-
Formation of Trapped Surfaces
in General Relativity
Xinliang An
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Mathematics
Adviser: Sergiu Klainerman
June 2014
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c© Copyright by Xinliang An, 2014.
All Rights Reserved
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Abstract
In this thesis we present two results regarding the formation of
trapped surfaces in general
relativity.
The first is a simplified approach to Christodoulou’s
breakthrough result which showed that
trapped surfaces can form dynamically by the focusing of
gravitational radiation from past null
infinity. We extend the methods of Klainerman-Rodnianski, who
gave a simplified proof of this
result in a finite region.
The second result extends the theorem of Christodoulou by
allowing for weaker initial data but
still guaranteeing that a trapped surface forms in the casual
domain. In particular, we show that a
trapped surface can form dynamically from initial data which is
merely “large” in a scale-invariant
way. The second result is obtained jointly with Luk.
iii
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Acknowledgements
First and foremost, I would like to convey my gratitude to my
advisor Professor Sergiu Klainerman
for bringing me into this field with great support and
encouragement. His vision and insight for
mathematics always motivate me to explore the mathematical
world.
I am very grateful to my collaborator and great friend, Jonathan
Luk for all those enlightening
discussions and help in my PhD years.
I would like to thank Professors Mihalis Dafermos, Jérémie
Szeftel, who shared their under-
standings and gave great suggestions and support to my
study.
Sincere thanks also go to Professors Spyros Alexakis, Kung-Ching
Chang, Peter Constantin,
Greg Galloway, Zheng-Chao Han, Gustav Holzegel, Alexandru
Ionescu, Michael Kiessling, Yanyan
Li, Elliot Lieb, Hans Lindblad, Mao-Kang Luo, Shige Peng, Frans
Pretorius, Jie Qing, Hans
Ringström, Igor Rodnianski, Avraham Soffer, Jared Speck, Shadi
Tahvildar-Zadeh, Gang Tian,
Mu-Tao Wang for many insightful conversations and for their
support.
I want to thank many friends in the analysis group at Princeton:
John Anderson, Stefanos
Aretakis, Tom Beck, Yu Deng, Ross Granowski, Georgios Moschidis,
Luc Nguyen, Hiro Oh, Sung-
Jin Oh, Oana Pocovnicu, Volker Schlue, Arick Shao, Yakov
Shlapentokh-Rothman, John Stogin,
Martin Taylor, Qian Wang, Xuecheng Wang, Willie Wong, Polam
Yang, Shiwu Yang, Pin Yu,
Ruixiang Zhang, Ruobing Zhang, who made my PhD years very joyful
and unforgettable.
I also would like to thank Shiguang Ma, Jinxing Xu, Bohua Zhan,
Ting Zhang and Xuefeng
Zhang for the friendship along the way.
I am grateful to our department Chair Professor Gabai, graduate
administrator Jill LeClair and
Jennifer Johnson for their help and support.
Last, but certainly not least, I would like to thank my family.
To my father Yuqing An and
my mother Pengqian Zhang for raising me and supporting me to
chase my dreams; To my sister
Xiaozhu An for always being a loyal friend; To my fiancée Chen
Chen, for her unconditional love
and company along these years.
iv
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to my family.
v
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Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . iv
1 Introduction 1
1.1 Formation of Trapped Surfaces from Past Null Infinity . . .
. . . . . . . . . . . . . . 3
1.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 3
1.1.2 Heuristic Argument . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 5
1.1.3 Main Results . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 8
1.2 Formation of Trapped Surfaces in Vacuum from Mild Incoming
Radiation
(Joint with Luk) . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 11
1.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 11
1.2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 16
1.2.3 Heuristic Argument with ESFS-S . . . . . . . . . . . . . .
. . . . . . . . . . . 19
2 Formation of Trapped Surfaces from Past Null Infinity 28
2.1 Setting, Equations and Notations . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 28
2.1.1 Double Null Foliation . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 29
2.1.2 The Coordinate System . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 30
2.1.3 Equations . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 31
2.1.4 Integration . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 35
2.1.5 Signatures . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 36
vi
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2.1.6 Scale Invariant Norms . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 38
2.1.7 Norms . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 39
2.2 Statement of Main Theorem . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 42
2.2.1 Structure of the Proof . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 44
2.2.2 Strategy of the Proof . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 45
2.3 The Preliminary Estimates . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 47
2.3.1 Estimates for Metric Components . . . . . . . . . . . . .
. . . . . . . . . . . 48
2.3.2 Estimates for Transport Equations . . . . . . . . . . . .
. . . . . . . . . . . . 50
2.3.3 Sobolev Embedding . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 54
2.3.4 Commutation Formulas . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 55
2.3.5 General Elliptic Estimates for Hodge Systems . . . . . . .
. . . . . . . . . . . 57
2.4 Ricci Coefficient Estimates . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 62
2.4.1 Estimates for Zero Derivatives of Ricci Coefficients . . .
. . . . . . . . . . . . 62
2.4.2 Estimates for First Derivatives of Ricci Coefficients . .
. . . . . . . . . . . . . 72
2.4.3 L∞(S) Estimates for Ricci Coefficients . . . . . . . . . .
. . . . . . . . . . . . 88
2.5 Elliptic Estimates . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 94
2.5.1 L4(S) Estimates for Curvature Components . . . . . . . . .
. . . . . . . . . . 95
2.5.2 L2(S) Estimates for First Derivatives of Curvature
Components . . . . . . . 98
2.5.3 L2(S) Estimates for Third Derivatives of Ricci
Coefficients . . . . . . . . . . 102
2.5.4 L4(S) Estimates for Second Derivatives of Ricci
Coefficients . . . . . . . . . . 135
2.6 Curvature Estimates . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 137
2.6.1 Curvature Estimates in the Scale Invariant Norms . . . . .
. . . . . . . . . . 138
2.6.2 Curvature Estimate I . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 142
2.6.3 Curvature Estimate II . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 146
2.6.4 Curvature Estimate III . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 156
2.7 Formation of Trapped Surfaces . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 169
2.8 Retrieving Christodoulou’s Estimates . . . . . . . . . . . .
. . . . . . . . . . . . . . . 174
2.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 179
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2.9.1 Norms in standard Lp form . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 179
2.9.2 Equations for Elliptic Estimates . . . . . . . . . . . . .
. . . . . . . . . . . . 180
2.9.3 Equations for Energy Estimates . . . . . . . . . . . . . .
. . . . . . . . . . . . 190
3 Formation of Trapped Surfaces in Vacuum from Mild Incoming
Radiation
(Joint with Luk) 197
3.1 Setting, Equations and Notations . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 197
3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 197
3.1.2 Double Null Foliation . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 198
3.1.3 The Coordinate System . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 199
3.1.4 Equations . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 200
3.1.5 Schematic Notation . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 204
3.1.6 Integration . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 205
3.1.7 Norms . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 206
3.2 Statement of main theorem . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 208
3.2.1 Structure of the Proof . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 209
3.2.2 Strategy of the Proof . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 210
3.3 The Preliminary Estimates . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 213
3.3.1 Estimates for Metric Components . . . . . . . . . . . . .
. . . . . . . . . . . 214
3.3.2 Estimates for Transport Equations . . . . . . . . . . . .
. . . . . . . . . . . . 216
3.3.3 Sobolev Embedding . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 219
3.3.4 Commutation Formula . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 221
3.3.5 General Elliptic Estimates for Hodge Systems . . . . . . .
. . . . . . . . . . . 223
3.4 Estimates for Ricci Coefficients . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 226
3.5 Elliptic estimates for the fifth derivatives of the Ricci
coefficients . . . . . . . . . . . 244
3.6 Estimates for Curvature . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 271
3.7 Formation of Trapped Surfaces . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 292
viii
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Chapter 1
Introduction
My thesis is motivated by a long-standing and central question
in general relativity: the dynamical
formation of black holes. This is a large data problem for the
Einstein field equations
Rµν −12gµνR = 8πTµν , (1.0.1)
where Rµν and R denote the Ricci and scalar curvatures of the
metric gµν , and Tµν denotes the
stress-energy-momentum tensor of matter. The Einstein field
equations are systems of supercrit-
ical quasilinear partial differential equations. The objects of
study are 4-dimensional Lorentzian
manifolds (M, g) called spacetimes. These equations describe the
gravitation that results from
spacetime being curved by matter and energy. When Tµν ≡ 0,
(1.0.1) is named the Einstein vac-
uum equations (EVE). One of the most fascinating features of
(1.0.1) is that it admits so called
black hole spacetimes. Informally, black holes are regions of
spacetime where the gravitational field
is so strong that even light cannot escape and hence invisible
to far-away observers. Famous explicit
solutions of EVE containing black hole regions include the
Schwarzschild and Kerr families. The
explicit properties of these solutions are well-known to
mathematicians and physicists. At present,
however, understanding the formation of black holes remains one
of the most important unfinished
tasks in the theory of relativity. Nevertheless, encouraging
progress has recently been made with
1
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respect to formation of trapped surfaces. A trapped surface is
defined as a two dimensional space-
like sphere whose outgoing and incoming expansions are negative.
This is a local object which,
heuristically, indicates the formation of a black hole to the
future.
In a 589-page breakthrough result [7] in 2008, Christodoulou
proves: without any symmetry
assumption, trapped surfaces form in the Cauchy development of
vacuum initial data which are
arbitrarily dispersed. In [7], Christodoulou presents a novel
technique called the short pulse method.
He introduces a certain large amplitude in the data, the pulse,
compensated by a small parameter
δ which controls the length of the outgoing direction. This
gives a hierarchy of large and small
quantities, which are coupled via the EVE. In [15], Klainerman
and Rodnianski extend the above
result and significantly simplify the proof in a finite region
of spacetime. In my works [1] and [2], I
first extend the result of Klainerman and Rodnianski to an
infinite region and use a new and more
direct way to re-prove the main theorem in Christodoulou’s
monograph [7]. Second, jointly with
Luk, I use the methods developed in [1] to provide a scale
invariant criterion for the formation of
trapped surfaces. This allows us to answer the following
question: what is the minimum initial
“mass density” required to guarantee that a trapped surface
forms in evolution?
In [1], I extend the results in [15] which are concerned with a
finite region of spacetime by
showing analogous results from an ideal conformal boundary at
infinity (so-called past null infinity).
One defines and studies gravitational waves from past null
infinity by understanding decay rates
of various geometric quantities. For the EVE there are fourteen
separate components (six from
curvature, eight from connection) which all behave differently.
To overcome the difficulties coming
from the complexity of the EVE and to capture the decay
properties of their components, I extend
the methods introduced by Klainerman and Rodnianski in [15] and
assign to each component a
number which carries the information of decay, called the
signature for decay rates. Using the
signature, I define a collection of scale invariant norms, which
enable me to explore the hidden
smallness of interactions between various geometric quantities
near past null infinity. Hence, except
for rare terms, which are called anomalies and left for further
analysis, I treat most terms in a
simple and direct manner. This reduces the workload
significantly.
Moreover, based on a relation between borderline (the most
anomalous) terms’ coefficients and
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their signatures for decay rates I extend a method employed by
Holzegel, Luk and Rodnianski and
offer a more direct and intuitive way to obtain energy
estimates. This approach is capable for
studying decay rates in far-from-symmetry cases.
The methods I exploit in [1] to study decay rates also fit well
for studying blow-up rates close
to the center of gravitational collapse. In [2], together with
Luk, I prove that with an initial
mass density of size δ, a trapped surface of radius δ forms
dynamically. By introducing two new
parameters and corresponding hierarchies and a number of
renormalizations, we extend the short
pulse method introduced by Christodoulou in [7].
The following subsections describe results in [1] and [2] in
more details.
1.1 Formation of Trapped Surfaces from Past Null Infinity
1.1.1 Background
A celebrated solution to the Einstein vacuum equations
Ric(g) = 0 (1.1.1)
is the Schwarzschild metric:
g = −(
1− 2Mr
)dt2 +
(1− 2M
r
)−1dt2 + r2dS,
where dS denotes the standard metric on unit sphere and M and r
represent mass and the radial
coordinate, respectively.
In the Schwarzschild spacetime, the three dimensional
hypersurface r = 2M represents the
boundary of an interior region. Every timelike or null geodesic
γ(s) starting from the interior
region (r < 2M) is confined in the region r ≤ 2M . The
interior region of Schwarzschild spacetime
is the most famous example of a black hole. Moreover, each γ(s)
is future incomplete. γ(s) will
reach r = 0 and RαβγδRαβγδ =∞ at r = 0.Therefore, the
Schwarzschild spacetime is future causally
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geodescially incomplete.
Of course, the Schwarzschild metric is a very special solution
to Einstein vacuum equations.
Thus, a fundamental question arises: do generic solutions to
Einstein vacuum equations possess
any of the singular features of the Schwarzschild metric?
To some extend, this question is answered by Penrose in [20]
with
Theorem 1.1.1. (Penrose’s Incompleteness Theorem) Let (M, g)
satisfy reasonable topolog-
ical conditions (M is globally hyperbolic with a noncompact
Cauchy hypersurface.) and physical
conditions (M satisfies Ric(V, V ) ≥ 0 for all null V .). It
follows that if M contains a closed trapped
surface, then it is future causally geodesically incomplete.
By definition, 2-sphere S is called trapped if both its future
expansions are negative. Let L,L
be null vector fields. L is the outgoing vectorfield and L is
the incoming vectorfield. Define χ, χ
to be the null second fundamental forms of the hypersurfaces
generated by L,L. If both trχ < 0
and trχ < 0 hold pointwise, then the 2-sphere S is called a
trapped surface. As a comparison, a
standard 2-sphere S with radius r in Minkowski space possesses
trχ = 2/r > 0 and trχ = −2/r < 0.
Thus, following Penrose’s Incompleteness Theorem, trapped
surface formation implies geodesi-
cally incomplete Hence, one may formally equate the existence of
a trapped surface with the exis-
tence of a black hole. This is very useful because trapped
surfaces are local and concrete objects.
To prove that trapped surfaces can form dynamically requires a
good understanding of the
solution to the EVE. Indeed, this problem was open for forty
years, until the recent breakthrough
result of Christodoulou. In the 589-page monograph [7] he
identifies an open set of smooth initial
data containing no trapped surfaces on a finite outgoing null
hypersurface, with trivial data on an
incoming null hypersurface, whose evolution must form a trapped
surface.
In [15], Klainerman and Rodnianski extend the above result and
simplify Christodoulou’s proof
significantly in a finite region of spacetime. They enlarge the
admissible set of initial conditions and
show that the corresponding propagation estimates are much
easier to derive than the ones in [7].
Meanwhile, this relaxation is still enough to guarantee that a
trapped surface forms dynamically.
Moreover, Klainerman and Rodnianski introduce a parabolic
scaling in [15] and corresponding scale
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invariant norms. These new norms allow them to capture the
hidden smallness of the nonlinear
interactions in the EVE.
In the first part of this thesis, we will extend Klainerman and
Rodnianski’s result to prove
the formation of trapped surfaces from past null infinity. We
will then retrieve Christodoulou’s
estimates and re-prove the main theorem in [7].
1.1.2 Heuristic Argument
In this subsection, we demonstrate the heuristic argument for
trapped surfaces formation. We
consider a region D = D(u, u) of a vacuum spacetime (M, g)
generated by optical functions (u, u),
which are increasing toward the future, where u∞ ≤ u ≤ −c ≤ −1
and 0 ≤ u ≤ δ. Here u∞ and
c are fixed constants and δ is to be determined. Our results are
independent of u∞. Later we will
set u∞ to go to −∞ and obtain theorems from past null infinity.
We denote by Hu the outgoing
null hypersurfaces generated by the level surfaces of u, and by
Hu the incoming null hypersurfaces
generated by the level surfaces of u. Hence Su,u = Hu ∩Hu is a
2-sphere.
H u∞(u
=u∞
)
H uHδ (u =
δ)H
0 (u =0)
e 4
e 4
e3
e3
• D(u, u) is the colored region on the left.
• The Optical functions (u, u) satisfy
gµν∂µu∂νu = 0,
gµν∂µu∂νu = 0.
• One point stands for a 2-sphere.
• (ea)a=1,2 is a frame tangent to the 2-sphere
Su,u.
• g(ea, eb) = δab for a, b = 1, 2.
• e3, e4 are a null pair.
• g(e3, e4) = −2.
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To create a trapped surface Su,u, we need that both trχ < 0
and trχ < 0 hold pointwise on
Su,u. For the initial data along H0, on each S(u, 0) we have
trχ(u, 0) = − 2|u|, trχ(u, 0) =
2|u|.
For the initial data along Hu∞ , we have
trχ(u∞, u) = −2|u∞|
+ l.o.t. < 0, trχ(u∞, u) =2|u∞|
+ l.o.t. > 0.
In the colored region D(u, u), trχ < 0 is always true due to
the Raychaudhuri Equation:
∇3trχ = −12
(trχ)2 − |χ̂|2 + l.o.t.
For χ, we have the following transport equations:
∇4trχ+12
(trχ)2 = −|χ̂|2 + l.o.t., (1.1.2)
and
∇3χ̂+12
trχχ̂ = l.o.t. (1.1.3)
Here χ̂ denotes the traceless part of χ.
Employing (1.1.2), we derive
∇3trχ ≤ −|χ̂|2.
Hence, it follows that
trχ(u, u) ≤ trχ(u, 0)−∫ u
0
|χ̂|2(u, u′)du′ = 2|u|−∫ u
0
|χ̂|2(u, u′)du′.
Using the fact that trχ = −2/|u|+ l.o.t. in D(u, u) as well as
(1.1.3), we obtain
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|u|2|χ̂|2(u, u) = |u∞|2|χ̂|2(u∞, u) + l.o.t.
Combining these together, along H−c we have
trχ(−c, u) ≤ trχ(−c, 0)−∫ u
0
|χ̂|2(−c, u′)du′ = 2|c|− |u∞|
2
|c|2
∫ u0
|χ̂|2(u∞, u′)du′ + l.o.t.
In order to create a trapped surface along the hypersurface
H(0,δ)c , and to avoid trapped surfaces
in the initial hypersurface H(0,δ)u∞ , it is sufficient to
require
4c|u∞|2
<
∫ u0
|χ̂|2(u∞, u′)du′ <1|u∞|
.
Thus, we expect
|u∞|‖χ̂‖L∞(Su∞,u) ≈ δ− 12 ,
which is very large.
To rigorously verify this heuristic argument, we encounter two
main difficulties.
1. With arbitrary dispersed initial data at past null infinity,
we use a focusing mechanism to
create a trapped surface of radius c. The gravitational
radiation needs to go sufficiently far inside.
Since this process takes a long time from past null infinity, we
will essentially need a semi-global
existence result for the Einstein vacuum equations without
symmetry assumption. We expect this
to be difficult because:
• with no symmetry assumptions, Einstein vacuum equations are
energy supercritical;
• it is a large data problem, since small data will lead to the
stability of Minkowski spacetime
(see [8]).
To deal with large data, in [7], Christodoulou introduces short
pulse ansatz and the correspond-
ing hierarchy, which is called the short pulse method. Based on
the smallness assumption on δ,
he establishes the initial data hierarchy for various geometric
quantities at Hu∞ . Then he proves
preservation of this hierarchy in the whole colored region D(u,
u) starting from Hu∞ . This enables
7
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him to obtain the desired semi-global existence result.
We will also use the short pulse method, but with a different
initial data hierarchy, and we will
employ a more direct and intuitive approach to the energy
estimates.
In the proof, we will obtain results independent of u∞. Thus, in
the end we will set u∞ to go
to −∞ to obtain formation of trapped surfaces from past null
infinity.
2. Explicitly refer to heuristic argument, we need to make sure
that all of the lower order terms
are truly negligible compared with the main terms. Since
Einstein vacuum equations are a coupled
system of many geometric quantities, this requires the
understanding of detailed information about
all of the geometric quantities and their interactions.
In Chapter 2 of this thesis, by following and extending the
ideas of Klainerman and Rodnianski
in [15], we introduce the notion of signatures. This associates
a pair of numbers (s1, s2) to every
quantities of interest, where s1 and s2 encode the information
about short pulse and decay rates,
respectively. This allows for a systematic treatment of many
terms and it significantly simplifies
the proof. There are however some terms which do not fall into
this framework and these must be
tracked carefully.
1.1.3 Main Results
Signature and Scale Invariant Norms
We now turn to the explicit definition of signature and
associate norms. We assign to each
geometric quantity φ two numbers s1(φ) and s2(φ). The former is
introduced by Klainerman and
Rodnianski in [15]. The latter is called the signature for decay
rates. Using them I define the scale
invariant norms:
‖φ‖L∞sc(Su,u) = δs1(φ)− 12 |u|2s2(φ)+1‖φ‖L∞(Su,u),
‖φ‖L2sc(Su,u) = δs1(φ)−1|u|2s2(φ)‖φ‖L2(Su,u).
Two identities follow from these definitions:
8
-
s1(φ1 · φ2) = s1(φ1) + s1(φ2),
s2(φ1 · φ2) = s2(φ1) + s2(φ2).
We therefore obtain Hölder’s inequality for scale invariant
norms,
‖φ1 · φ2‖L2sc(Su,u) ≤δ
12
|u|‖φ1‖L∞sc(Su,u)‖φ2‖L2sc(Su,u).
This inequality tells us that, if all terms are normal, then the
nonlinear interactions can be
treated as lower order terms. Hence, only rare anomalous terms
are left for further analysis. By
adapting the techniques in [8] and [12], I deal with all of the
borderline terms for decay rate without
encountering a logarithmic divergence.
Energy Estimates without Bel-Robinson Tensor
Based on the relation between the new signatures and the
coefficient in front of the borderline
terms I extend the methods used by Holzegel in [11] and
Luk-Rodnianski in [18]. Without using the
Bel-Robinson tensor, I employ a more direct and intuitive
approach to establish energy estimates.
Denote Ψ to be Curvature component and ψ to be Ricci
coefficient. We observe that, by
separating Ψ into proper pairs, the pair Ψ(s,s′) and Ψ(s−
12 ,s′+ 12 ) satisfy
∇3Ψ(s,s′) + (
12
+ s′)trχΨ(s,s′) = ∇Ψ(s− 12 ,s
′+ 12 ) +∑
s1+s2=s,s′1+s
′2=s′+1
ψ(s1,s′1) ·Ψ(s2,s
′2),
and
∇4Ψ(s−12 ,s′+ 12 ) = ∇Ψ(s,s
′) +∑
s̃1+s̃2=s+12 ,
s̃′1+s̃′2=s′+ 12
ψ(s̃1,s̃′1) ·Ψ(s̃2,s̃
′2).
Here Ψ(s,s′) and ψ(s,s
′) stand for S-tangent tensors Ψ and ψ with signatures s1(Ψ) =
s, s2(Ψ) =
s′ and s1(ψ) = s, s2(ψ) = s′, respectively. In the above
equations, (1/2 + s′)trχΨ(s,s′) is the
borderline term. Taking the fact trχ = −2/|u|+ l.o.t in D(u, u)
and using the connection between
the coefficient 1/2 + s′ and Ψ(s,s′)’s signature for decay rates
s2(Ψ(s,s
′)) = s′, we will cancel this
9
-
borderline term in our newly defined scale invariant norms. This
observation avoids employing the
Bel-Robinson tensor and gives us a more direct and intuitive
approach for energy estimates. Since
we do not need the symmetry property of the Bel-Robinson tensor,
this approach can be used to
remove the Minkowskian initial data assumption along the
incoming cone in [7].
Retrieving Christodoulou’s Estimates
The norms we use are consistent with the norms in [15], which
are weaker than the norms in
[7]. With these weaker norms, we encounter many fewer borderline
terms and it is less difficult to
establish the semi-global existence result. Furthermore, based
on the existence results obtained in
weak norms together with initial data in strong norms, we can
improve the estimates derived in
weak norms to strong norms and thus retrieve Christodoulou’s
estimates in [7].
Let χ be the second fundamental form for Su,u with respect to Hu
and let χ̂ be the traceless
part of χ. In Chapter 2 of this thesis, we re-prove the main
theorem in Christodoulou’s monograph:
Theorem 1.1.2. (Christodoulou [7], 2008; A. [1], 2012)
Given c and B, there exists δ0 = δ0(c,B) sufficiently small,
such that for 0 < δ < δ0, with initial
data:
•∑i≤5,k≤3 δ
12 +k|u∞|1+i‖∇k4∇iχ̂∞‖L∞(Su∞,u) ≤ B along u = u∞
• Minkowskian initial data along u = 0
•∫ δ
0u2∞|χ̂∞|2 ≥ 4c for every direction along u = u∞
we have that S(c, δ) is a trapped surface.
Moreover, all of our proofs are independent of u∞. Letting u∞ go
to −∞, we obtain
Theorem 1.1.3. (Christodoulou [7], 2008; A. [1], 2012)
Trapped surfaces can form dynamically for Einstein vacuum
equations with initial data which
are dispersed at past null infinity.
10
-
1.2 Formation of Trapped Surfaces in Vacuum from Mild
Incoming Radiation
(Joint with Luk)
1.2.1 Background
A famous conjecture in general relativity is the Weak Cosmic
Censorship Conjecture:
For generic asymptotically flat initial data, the maximal
development of the Einstein field equa-
tions possesses a complete future null infinity.
Heuristically this conjecture states that a generic
gravitational singularity is hidden from far-
away observers (Cosmic Censorship Principle).
In [3]-[6] Christodoulou proves this conjecture for the
Einstein-scalar field system under spherical
symmetry (ESFS-S):
Rµν −12gµνR = 8πTµν ,
�gψ = 0,
Tµν = ∂µψ∂νψ −12gµν∂
γψ∂γψ.
We employ u and u as coordinates. Denote r(u, u) to be the
radius of 2-sphere Su,u and define
Ω through
Ω :=
√−g( ∂
∂u,∂
∂u).
ESFS-S reduces to the following 1 + 1 dimensional partial
differential equation system for the
functions r, Ω and ψ (see [3]):
∂u(ruΩ2
) = −4πr φ2u
Ω2, (1.2.1)
11
-
∂u(ruΩ2
) = −4πrφ2uΩ2, (1.2.2)
ruu = −Ω2
4r−rurur
, (1.2.3)
(log Ω)uu =Ω2
4r2+rurur2− 4πφuφu, (1.2.4)
φuu = −rurφu −
rurφu. (1.2.5)
We also define Hawking mass m through
m := r(1 + ∂ur∂ur)/2. (1.2.6)
And Hawking mass is an important quantity for ESFS-S.
Here, we briefly outline Christodoulou’s proof strategy. A
spacetime with an incomplete future
null infinity corresponds to a naked singularity. In [5]
Christodoulou shows the existence of regular
asymptotically flat initial data whose future developments have
an incomplete future null infinity.
Hence, he has constructed an example of naked singularity in
[5]. In [6] it is proved that the naked
singularity in [5] is unstable, that is, a small perturbation of
the initial data corresponding to the
naked singularity will lead to the formation of trapped surfaces
in the future. And moreover he
proves that all naked singularities are unstable (see also [4]),
thus verifying weak cosmic censorship
for ESFS-S.
One essential step in Christodoulou’s proof of the instability
of naked singularities is to establish
a sharp trapped surface formation criterion. This criterion
gives sufficient condition on the “initial
mass density” such that a trapped surface forms in the
future.
Christodoulou’s criterion also addresses a natural question:
what is the least “initial mass den-
sity” (size of initial data) required to guarantee that a
trapped surface forms in evolution?
12
-
In the below, we will list the known results on formation of
trapped surfaces and carry out a
comparison.
Here is the specific criterion in [3]:
Christodoulou’s trapped surface formation criterion for
ESFS-S
Theorem 1.2.1. (Christodoulou [3],1991)
O
A
B
For ESFS-S we prescribe characteris-
tic initial data along OA and AB. A
and B are 2-spheres with radius r1, r2
and Hawking mass m1,m2 respectively,
where m := r(1 + ∂ur∂ur)/2.For data along AB we define δ := (r2
− r1)/r2 and η := 2(m2 −m1)/r2. We call η the initial
mass density. For some fixed C0, if η ≥ C0δ log(1/δ), then there
exists a trapped surface in the
causal future of the prescribed initial data.
In [7], [15], [1], we have:
Trapped Surface formation criterion for EVE
Theorem 1.2.2. (see [7], [15], [1])
H 1(u
=1)
Hδ (u =
δ)H
0 (u =0)
H c
L
L
L
L
13
-
Given c and B, there exists δ0 = δ0(c,B) sufficiently small,
such that for 0 < δ < δ0, with initial
data:
•∑i≤5,k≤3 δ
12 +k‖∇k4∇iχ̂0‖L∞(S1,u) ≤ B along u = 1,
• Minkowskian initial data along u = 0,
•∫ δ
0|χ̂0|2 ≥ 4c for every direction along u = 1,
we have that S(c, δ) is a trapped surface.
Comparison with Hs norms
We employ Hs norms to carry out the comparison between ESFS-S
and EVE.
ESFS-S
In [3] for ESFS-S by using the following equation for Hawking
mass m
∂um = r2(∂uψ)2ru,
we have
η =2(m2 −m1)
r2=
∫ BA
2r2(∂uψ)2rur2
≈ r2(∂uψ)2∫ BAru
r2≈ r2(∂uψ)2
r2 − r1r2
≈ r2(∂uψ)2δ.
To satisfy condition η ≥ C0δ log(1/δ), we expect ∂uψ is of size
log12 (1/δ)/r along AB. When
trying to acquire intuition for the EVE with no symmetry
assumption, via results for ESFS-S, it is
helpful to use a formal correspondence between ψ and the metric
g. ∂uψ should correspond to ∂ug.
Thus, one would expect that the analogous criterion for EVE
would require ∂ug ∼ log12 (1/δ)/r.
Scaling consideration leads one formally to ∂u ∼ δ−1 and ∂12u ∼
δ−
12 . Therefore, we have
∫HABu
|∂ug|2 =∫ δ
0
∫Su,u
|∂ug(u, u′)|2dθ1dθ2du′ ≈ δ · log12 (1/δ) · log
12 (1/δ) ≈ δ log(1/δ),
14
-
and
∫HABu
|∂32u g|2 =
∫ δ0
∫Su,u
|∂12u ∂ug(u, u′)|2dθ1dθ2du′ ≈ δ · δ−
12 · log
12 (1/δ) · δ− 12 · log
12 (1/δ) ≈ log(1/δ).
Thus, we expect the H1 norm of g along AB to be of size δ
log(1/δ) and H32 norm along AB to be
bounded by log(1/δ).
EVE
In the nonsymmetric case, χ̂ is considered to be ∂ug. To make
trapped surfaces form dynamically,
in [7], [15] and [1], it is required that χ̂ is of size δ−12 /r
along AB, which formally means ∂ug is of
size δ−12 /r along AB. Recalling ∂
12u ∼ δ−
12 , hence we have
∫HABu
|∂ug|2 =∫ δ
0
∫Su,u
|∂ug(u, u′)|2dθ1dθ2du′ ≈ δ · δ−12 · δ− 12 ≈ 1,
and
∫HABu
|∂32u g|2 =
∫ δ0
∫Su,u
|∂12u ∂ug(u, u′)|2dθ1dθ2du′ ≈ δ · δ−
12 · δ− 12 · δ− 12 · δ− 12 ≈ δ−1.
The works in [7], [15] and [1] prescribe Minkowskian initial
data along OA. Since ∂ug is the largest
component among all the 1st derivatives of g, the trapped
surface formation criteria of [7], [15],
[1] require the H1 norm and the H32 norm of characteristic
initial data are of size 1 and δ−1,
respectively.
Motivation
It is natural to ask whether in the context of δ-pulsed initial
data, for EVE one can derive a
similar trapped surface formation criterion as in [3]. However,
the proof of Theorem 1.2.1 in [3]
depends on a proof by contradiction argument. We do not know the
precise location of trapped
surfaces and it requires a crucial monotonic property of Hawking
mass for ESFS-S. Without spher-
ical symmetry, this nice monotonic property fails. Hence, the
method of proof in [3] cannot be
generalized.
In the work [2], jointly with Luk, we overcome these
difficulties and, without any symmetry
15
-
assumption, but still prescribing Minkowskian data along OA, we
derive, in fact improve, a trapped
surface formation criterion similar to the one in [3]
corresponding to mild incoming radiation. The
precise statement will be described in the next subsection.
1.2.2 Main Results
By taking the techniques developed in [1] for studying decay
rates and instead using them to study
blow-up rates close to the center, in [2] jointly with Luk, we
establish the formal analogous of
Theorem 1.2.1 for the EVE with no symmetry assumptions. We
extend the short pulse method
introduced by Christodoulou in [7] as well by introducing two
new parameters and corresponding
hierarchies and a number of renormalizations.
H 1(u
=1)
H bδa12
Hδ (u =
δ)H
0 (u =0)
H δa
L
L
L
L
H 1(u
=1)
Hδ (u =
δ)H
0 (u =0)
H δa
L
L
L
L
We study the colored regions above. We will state one theorem
and two corollaries. The left
picture is for Theorem 1.2.3. The right picture is for Corollary
1.2.4. Recall that in [7] one small
parameter δ is exploited, we will use two more parameters a and
b, where 1 ≤ b ≤ a 12 ≤ δ− 12 .
Theorem 1.2.3. (A.- Luk [2], 2014)
Given B, there exists a0 = a0(B) and b0 = b0(B) sufficiently
large, such that for a0 ≤ a ≤ δ−1
and b0 ≤ b ≤ a12 ≤ δ− 12 with initial data:
16
-
•∑i≤5,k≤3 δ
ka−12 ‖∇k4∇iχ̂0‖L∞(S1,u) ≤ B along u = 1
• Minkowskian initial data along u = 0
•∫ δ
0|χ̂0|2 ≥ 4bδa
12 for every direction along u = 1
we have that S(bδa12 , δ) is a trapped surface.
For the comparison with [3], [7], [15] and [1], it suffices to
set b = ca12 , which gives
Corollary 1.2.4. (A.- Luk [2], 2014)
Given c and B, there exists a0 = a0(c,B) sufficiently large,
such that for a > a0, with initial
data:
•∑i≤5,k≤3 δ
ka−12 ‖∇k4∇iχ̂0‖L∞(S1,u) ≤ B along u = 1
• Minkowskian initial data along u = 0
•∫ δ
0|χ̂0|2 ≥ 4cδa for every direction along u = 1
we have that S(cδa, δ) is a trapped surface.
Remark 1. In [2], note that ∂ug ∼ χ̂ and it is of size a12 /r.
By dimensional analysis ∂u ∼ δ−1,
∂s−1u ∼ δ−s+1 with 0 < s < 32 and ∂12u ∼ δ−
12 . Hence, we have
∫H
(0,δ)u=1
|∂ug|2 =∫ δ
0
∫S1,u′
|∂ug(u, u′)|2dθ1dθ2du′ ≈ δa12 a
12 ≈ δa,
∫H
(0,δ)u=1
|∂sug|2 =∫ δ
0
∫S1,u′
|∂s−1u ∂ug(u, u′)|2dθ1dθ2du′ ≈ δδ−s+1a12 δ−s+1a
12 ≈ δ3−2sa,
and ∫H
(0,δ)u=1
|∂32u g|2 =
∫ δ0
∫S1,u′
|∂12u ∂ug(u, u′)|2dθ1dθ2du′ ≈ δδ−
12 a
12 δ−
12 a
12 ≈ a.
Here a is a large universal constant, which is independent of δ.
Therefore, to make trapped surfaces
form dynamically, it suffices to choose initial data for which
the H32 norm of the characteristic
initial data is bounded below and above by a large universal
constant a and all Hs(s < 3/2) norms
17
-
are small. In particular, H1 norm is of size δa, where a is a
universal large constant independent
of δ. Note that formally this is an improvement of [3].
Remark 2. Recalling that H32 norm is the critical norm for
Einstein vacuum equation in 3 + 1
dimensions. To get formation of trapped surfaces, we only need
the critical norm of initial data is of
size a universal large constant a. Therefore, we also call our
improved trapped surfaces formation
criterion as a scale invariant trapped surfaces formation
criterion.
We can further specialize the statement by coupling the choice
of a and δ.
Corollary 1.2.5. (A.- Luk [2], 2014)
For a = a(δ), satisfying a(δ)→ +∞ as δ → 0 and fixed c and B,
there exists δ0 = δ0(c,B), such
that for 0 ≤ δ ≤ δ0, given initial data:
•∑i≤5,k≤3 δ
k(a(δ))−12 ‖∇k4∇iχ̂0‖L∞(S1,u) ≤ B along u = 1
• Minkowskian initial data along u = 0
•∫ δ
0|χ̂0|2 ≥ 4cδa(δ) for every direction along u = 1
we have that S(cδa(δ), δ) is a trapped surface.
Setting a to δ−1, log(1/δ) and log log(1/δ), we compare
Corollary 1.2.4 with all the known
results:
Remark 3. Let a = δ−1. By employing the corollary above, we
reproduce the main results in [7],
where the initial mass density for each angle is of size c.
Remark 4. If we set a = log 1/δ, by using Corollary 1.2.4, we
derive that S(cδ log 1/δ, δ) is a
trapped surface. In this case our corresponding initial mass
density η ∼∫ δ
0|χ̂0|2 and it is of size
cδ log 1/δ. In [3] for ESFS-S Christodoulou has proved that if
the initial mass density η is larger
than C0δ log 1/δ for some fixed C0, then a trapped surface will
form in future. These two results
are consistent with each other.
18
-
Remark 5. In order to go further, by employing Corollary 1.2.4
and taking a = log log 1/δ we
obtain that S(cδ log log 1/δ, δ) is a trapped surface. Under
this situation, the corresponding initial
mass density η is of size cδ log log 1/δ. And for ESFS-S, to
make a trapped surface form dynamically
with initial mass density of size cδ log log 1/δ is already
beyond the conclusions in [3].
Remark 6. In Corollary 1.2.4, the parameter a can be chosen as a
fixed large constant, which is
independent of δ. Hence, as δ → 0, the initial mass density is
of size cδa, which is even smaller
than cδ log log(1/δ). In fact, Corollary 1.2.4 is sharp for δ
with respect to the methods.
Remark 7. Among all the known results for Einstein’s equation,
our result is of the least “initial
mass density” (size of initial data) to guarantee that a trapped
surface forms in evolution.
Being back to Theorem 1.2.3, we have two more remarks.
Remark 8. Compared with [7], [15] and [1], we are able to reach
the region u ≈ bδa 12 . This is
very close to the center of gravitational collapse (u = 0). And
we have established sharp results for
both δ and a with respect to the methods.
Remark 9. The extra generality of Theorem 1.2.3 can be used to
obtain a trapped surface for-
mation criterion with different bounds for χ̂0 from above and
from below. Indeed, compared with
Corollary 1.2.4, under the same upper bound of initial data
along u = 1, we obtain a larger region
for the existence of Einstein vacuum equations. Hence, we can
make trapped surfaces form dynam-
ically with different upper and lower bounds of χ̂0 along u = 1.
χ̂0 can be chosen in the range
2b12 a
14 ≤ |χ̂0| ≤ a
12 , where 1 ≤ b ≤ a 12 ≤ δ− 12 .
1.2.3 Heuristic Argument with ESFS-S
In this subsection we demonstrate the heuristic argument in [2]
by establishing a corresponding
result for ESFS-S.
Let a and b be two large constants satisfying 1 ≤ b ≤ a 12 ≤ δ−
12 . In this subsection, we will
prove:
19
-
Theorem 1.2.6. (Formation of Trapped Surfaces for ESFS-S)
H 1(u
=−1
)
Hδ (u =
δ)H
0 (u =0)
H −bδa12
L
L
L
L
We consider the characteristic initial value problem for
ESFS-S.
• Along H0 we prescribe trivial initial data, which corresponds
to
r = −1/2u,
ru(u, 0) = −1/2, ru(u, 0) = 1/2, Ω(u, 0) = 1,
φ(u, 0) = 0, φu(u, 0) = 0, φu(u, 0) = 0.
• Along H−1 where u = −1 and u′ ∈ [0, δ], for Ω and φu, we
set
Ω(−1, u′) = 1, (1.2.7)
|φu(−1, u′)| ≤ a12 , (1.2.8)∫ δ
0
φ2u(−1, u′)du′ ≥ 4δa12 b. (1.2.9)
Then S−bδa
12 ,δ
is a trapped surface.
20
-
We first establish the existence of solutions to ESFS-S in the
colored region.
Bootstrap Argument
Following the ideas in [2], in the colored region where bδa12 ≤
|u| ≤ 1 and 0 ≤ u ≤ δ, for ESFS-S
we make bootstrap assumptions:
|Ω− 1| < 12,
|ru +12| ≤ 1
b, |ru| ≤ a
12 ,
|φu(u, u)| ≤δa
12
|u|2b
14 , |φu(u, u)| ≤
6a12
|u|.
We prove the existence result by closing a bootstrap argument,
which requires improving all the
bounds above.
We begin with a useful remark about r.
Remark 10. Under the bootstrap assumptions above, in the colored
region, we have 14 |u| ≤ r ≤ |u|.
This is because for the initial data along H0, r = −1/2u. With
bootstrap assumption |ru| ≤ a12 ,
we derive
|r(u, u) + 12u| ≤ δa 12 ≤ 1
4|u|.
Therefore, we obtain 1/4|u| ≤ r ≤ |u|.
Using this remark, we can substitute |u| for r in the proofs
below.
We then start from deriving better estimates for φu and φu with
Equation 1.2.5:
φuu = −rurφu −
rurφu.
For φu, we have
(φu)u = −rurφu −
rurφu.
Using Gronwall’s inequality and bootstrap assumptions for ru, ru
and φu, we get
|φu(u, u)| ≤18δa
12
|u|2. (1.2.10)
21
-
For φu, we employ
(φu)u = −rurφu −
rurφu.
This is equivalent to
(rφu)u = ruφu + rφuu = −ruφu.
Using bootstrap assumptions for ru and φu, we obtain
|(rφu)u(u, u)| ≤δab
14
|u|2.
It follows
|rφu(u, u)− φu(−1, u)| ≤δa
12
|u|a
12 b
14 ≤ a
12
b34.
With the initial data condition
|φu(−1, u)| ≤ a12 ,
we derive
|φu(u, u)| ≤5a
12
|u|. (1.2.11)
We move to Ω. Ω obeys Equation 1.2.4:
(log Ω)uu =Ω2
4r2+rurur2− 4πφuφu.
With the fact that Ω = 1 along both H−1 and H0 and the bootstrap
assumptions above, we
have
|(log Ω)u(u, u)| ≤16δ|u|2
+16δa
12
|u|2+
4πδa12
|u|2δa
12
|u|b
14 .
Since (log Ω)u = Ω−1∂uΩ, we derive
|∂uΩ(u, u)| ≤32δ|u|2
+32δa
12
|u|2+
4πδa12
|u|2δa
12
|u|b
14 .
22
-
Therefore, we get
|Ω(u, u)− 1| ≤ 32δ|u|
+32δa
12
|u|+
4πδa12
|u|δa
12
|u|b
14 ≤ 32
a12 b
+32b
+4πb
74.
We now focus on ru and ru. ru satisfies Equation 1.2.3:
(ru)u = −Ω2
4r−rurur
.
Using initial data condition
ru(u, 0) = −12,
together with bootstrap assumptions for Ω and ru and Gronwall’s
inequality, we arrive at
|ru(u, u) +12| ≤ 2δ|u|≤ 2a
12 b.
For ru, by employing Equation 1.2.2:
∂u(ruΩ2
) = −4πrφ2uΩ2,
we have
|ruΩ2
(u, u)− 12| ≤
∫ δ0
4π(rφ2uΩ2
)(u, u′)du′. (1.2.12)
Using bootstrap assumptions for Ω and φu, it follows that
|ruΩ2
(u, u)− 12| ≤ 10000πδa
|u|≤ 1
2+
10000πa12
b.
Recalling the bootstrap assumption |Ω(u, u)− 1| < 12 , we
obtain
|ru(u, u)| ≤ 2 +40000πa
12
b.
Hence by choosing a, b sufficiently large, we have improved all
the estimates in bootstrap as-
23
-
sumptions. Thus, we establish the existence of ESFS-S in the
whole colored region.
Formation of Trapped Surfaces
We want to show both ru(−bδa12 , δ) and ru(−bδa
12 , d) are negative and then S
−bδa12 ,δ
is a
trapped surface.
It is true that ru(−bδa12 , δ) < 0, because we have already
proved |ru + 1/2| ≤ 1/b.
For ru(−bδa12 , δ), we first prove an useful lemma
Lemma 1.2.7. Under the same initial data condition as in Theorem
1.2.6, we have
∫ δ0
(r2φ2u)(−bδa12 , u′)du′ ≥ 1
2
∫ δ0
φ2u(−1, u′)du′.
To prove this lemma, we make a new bootstrap assumption:
sup|u|∈[bδa
12 ,1]
∫ δ0
(r2φ2u)(u, u′)du′ ≤ 2
∫ δ0
φ2u(−1, u′)du′. (1.2.13)
We then improve this bound.
Using Equation 1.2.5:
φuu = −rurφu −
rurφu.
we derive ((rφu)2
)u
= −2rφuruφu.
Hence, it follows
|∫ δ
0
(r2φ2u)(u, u′)du′ −
∫ δ0
φ2u(−1, u′)du′| ≤∫ u−1
∫ δ0
2|rφuruφu|(u′, u′)du′du′.
For ru, with the bound in (1.2.12) and bootstrap assumption
(1.2.13), we have
|ruΩ2
(u, u)− 12| ≤
∫ δ0
4π(rφ2uΩ2
)(u, u′)du′ ≤ 64π|u|
∫ δ0
(r2φ2u)(u, u′)du′ ≤ 128π
|u|
∫ δ0
φ2u(−1, u′)du′.
24
-
Together with the derived estimates for φu in (1.2.10) and φu in
(1.2.11):
|φu| ≤5a
12
|u|, |φu| ≤
18δa12
|u|2,
we get
|∫ δ
0
(r2φ2u)(u, u′)du′ −
∫ δ0
φ2u(−1, u′)du′|
≤∫ u
1
∫ δ0
2|rφuruφu|(u′, u′)du′du′
≤1000δa12
|u|δa
12 +
60000πδa12
|u|δa
12
|u|
∫ u0
φ2u(−1, u′)du′
≤1000b
δa12 +
60000πb2
∫ u0
φ2u(−1, u′)du′.
Recall for the initial condition we have
∫ u0
φ2u(−1, u′)du′ ≥ 4bδa12 .
Hence, for sufficiently large b, we arrive at
−12
∫ δ0
φ2u(−1, u′)du′ ≤∫ δ
0
(r2φ2u)(u, u′)du′ −
∫ δ0
φ2u(−1, u′)du′ ≤12
∫ δ0
φ2u(−1, u′)du′,
which is equivalent to
∫ δ0
(r2φ2u)(u, u′)du′ ≤ 3
2
∫ δ0
φ2u(−1, u′)du′,
and ∫ δ0
(r2φ2u)(u, u′)du′ ≥ 1
2
∫ δ0
φ2u(−1, u′)du′.
Therefore, we have improved bootstrap assumption (1.2.13) and
have also proved Lemma 1.2.7.
We are now ready to prove Theorem 1.2.6.
25
-
Employing Equation 1.2.2:
∂u(ruΩ2
) = −4πrφ2uΩ2,
and together with the fact
ru(−bδa12 , 0) =
12, Ω(−bδa 12 , 0) = 1,
we derive
ruΩ2
(−bδa 12 , δ) =12−∫ δ
0
4π(rφ2uΩ2
)(−bδa 12 , u′)du′
≤12− π
4bδa12
∫ δ0
(r2φ2u)(−bδa12 , u′)du′
≤12− π
8bδa12
∫ δ0
φ2u(−1, u′)du′
≤12− π
8bδa12
4bδa12
-
design and derive correct hierarchies with respect to δ, a, b
for all of them. Especially, we will have
curvature components. We control curvature components and their
derivatives through L2 type
energy estimates.
In Chapter 3, we will demonstrate the detailed proofs of Theorem
1.2.3 for EVE.
27
-
Chapter 2
Formation of Trapped Surfaces
from Past Null Infinity
2.1 Setting, Equations and Notations
Our setting is the characteristic initial value problem with
data given on the two characteristic
hypersurfaces Hu∞ and H0 intersecting at the sphere Su∞,0. The
spacetime will be a solution to
the Einstein equations constructed in a neighborhood of Hu∞ and
H0 containing Su∞,0. Here u∞
is a fixed constant. Our proofs are independent of u∞. In the
end of the paper, we will set u∞ to
go −∞ and obtain results from past null infinity.
28
-
2.1.1 Double Null Foliation
H u∞(u
=u∞
)
Hδ (u =
δ)H
0 (u =0)
H u
L
L
L
L
For a spacetime in a neighborhood of Su∞,0, we define a double
null foliation as follows: Let u
and u be solutions to the eikonal equation
gµν∂µu∂νu = 0, gµν∂µu∂νu = 0,
satisfying the initial conditions u = u∞ (u∞
-
Define
e3 = ΩL′, e4 = ΩL′
to be the normalized null pair such that
g(e3, e4) = −2
and
L = Ω2L′, L = Ω2L′
to be the so-called equivariant vector fields.
In the sequel, we will consider spacetime solutions to the
vacuum Einstein equations in the
gauge such that
Ω = 1, on Hu∞ and H0.
We denote the level sets of u as Hu and the level sets of u and
Hu. By virtue of the eikonal
equations, Hu and Hu are null hypersurface. The sets defined by
the intersections of the hyper-
surfaces Hu and Hu are topologically 2-spheres, which we denote
by Su,u. Notice that the integral
flows of L and L respect the foliation Su,u.
2.1.2 The Coordinate System
On a spacetime in a neighborhood of Su∞,0, we define a
coordinate system (u, u, θ1, θ2) as follows:
On the sphere Su∞,0, define a coordinate system (θ1, θ2) such
that on each coordinate patch the
metric γ is smooth, bounded and positive definite. Then we
define the coordinates on the initial
hypersurfaces by requiring
∂
∂uθA = 0 on H0, and
∂
∂uθA = 0 on Hu∞ .
30
-
We now define the coordinate system in the spacetime in a
neighborhood of Su∞,0 by letting u and
u to be solutions to the eikonal equations:
gµν∂µu∂νu = 0, gµν∂µu∂νu = 0,
and define θ1, θ2 by
L/ LθA = 0,
where L/ L denote the restriction of the Lie derivative to TSu,u
(See [7], Chapter 1). Relative to the
coordinate system (u, u, θ1, θ2), the null pair e3 and e4 can be
expressed as
e3 = Ω−1(∂
∂u+ bA
∂
∂θA
), e4 = Ω−1
∂
∂u,
for some bA such that bA = 0 on H0, while the metric g takes the
form
g = −2Ω2(du⊗ du+ du⊗ du) + γAB(dθA − bAdu)⊗ (dθB − bBdu).
Consider a coordinate patch U on Su∞,0 and define Uu,0 to be a
coordinate patch on Su,0 given
by the one-parameter diffeomorphism generated by L. Define Uu,u
to be the image of Uu,0 under
the one-parameter diffeomorphism generated by L. Define D
=⋃u∞≤u≤−1,0≤u≤δ Uu,u and it is the
shadowed region.
2.1.3 Equations
We will recast the Einstein equations as a system for Ricci
coefficients and curvature components
associated to a null frame e3, e4 defined above and an
orthonormal frame e1, e2 tangent to the
2-spheres Su,u. Using the indices A,B to denote 1, 2, we define
the Ricci coefficients relative to the
31
-
null fame:
χAB = g(DAe4, eB), χAB = g(DAe3, eB),
ηA = −12g(D3eA, e4), ηA = −
12g(D4eA, e3)
ω = −14g(D4e3, e4), ω = −
14g(D3e4, e3),
ζA =12g(DAe4, e3)
(2.1.1)
where DA = De(A) . We also introduce the null curvature
components,
αAB = R(eA, e4, eB , e4), αAB = R(eA, e3, eB , e3),
βA =12R(eA, e4, e3, e4), βA =
12R(eA, e3, e3, e4),
ρ =14R(e4, e3, e4, e3), σ =
14∗R(e4, e3, e4, e3).
(2.1.2)
Here ∗R denotes the Hodge dual of R. We denote by ∇ the induced
covariant derivative operator
on Su,u and by ∇3, ∇4 the projections to Su,u of the covariant
derivatives D3, D4 (see precise
definitions in [12]).
Observe that,
ω = −12∇4(log Ω), ω = −
12∇3(log Ω),
ηA = ζA +∇A(log Ω), ηA = −ζA +∇A(log Ω).(2.1.3)
Let S-tensor φ(1) · φ(2) denote an arbitrary contraction of the
tensor product of φ(1) and φ(2)
with respect to the metric γ. We also define
(φ(1)⊗̂φ(2))AB := φ(1)A φ(2)B + φ
(1)B φ
(2)A − δAB(φ
(1) · φ(2)) for one forms φ(1)A , φ(2)A ,
(φ(1) ∧ φ(2))AB := �/AB(γ−1)CDφ(1)ACφ(2)BD for symmetric two
tensors φ
(1)AB , φ
(2)AB ,
where �/ is the volume form associated to the metric γ. For
totally symmetric tensors, the div and
32
-
curl operators are defined by the formulas
(div φ)A1...Ar := ∇BφBA1...Ar ,
(curl φ)A1...Ar := �/BC∇BφCA1...Ar .
Define also the trace to be
(trφ)A1...Ar−1 := (γ−1)BCφBCA1...Ar−1 .
We separate the trace and traceless part of χ and χ. Let χ̂ and
χ̂ be the traceless parts of χ
and χ respectively. Then χ and χ satisfy the following null
structure equations:
∇4trχ+12
(trχ)2 = −|χ̂|2 − 2ωtrχ,
∇4χ̂+ trχχ̂ = −2ωχ̂− α,
∇3trχ+12
(trχ)2 = −2ωtrχ− |χ̂|2,
∇3χ̂+ trχ χ̂ = −2ωχ̂− α,
∇4trχ+12
trχtrχ = 2ωtrχ+ 2ρ− χ̂ · χ̂+ 2div η + 2|η|2,
∇4χ̂+12
trχχ̂ = ∇⊗̂η + 2ωχ̂− 12
trχχ̂+ η⊗̂η,
∇3trχ+12
trχtrχ = 2ωtrχ+ 2ρ− χ̂ · χ̂+ 2div η + 2|η|2,
∇3χ̂+12
trχχ̂ = ∇⊗̂η + 2ωχ̂− 12
trχχ̂+ η⊗̂η.
(2.1.4)
The other Ricci coefficients satisfy the following null
structure equations:
∇4η = −χ · (η − η)− β,
∇3η = −χ · (η − η) + β,
∇4ω = 2ωω +34|η − η|2 − 1
4(η − η) · (η + η)− 1
8|η + η|2 + 1
2ρ,
∇3ω = 2ωω +34|η − η|2 + 1
4(η − η) · (η + η)− 1
8|η + η|2 + 1
2ρ.
(2.1.5)
33
-
The Ricci coefficients also satisfy the following constraint
equations
div χ̂ =12∇trχ− 1
2(η − η) · (χ̂− 1
2trχ)− β,
div χ̂ =12∇trχ+ 1
2(η − η) · (χ̂− 1
2trχ) + β,
curl η = −curl η = σ + 12χ̂ ∧ χ̂,
K = −ρ+ 12χ̂ · χ̂− 1
4trχtrχ.
(2.1.6)
with K the Gauss curvature of the spheres Su,u. The null
curvature components satisfy the following
null Bianchi equations:
∇3α+12
trχα = ∇⊗̂β + 4ωα− 3(χ̂ρ+∗ χ̂σ) + (ζ + 4η)⊗̂β,
∇4β + 2trχβ = div α− 2ωβ + ηα,
∇3β + trχβ = ∇ρ+ 2ωβ +∗ ∇σ + 2χ̂ · β + 3(ηρ+∗ ησ),
∇4σ +32
trχσ = −div ∗β + 12χ̂ ·∗ α− ζ ·∗ β − 2η ·∗ β,
∇3σ +32
trχσ = −div ∗β + 12χ̂ ·∗ α− ζ ·∗ β − 2η ·∗ β,
∇4ρ+32
trχρ = div β − 12χ̂ · α+ ζ · β + 2η · β,
∇3ρ+32
trχρ = −div β − 12χ̂ · α+ ζ · β − 2η · β,
∇4β + trχβ = −∇ρ+∗ ∇σ + 2ωβ + 2χ̂ · β − 3(ηρ−∗ ησ),
∇3β + 2trχβ = −div α− 2ωβ + η · α,
∇4α+12
trχα = −∇⊗̂β + 4ωα− 3(χ̂ρ−∗ χ̂σ) + (ζ − 4η)⊗̂β
(2.1.7)
where ∗ denotes the Hodge dual on Su,u.
In the sequel, we will use capital Latin letters A ∈ {1, 2} for
indices on the spheres Su,u and
Greek letters µ ∈ {1, 2, 3, 4} for indices in the whole
spacetime.
34
-
2.1.4 Integration
Let U be a coordinate patch on Su∞,0 and pU be a partition of
unity in DU such that pU is supported
in DU . Given a function φ, the integration on Su,u is given by
the formula:
∫Su,u
φ :=∑U
∫ ∞−∞
∫ ∞−∞
φpU√
det γdθ1dθ2.
Let Du′,u′ by the region u∞ ≤ u ≤ u′, 0 ≤ u ≤ u′. The
integration on Du,u is given by the formula
∫Du,u
φ :=∑U
∫ u0
∫ u0
∫ ∞−∞
∫ ∞−∞
φpU√− det gdθ1dθ2dudu
=2∑U
∫ u0
∫ u0
∫ ∞−∞
∫ ∞−∞
φpUΩ2√
det γdθ1dθ2dudu.
Since there are no canonical volume forms on Hu and Hu, we
define integration by
∫Hu
φ :=∑U
∫ δ0
∫ ∞−∞
∫ ∞−∞
φ2pUΩ√
det γdθ1dθ2du,
and ∫Hu
φ :=∑U
∫ uu∞
∫ ∞−∞
∫ ∞−∞
φ2pUΩ√
det γdθ1dθ2du.
With these notions of integration, we can define the norms that
we will use. Let φ be an
arbitrary tensorfield. For 1 ≤ p p/2γ ,
||φ||pLp(Hu) :=∫Hu
< φ, φ >p/2γ ,
||φ||pLp(Hu) :=∫Hu
< φ, φ >p/2γ .
35
-
Define also the L∞ norm by
||φ||L∞(Su,u) := supθ∈Su,u
< φ, φ >1/2γ (θ).
We will also use mixed norms defined by
||φ||L2uL∞u Lp(S) =
(∫ u∗0
( supu∈[0,u∗]
||φ||Lp(Su,u))2du
) 12
,
||φ||L2uL∞u Lp(S) =
(∫ u∗0
( supu∈[0,�]
||∇iφ||Lp(Su,u))2du
) 12
.
Note that L∞Lp is taken before taking L2. In the sequel, we will
frequently use
|| · ||L∞u L2uLp(S) ≤ || · ||L2uL∞u Lp(S).
With the above definition, ||φ||L2uL2(Su,u) and ||φ||L2(Hu)
differ by a factor of Ω. Nevertheless,
in view of Proposition 2.3.1, these norms are equivalent up to a
factor of 2.
2.1.5 Signatures
To capture the structure of Einstein’s equation, we introduce
the following definitions below.
Definition of signatures
To φ ∈ {α, β, ρ, σ,K, β, α, χ, χ, ζ, η, η, ω, ω, γ}, we assign
signatures s(φ) according to the follow-
ing rules:
s(φ) := (s1(φ), s2(φ)),
where
s1(φ) := 1 ·N4(φ) +12·Na(φ) + 0 ·N3(φ)− 1,
36
-
and
s2(φ) := 0 ·N4(φ) +12·Na(φ) + 1 ·N3(φ)− 1.
N4(φ) is the number of times e4 appears in the definition of φ.
Similarly we define N3(φ) and
Na(φ) where a = 1, 2.
By the definition above, we have
Signature table
α β ρ σ K β α χ ω ζ η η trχ χ̂ ω γ
s1 2 1.5 1 1 1 0.5 0 1 1 0.5 0.5 0.5 0 0 0 0
s2 0 0.5 1 1 1 1.5 2 0 0 0.5 0.5 0.5 1 1 1 0
Properties of signatures
s1(∇4φ) = s1(φ) + 1, s2(∇4φ) = s2(φ),
s1(∇φ) = s1(φ) +12, s2(∇φ) = s2(φ) +
12,
s1(∇3φ) = s1(φ), s2(∇3φ) = s2(φ) + 1.
Conservation of signatures
s1(φ1 · φ2) = s1(φ1) + s1(φ2),
s2(φ1 · φ2) = s2(φ1) + s1(φ2),
s(φ1 · φ2) = (s1(φ1 · φ2), s2(φ1 · φ2)) = (s1(φ1) + s1(φ2),
s2(φ1) + s1(φ2)) = s(φ1) + s(φ2).
37
-
Remark: s1 is the same signature used in [15] and s2 is
introduced to study the decay rate
near past null infinity.
2.1.6 Scale Invariant Norms
For any horizontal tensor-field φ with signature s(φ) = (s1(φ),
s2(φ)), we give
Definition of scale invariant norms on Su,u
‖φ‖L∞sc(Su,u) := δs1(φ)− 12 |u|2s2(φ)+1‖φ‖L∞(Su,u),
‖φ‖L4sc(Su,u) := δs1(φ)− 34 |u|2s2(φ)+ 12 ‖φ‖L4(Su,u),
‖φ‖L2sc(Su,u) := δs1(φ)−1|u|2s2(φ)‖φ‖L2(Su,u),
‖φ‖L1sc(Su,u) := δs1(φ)− 32 |u|2s2(φ)−1‖φ‖L1(Su,u).
More generally, for 1 ≤ p ≤ ∞, we define
‖φ‖Lpsc(Su,u) := δs1(φ)− 12−
1p |u|2s2(φ)+1−
2p ‖φ‖Lp(Su,u).
In scale invariant norms, we have
Hölder’s inequalities
‖φ1 · φ2‖L1sc(Su,u) ≤δ
12
|u|‖φ1‖L∞sc(Su,u)‖φ2‖L1sc(Su,u),
‖φ1 · φ2‖L1sc(Su,u) ≤δ
12
|u|‖φ1‖L2sc(Su,u)‖φ2‖L2sc(Su,u),
38
-
‖φ1 · φ2‖L2sc(Su,u) ≤δ
12
|u|‖φ1‖L∞sc(Su,u)‖φ2‖L2sc(Su,u),
‖φ1 · φ2‖L2sc(Su,u) ≤δ
12
|u|‖φ1‖L4sc(Su,u)‖φ2‖L4sc(Su,u),
‖φ1 · φ2‖L4sc(Su,u) ≤δ
12
|u|‖φ1‖L∞sc(Su,u)‖φ2‖L4sc(Su,u).
For convenience, along the null hypersurfaces H(0,u)u and
H(u∞,u)u we also define
Scale invariant norms along a hypersurface
‖φ‖2L2sc(H
(0,u)u )
:= δ−1∫ u
0
‖φ‖2L2sc(Su,u′ )du′,
‖φ‖2L2sc(H
(u∞,u)u )
:=∫ uu∞
1|u′|2‖φ‖2L2sc(Su′,u)du
′.
2.1.7 Norms
We now define the following norms that we will work with
Ricci coefficient norms:
For any Su,u, we introduce Os,p(u, u).
O0,∞(u, u) :=‖ω‖L∞sc(Su,u) + ‖χ̂‖L∞sc(Su,u) + ‖trχ‖L∞sc(Su,u) +
‖η‖L∞sc(Su,u)
+ ‖η‖L∞sc(Su,u) +1|u|‖χ̂‖L∞sc(Su,u) +
δ12
|u|2‖trχ‖L∞sc(Su,u) + ‖ω‖L∞sc(Su,u),
O0,4(u, u) :=‖ω‖L4sc(Su,u) + δ14 ‖χ̂‖L4sc(Su,u) +
‖trχ‖L4sc(Su,u) + ‖η‖L4sc(Su,u)
+ ‖η‖L4sc(Su,u) +δ
14
|u|‖χ̂‖L4sc(Su,u) +
δ34
|u|2‖trχ‖L4sc(Su,u) + ‖ω‖L4sc(Su,u),
39
-
O1,4(u, u) :=‖∇ω‖L4sc(Su,u) + ‖∇χ̂‖L4sc(Su,u) +
‖∇trχ‖L4sc(Su,u)
+ ‖∇η‖L4sc(Su,u) + ‖∇η‖L4sc(Su,u) +1|u|‖∇χ̂‖L4sc(Su,u)
+ ‖∇trχ‖L4sc(Su,u) + ‖∇ω‖L4sc(Su,u),
O2,4(u, u) :=‖∇2ω‖L4sc(Su,u) + ‖∇2χ̂‖L4sc(Su,u) + ‖∇
2trχ‖L4sc(Su,u)
+ ‖∇2η‖L4sc(Su,u) + ‖∇2η‖L4sc(Su,u) +
1|u|‖∇2χ̂‖L4sc(Su,u)
+ ‖∇2trχ‖L4sc(Su,u) + ‖∇2ω‖L4sc(Su,u),
O3,2(u, u) :=‖∇3ω‖L2sc(Su,u) + ‖∇3χ̂‖L2sc(Su,u) + ‖∇
3trχ‖L2sc(Su,u)
+ ‖∇3η‖L2sc(Su,u) + ‖∇3η‖L2sc(Su,u) +
1|u|‖∇3χ̂‖L2sc(Su,u)
+ ‖∇3trχ‖L2sc(Su,u) + ‖∇3ω‖L2sc(Su,u).
We denote O0,4, O0,∞, O1,4, O2,4 and O3,2 to be the supremum of
the corresponding norms
over all values of u, u in our slab . Finally, we define the
total Ricci norm O
O := O0,4 +O0,∞ +O1,4 +O2,4 +O3,2
and let O(0) be the corresponding norm of the initial
hypersurface Hu∞ .
Curvature norms:
Along the null hypersurfaces H = H(0,u)u and H = H(u∞,u)u , we
introduce
R0(u, u) := δ12 ‖α‖L2sc(H) + ‖β‖L2sc(H) + ‖ρ‖L2sc(H) +
‖σ‖L2sc(H) + ‖β‖L2sc(H),
40
-
R0(u, u) := δ12 ‖β‖L2sc(H) + ‖ρ‖L2sc(H) + ‖σ‖L2sc(H) +
‖β‖L2sc(H) + ‖α‖L2sc(H),
R1(u, u) := ‖∇α‖L2sc(H) + ‖∇β‖L2sc(H) + ‖∇ρ‖L2sc(H) +
‖∇σ‖L2sc(H) + ‖∇β‖L2sc(H),
R1(u, u) := ‖∇β‖L2sc(H) + ‖∇ρ‖L2sc(H) + ‖∇σ‖L2sc(H) +
‖∇β‖L2sc(H) + ‖∇α‖L2sc(H),
R2(u, u) := ‖∇2α‖L2sc(H) + ‖∇2β‖L2sc(H) + ‖∇
2ρ‖L2sc(H) + ‖∇2σ‖L2sc(H) + ‖∇
2β‖L2sc(H),
R2(u, u) := ‖∇2β‖L2sc(H) + ‖∇2ρ‖L2sc(H) + ‖∇
2σ‖L2sc(H) + ‖∇2β‖L2sc(H) + ‖∇
2α‖L2sc(H),
We set R0, R1, R2 to be the supremum over u, u in our spacetime
slab of R0(u, u), R1(u, u)
and R2(u, u), respectively. Similarly, we define R0, R1 and R2.
We write R := R0 +R1 +R2 and
R := R0 +R1 +R2. Finally, we denote R(0) as the initial value
for the norm R, i.e.,
R(0) := sup0≤u≤δ
(R0(u∞, u) +R1(u∞, u) +R2(u∞, u)
).
Initial data assumptions:
We define the initial data quantity
I(0) := sup0≤u≤δ
I(0)(u),
41
-
where
I(0)(u) :=δ 12 |u∞|‖χ̂∞‖L∞(Su∞,u) +∑
0≤k≤2
δ12 ‖(δ∇4)kχ̂∞‖L2(Su∞,u)
+∑
0≤k≤1
∑1≤m≤4
δ12 |u∞|‖(δ
12 |u∞|∇)m−1(δ∇4)k∇χ̂∞‖L2(Su∞,u).
Here χ̂∞ denotes χ̂ along H(0,u)u∞ .
Norms in standard L∞ form:
For convenience, we list the norms in the standard L∞ form:
O0,∞(u, u) =δ12 |u|‖χ̂‖L∞(Su,u) + δ
12 |u|‖ω‖L∞(Su,u) + δ
12 |u|‖trχ‖L∞(Su,u)
+ |u|2‖η, η‖L∞(Su,u) + δ− 12 |u|2‖χ̂‖L∞(Su,u)
+ |u|‖trχ‖L∞(Su,u) + δ− 12 |u|3‖ω‖L∞(Su,u).
The norms in the standard Lp form p = 2, 4 are listed in the
appendix.
2.2 Statement of Main Theorem
We are now ready to state our main theorem
Theorem 2.2.1. (Main Theorem)
Consider the following characteristic initial value problem for
the Einstein vacuum equations.
The initial incoming hypersurface H0 is required to coincide
with a backwards light cone in
Minkowski space with u∞ ≤ u ≤ 0. On the initial outgoing
hypersurface Hu∞ , the data are
smooth and I(0)(u) is bounded by an arbitrary constant I(0)
uniformly. Given I(0) and another
arbitrary constant c, there exists δ0 = δ0(I(0), c) > 0
sufficiently small, such that, for 0 < δ < δ0,
in the region u∞ ≤ u ≤ c, 0 ≤ u ≤ δ, we have
42
-
R+R+O ≤ I(0).
Hence there exists a unique smooth spacetime solution (M, g)
endowed with a double null
foliation (u, u) which solves the characteristic initial value
problem to the vacuum Einstein equations
in the region u∞ ≤ u ≤ c, 0 ≤ u ≤ δ.
Remark 13. In the following, we will only prove the a priori
estimates for O, Õ5,2 andR. The exis-
tence and uniqueness of solution and the propagation of
regularity follow from standard arguments
(see, for example, [7]).
Remark 14. Following [7], one can solve the constraint ODEs and
obtain bounds for the initial
data on Hu∞ from that of the initial shear. In particular, under
the assumption of Theorem 2.2.1,
we have the following initial bounds for the Ricci coefficients
and curvature components
R(0) +O(0) ≤ I(0).
Once the existence theorem is established, the actual formation
of trapped surfaces follows from
a simple ODE argument as in [7]:
Theorem 2.2.2. (Formation of Trapped Surfaces)
Given I(0) and c, there exist δ0 = δ0(I(0), c) sufficiently
small, such that for 0 < δ < δ0, with
initial data:
•∑i≤5,k≤3 δ
kδ12 |u∞|‖∇k4(|u∞|∇)iχ̂∞‖L∞(Su∞,u) ≤ I
(0) along u = u∞
•∑
2≤j≤7 δ12 ‖(δ 12 |u∞|∇)jχ̂∞‖L2(Su∞,u) ≤ � along u = u∞
• Minkowski initial data along u = 0
•∫ δ
0|u∞|2|χ̂∞|2 ≥ 4c for every direction along u = u∞
we have that Sc,δ is a trapped surface.
Moreover, our proofs are independent of u∞. After let u∞ go to
−∞, we obtain
43
-
Theorem 2.2.3. (Christodoulou [7], 2008; A. [1], 2012)
Trapped surfaces form in the Cauchy development of Einstein
vacuum equations with initial
data which are arbitrarily dispersed at past null infinity.
2.2.1 Structure of the Proof
Let C denote a large fixed constant and R denote a constant to
be determined. We briefly outline
the proof of Theorem 2.2.1:
STEP 0: Assuming that O0,∞
-
2.2.2 Strategy of the Proof
In this subsection, we explain some strategies that we use in
the proof.
Employing Scale Invariant Norms
Without symmetry assumptions, proving the existence and the
uniqueness of solutions to Ein-
stein’s equation relies on L2-type estimates for the curvature
component and its derivatives. And
we are required to control many different curvature components
and Ricci coefficients. Following
Klainerman-Rodnianski’s method in [15], we extend their
definitions for scale invariant norms to
include decay rate u and we employ a systematical approach to
control all the terms. The following
Hölder’s inequality holds in scale invariant norms
‖φ1 · φ2‖L4sc(Su,u) ≤δ
12
|u|‖φ1‖L∞sc(Su,u)‖φ2‖L4sc(Su,u). (2.2.1)
This implies that, if all the terms are of size 1 in scale
invariant norms (we call them normal
terms), the nonlinear terms behave as lower order terms. And
therefore, we can treat nonlinear
terms uniformly. In fact, most of the Ricci coefficients and
curvature components are normal terms.
Hence we only need to track a few anomalous terms.
Controlling Anomalous Terms
In this section, α, χ̂ and trχ are the anomalous terms, which
can be observed via the definition
of norms and the expectation that all norms are of size 1.
Therefore, through the definition of
R0(u, u) and O0,∞, we only expect that δ12 ‖α‖L2sc(H),
1|u|‖χ̂‖L∞sc(Su,u) and
δ12
|u|2 ‖trχ‖L∞sc(Su,u) are
bounded by a fixed constant.
Here we list three typical cases about how to control anomalous
terms.
Case 1: We constantly exploit Hölder’s inequalities in scale
invariant norms, such as Hölder’s
inequality (2.2.1), to gain smallness in δ and 1|u| . We use the
smallness to compensate for the
largeness arising from the anomalous terms. For example, to
control ‖∇η‖L4sc(Su,u), we would like
to show that ∫ uu∞
|u′|−3‖trχηη‖L4sc(Su′,u)du′ ≤ 1|u|.
45
-
Here trχ is anomalous. By employing Hölder’s inequality (2.2.1)
twice and using the definition of
O0,∞(u, u) and O0,4(u, u), we derive the following
inequality:
∫ uu∞
|u′|−3‖trχηη‖L4sc(Su′,u)du′
≤∫ uu∞
δ
|u′|5‖trχ‖L∞sc(Su′,u)‖η‖L∞sc(Su′,u)‖η‖L4sc(Su′,u)du
′
≤∫ uu∞
δ12
|u′|3δ
12
|u′|2‖trχ‖L∞sc(Su′,u)‖η‖L∞sc(Su′,u)‖η‖L4sc(Su′,u)du
′
≤ δ12
|u|2O20,∞O0,4
≤ 1|u|,
where δ12
|u|2 is used to eliminate the largeness from
‖trχ‖L∞sc(Su,u).
Case 2: Let ψ denote Ricci coefficients and Ψ denote curvature
components. ψ and Ψ obey null
structure equations
∇3ψ + c[ψ]trχψ = Ψ + ψ′ψ′, ∇4ψ = Ψ + ψ′ψ′,
and null Bianchi equations
∇3Ψ + c[Ψ]trχΨ = ∇Ψ′ + ψΨ′, ∇4Ψ = ∇Ψ′ + ψΨ′.
In these equations c[ψ]trχψ and c[Ψ]trχΨ are borderline terms
and may lead to∫ uu∞
1|u′|du
′, which
results in a logarithmic divergence when u∞ goes to −∞. We
overcome this difficulty by using the
fact
trχ = − 2|u|
+ l.o.t.,
and exploiting integrating factors to cancel − 2|u| terms. Thus,
we get rid of the borderline terms.
Case 3: When deriving estimates for ∇trχ and ∇3trχ, if we employ
equations for ∇3∇trχ
and ∇3∇3trχ, the borderline terms (η, η)trχtrχ and
(∇2η,∇2η)trχtrχ will appear. Those two
terms cannot be controlled by using integrating factors and will
lead to a logarithmic divergence.
We circumvent the divergence by deducing equations for ∇3∇(Ωtrχ
− 2u ) and ∇3∇3(Ωtrχ − 2u ).
46
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The borderline terms (η, η)trχtrχ and (∇2η,∇2η)trχtrχ are
replaced by (η, η)(trχ + 2|u| )trχ and
(∇2,∇2)(trχ + 2|u| )trχ, respectively. Meanwhile, by obtaining
bounds for ∇Ω, ∇2Ω and ∇3Ω, we
establish desired estimates for ∇trχ and ∇3trχ.
Retrieving Christodoulou’s Estimates
Using the strategies above, under the assumptions for the
initial data in Theorem 2.2.1, we can
establish the existence of solutions to Einstein vacuum
equations in c ≤ |u| ≤ |u∞| and 0 ≤ u ≤ δ.
An interesting question is, with smaller initial data as
Christodoulou prescribes in [7], can we
retrieve Christodoulou’s estimates in [7]?
The answer is yes. The proof is outlined in Section 2.8. We
demonstrate the following ideas:
first, with scale invariant norms in this section, we establish
Theorem 2.2.1. As a consequence, we
derive bounds for Ricci coefficients in Section 2.4. With these
bounds for Ricci coefficients and
the new initial data in [7], we improve L2 bounds for curvature
components via the established
controls in energy estimates in Section 2.6. Second, by
employing Sobolev Embedding, we improve
L∞ control of curvature components. Third, with null structure
equations, we improve the bounds
for Ricci coefficients to retrieve all the estimates in [7].
We are ready to start the proof.
2.3 The Preliminary Estimates
We will make the following bootstrap assumptions on the Ricci
coefficients:
O0,∞ ≤ ∆0, (2.3.1)
O0,4 ≤ ∆1, (2.3.2)
O1,4 ≤ ∆2. (2.3.3)
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Moreover, we set a bootstrap assumption on curvature components
and their derivatives:
R+R ≤ R. (2.3.4)
Here, and in the rest of this section, we use the convention
that A ≤ B denotes the
inequality A ≤ C0B for some universal constant C0 that is
independent of δ and u∞.
2.3.1 Estimates for Metric Components
We first show that we can control Ω under the bootstrap
assumption (2.3.1):
Proposition 2.3.1. Under the assumptions of Theorem 2.2.1 and
bootstrap assumption (2.3.1),
we have
‖Ω− 1‖L∞(Su,u) ≤δ
12
|u|∆0.
Proof. Consider the equation
ω = −12∇4 log Ω =
12
Ω∇4Ω−1 =12∂
∂uΩ−1. (2.3.5)
Fix u. Notice that both ω and Ω are scalars and therefore the L∞
norm is independent of the
metric. We can integrate equation (2.3.5) using the fact that
Ω−1 = 1 on H0 to obtain
||Ω−1 − 1||L∞(Su,u) ≤∫ u
0
||ω||L∞(Su,u′ )du′ ≤ δ
12
|u|∆0,
where we have used the bootstrap assumption (2.3.1). Finally,
notice that
‖Ω− 1‖L∞(Su,u) = ‖Ω‖L∞(Su,u)‖Ω−1 − 1‖L∞(Su,u) ≤
δ12
|u|∆0.
We then show that we can control γ under the bootstrap
assumption (2.3.1):
Proposition 2.3.2. Under the assumptions of Theorem 2.2.1 and
the bootstrap assumptions
(2.3.1), (2.3.2) and (2.3.3), we have C and c depending only on
initial data such that the following
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pointwise bounds for γ in D hold:
c ≤ det γ ≤ C.
Moreover, in D,
|γAB |, |(γ−1)AB | ≤ C.
Proof. The first variation formula states that
L/ Lγ = 2Ωχ.
In coordinates, this means∂
∂uγAB = 2ΩχAB .
From this we derive that∂
∂ulog(det γ) = Ωtrχ.
Define γ0(u, u, θ1, θ2) = γ(u, 0, θ1, θ2). Then
|det γ − det(γ0)| ≤ C∫ u
0
|trχ|du′ ≤ C δ12
|u|∆0. (2.3.6)
This implies that the det γ is bounded above and below. Let Λ be
the larger eigenvalue of γ.
Clearly,
Λ ≤ C supA,B=1,2
γAB , (2.3.7)
and ∑A,B=1,2
|χAB |2 ≤ CΛ||χ||L∞(Su,u).
Then
|γAB − (γ0)AB | ≤ C∫ u
0
|χAB |du′ ≤ CΛδ
12
|u|∆0.
Using the upper bound (2.3.7), we thus obtain the upper bound
for |γAB |. The upper bound for
|(γ−1)AB | follows from the upper bound for |γAB | and the lower
bound for det γ.
49
-
A consequence of the previous proposition is an estimate on the
surface area of the two sphere
Su,u.
Proposition 2.3.3. In D we have
supu|Area(Su,u)−Area(Su,0)| ≤ C∆
120
δ14
|u| 12|u|2.
Proof. This follows from (2.3.6).
With the estimate on the volume form, we can now show that the
Lp norms defined with respect
to the metric and the Lp norms defined with respect to the
coordinate system are equivalent.
Proposition 2.3.4. Given a covariant tensor φA1...Ar on Su,u, we
have
∫Su,u
< φ, φ >p/2γ ∼r∑i=1
∑Ai=1,2
∫∫|φA1...Ar |p
√det γdθ1dθ2.
2.3.2 Estimates for Transport Equations
In latter sections of the paper, we will derive the estimates
for the Ricci coefficients and the null
curvature components from the null structure equations and the
null Bianchi equations respectively.
These will be viewed as transport equations and we will need a
way to obtain estimates from the
covariant null transport equations. For the transport equation
in the e4 direction, we will need the
smallness of ‖trχ‖L∞u L1uL∞(Su,u), which is a consequence of our
bootstrap assumption (2.3.1). More
precisely, we have
Proposition 2.3.5. Under the assumptions of Theorem 2.2.1 and
the bootstrap assumptions
(2.3.1), we have
||φ||L2(Su,u) ≤ ||φ||L2(Su,u′ ) +∫ uu′||∇4φ||L2(Su,u′′ )du
′′, (2.3.8)
||φ||L4(Su,u) ≤ ||φ||L4(Su,u′ ) +∫ uu′||∇4φ||L4(Su,u′′ )du
′′ (2.3.9)
for an Su,u tangent tensor φ of arbitrary rank.
50
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Proof. We first note that the following identity holds for any
scalar f :
d
du
∫Su,u
f =∫Su,u
(df
du+ Ωtrχf
)=∫Su,u
Ω (e4(f) + trχf) .
Hence, taking f = |φ|2γ , we have
||φ||2L2(Su,u) =||φ||2L2(Su,u′ )
+∫ uu′
∫Su,u′′
2Ω(< φ,∇4φ >γ +
12
trχ|φ|2γ)du′′.
The inequality (2.3.8) can be concluded using Cauchy-Schwarz on
the sphere and the L∞ bounds for
Ω and trχ which are provided by Proposition 2.3.1 and the
bootstrap assumption (2.3.1) respectively.
With the same method, taking f = |φ|4γ , we obtain inequality
(2.3.9).
Proposition 2.3.6. Under the assumptions of Theorem 2.2.1 and
the bootstrap assumptions
(2.3.1), we have
||φ||L2(Su,u) ≤ ||φ||L2(Su′,u) +∫ uu′||∇3φ||L2(Su′′,u)du
′′, (2.3.10)
||φ||L4(Su,u) ≤ ||φ||L4(Su′,u) +∫ uu′||∇3φ||L4(Su′′,u)du
′′ (2.3.11)
for an Su,u tangent tensor φ of arbitrary rank.
Proof. We first note that the following identity holds for any
scalar f :
d
du
∫Su,u
f =∫Su,u
(df
du+ Ωtrχf
)=∫Su,u
Ω(e3(f) + trχf
).
Hence, taking f = |φ|2γ , we have
||φ||2L2(Su,u) =||φ||2L2(Su′,u)
+∫ uu′
∫Su′′,u
2Ω(< φ,∇3φ >γ +
12
trχ|φ|2γ)du′′.
The inequality (2.3.10) can be concluded using Cauchy-Schwarz on
the sphere and the L∞ bounds
for Ω and trχ which are provided by Proposition 2.3.1 and the
bootstrap assumption (2.3.1) re-
51
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spectively. With the same method, taking f = |φ|4γ , we obtain
inequality (2.3.11).
We rewrite the above inequalities in scale invariant norms as
follows:
Proposition 2.3.7. For an Su,u tangent tensor φ of arbitrary
rank, we have
‖φ‖L2sc(Su,u) ≤ ‖φ‖L2sc(Su,0) +∫ u
0
δ−1‖∇4φ‖L2sc(Su,u′ )du′,
‖φ‖L2sc(Su,u) ≤ ‖φ‖L2sc(Su∞,u) +∫ uu∞
1|u′|2‖∇3φ‖L2sc(Su′,u)du
′,
‖φ‖L4sc(Su,u) ≤ ‖φ‖L4sc(Su,0) +∫ u
0
δ−1‖∇4φ‖L4sc(Su,u′ )du′,
‖φ‖L4sc(Su,u) ≤ ‖φ‖L4sc(Su∞,u) +∫ uu∞
1|u′|2‖∇3φ‖L4sc(Su′,u)du
′.
For the ∇3 equation, we can get more precise estimates by
incorporating the weights in the
norms. These weights depend on the coefficients in front of the
linear term with a trχ factor. The
main observation is that under the bootstrap assumption (2.3.1),
trχ can be viewed essentially as
− 2|u| . More clearly, we have
Proposition 2.3.8 (Evolution Lemma). We continue to work under
the assumptions of Theorem
2.2.1 and the bootstrap assumptions (2.3.1). Let φ and F be
Su,u-tangent tensor fields of rank k
satisfying the following transport equation:
∇3φA1...Ak + λ0trχφA1...Ak = FA1...Ak .
Set p ∈ {2, 4}. Denoting λ1 = 2(λ0 − 1p ), for φ we have
|u|λ1‖φ‖Lp(Su,u) . |u∞|λ1‖φ‖Lp(Su∞,u) +
∫ uu∞
|u′|λ1‖F‖Lp(Su′,u)du′.
where the implicit constant is allowed to depend on λ0.
52
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Proof. To begin, we have the following identity for any scalar
function f :
d
du
∫Su,u
f =∫Su,u
(df
du+ Ωtrχf
)=∫Su,u
Ω(e3(f) + trχf
).
Using this identity, we obtain
d
du(∫Su,u
|u|λ1p|φ|p)
=∫Su,u
Ω(− λ1p|u|λ1p−1(e3u)|φ|p + p|u|λ1p < φp−1,∇3φ >
+trχ|u|λ1p|φ|p
)=∫Su,u
Ω(p|u|λ1p < φp−1,∇3φ+ λ0trχφ >
)+∫Su,u
Ω|u|λ1p(− λ1p(e3u)
|u|+ (1− λ0p)trχ
)|φ|p.
Observe that we have
− λ1p(e3u)|u|
+ (1− λ0p)trχ
=− λ1pΩ−1
|u|+ (1− λ0p)trχ
=− λ1p(Ω−1 − 1)|u|
+ (1− λ0p)(trχ+2|u|
)− λ1p+ 2− 2λ0p|u|
≤ δ12
|u|2∆0.
For the last inequality, we employ the bound for Ω in
Proposition 2.3.1 and the bootstrap assumption
(2.3.1) together with the chosen parameters to satisfy λ1p+ 2−
2λ0p = 0.
Therefore,
| ddu
(∫Su,u
|u|λ1p|φ|p)| ≤∫Su,u
(p|u|λ1p|φ|p−1|F |+ |u|2λ1−2δ 12 ∆0|φ|p
).
Using Cauchy-Schwarz for the first term and applying Gronwall’s
inequality for the second term,
53
-
we obtain
|u|λ1‖φ‖Lp(Su,u)
≤eδ12 ∆0‖u−2‖L1u
(|u∞|λ1‖φ‖Lp(Su∞,u) +
∫ uu∞
|u′|λ1‖F‖Lp(Su′,u)du′)
≤|u∞|λ1‖φ‖Lp(Su∞,u) +∫ uu∞
|u′|λ1‖F‖Lp(Su′,u)du′.
since δ12 ∆0‖u−2‖L1u ≤ 1, when δ
12 ∆0 is small.
2.3.3 Sobolev Embedding
Using the upper and lower bounds of the volume form, Sobolev
embedding theorems in our setting
follow from standard Sobolev embedding theorems (see [18]).
Proposition 2.3.9. (L4 Sobolev Embedding) Under the assumptions
of Theorem 2.2.1 and the
bootstrap assumptions (2.3.1), (2.3.2) and (2.3.3), we have
||φ||L4(Su,u) ≤ C(‖φ‖
12L2(Su,u)
‖∇φ‖12L2(Su,u)
+1|u| 12‖φ‖L2(Su,u)
),
and in scale invariant norms
‖φ‖L4sc(Su,u) ≤ C(‖φ‖
12L2sc(Su,u)
‖∇φ‖12L2sc(Su,u)
+ δ14 ‖φ‖L2sc(Su,u)
).
Similarly, for L∞ norm we obtain
Proposition 2.3.10. (L∞ Sobolev Embedding) Under the assumptions
of Theorem 2.2.1 and the
bootstrap assumptions (2.3.1), (2.3.2) and (2.3.3), we have
‖φ‖L∞(Su,u) ≤ C(‖φ‖
12L4(Su,u)
‖∇φ‖12L4(Su,u)
+1|u| 12‖φ‖L4(Su,u)
),
and in scale invariant norms
54
-
‖φ‖L∞sc(Su,u) ≤ C(‖φ‖
12L4sc(Su,u)
‖∇φ‖12L4sc(Su,u)
+ δ14 ‖φ‖L4sc(Su,u)
).
In the same manner, we derive
Proposition 2.3.11. Under the assumptions of Theorem 2.2.1 and
the bootstrap assumptions
(2.3.1), (2.3.2) and (2.3.3), let δSu,u ⊂ Su,u denote a disk of
radius δ12 |u| relative to either θ or θ
coordinate system. There exist δ0 = δ0(∆0), such that whenever δ
≤ δ0, in D for any horizontal
tensor φ we have
‖φ‖L∞(Su,u) . supδSu,u⊂Su,u
(δ
14 |u| 12 ‖∇φ‖L4(δSu,u) +
δ−14
|u| 12‖φ‖L4(δSu,u)
),
and in scale invariant norm
‖φ‖L∞sc (Su,u) . supδSu,u⊂Su,u
(‖∇φ‖L4sc(δSu,u) + ‖φ‖L4sc(δSu,u)
).
2.3.4 Commutation Formulas
We list the following formulas from [12]:
Proposition 2.3.12. For a scalar function f , we have
[∇4,∇]f =12
(η + η)D4f − χ · ∇f,
[∇3,∇]f =12
(η + η)D3f − χ · ∇f.
Proposition 2.3.13. For a 1-form Ub tangent to Su,u, we have
[D4,∇a]Ub = −χac∇cUb + �ac∗βbUc +12
(ηa +