Formation of Trapped Surfaces in General Relativity thesis.pdfLi, Elliot Lieb, Hans Lindblad,...

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

c© Copyright by Xinliang An, 2014.

All Rights Reserved

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

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

to my family.

v

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

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

vii

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

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

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

2

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

3

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

4

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.

5

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

6

|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

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


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


•∫ 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


•∫ 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


•∫ 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:


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.


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:


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


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


12

|u|‖φ1‖L∞sc(Su,u)‖φ2‖L1sc(Su,u),


12

|u|‖φ1‖L2sc(Su,u)‖φ2‖L2sc(Su,u),

38


12

|u|‖φ1‖L∞sc(Su,u)‖φ2‖L2sc(Su,u),


12

|u|‖φ1‖L4sc(Su,u)‖φ2‖L4sc(Su,u),


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


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


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

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)

47

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

48

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


(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

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).


(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

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

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


(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 +

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