In particular, the name weak null singularity is in some sense unfortunate!
Black holes in general relativity 23

5.3.2. The global stability of the Kerr Cauchy horizon. Putting together
essentially all the ideas form Sections 5.25.3.1, we have very recently obtained the
following result in upcoming joint work with J. Luk.
Theorem 5.8 (Global stability of the Kerr Cauchy horizon [31]). Consider char-
acteristic initial data for (2) on a bifurcate null hypersurface H+ H+ , where
H have future-affine complete null generators and their induced geometry is glob-
ally close to and dynamically approaches that of the event horizon of Kerr with
0 < |a| < M at a sufficiently fast polynomial rate. Then the maximal development
can be extended beyond a bifurcate Cauchy horizon CH+ as a Lorentzian manifold
with C 0 metric. All finitely-living observers pass into the extension.
Let us note explicitly that a corollary of the above theorem together with a
successful resolution of Conjecture 4.1 would be the following definitive statement
Corollary 5.9. If Conjecture is 4.1 is true then the Cauchy horizon of the Kerr
solution is globally stable and the C 0 -inextendibility formulation and the generi-
cally, spacetime singularities are spacelike formulation of strong cosmic censorship
are both false.

5.3.3. The future for strong cosmic censorship. In view of the toy-model
results of Theorem 5.3 and Corollary 5.5, all is not lost for strong cosmic censorship.
A version of the inextendibility requirement in the formulation of strong cosmic
censorship which is compatible with the result of Theorem 5.3 for the toy problem
and may still be true for the vacuum without symmetry is the statement that
(M, g) be inextendible as a Lorentzian manifold with locally square integrable
Christoffel symbols. This formulation is due to Christodoulou [20] and would
guarantee that there be no extension which can be interpreted as a weak solution
of (2). It is an interesting open problem to obtain this in a neighbourhood of the
Kerr family. This naturally separates into the following two statements:
Conjecture 5.10. 1. Under a suitable assumption on the data on H+ in The-
orem 5.8, then CH+ is a weak null singularity, across which the metric is inex-
tendible as a Lorentizian manifold with locally square integrable Christoffel symbols.
2. The above assumption on H+ holds for the data of Conjecture 4.1, provided the
latter are generic.
One can in fact localise the result of Theorem 5.8 to apply to spacetimes with
one asympotically flat end, provided they satisfy the assumption on H+ , and one
can infer again a non-empty piece of null singular boundary CH+ . Thus, all black
holes which asymptotically settle down in their exterior region to Kerr with 0 <
|a| < M will have a non-empty C 0 -Cauchy horizon, which, assuming a positive
resolution to Conjecture 5.10, will correspond to a weak null singularity.
Do the above Cauchy horizons/weak null singularities close up the whole
maximal development as in the above two-ended case? Or will they give way to
a spacelike (or even more complicated) singularity? These questions may hold
the key to understanding strong cosmic censorship beyond a neighbourhood of the
Kerr family.
24 Mihalis Dafermos

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DPMMS, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK and

Fine Hall, Department of Mathematics, Washington Road, Princeton NJ 08544 USA
E-mail: dafermos@math.princeton.edu