Proof of a Null Penrose Conjecture Using a New Quasi-Local Mass

Proof of a Null Penrose Conjecture Using a New Quasi-local Mass

Henri P. Roesch

Department of Mathematics Duke University

Date: Approved:

Hubert L. Bray, Supervisor

Paul S. Aspinwall

Robert L. Bryant

Mark A. Stern

Dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2017 Abstract Proof of a Null Penrose Conjecture Using a New Quasi-local Mass

Henri P. Roesch

Department of Mathematics Duke University

Date: Approved:

Hubert L. Bray, Supervisor

Paul S. Aspinwall

Robert L. Bryant

Mark A. Stern

An abstract of a dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2017 Copyright c 2017 by Henri P. Roesch All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial Licence Abstract

In the theory of general relativity, the Penrose conjecture claims a lower bound for the mass of a spacetime in terms of the area of an outermost horizon, if one exists. In physical terms, this conjecture is a geometric formulation of the statement that the total mass of a spacetime is at least the mass of any black holes that are present, assuming non-negative energy density. For the geometry of light-rays emanating oﬀ of a black hole horizon (called a nullcone), the Penrose conjecture can be reformulated to the so-called Null Penrose Conjecture (NPC). In this thesis, we deﬁne an explicit quasi-local mass functional that is non-decreasing along all foliations (satisfying a convexity assumption) of nullcones. We use this new functional to prove the NPC under fairly generic conditions.

iv Dedicated to God, my wife, and my parents.

v Contents

Abstract iv

List of Figures ix

Acknowledgementsx

1 Introduction1

1.1 The Penrose conjecture...... 2

1.2 Minkowski spacetime...... 3

1.2.1 Particles and Observers...... 4

1.2.2 Nullcones...... 5

1.3 Formulations of the Penrose conjecture...... 9

1.3.1 The spacelike setting...... 10

1.3.2 The null setting...... 14

1.4 Schwarzschild Spacetime...... 18

1.4.1 Time-symmetric slices of Schwarzschild...... 19

1.4.2 Standard Nullcones of Schwarzschild...... 20

1.5 Mass rather than Energy...... 22

1.5.1 Overview of Thesis...... 24

2 Technical Background 26

2.1 Preliminaries and Main Results...... 26

2.2 Final introductory remarks...... 33

vi 3 Motivation 35

3.1 Nullcone of a point in a Space Form...... 37

3.2 Standard Nullcones of Schwarzschild Revisited...... 45

3.3 Variation of EH ...... 50

4 Propagation of ρ 56

4.1 Setup...... 56

4.2 The Structure Equations...... 58

4.3 Propagation of ρ ...... 67

4.3.1 Case of Equality...... 71

5 Foliation Comparison 76

5.1 Additional Setup...... 76

5.2 Flux Comparison...... 79

5.3 Asymptotic ﬂatness...... 87

5.3.1 Asymptotic Mass and Energy...... 96

6 Perturbing Spherical Symmetry 107

6.1 Spherical Symmetry...... 107

6.1.1 The metric...... 107

6.1.2 Calculating ρ ...... 108

6.1.3 Asymptotic ﬂatness...... 113

6.2 Perturbing Spherical Symmetry...... 115

6.2.1 A geodesic foliation...... 117

6.2.2 Calculating ρ ...... 118

6.2.3 Smoothing to Spherical Symmetry...... 125

A Bartnik data of coordinate spheres in boosted Schwarzschild 130

2 A.1 The Fundamental Forms of Sr¯ ...... 134

vii A.2 Curvature and Energy...... 139

A.3 Normal Connection and Momentum...... 149

Bibliography 158

Biography 161

viii List of Figures

4 1.1 Particle in R1 ...... 7 1.2 Isolated Black Hole...... 16

1.3 The time-symmetric slice N ...... 19

1.4 Standard Nullcone of Schwarzschild Ω...... 21

1.5 Boosts...... 23

2.1 About a 2-sphere in spacetime...... 27

2.2 An expanding nullcone...... 28

3.1 Nullcones at ~o and o ...... 41

4.1 Cross-sections: s3 ă s2 ă S´ ă s0 ă S` ă s1 ...... 57

5.1 Σω as a graph over Σs0 ...... 77 6.1 Perturbing Spherical Symmetry...... 126

ix Acknowledgements

I would like to express deep gratitude to my adviser, Dr. Hubert L. Bray. Apart from being a truly outstanding mentor, Dr. Bray has consistently shown his support, generosity, and friendship throughout my time at Duke. His influence will undoubt- edly enrich my future endeavors and in no small measure was this the case for the problems dealt with in this thesis. I am also indebted to Dr. Marc Mars for his invaluable suggestions toward my completion of this project as well as Drs. Richard Schoen and Gregory Galloway for their support and encouragement. Next, I would like to thank Dr. Michael Reed for his wisdom in the early stages of my doctoral work as well as my committee mem- bers, Drs. Paul Aspinwall, Robert Bryant, and Mark Stern, for their many valuable contributions. I am also truly grateful to my friends and office mates with whom I have had so many formative and exhilarating mathematical discussions, including Andrew Goetz, Tatsunari Watanabe, Michael Casey, Akihiko Nishimura, Kevin Ko- rdek, Hangjun Xu, Christopher O’Neill, Ioannis Sgouralis, and Benjamin Gaines. I acknowledge and sincerely thank my undergraduate mentor, Dr. Yiannis Petridis, and the University College London Department of Mathematics as well as my post- graduate mentor Dr. Tadashi Tokieda who first nurtured my love of mathematics. I am also indebted to the National Science Foundation for its financial support of my final year in graduate school and travel with the grant DMS-1406396. Addi- tionally, I would like to thank the Graduate School of Duke University for funding

x my third summer with a Clinical Faculty Arts and Sciences Research Fellowship. Most of all, I wish to thank my best friend and wife, Erin Cloninger Roesch, for her endless love and support. My grandfather, Nick Wolmarans, for his inspiration. As well as my parents, Elize and Henning, whose strength and countless sacriﬁces for me and my brother Stefan cleared the path that made this work possible.

xi 1

Introduction

In the framework of Albert Einstein’s theory of General Relativity, the presence of matter induces curvature of the four dimensional fabric we inhabit called spacetime. Our perception of gravity is a direct consequence of this curvature and the reaction of entities to its presence. Mathematically, we recognize a spacetime as a semi-Riemannian manifold pM4, gq, where M is a four dimensional manifold with metric gp¨, ¨q of signature p´, `, `, `q. Throughout this thesis, we will be setting the universal gravitational constant and the speed of light to unity. The fundamental bridge between our geometric and physical interpretations of the framework is given by the Einstein equation,

G “ 8πT

1 where T represents the physical stress-energy tensor, and G “ Ricg ´ 2 Rgg is the Einstein tensor. From a physical perspective, the tensor T p¨, ¨q signals the presence of matter in space-time by the measurement of energy-momentum at any given point. With regards to geometry, since the Einstein tensor G is constructed from the Ricci curvature tensor Ricgp¨, ¨q and scalar curvature Rg (of the metric g), it measures geometric curvature. The Einstein equation therefore conﬂates both these interpretive

1 lenses providing a theory whereby matter curves spacetime.

1.1 The Penrose conjecture

In 1973, Sir Roger Penrose ([23]) conjectured that the mass contributed by a collec-

|Σ| tion of black holes should be no less than 16π , where |Σ| is the total combined area b of all black hole horizons Σ in our spacetime. Alternatively,

|Σ| M ě , (1.1) c16π

where M is the total mass. One of the fundamental ingredients of Penrose’s heuris- tic argument was the use of cosmic censorship. As a statement on the global future evolution of a system, cosmic censorship is essential for an existence theorem in general relativity. This hypothesis asserts that generically all spacetime singularities are hidden from the rest of the universe by black holes. Since singularities are shown to exist for physically reasonable spacetimes (by the famous work of Hawking and Penrose [13]), these semi-permeable information barriers serve to prevent their chaotic physical implications from inﬂuencing our deterministic system. Finding a counterexample to (1.1) would likely indicate a failure of cosmic censorship (in fact this was Penrose’s original motivation for studying (1.1)), and, conversely, a proof of (1.1) would provide indirect support to its validity. Another important ingredient in Penrose’s argument was to assume the Dominant Energy Condition. This condition, via the energy-momentum tensor T , imposes local curvature constraints to model a spacetime with non-negative energy density. From this perspective, (1.1) in essence claims that this non-negative energy density must aggregate (in analogy to our classical understanding of mass) to at least the black hole contributions. This would also reﬁne the famous positive mass theorem of Schoen and Yau ([26],[27]) discussed in Section 1.3.

2 Recognizing the implications of the Penrose conjecture on the immensely successful framework laid out by Einstein, mathematicians and physicists have spent consider- able eﬀort in the last forty years towards a rigorous proof. Thus far, attempts have centered on two main approaches. The traditional approach centers on analyzing initial data pN 3, g,¯ Kq where pN , g¯q denotes a Riemannian slice of the spacetime M of extrinsic curvature (or second fundamental form) K. We brieﬂy discuss this setting and the beautiful results that answer special cases in Section 1.3. The study of this thesis however, involves an approach involving null slices of M.A null hypersurface

Ω ãÑ M is represented by a three-dimensional manifold Ω on which the induced metric γ “ g|Ω is degenerate. A major appeal of this setting is the existence of a null tangent vector L P ΓpT Ωq generating null geodesics that rule Ω. For this very rea- son, many approaches in the Riemannian setting involving difficult geometric partial differential equations reduce in the null setting to an analysis of ordinary differential equations. To enrich the discussion of these two approaches and to help build an intuition for the fundamental ideas that underpin this thesis we spend some time in the setting of Einstein’s theory of Special Relativity.

1.2 Minkowski spacetime

In a vacuum, namely G “ 8πT “ 0, our simplest model is given by the ﬂat Minkowski

4 4 space R1 :“ pR , gq whereby g is given by the quadratic form

ds2 “ ´dt2 ` dx2 ` dy2 ` dz2 in a global chart pt, x, y, zq. Although highly restrictive, Minkowski spacetime orig- inally served as the springboard for the development of Albert Einstein’s general theory as well as an initial gateway to some of the most fundamental discoveries in twentieth century physics. In this section, we invest some time introducing key aspects of the Minkowski spacetime that will serve as a backdrop throughout this

3 thesis.

1.2.1 Particles and Observers

An immediate consequence of the vector space structure is the canonical isomorphism

4 4 between the spacetime and its tangent spaces, whereby R1 – T~qR1 for any point 4 ~q P R1. This warrants our ability to unambiguously pull-back the quadratic form 4 associated to g “ x¨, ¨y. Therefore, given a non-trivial vector v P R1, we immediately notice its existence within one of three categories. If xv, vy ă 0 we say v is timelike, if xv, vy ą 0 we say v is spacelike, and if xv, vy “ 0 we say v is null. We deﬁne the length

4 of v P R1 with |v| :“ |xv, vy|. It will also be useful to distinguish spacelike vectors from non-spacelike vectorsa by referring to the latter as causal vectors. Taking the

globally deﬁned timelike vector Bt to deﬁne a pointwise ‘direction of time’, any causal

vector v can then be identiﬁed as either future pointing if xv, Bty ă 0 or past pointing

4 4 if xv, Bty ą 0. A material particle is then represented in R1 by a curve αpτq : I Ñ R1 which has a unit future-pointing timelike velocity α1 (i.e. such that xα1, α1y “ ´1).

Given the frame tBt, Bx, By, Bzu “ tBt, Biu the energy-momentum for a particle of mass M is given by

1 i T :“ Mα “ EBt ` P Bi.

For convenience we may denote T “ pE, P~ q, where E is the measured energy of the

particle and P~ its linear momentum. We see that M 2 “ ´xT , T y “ E2 ´ |P~ |2, which

4 4 is independent of our choice of frame (i.e. under an arbitrary isometry φ : R1 Ñ R1), whereas E and P~ are not. In General Relativity, ‘free-falling’ particles are characterized by the restriction that

2 4 α be geodesic i.e. α “ 0. In R1 we conclude therefore that free falling particles have constant energy-momentum.

4 Within the Lie group of isometries of R1, we will concentrate on the subgroup ﬁxing the origin called the Lorentz group. In particular, the connected component of the

4 identity (or the restricted Lorentz group), is generated by pure rotations of space

4 4 (SOp3q) and the Lorentz boosts φ~v : R1 Ñ R1:

φ pγ ´ 1qp~p ¨ ~vq pE, ~pq ÝÑ~v µ pE ´ ~v ¨ ~pq, ~p ` pµ E ` q~v (1.2) ~v ~v |~v|2 ´ ¯

8 3 3 ´1 2 for some ~v P B Ă R and µ~v :“ 1 ´ |~v| . It’s easily seen that all future directed unit timelike vectors are uniquelya reached by a boost of Bt. As a result, given any free falling future directed trajectory α we are able to boost to a frame where it remains at rest, i.e. αi “ 0. Equivalently, we may identify these trajectories as co-observers within Minkowski spacetime each with an associated boosted frame of reference. A free falling particle is therefore viewed from an observer at rest (relative to the motion of the particle) to have energy-momentum

T “ pM,~0q giving Einstein’s famous identity that E “ M (or E “ Mc2 when the speed of light, c, is not set to unity).

1.2.2 Nullcones

4 4 Given a linear map φ : R1 Ñ R1 it’s an easy exercise to show (forgiving the abuse of notation) that φ “ dφ. Since the Lorentz group is induced by linear maps we

4 4 therefore observe for any w P R1 – Tφ~vpwqR1 that xdφ~vpwq, dφ~vpwqy “ xw, wy. As a 4 result, the hyperquadrics of R1

4 HC :“ tv P R1|xv, vy “ Cu

are fixed under the action of φ~v, which therefore restricts to a diffeomorphism of submanifolds. Whenever C ‰ 0, φ~v furthermore restricts to an isometry of the induced semi-Riemannian submanifold HC . Considering the hyperquadric given by C “ 0, we are finally led to one of the

5 fundamental entities in this thesis and the notion of light in General Relativity. A ray of light (similarly to a free falling particle) is identiﬁed by a geodesic trajectory.

While a particle is restricted to having a timelike velocity, a light ray βpλq instead must satisfy xβ1, β1y “ 0. As a result, for a ray of light emanating from the origin

4 in R1 we are handed some βpλq “ λv whereby xv, vy “ 0. The collection of all 4 past-pointing null geodesics form a null hypersurface Ω ãÑ R1

4 Ω:“ tv P R1 | xv, vy “ 0 ă xv, Btyu called the past nullcone of o. Interestingly, given the characterization of particles and light, we notice that the set Ω ` αpτq bounds the causal history of an observer at αpτq since no matter or light to the future of this boundary can reach the event αpτq. Moreover, by time symmetry, no particle released to the future of αpτq will be able to escape the future nullcone. The reader may recognize this as a geometric realization of the fact that matter cannot travel faster than the speed of light (which is ﬁxed at c “ 1). We conclude this section by showing a one-to-one correspondence between round

4 foliations of Ω and the boosts φ~v. Since any slicing of R1 by parallel spacelike hyper- planes is uniquely identiﬁable by their common timelike future normal (and therefore to a boost), they are given as level sets of a boosted ‘time function’ t¯. We start therefore with the trivial slicing associated to our coordinate function t and observe that it induces a foliation of Ω by round spheres (any two of which are homothetic under a radius re-scaling). Since the diﬀeomorphism φ~v|Ω is induced by an isometry that isometrically ‘tilts’ the trivial slicing, intersection with Ω leads to an isometric tilting of the initial foliation. In fact, in spherical polar coordinates we see that the image

2 under φ~v of the unit sphere is given by Σ :“ tt “ r “ ωu for some function ω P FpS q

only if the inverse map gives φ´~vp´ωpϑ, ϕq, ωpϑ, ϕq, ϑ, ϕq “ p´1, 1, fpϑ, ϕq, gpϑ, ϕqq.

6 t¯

4 R1 y¯

x¯,¯z

x, z 4 Figure 1.1: Particle in R1

From this, (1.2) gives 1 ωpϑ, ϕq “ (1.3) µ~vp1 ´ ~v ¨ ~npϑ, ϕqq for ~npϑ, ϕq the unit position vector in R3. From the quadratic form in spherical polar coordinates

ds2 “ ´dt2 ` dr2 ` r2pdϑ2 ` psin ϑq2dϕ2q,

2 we conclude that Σ has induced round metric ψ ˚γ (i.e. Gaussian curvature Kψ2˚γ “ 1) where ˚γ is the standard round metric of S2. Equivalently, for ∆˚ the Laplacian associated to ˚γ, 1 1 “ p1 ´ ∆˚ log ωq. (1.4) ω2

The converse, namely that any function solving (1.4) must be of the form (1.3) is

7 known ([18]) and follows from the following geometric result (the reader may wish to skip the proof, which we include for completeness):

Lemma 1.2.1. Σ ãÑ Ω is a round sphere only if Σ is the intersection of Ω with a spacelike hyperplane.

Proof. Utilizing the spherical polar form of the metric of Minkowski space we con-

clude as before that Σ must have induced metric γ “ ω2˚γ for some function ω P FpS2q.

The assumption that Σ be round also gives us that Kω2˚γ “ 1 (and that ω solves (1.3)). Our result therefore follows as soon as we show the existence of a constant unit time-

like vector N P ΓpT KΣq for Σ. In the Minkowski spacetime we have the position

i 4 vector ﬁeld P :“ tBt ` x Bi such that DX P “ X for any X P ΓpT R1q, where D is the Levi-Civita connection. From the associated second fundamental form and mean curvature of Σ

K IIpV,W q “ DV W ~ H “ trΣ II

for V,W P ΓpT Σq, we conclude from metric compatibility that xII,P y|Σ “ ´γ implies ~ xH,P y “ ´2. From construction of Ω it also follows that xP,P y|Ω “ 0 so that

1 xV,P y “ 2 V xP,P y “ 0, for any V P ΓpT Σq. We conclude that Σ has trivial normal bundle with basis tP, H~ u Ă T KΣ. Since II “ aP ` bH~ for two symmetric 2-tensors a, b we have ´2b “ xII,P y “ ´γ. So for the traceless part of II it follows that ˆ 4 II “ aPˆ . From the Gauss equation ([21], pg.100) for Σ ãÑ R1 we see

1 1 K ´ xH,~ H~ y ` xIIˆ, IIˆy “ 0, 4 2

~ ~ 1 ~ so we immediately conclude that xH, Hy “ 4. We now show that N “ P ` 2 H suﬃces as our choice of timelike unit normal.

8 ~ ~ ~ ~ 1 ~ ~ Since xDV H,P y “ V xH,P y “ 0 and xDV H, Hy “ 2 V xH, Hy “ 0, we conclude that

K ~ DV H “ 0. From the Codazzi equation ([21],pg.115)

K K pDV IIqpW, Uq “ pDW IIqpV,Uq

K ~ the fact that D H “ 0 implies that p∇V aˆqpW, Uq “ p∇W aˆqpV,Uq for ∇ the induced connection on Σ. As a result, taking a trace over V,U and using the fact that contraction commutes with covariant diﬀerentiation, we have

p∇ ¨ aˆqpW q “ trΣp∇W aˆq “ W trΣ aˆ “ 0.

It is a well known consequence of the Uniformization Theorem that the divergence operator on a spacelike 2-sphere is injective when restricted to symmetric tracefree 2- tensors (see for example, [25]), so we conclude thata ˆ “ 0 which implies IIˆ “ 0. From ~ ~ 1 ~ ~ this it follows that xDV H,W y “ ´xH, IIpV,W qy “ ´ 2 xH, HyxV,W y “ ´2xV,W y ~ and therefore DV H “ ´2V . Finally, this gives us that

1 D N “ D P ` D H~ “ V ´ V “ 0, V V 2 V

as desired.

From Lemma 1.1.1, the one-to-one correspondence between boosts φ~v and round foliations of Ω is therefore evident via (1.3).

1.3 Formulations of the Penrose conjecture

An immediate diﬃculty we face when we delve into the general theory is to un- derstand the nature and behavior of energy and momentum, and, by extension, a meaningful measurement of mass. Fortunately, a means of measuring total energy and momentum is aﬀorded by certain spacetimes.

For matter that is isolated and locally concentrated within a spacetime pM, gq,

9 the ‘farther away’ one is from the matter content the more the geometry settles towards ‘ﬂatness’. With regards to the metric g, this translates into certain decay conditions on the associated curvature tensors, and such spacetimes are called asymptotically ﬂat.

1.3.1 The spacelike setting

A hypersurface N 3 Ă M with induced Riemannian metricg ¯ is said to be asymptotically Euclidean ifg ¯ decays sufficiently fast to the flat metric δ on R3 and, up to a compact region C (surrounding the matter content), the sub-structure of N Ć is also

3 k in agreement with (possibly more than one copy of) R . Specifically, N Ć “ i“1 Ni 3 3 where each end Ni is diffeomorphic to R ´ B , for B the unit ball. Moreover,Ť in 3 2 3 2 the chart txiu for R (with radius r :“ i“1 xi ) we have thatg ¯ij “ δij ` ij and 2 3 r|ij|`r p|Bij|`|Kij|q`r p|BBij|`|BKij|qř ď C, where K is the second fundamental form of N . The pair N Ă M therefore become asymptotically characteristic of a

4 4 Euclidean slice E Ă R1. Within these isolated systems, similarly to a particle in R1, we observe an abstract ADM energy-momentum pE, P~ q ([2]) that is given (assuming summation over repeated indices) by

1 E “ lim pBjg¯ij ´ Big¯jjq~nidσ (1.5) rÑ8 16π żSr

i 1 i i and P “ lim pKj ´ δjKkkq~njdσ (1.6) rÑ8 8π żSr where Sr represents a coordinate sphere of radius r with unit normal ~n. It can be

i shown that E and P do not depend on our choice of chart txiu ([3]), and two distinct asymptotically Euclidean slicings of M diﬀer by a Minkowskian boost (1.2) in their respective measurements of pE, P~ q (see [24]). We are now in a position to state a beautiful result ﬁrst proved by Schoen and Yau using minimal surface techniques ([26],[27]) and then by Witten using spinors

10 ([32]).

Theorem 1.3.1 (Positive Mass Theorem). Let pN , g¯q ãÑ pM, gq be an asymptotically Euclidean hypersurface, where the metric g satisﬁes the Dominant Energy Con- dition. Then for each end Ni we have that

E2 ě |P~ |2.

Moreover, if E “ 0 for some i, then pN , g¯q “ pR3, δq.

It follows therefore that the mass of the spacetime satisﬁes

M “ E2 ´ |P~ |2 ě 0 b as expected. The Penrose conjecture further predicts a stricter lower bound for the total mass M whenever spacetimes contain any black hole horizons, namely, the inequality (1.1).

A natural special case to consider is when the slice pN , g¯q represents a rest- frame P i “ 0, E “ M. We see from (1.6) that this holds in the event that N is totally geodesic, i.e. K “ 0. Since the tensor K is given by a normal variation of g¯ oﬀ of N (by ﬁrst variation of area) we would obtain this condition from a time- symmetric slicing of the spacetime M. From the Gauss equation, it then follows that the Dominant Energy Condition is equivalent to the statement that N have non- negative scalar curvature Rg¯ ě 0. It is also known in this setting that the black hole horizon is represented by an ‘outermost’ minimal surface Σ0 possibly with multiple components; each component a sphere and no two spheres intersecting (see [20, 15]). From this the Penrose conjecture completely reduces to a statement relating the total mass of the time-symmetric three-dimensional Riemannian manifold N and the area of Σ0 ãÑ N , called the Riemannian Penrose Inequality (or RPI). A fundamental breakthrough came at the very end of the twentieth century with a complete proof of

11 the RPI. Whenever Σ0 is a single connected component Huisken and Ilmanen proved the RPI in 1997 ([15]) which was subsequently generalized to multiple components by Bray in 1999 ([6]) using a completely new approach.

Theorem 1.3.2. (Bray) Suppose pN , g¯q is complete, has non-negative scalar curva-

n ture and contains an outermost minimal surface Σ0 “ i“1 Si. Then Ť |S | M ě i i 16π cř with equality only in the case that pN , g¯q is isometric to a time-symmetric slice of the Schwarzschild spacetime (see Section 1.4.1 below) of mass M.

An interesting energy functional for any closed spacelike surface Σ introduced by Hawking ([12]) is deﬁned by

|Σ| 1 ~ ~ EH pΣq “ 1 ´ xH, HydA c16π 16π Σ ´ ż ¯ for H~ the mean curvature of Σ. This Hawking Energy was the deﬁnitive tool used by Huisken and Ilmanen to prove the Riemannian Penrose Inequality. An observation due to Geroch ([10]) shows that under inverse mean curvature ﬂow, EH is non- decreasing within slices of non-negative scalar curvature. It also follows for coordinate spheres Sr of an asymptotically Euclidean slice that lim EH pSrq “ E. So if the rÑ8

slice N is time-symmetric, not only do we conclude that lim EH pSrq “ M but on rÑ8

~ |Σ0| a minimal surface Σ0 (where H “ 0) it also follows that EH pΣ0q “ . From c 16π Geroch’s observations, Jang and Wald ([16]) observed a potential application towards the Riemannian Penrose Inequality given the existence of a smooth flow from the vicinity of a horizon to round spheres at infinity. It was soon realized however that this flow would not remain smooth in general, which is where Huisken and Ilmanen’s

12 work was needed to close the argument. They were able to show the existence of a weakly deﬁned inverse mean curvature ﬂow to overcome possible singularities

and found that the leaves Σs approached round spheres asymptotically as needed. Unfortunately, the monotonicity of the ﬂow is dependent on the topology of the initial

surface, requiring that the ﬂow begin on a connected component of Σ0 “ i Si. As a result, they were able to show Ť

|Si| max “ EH p0q ď lim EH psq “ M. i c16π sÑ8

Rather than a flow of surfaces within the fixed geometry of the slice, Bray’s insight was instead to construct a flow of metrics that vary the geometry of the slice via a one-parameter family of conformal factors. Bray was able to construct this flow so that the area of the outermost minimal surface was unchanging, the scalar cur-

vature remained non-negative (enforcing Mptq ě 0 by the Positive Mass Theorem) and the ADM mass was non-increasing. Moreover, the ﬂow approaches spherical symmetry, so combined with positive mass this ensures the limit is a time-symmetric slice of the Schwarzschild spacetime. Since (1.1) achieves equality for these slices in Schwarzschild we conclude that:

|Σ | |Σ | 0 “ lim t “ lim Mptq ď M. c 16π tÑ8 c16π tÑ8

Where Huisken and Ilmanen’s proof is strongly dependent on the dimension of the slice, Bray’s argument was shown by Bray and Lee ([8]) to generalize to dimensions less than eight. In either instance the case of equality for (1.1) enforces N to be the time-symmetric slice of Schwarzschild.

13 1.3.2 The null setting

In Minkowski spacetime, upon changing our coordinates to ‘ingoing null coordinates’

pt, rq Ñ pv, rq where v “ t ` r, the associated quadratic form is given by

ds2 “ ´dv2 ` 2dvdr ` r2pdϑ2 ` psin ϑq2dϕ2q,

and we recognize the past null cones along the time-axis as the slices Ω “ tv “ v0u.

K We ﬁnd that Bv “Bt and the gradient of v satisﬁes Dv “Br P ΓpT Ωq X ΓpT Ωq,

1 2 which is a past-pointing null vector. From the identity DDf Df “ 2 D|Df| , we conclude that integral curves of Br are geodesic, so Br generates the light rays that rule Ω. In fact, any spherical cross-section Σ ãÑ Ω is uniquely identiﬁed as a graph

2 over S by specifying r|Σ. For a cross-section Σ :“ tr “ ωpϑ, ϕqu, we may then assign

K a normal null basis tL,Lu P T Σ such that L “Br and xBr,Ly “ 2, from which we

decompose the second fundamental form II of Σ accordingly to χ :“ ´xII, Bry and

χ :“ ´xII,Ly. For Σ, as shown in Section 6.1 (with β “ M “ 0), denoting by ∇{

1 the induced connection, ζpV q :“ 2 xDV Br,Ly the connection 1-form for the normal bundle, and ˚γ the standard round metric on S2, one has 1 1 2 χ “ γ, χ “ p1 ` |∇{ ω|2qγ ´ 2Hω, γ “ ω ˚γ, ω ω ζ d{ log ω, 2 2 “ ´ tr χ “ , tr χ “ p1 ´ ∆˚ log ωq. ω ω

Therefore, for any geodesically induced foliation of Ω along Bs “ ψBr (for some

ψ P FpS2q), dependence of the data on the aﬃne parameter s is given via ωps, ϑ, ϕq “ 1 sψpϑ, ϕq. According to the null basis tψBr, ψ Lu we therefore get 1 2 2 χ “ ψχ “ γ, γs “ s ψ ˚γ, s s tr χ 2 tr χ “ “ K 2 . ζ ζ d{ log ψ 0, 2 s ψ s ψ ˚γ s “ ` “ tr χ “ ψ tr χ “ , s s

For asymptotically ﬂat spactimes, one obtains the notion of an asymptotically ﬂat nullcone Ω by specifying decay along a geodesic foliation that mirrors the above

14 dependence on the aﬃne parameter s to leading order (see Section 5.3 for precise

deﬁnitions). The work of Mars and Soria ([18]) then shows that lim EH pΣs‹ q ă 8 s‹Ñ8

for any asymptotically geodesic foliation tΣs‹ u satisfying s “ ψs‹ `ξ, where ξ decays ‘suﬃciently fast’. As was the case for asymptotically Euclidean slices, we ﬁnd that

EH pΣs‹ q approaches a measure of energy called the Bondi-energy EB if the leaves of

our foliation tΣs‹ u approach asymptotically round spheres. This coupling between energy measurements and asymptotically round foliations of Ω should be familiar given our analysis of the Minkowski spacetime, where we showed a one-to-one correspondence between the boosts (1.2) and round foliations of the past null cone of a point. Heuristically, we imagine an asymptotically round foliation of Ω induced by intersection with an asymptotically Euclidean slicing of M (see Figure 1.2 below). As these intersections asymptotically coincide with coordinate spheres in the slicing (as is the case in Minkowski spacetime) the Hawking energy becomes comparable to the energy associated with the slicing. Minimizing over all possible energies, we there-

fore obtain the Bondi-mass mB. By taking a Riemannian hypersurface asymptotic to Ω, Schoen and Yau (see [28]) were able to construct an asymptotically Euclidean

manifold (not necessarily embeddable in M) whose ADM mass is no larger than mB. From this the positivity of the Bondi-mass follows from the Positive Mass Theorem. In order to formulate the Penrose conjecture in this setting, we need to ﬁrst mo-

tivate a deﬁnition of a horizon Σ0. For any cross-section Σ ãÑ Ω, it follows that we recover Ω ‘outside Σ’ by emitting null geodesics along our past-pointing null

geodesic generator L P ΓpT Ωq X ΓpT KΩq. If Ω is asymptotically ﬂat, we also conclude that any foliation along L has expanding area (see Lemma 5.3.1), so L points ‘outwards and to the past’ of Σ. It follows, by rescaling L, that Σ admits a null

´ L ` ´ ` ` basis tL “ tr χ ,L “ tr χLu such that xL ,L y “ 2 (i.e. L is ‘outward and to the future’ of Σ) and therefore ´2H~ “ L` ´ xH,L~ `yL´. For an inﬁnitesimal ﬂow

15 t¯ y¯

x¯,¯z

EH pΣs‹ q Ñ E

x,z

EH pΣsq Ñ M

Figure 1.2: Isolated Black Hole along L` off of our initial surface Σ, we have from a first variation of the area form that dA9 “ ´xH,L~ `ydA “ xH,~ H~ ydA. Thus, if we’re able to locate a Σ ãÑ Ω for which H~ is a timelike vector, then Σ is ‘trapped’ in the sense that light rays emitted off of the surface of Σ yield a collapsing sphere. A horizon is therefore identified ~ by a marginally outer trapped surface Σ0 where the mean curvature H is null, i.e. xH,~ H~ y “ 0.

16 |Σ0| We notice immediately for a horizon Σ0 ãÑ Ω that EH pΣ0q “ 16π . As was the case b for Huisken and Ilmanen’s proof of the RPI, we ask whether we’re able to interpolate the Hawking energy along Ω from the horizon to null inﬁnity. In his PhD thesis,

Johannes Sauter ([25]) showed that, for a shear free nullcone Ω (i.e.χ ˆ “ 0) and M a vacuum spacetime, one is able to solve a system of ODEs to yield explicitly the data on Ω. This then enables a direct analysis of EH at null inﬁnity that allowed Sauter to prove the Penrose conjecture in this setting. From an observation attributed to

Christodoulou (see [25]), monotonicity of EH follows for foliations tΣs‹ u if either the 1 ~ ~ { mass aspect function µ :“ K ´ 4 xH, Hy ´ ∇ ¨ ζ or tr χ remain constant on each Σs‹ . Interestingly, the latter flow is in fact a null ‘inverse mean curvature’ flow since the ~ flow vector L satisfies ´xL, Hy “ tr χ “ const. on each Σs‹ . Sauter was able to show that, for small pertubations of Ω off of the shear free condition, one obtains global

existence of either ﬂow and that EH converges. Unfortunately though, unlike the Huisken and Ilmanen case, one is unable to conclude that the foliating 2-spheres become round asymptotically. In fact, Bergqvist ([4]) noticed this exact diﬃculty had been overlooked in an earlier work of Ludvigsen and Vickers ([17]) towards proving

|Σ0| the weak null Penrose conjecture, namely 16π ď EB. b Alexakis ([1]) was able to prove the null Penrose conjecture for vacuum perturbations of the black hole exterior in Schwarzschild spacetime by successfully using the mono-

tonicity of EH along the null inverse mean curvature flow. Here the author was once again aided by an explicit analysis of EH at null infinity. Work by Mars and Soria ([18]) followed soon after that identified the necessary conditions on Ω (inside general ambient spacetimes) needed to free up an explicit analysis of lim EH pΣsq along sÑ8 geodesic foliations. With their notion of an asymptotically flat null hypersurface Ω,

the authors were able to show an explicit limit of EH at null inﬁnity along various asymptotically geodesic foliations. In a later work ([19]), Mars and Soria constructed

17 a new functional on 2-spheres and showed for a special foliation tΣλu oﬀ of the horizon

|Σ0| Σ0 called geodesic asymptotically Bondi (or GAB) that, ď lim EH pΣλq ă 8. c 16π λÑ8 Thus, for GAB foliations that approach round spheres, the authors reproduce the weak null Penrose conjecture of Bergqvist ([4]), Ludvigsen and Vickers ([17]). Unfor- tunately, as in the aforementioned work of Sauter, Bergqvist, Ludvigsen and Vickers there is no guarantee of asymptotic roundness.

1.4 Schwarzschild Spacetime

Relaxing our restriction from ﬂat to spherically symmetric solutions in vacuum

(G “ 0 or, equivalently Ric “ 0), we extend beyond Minkowski to the 1-parameter family of Schwarzschild solutions. Modeling a static black hole of mass M, the Schwarzschild spacetime is characterized in spherical polar coordinates by the quadratic form 2M dr2 ds2 “ ´p1 ´ qdt2 ` ` r2pdϑ2 ` psin ϑq2dϕ2q r 2M p1 ´ r q

valid for 2M ą r ą 0, r ą 2M. From this we conﬁrm that M “ 0 reproduces the quadratic form of Minkowski spacetime. The maximal extension of this geometry is

2 called the Kruskal spacetime pP ˆr S , gK q which is given by the warped product of the Kruskal Plane P :“ tUV ą ´2Me´1u and the standard round S2 with warping

´1 r ´1 function r “ g pUV q for gprq “ pr ´ 2Mqe 2M , r ą 0. Therefore, the associated quadratic form that extends the Schwarzschild metric is given by:

ds2 “ 2F prqdV dU ` r2pdϑ2 ` psin ϑq2dϕ2q

8M 2 r 1´ 2M where F prq “ r e . We recover the Schwarzschild spacetime on V ą 0,U ‰ 0 V 2 2 with the coordinate change t “ 2M log | U | ([21]). Each round S has area 4πr so we interpret r as a ‘radius function’, the curvature singularity at r “ 0 we trace back to the function F prq, known as the ‘black hole’ singularity.

18 1.4.1 Time-symmetric slices of Schwarzschild

The time-symmetric slices of the Riemannian Penrose inequality are given by N :“

t0 tV “ e 2M Uu or t “ t0 in the Schwarzschild region. The fact that N is asymptotically Euclidean is evident under a change to isotropic coordinates represented by

M 2 pt, rq Ñ pt, Rq for r “ Rp1 ` 2R q . The metricg ¯ is represented by:

dr2 M 4 ` r2pdϑ2 ` psin ϑq2dϕ2q “ 1 ` pdx2 ` dy2 ` dz2q 1 ´ 2M 2R r ´ ¯

where R2 “ x2 ` y2 ` z2. We see that the coordinate singularity at r “ 2M has been removed in the change to isotropic coordinates. Moreover, for the diﬀeomorphism

M 2 1 φ : p0, 8q Ñ p0, 8q given by φpRq “ p 2 q R inspection of isotropic coordinates in spherical polar form identiﬁes an isometry of N . One can also show that φ is the

restriction to N of the isometry given by U Ñ ´U, V Ñ ´V . From this we conclude

M that N has two asymptotically Euclidean ends joined at the minimal sphere R “ 2 .

P N

V r “ 4M r “ 5M

r “ 0

M R “ 2

r “ 0

U Figure 1.3: The time-symmetric slice N

This minimal sphere corresponds to the black hole horizon at r “ 2M. We also show that the second fundamental form K vanishes. Since the components of the

Schwarzschild metric are independent of the coordinate t, we know Bt is a Killing

19 vector i.e. xDX Bt,Y y ` xDY Bt,Xy “ 0 for all vector ﬁelds X,Y . So restricting to X,Y P ΓpT N q we conclude that 1 KpX,Y q9 ´ xB ,D Y y “ ´ xB ,D Y y ` xB ,D Xy ` xB , rX,Y sy t X 2 t X t Y t ´ ¯ 1 “ xD B ,Y y ` xD B ,Xy “ 0 2 X t Y t ´ ¯ as expected. In the Appendix, we further explicitly analyze the eﬀect on Bartnik data for coordinate spheres of a time-symmetric slice that undergoes a boost of its asymptotic frame of reference (recall Figure 1.2). As a consequence, we expand calculations of

Wang and Yau ([31]) in obtaining the energy-momentum (according to EH and (1.6)). Speciﬁcally, we show, under a boost of isotropic coordinates

t cosh ψ ´ sinh ψ t Ñ z ´ sinh ψ cosh ψ z ˆ ˙ ˆ ˙ ˆ ˙

(here ψ represents a constant ‘rapidity’), that

pE, P~ q : pM, 0, 0, 0q Ñ pM cosh ψ, 0, 0,M sinh ψq.

1.4.2 Standard Nullcones of Schwarzschild

The past nullcone of a point in Minkowski spacetime also has a counterpart in Schwarzschild spacetime. In order to show this, we make yet another change of coordinates to the so called ingoing Eddington-Finkelstein coordinates, pt, rq Ñ pv, rq

dr whereby dv “ dt ` 2M : 1´ r

2M ds2 “ ´p1 ´ qdv2 ` 2dvdr ` r2pdϑ2 ` psin ϑq2dϕ2q. r

Once again, setting M “ 0 the resemblance with Minkowski is apparent. The stan-

dard nullcone of Schwarzschild is therefore given by slices of the form Ω :“ tv “ v0u. 20 v Up to a positive constant multiple we ﬁnd that V “ e 4M , so that the coordinates pv, r, ϑ, ϕq cover the whole Schwarzschild region V ą 0, including the horizon at U “ 0. Arguing identically as in the case of Minkowski, we ﬁnd our null geodesic

BU generator to be Br “ F prq , and any cross section Σ ãÑ Ω is uniquely identiﬁed by specifying r|Σ “ ωpϑ, ϕq.

U “ 0 P Ω r “ 0 U “ gp4Mq V0 2B V V r “ 2M r “ 4M BU F r “ 5M U “ gp5Mq V0 r “ 0 r “ 4M

Ω:“ tV “ V0u r “ 5M r “ 0 L Σ

U ω “ r|Σ L

Figure 1.4: Standard Nullcone of Schwarzschild Ω

K Endowed with a null normal basis tL “Br,Lu Ă T Σ such that xBr,Ly “ 2, the data on Σ is found in Section 3.2 (or Section 6.1 with β “ 0) 1 1 2M 2 χ “ γ, χ “ p1 ´ ` |∇{ ω|2qγ ´ 2Hω, γ “ ω ˚γ, ω ω ω ζ “ ´d{ log ω, 2 2 2M tr χ “ , tr χ “ p1 ´ ´ ∆˚ log ωq. ω ω ω

~ ~ 4 2M ˚ It follows that tr χ tr χ “ xH, Hy “ ω2 p1 ´ ω ´ ∆ log ωq and, by the Maximum ~ ~ Principle, the cross-section of Ω given by Σ0 :“ tr “ 2Mu uniquely satisﬁes xH, Hy “

0. So Σ0 is a horizon, in fact, from either the ingoing Eddington-Finkelstein or

21 Kruskal metric, we see that U “ 0 if and only if r “ 2M is a null hypersurface with spherical cross-sections, all of area 16πM 2. This hypersurface represents the outermost boundary of the black hole singularity called the event horizon. Regarding

asymptotic ﬂatness, from a geodesic foliation Σs :“ tr “ sψpϑ, ϕqu of Ω we see that the only distinction from the case in Minkowski spacetime is higher order decay in

2 1 2M the datum tr χs “ s Kψ2˚γ ´ s2 ψ2 . In fact, for any cross-section Σ we observe that 1 ~ ~ 2M K ´ 4 xH, Hy “ ω3 . Thus, from the Gauss-Bonnet Theorem followed by Jensen’s inequality, we conclude, as did Sauter (see [25]), that,

ω2dσ 2M E pΣq “ dσ ě M, H 16π ω cş ż where integration is over the standard round S2. We have equality only if ωpϑ, ϕq “

ω0, which is given by intersections of the time-symmetric slices N with Ω (or a rest frame measurement with respect to our black hole). Moreover, for round spheres

M given by (1.3), we see EH pΣsψq “ ? , in exact agreement with the boosted 1´|~v|2 energies of a particle (here the black hole) in special relativity.

1.5 Mass rather than Energy

Although the Hawking Energy enjoys monotonicity and convergence along certain

flows, difficulty remains in assigning physical significance to the convergence of EH due to the lack of control on the asymptotics of such flows. We expect these difficul- ties may very well be symptomatic of the fact that an energy functional is particularly susceptible to the plethora of ways boosts can develop along any given flow.

Analogous to the addition of 4-velocities in special relativity, P1 ` P2 “ P3 which

gives E3 “ E1 ` E2 (see Figure 1.5), we expect an infinitesimal null flow of Σ within a fixed reference frame to raise energy due to an influx of matter. However, with no a priori knowledge of the flow, we have no way to fix or even identify a reference frame.

22 ~ P2 “ pE2, P2q

~ P3 “ pE3, P3q 1 P P

~ P1 “ pE1, P1q

Figure 1.5: Boosts

So it is likely that ‘phantom energy’ will accumulate from infinitesimal boosts along the flow, in analogy with special relativistic boosts, P Ñ P 1 (i.e energy increases), or P 1 Ñ P (i.e. energy decreases) as shown in Figure 1.5. Geometrically, we expect this to manifest along the flow in a (local) ‘tilting’ of Σ. One may even expect a net decrease in energy. For the standard nullcone Ω of Schwarzschild spacetime, if we insist upon the use of EH , we observe as in the previous section that our only choice of foliation increasing to the mass M is to foliate Ω with time-symmetric slice intersections (i.e. ω “ const.). Not only is this flow highly specialized, it dictates strong restrictions on our initial choice of Σ from which to begin the flow. This is to be expected of a quasi-local energy due to its inherent sensitivity to boosts of our abstract reference frame along the flow. This is not a problem, however, if we appeal instead to mass rather than energy since boosts leave mass invariant, M 2 “ E2 ´|P~ |2 “ pE1q2 ´|P~ 1|2 “ pM 1q2. Moreover, by virtue of the Lorentzian triangle inequality (provided all vectors are timelike and either all future or all past-pointing), along any given flow the mass should always increase:

~ ~ ~ ~ M3 “ |pE1 ` E2, P1 ` P2q| ě |pE1, P1q| ` |pE2, P2q| “ M1 ` M2.

We hope therefore that, by appealing instead to a quasi local mass functional, a

23 larger class of valid flows and more generic monotonicity should arise. In this thesis, we construct such a mass functional by first finding an optimal choice of flux function for EH .

1.5.1 Overview of Thesis

In Chapter 2, we develop the necessary technical preliminaries in order to provide a canonical choice of null frame tL´,L`u P T KΣ for 2-spheres Σ in spacetime. From this, we introduce a new ﬂux function ρ and use it to construct a new quasi-local mass mpΣq. We also state and provide short discussions of our two main results, Theorem 2.1.1 and Theorem 2.1.2, motivating the notion of a (P) (or (SP))-foliation of a nullcone.

In Chapter 3 we motivate our construction of ρ and mpΣq from some of our simplest example spacetimes. Specifically, in Section 3.1, we identify an interesting relation- ship between the canonically defined connection 1-form τ of T KΣ, the shear tensor χˆ´, and the underlying null geometry of Σ in a space form M. By way of the Gauss equation, this yields a simple expression for our flux function ρ and mass mpΣq. Moreover, revisiting the standard nullcone of Schwarzschild, we show that mpΣq is chosen precisely to yield the mass M of the black hole irrespective of the choice of cross-section Σ.

In Chapter 4 we expand to general spacetimes and calculate the propagation of ρ along an arbitrary foliation of a nullcone Ω. From this, we prove Theorem 2.1.1, which indicates fairly generic monotonicity of our mass functional mpΣq. Speciﬁcally, we have monotonicity along any (P) (or (SP))-foliation.

In Chapter 5, we decompose the ﬂux ρ of any cross-section Σ ãÑ Ω in terms of

24 the data for some background foliation of our nullcone Ω. We then introduce the necessary asymptotics to explicitly formulate the limit of our mass functional mpΣq.

We ﬁnd a new notion of mass for Ω, namely M :“ limsÑ8 mpΣsq, which is completely independent of our choice of asymptotically geodesic foliation. This allows us to prove Theorem 2.1.2, yielding the Null Penrose Inequality under fairly generic conditions.

In Chapter 6, we investigate spherically symmetric spacetimes and identify a class of perturbations of the black hole exterior admitting asymptotically ﬂat nullcones of strong ﬂux decay with an (SP)-foliation. As a result, the existence of such perturbations satisfying the dominant energy condition gives the Null Penrose Inequality (1.1).

25 2

Technical Background

2.1 Preliminaries and Main Results

A spacetime pM, gq is defined to be a four dimensional smooth manifold M equipped with a metric gp¨, ¨q (or x¨, ÿ) of Lorentzian signature p´, `, `, `q. We assume that the spacetime is both orientable and time orientable, i.e. admits a nowhere vanishing timelike vector field, defined to be future-pointing. Throughout this paper, we will denote by Σ a spacelike embedding of a sphere in M with induced metric γ. It is well known that Σ has trivial normal bundle T KΣ

with induced metric of signature p´, `q. From any choice of null section L P ΓpT KΣq, we have a unique null partner section L P ΓpT KΣq satisfying xL,Ly “ 2, providing

K T Σ with a null basis tL, Lu. We also notice that any ‘boost’ tL,Lu Ñ tLa,Lau given by: 1 L :“ aL,L :“ L a a a

(for a P FpΣq a non-vanishing smooth function on Σ) gives xLa,Lay “ xL,Ly “ 2 as well. Our convention for the second fundamental form II and mean curvature H~ of Σ

26 are K ~ IIpV,W q “ DV W, H “ trΣ II for V,W P ΓpT Σq and D the Levi-Civita connection of the spacetime.

M T KΣ L

Σ H~

V,W P ΓpT Σq L

Figure 2.1: About a 2-sphere in spacetime

Deﬁnition 2.1.1. Given a choice of null basis tL,Lu, following the conventions of Sauter [25], we deﬁne the associated symmetric 2-tensors χ, χ and torsion (connection 1-form) ζ by

χpV,W q :“ xDV L,W y “ ´xL, IIpV,W qy

χpV,W q :“ xDV L, W y “ ´xL, IIpV,W qy

1 1 ζpV q :“ xD L,Ly “ ´ xD L, Ly 2 V 2 V

where V,W P ΓpT Σq.

Denoting the exterior derivative on Σ by d{, any boosted basis tLa,Lau produces the associated tensors of Deﬁnition 2.1.1:

χ V,W : D aL ,W aχ V,W ap q “ x V p q y “ p q 1 1 χ pV,W q :“ xD p Lq,W y “ χpV,W q a V a a 1 1 ζ pV q :“ xD paLq, Ly “ ζpV q ` V log |a| “ pζ ` d{ log |a|qpV q. a 2 V a 27 For a symmetric 2-tensor T on Σ its trace-free (or trace-less) part is given by

1 Tˆ :“ T ´ ptr T qγ 2 γ allowing us to decompose χ into its shear and expansion components respectively:

1 χ “ χˆ ` ptr χqγ. 2

Deﬁnition 2.1.2. We say Σ is expanding along L for some null section L P ΓpT KΣq provided that,

x´H,L~ y “ tr χ ą 0 (†) on all of Σ.

Any infinitesimal flow of Σ along L gives, by first variation of area,

dA9 “ x´H,L~ ydA “ tr χdA.

So the ﬂow is locally area expanding (dA9 ą 0) only if Σ “is expanding along L”:

Σ Ω 0 L L

Σs

Figure 2.2: An expanding nullcone

Remark 2.1.1. In Section 5.3 we will show (Lemma 5.3.1), whenever Ω is past asymptotically ﬂat inside a spacetime satisfying the dominant energy condition, a

28 consequence of the famous Raychaudhuri equation ((4.4), Section 4.2) is that any

K Σ ãÑ Ω is expanding along L P ΓpT Σq X ΓpT Ωq|Σ the past pointing null section.

So inequality (:) holds for any foliation of Ω along La where a ą 0 and we have an expanding nullcone (as illustrated in the ﬁgure above).

For Σ expanding along some L P ΓpT KΣq we are able to choose a canonical null basis tL´,L`u by requiring that our ﬂow along L´ “ aL be uniformly area expanding

(dA9 “ dA). From ﬁrst variation of area, ﬂowing along aL gives

dA9 “ ´xH,~ aLydA “ a tr χdA.

1 So we achieve a uniformly area expanding null ﬂow when a “ tr χ giving:

Deﬁnition 2.1.3. For Σ expanding along some L P ΓpT KΣq we call the associated canonical uniformly area expanding null basis tL´,L`u given by

L L´ :“ ,L` :“ tr χL tr χ

the null inﬂation basis.

We also deﬁne χ´p`q :“ ´xII,L´p`qy. It follows from the comments proceeding Deﬁ- nition 2.1.1 that

tr χ´ “ 1

tr χ` “ tr χ tr χ “ xH,~ H~ y and for V P ΓpT Σq the torsion associated to this basis is given by

1 τpV q “ xD L´,L`y “ pζ ´ d{ log tr χqpV q. 2 V

We will denote the induced covariant derivative on Σ by ∇{ .

29 Definition 2.1.4. Assuming Σ is expanding along L, for some L P ΓpT KΣq, we define the geometric flux function

1 ρ “ K ´ xH,~ H~ y ` ∇{ ¨ τ (2.1) 4 where K represents the Gaussian curvature of Σ. This allows us to deﬁne the associated quasi-local mass

3 1 1 2 2 mpΣq “ ρ 3 dA . (2.2) 2 4π Σ ´ ż ¯ For the induced covariant derivative ∇{ we denote the associated Laplacian on Σ by ∆.{

Remark 2.1.2. Whenever tr χ` “ xH,~ H~ y ‰ 0, Σ has two null inﬂation bases given

´ ` L` ` ´ by tL ,L u and t tr χ` , tr χ L u. As a result we typically have two distinct ﬂux functions 1 ρ “ K ´ xH,~ H~ y ` ∇{ ¨ τ ´ 4 1 ρ “ K ´ xH,~ H~ y ´ ∇{ ¨ τ ´ ∆{ log |xH,~ H~ y| ` 4

with associated mass functionals m˘. For the Bartnik datum αH (see Deﬁnition

3.1.1), we will see for a past pointing L that ρ´ ´ ρ` “ 2∇{ ¨ αH (Lemma 3.1.3). ~ ~ For xH, Hy ‰ 0, whenever Σ is ‘time-ﬂat’ (i.e. ∇{ ¨ αH “ 0) it follows that ρ´ “ ρ` implies m´ “ m`.

For a normal null flow off of some Σ with null flow vector L, technically the flow speed is zero since xL,Ly “ 0. In the case the Σ expands along L we define the expansion speed, σ, according to L “ σL´. We notice that σ “ tr χ. We are now ready to state our first result.

30 Theorem 2.1.1. Let Ω be a null hypersurface foliated by spacelike spheres tΣsu expanding along the null ﬂow direction L such that |ρpsq| ą 0 for each s. Then the

mass mpsq :“ mpΣsq has rate of change

1 dm p2mq 3 σ 1 1 1 ´ 2 ´ ´ ~ ~ 3 2 ´ “ 1 p|χˆ | `GpL ,L qqp xH, Hy´∆{ log |ρ| q` |ηρ| `GpL ,Nq dA ds 8π Σs ρ 3 4 2 ż ´ ¯ where

• G is the Einstein tensor for the ambient metric g

• σ “ tr χ is the expansion speed

´ 1 • ηρ :“ 2ˆχ ¨ d{ log |ρ| 3 ´ τ

1 1 1 3 2 ´ 3 ` • N :“ |d{ log |ρ| | L ` ∇{ log |ρ| ´ 4 L

If we assume therefore that our spacetime M satisﬁes the dominant energy con-

dition we can show our mass functional mpΣsq is non-decreasing for foliations tΣsu satisfying the following convexity condition:

Deﬁnition 2.1.5. Given a foliation of 2-spheres tΣsusě0 we say it is a (P)-foliation provided:

ρ ą 0

1 1 xH,~ H~ y ě ∆{ log ρ 3 4

is satisﬁed on each Σs. We say tΣsusě0 is a strict (P)-foliation or (SP)-foliation if additionally:

1 1 xH,~ H~ y “ ∆{ log ρ 3 , for s “ 0 4

1 1 xH,~ H~ y ą ∆{ log ρ 3 , for s ą 0. 4

31 So for a (P)-foliation the dominant energy condition ensures the product of the ﬁrst two terms of the integrand in Theorem 2.1.1 be non-negative. The second is

non-negative since each Σs is spacelike and the last term is non-negative from the

´ 1 dominant energy condition since xN,Ny “ 0 and xN,L y “ ´ 2 ă 0 (i.e. N is null and at every point p P Σ lies inside the same connected component of the nullcone

´ in TpM as L ). We will assume in Chapters 5 and 6 that L is past pointing. Adopting the same deﬁnitions as Mars and Soria [18] (see Section 5.3) we have our second main result:

Theorem 2.1.2. Let Ω be a null hypersurface in a spacetime satisfying the dominant energy condition that extends to past null inﬁnity. Then given the existence of a (P)-

foliation tΣsu we have

mp0q ď lim mpΣsq “: M sÑ8

(for M ď 8). If, in addition, Ω is past asymptotically ﬂat with strong ﬂux decay and

tΣsu asymptotically geodesic (see Section 5.3) then

M ď mB

where mB is the Bondi mass of Ω. Moreover, in the case that tr χ|Σ0 “ 0 we have the null Penrose inequality

|Σ0| ď mB. c 16π Furthermore, when equality holds for an (SP)-foliation we conclude that equality holds for all foliations of Ω and the data (γ, χ, tr χ and ζ) agree with some foliation of the standard nullcone of the Schwarzschild spacetime.

32 2.2 Final introductory remarks

Recalling the Hawking Energy for a closed spacelike surface Σ

|Σ| 1 ~ ~ EH pΣq “ 1 ´ xH, HydA c16π 16π Σ ´ ż ¯ we notice by the Gauss-Bonnet and Divergence Theorems that

E pΣq ρdA “ 8π H Σ |Σ| ż 4π b motivating in part why we call ρ a ﬂux function. As mentioned previously, one such ﬂux introduced by Christodoulou [9], is the ‘mass aspect function’ 1 µ “ K ´ xH,~ H~ y ´ ∇{ ¨ ζ 4

associated to an arbitrary null basis tL,Lu Ă ΓpT KΣq. Using µ in his Ph.D thesis [25],

Sauter showed the existence of ﬂows on past nullcones that render EH non-decreasing making explicit use of the fact that under a boost this mass aspect function changes via ζ according to

1 ζ Ñ ζ “ ζ ` d{ log |a| ùñ µ Ñ µ “ K ´ xH,~ H~ y ´ ∇{ ¨ ζ ´ ∆{ log |a|. a a 4

From these observations, the divergence term in (2.1) (up to a sign) is somewhat motivated by an attempt to ﬁnd a ﬂux function independent of boosts. In fact, it

follows in the case that 0 ă xH,~ H~ y “: H2 and L is past pointing, that ρ can be given in terms of the Bartnik data of Σ as

1 ρ “ K ´ xH,~ H~ y ` ∇{ ¨ α ´ ∆{ log H 4 H

(we refer the reader to Chapter 3 for deﬁnitions and proof). Moreover, in our two simplest cases, namely spherical cross-sections of the nullcone of a point in a space

33 form or the standard nullcone of Schwarzchild spacetime, the last two terms cancel identically. Interestingly, work of Wang, Wang and Zhang [30] show deep connection

between the 1-form αH ´ d{ log H and the underlying null geometry of a closed, co-

dimension 2 surface Σ. For Σ satisfying αH “ d{ log H they show for various ambient structures that Σ must be constrained to a shear-free (ˆχ “ 0) null-hypersurface of spherical symmetry. In Chapter 3 (Proposition 3.1.1) we show for a connected Σ of arbitrary co-dimension inside a space form, if Σ is expanding along some null section

L P ΓpT KΣq such that DKL9L, then it must be constrained to the nullcone of point wheneverχ ˆ “ 0. Leaning on work by Bray, Jauregui and Mars ([7]) we also ﬁnd

direct motivation for (2.1) showing that ρ arises naturally from variation of EH along null ﬂows.

34 3

Motivation

In this chapter we further develop our motivation for ρ based on an analysis of nullcones within space forms. We then provide analysis of standard nullcones in

Kruskal spacetime to motivate mpΣq and for comparison with (SP)-foliations satis-

dm fying ds “ 0. We also show how an arbitrary variation of EH building on work of Bray, Jauregui and Mars [7], points toward ρ being the optimal choice of ﬂux function in the case of null ﬂows. In this thesis we will be using the following convention to construct the Riemann curvature tensor:

RXY Z :“ DrX,Y sZ ´ rDX ,DY sZ.

From this will have need of the following versions of the Gauss and Codazzi equations:

Proposition 3.0.1. Suppose Σ is a co-dimension 2 semi-Riemannian submanifold

of Mn`1 that locally admits a normal null basis tL,Lu such that xL,Ly “ 2. Then, n ´ 2 1 pn ´ 1qK ´ xH,~ H~ y ` χˆ ¨ χˆ “ ´R ´ 2GpL,Lq ´ xR L,Ly (3.1) n ´ 1 2 LL n ´ 2 n ´ 2 1 ∇{ ¨ χˆpV q ´ χˆpV, ζ~q ` tr χζpV q ´ V trχ “ GpV,Lq ´ xR L, Ly (3.2) n ´ 1 n ´ 1 2 LV

for V P ΓpT Σq and pn ´ 1qK the scalar curvature of Σ.

35 Proof. From the Gauss equation ([21],pg.100) we have,

~ ~ ~ ~ xR{ V,W U, Sy “ xRV,W U, Sy ` xIIpV,Uq, IIpW, Sqy ´ xIIpV,Sq, IIpW, Uqy for R,R{ the Riemann tensors of Σ, M respectively and V, W, U, S P ΓpT Σq. Re- stricted to Σ the ambient metric has inverse

1 1 g´1| “ L b L ` L b L ` γ´1 Σ 2 2

so taking a trace over V,U then W, S in Σ we have:

pV,Uq pW,Sq xR{ VW U, Sy ÝÝÝÑ Ric{ pW, Sq ÝÝÝÑpn ´ 1qK

pV,Uq 1 xR U, Sy ÝÝÝÑ RicpW, Sq ´ pxR L, Sy ` xR L,Syq VW 2 LW LW

pW,Sq 1 ÝÝÝÑ R ´ 2RicpL,Lq ´ xR L,Ly. 2 LL

~ ˆ 1 ~ 1 1 1 ~ Since II “ II ` n´1 Hγ “ ´ 2 χˆL ´ 2 χLˆ ` n´1 Hγ we have

1 xH,L~ y xH,L~ y xII~ , II~ y “ pχˆ b χˆ ` χˆ b χˆq ´ pχˆ b γ ` γ b χˆq ´ pχˆ b γ ` γ b χˆq 2 2pn ´ 1q 2pn ´ 1q

1 2 ` xH,~ H~ yγ b γ n ´ 1 ´ ¯ so returning to our trace

pV,Uq,pW,Sq xII~ pV,Uq, II~ pW, Sqy ÝÝÝÝÝÝÝÑxH,~ H~ y

pV,Uq,pW,Sq 1 xII~ pV,Sq, II~ pW, Uqy ÝÝÝÝÝÝÝÑ χˆ ¨ χˆ ` xH,~ H~ y. n ´ 1

Equating terms according to the Gauss equation we have 1 1 pn ´ 1qK “ R ´ 2RicpL,Lq ´ xR L,Ly ´ χˆ ¨ χˆ ` p1 ´ qxH,~ H~ y 2 LL n ´ 1 1 n ´ 2 “ ´R ´ 2GpL,Lq ´ xR L,Ly ´ χˆ ¨ χˆ ` xH,~ H~ y 2 LL n ´ 1

36 1 having used Gp¨, ¨q “ Ricp¨, ¨q ´ 2 Rx¨, ¨y, (3.1) follows. From the Codazzi equation ([21],pg.115), for any V, W, U P ΓpT Σq,

K RVW U “ ´p∇{ V IIqpW, Uq ` p∇{ W IIqpV,Uq

where

K p∇{ V IIqpW, Uq :“ DV pIIpW, Uqq ´ IIp∇{ V W, Uq ´ IIpW, ∇{ V Uq.

So given our choice of null normal L we see that

K xDV pIIpW, Uqq,Ly “ ´V pχpW, Uqq ´ xIIpW, Uq,DV Ly 1 “ ´V pχpW, Uqq ´ xIIpW, Uq,LyxL, D Ly 2 V “ ´V pχpW, Uqq ` χpW, UqζpV q

xp∇{ V IIqpW, Uq,Ly “ ´V pχpW, Uqq ` χpW, UqζpV q ` χp∇{ V W, Uq ` χpW, ∇{ V Uq

“ ζpV qχpW, Uq ´ p∇{ V χqpW, Uq.

Therefore,

xRVW U, Ly “ p∇{ V χqpW, Uq ´ p∇{ W χqpV,Uq ´ ζpV qχpW, Uq ` ζpW qχpV,Uq.

Taking a trace over V,U we conclude, 1 RicpW, Lq ´ xR L, Ly “ ∇{ ¨ χpW q ´ W trχ ´ χpW, ζ~q ` trχζpW q 2 L,W n ´ 2 n ´ 2 “ ∇{ ¨ χˆpW q ´ W trχ ´ χˆpW, ζ~q ` trχζpW q n ´ 1 n ´ 1

and notice that GpW, Lq “ RicpW, Lq since xL,W y “ 0.

3.1 Nullcone of a point in a Space Form

In this section we spend some time studying ρ and mpΣq on cross-sections of the

n nullcone of a point in a space form. We adopt the notation as in [21] where Rν corresponds to the manifold Rn endowed with the standard inner product of index ν.

37 k n Lemma 3.1.1. Suppose Σ ãÑ Rν (k ě 2) is a connected semi-Riemannian submanifold admitting a non-trivial section ~n P ΓpT KΣq such that DK~n “ η~n for some 1-form η. Then the following are equivalent

1. p ÞÑ expp´~n|pq is constant

2. η “ 0 and xII, ´~ny “ γ

3. xII, ´~ny “ γ

n n where γ “ x¨, ¨y|Σ and exp : T Rν Ñ Rν is the exponential map.

n Proof. Choosing an origin ~o for Rν with associated position vector ﬁeld given by i n P “ x Bi P ΓpT Rν ) it follows that

expp´~n|~pq “ pP ´ ~nq|~p

where, by an abuse of notation, we have omitted the composition of canonical isome-

n n n tries T~pRν Ñ T~oRν Ñ Rν identifying ~p with P |~p. As a result, for any V P ΓpT Σq:

dpexpp´~nqqpV q “ DV pP ´ ~nq

“ V ´ DV ~n

} “ pV ´ DV ~nq ´ ηpV q~n

and we conclude that exp ˝p´~nq is locally constant (or constant when Σ is connected)

} } if and only if both V “ DV ~n for any V P ΓpT Σq and η “ 0. Since DV ~n “ V for

any V P ΓpT Σq is equivalent to ´xIIpV,W q, ~nyp“ xW, DV ~nyq “ xV,W y for any V,W P ΓpT Σq we have that 1. ðñ 2. 2. ùñ 3. is trivial. To show 3. ùñ 2. we start by taking any U, V, W P ΓpT Σq so that the Codazzi equation gives

xp∇{ V IIqpW, Uq, ~ny “ xp∇{ W IIqpV,Uq, ~ny

38 where

K xp∇{ V IIqpW, Uq, ~ny :“ xDV pIIpW, Uqq ´ IIp∇{ V W, Uq ´ IIpW, ∇{ V Uq, ~ny

“ ´p∇{ V γqpW, Uq ´ xIIpW, Uq,DV ~ny

“ ηpV qxW, Uy

and therefore ηpV qxW, Uy “ ηpW qxV,Uy. Taking a trace over V,U we conclude that ηpW q “ kηpW q so that k ě 2 forces η “ 0 as desired.

n The Hyperquadrics of Rν correspond to the complete, totally umbilic hypersurfaces HC of constant curvature C (provided C ‰ 0) given by

n HC :“ t~v P Rν |x~v,~vy “ Cu

where C runs through all values in R. When C “ 0, Ω “ H0 is the collection of all null geodesics emanating from the origin called the nullcone centered at the origin.

Consequently Ω ` ~p corresponds to the nullcone at the point ~p. Similarly for a space form M we will deﬁne the nullcone of a point p P M as the collection of all null geodesics emanating from p.

Proposition 3.1.1. Suppose Σk ãÑ Mn´1 (k ě 2) is a connected semi-Riemannian submanifold of a space form M. Suppose that Σ is expanding along some null section

L satisfying DKL “ ζL for some 1-form (or torsion) ζ. Then the following are equivalent

kL 1. p ÞÑ expp´ tr χ |pq is constant

2. τ :“ ζ ´ d{ log tr χ “ 0 and χˆ “ 0

3. χˆ “ 0

where χ :“ ´xL, IIy.

39 n Proof. If M has constant curvature C ‰ 0 we find a Hyperquadric HC of Rν (for some ν) of the same dimension and index as M. It’s a well known fact that M and HC have isometric semi-Riemannian coverings ([21],pg.224 Theorem 17) which we identify and denote by O. As a result, for any q P Σ Ă M we find a ~q P HC with isometric neighborhoods. Moreover, we find an open set Uq Ă Σ of q which is

K isometric onto some V~q Ă HC . Without loss of generality we will also identify T pUqq

K n ¯ and T pV~qq. Denoting the ambient connection on Rν by D and the unit normal of n´1 n ~ ´ L HC Ă Rν by N we conclude, for the null section L “ tr χ , with the decomposition

¯ K ´ ´ ~ ~ ¯ ´ ~ ~ DV L “ τpV qL `xN, NyxDV L , NyN. So given that all Hyperquadrics are totally

¯ ´ ~ ´ ¯ K ´ ´ umbilic it follows that xDV L , Ny9xV,L y “ 0 and therefore DV L “ τpV qL . 1 We wish to show 3. ùñ 1. From the hypothesis we have that χ “ k tr χγ ´ ´ n so it follows that kχ “ ´xkL , IIy “ γ and Lemma 3.1.1 applies for V~q Ă Rν . n We conclude that V~q is contained inside the nullcone of a point ~o P Rν , where

´ n ~o “ expn,νp´kL q|V~q , and every ~p P V~q is connected to ~o by a null geodesic in Rν

´ kL K along kL |~p “ tr χ P T~p V~q Ă T~pHC . Since HC is complete and totally umbilic these ~p ˇ ˇ null geodesics mustˇ remain within HC . Up to a possible shrinkage of V~q we may lift a neighborhood of the geodesic ~q Ñ ~o to a neighborhood of some null geodesicq ˜ Ñ o˜

in O concluding that the isometric image Vq˜ of V~q contracts too ˜ along null geodesics. Since M is complete the null geodesicq ˜ Ñ o˜ in turn gives rise to a null geodesic

q Ñ o in M and up to an additional shrinkage we conclude that Uq contracts along null geodesics onto o: In fact our argument shows that the union of all points in Σ that get transported to o must form an open subset of Σ. Conversely, if any point in Σ gets transported to a point other than o the same follows for a neighborhood around that point in Σ. By connectedness, all of Σ must be transported to o along null geodesics as desired.

´ kL For 1. ùñ 3. we take a null geodesic from q P Σ along kL “ tr χ to the focal

40 HC

N~ Uq V~q ~q q

kL M tr χ

~o Figure 3.1: Nullcones at ~o and o

point, at say, o P M. Similarly as before this gives rise to a tubular neighborhood

around some null geodesic ~q Ñ ~o in HC within which V~q is contracted along null geodesics onto ~o. Since HC is totally umbilic V~q is transported to ~o along null

n ´ geodesics in Rν forming part of the nullcone at ~o “ expn,νp´kL q|V~q . Lemma 3.1.1

´ applies once again and we conclude that kχ “ γ impliesχ ˆ “ 0 on V~q hence on Uq (since they have isometric neighborhoods). Since q was arbitrary chosen the result follows.

Once again 2. ùñ 3. is trivial. To show 3. ùñ 2. we have similarly as in Lemma 3.1.2 from the Codazzi equation for Σ ãÑ M and M of constant curvature that:

τpV qxW, Uy “ τpW qxV,Uy so that a trace over V,U yields again τpW q “ kτpW q and therefore τ “ 0.

For any connected, co-dimension 2 surface admitting a null section L P ΓpT KΣq,

1 K since xDV L,Ly “ 2 V xL,Ly “ 0, it necessarily follows that D L “ ζL for some associated 1-form ζ. In particular, Σ will be contained inside the nullcone of a point

41 inside a space form M if we’re able to ﬁnd a null section L along which Σ is expanding

and shear-free. Along such L it follows from Lemma 3.1.1 for C “ 0 and Proposition 3.1.1 for C ‰ 0 that τ “ 0. So for Lorentzian space forms of dimension-4 (i.e. ‘Minkowski spacetime’ for C “ 0, ‘de Sitter spacetime’ for C ą 0 and ‘anti-de Sitter 1 ~ ~ spacetime’ for C ă 0) it follows from (3.1) that Σ has ﬂux ρ “ K ´ 4 xH, Hy “ C. When Σ is a 2-sphere, by the Gauss-Bonnet Theorem, we conclude that

3 |C| |Σ| 2 mpΣq “ |E pΣq| “ . H 2 4π ´ ¯ The reader may be wondering why the need for the divergence term in (2.1) when it

vanishes altogether. We take as our ﬁrst hint the fact that vanishing τ “ ζ ´d{ log tr χ is characteristic of spherical cross-sections of Ω which subsequently may obscure it’s contribution. In the paper by Wang, Wang and Zhang ([30] Theorem 3.13, Theorem

5.2) the authors prove τ “ 0 to be suﬃcient in spacetimes of constant curvature to constrain a closed, co-dimension 2 surface Σ to a shear-free null hypersurface of spherical symmetry. Proof follows from the following Lemma and Proposition 3.1.1 when Σ is a 2-sphere:

Lemma 3.1.2. Suppose Σ is a spacelike 2-sphere expanding along some L inside a

space form M. Suppose also DKL “ ζL for some 1-form ζ then

τ :“ ζ ´ d{ log tr χ “ 0 ùñ χˆ “ 0.

Proof. As used in Proposition 3.1.1 to prove the implication in the opposite direction,

´ L K ´ ´ we start with the Codazzi equation. For L “ tr χ we recall that D L “ τL “ 0 and tr χ´ “ 1 so we have:

´ K ´ x∇{ V IIpW, Uq,L y “ xDV pIIpW, Uqq ´ IIp∇{ V W, Uq ´ IIpW, ∇{ V Uq,L y

´ K ´ “ ´p∇{ V χ qpW, Uq ´ xIIpW, Uq,DV L y

´ “ ´p∇{ V χ qpW, Uq 42 ´ “ ´p∇{ V χˆ qpW, Uq

K ´ ´ 0 “ xRVW U, L y “ x´∇{ V IIpW, Uq ` ∇{ W IIpV,Uq,L y

´ ´ “ p∇{ V χˆ qpW, Uq ´ p∇{ W χˆ qpV,Uq.

Taking a trace over V,U this implies ∇{ ¨ χˆ´ “ 0. Since Σ is a topological 2-sphere it’s a well known consequence of the Uniformization Theorem (see for example [9]) that the divergence operator on symmetric trace-free 2-tensors is injective so that

χˆ “ tr χχˆ´ “ 0.

Deﬁnition 3.1.1. We say a 2-sphere Σ is admissible if

xH,~ H~ y “ H2 ą 0.

In the case that Σ is admissible we’re able to construct the orthonormal frame ﬁeld H~ te “ ´ , e u r H t for et future pointing. The associated connection 1-form is given by

αH pV q :“ xDV er, ety.

From the following known Lemma ([30]), Proposition 3.1.1 and Lemma 3.1.2, a necessary and suﬃcient condition for an admissible sphere Σ to be constrained to the past(future) light-cone of a point in a space form is given by αH “ ˘d{ log H:

Lemma 3.1.3. For Σ admissible

τ “ ˘αH ´ d{ log H

from which we conclude that

1 ρ “ K ´ xH,~ H~ y ˘ ∇{ ¨ α ´ ∆{ log H ¯ 4 H

where `{´ indicates whether L´ is past/future pointing.

43 ~ 1 2 Proof. Since ´H “ 2 ptr χL ` tr χLq we see H “ tr χ tr χ so that the inverse mean curvature vector is given by

H~ 1 L L I~ :“ ´ “ ` . H2 2 tr χ tr χ ´ ¯ As a result,

αH pV q “ xDV er, ety e “ xD r , He y V H t 1 L L 1 “ xD ` , ¯ ptr χL ´ tr χLqy V 2 tr χ tr χ 2 ´ ¯ 1 L tr χL L “ ˘ xD , tr χLy ´ xD ,H2 y 4 V tr χ V H2 tr χ ´ ¯ 1 L L tr χL “ ˘ xD , tr χLy ` xD pH2 q, y 4 V tr χ V tr χ H2 ´ ¯ 1 L “ ˘ 2xD , tr χLy ` 2V log H2 4 V tr χ ´ ¯ “ ˘ ζpV q ´ V log tr χ ` V log H ´ ¯

Wang, Wang and Zhang ([30] Theorem B’) also extend their result to expand-

ing, co-dimension 2 surfaces Σ in n-dimensional Schwarzschild spacetime (n ě 4).

Namely, that any such Σ satisfying αH “ d log H must be inside a shear-free null hypersurfaces of symmetry, or the ‘standard nullcone’ in this geometry. So with the hopes of further illuminating modiﬁcation of EH by way of the ﬂux function ρ we move on to this setting in dimension 4.

44 3.2 Standard Nullcones of Schwarzschild Revisited

2 Recall the Kruskal spacetime P ˆr S with associated quadratic form:

ds2 “ 2F prqdvdu ` r2pdϑ2 ` psin ϑq2dϕ2q.

A standard past nullcone of Schwarzchild spacetime Ω is the hypersurface given by

ﬁxing the coordinate at v “ v0 ą 0. Denoting the gradient of a function f by Df

Bu we recognize the null vector ﬁeld F “ Dv restricts to Ω as both a tangent (since K Bupvq “ 0) and normal (since Dv K T Ω) vector ﬁeld. It follows that Dv P T Ω X T Ω and the induced metric on Ω degenerate, so Ω is an example of a null hypersurface.

1 2 Bu From the identity DDf Df “ 2 D|Df| we see F is geodesic and Ω is realized as the past light cone of a section of the event horizon at r “ 2M (see Figure 1.4).

4M Bu 4M ru 4M v 4M v Setting L “ Dp4M log vq “ v F we see Lprq “ v F “ v g1prqF “ v 4M “ 1. We conclude that r restricts to an aﬃne parameter along the geodesics generating Ω and

therefore any cross-section Σ can be given as a graph over S2 in Ω with graph function

ω “ r|Σ. We extend ω to the rest of Ω by assigning Lpωq “ 0 and to a neighborhood

of Ω by assigning Bvω “ 0. From the canonical, homothetic embedding onto the

2 2 2 2 leaves S ãÑ P ˆr S we obtain the lifted vector ﬁelds V P LpS q Ă ΓpT pP ˆr S qq 2 such that xBupBvq,V y “ rBvpBuq,V s “ 0. It follows that LpS q|Σr “ ΓpT Σrq (for ˜ Σr :“ tr “ const., v “ v0u) and therefore V :“ V ` V ωL P ΓpT Σq since

V˜ pr ´ ωq “ ´V ω ` V ωLprq “ ´V ω ` V ω “ 0.

Since Σ “ t4M log v “ 0, r “ ωu we have L,Dpr ´ ωq P ΓpT KΣq are linearly v0 independent so that L “ aL ` bDpr ´ ωq and we wish to solve for a, b. Also, we

Bv Bu v have Dr “ ru F ` rv F “ 4M pBv ` rvLq and Dω “ ∇ω for ∇ the induced covariant

derivative on Σr giving

v bv L “ pa ` b 0 r qL ` 0 B ´ b∇ω. 4M v 4M v 45 v0 v0 For simplicity we set A “ a ` b 4M rv and solve for A, b in L “ AL ` bp 4M Bvq ´ b∇ω: v 2 “ xL, Ly “ bx 0 B ,Ly “ b 4M v 0 “ xL, Ly “ 2Ab ` b2|∇ω|2 “ 4pA ` |∇{ ω|2q having used ∇ω “ ∇{ ω ´ |∇{ ω|2L in the second equality. We conclude that v L “ 0 B ´ |∇{ ω|2L ´ 2p∇{ ω ´ |∇{ ω|2Lq 2M v v “ 0 B ` |∇{ ω|2L ´ 2∇{ ω. 2M v

Lemma 3.2.1. Given a cross-section Σ:“ tr “ ωu of the standard nullcone Ω, given by Ω:“ tv “ v0u in Kruskal spacetime, we have for the generator L satisfying Lprq “ 1 that

xV,˜ W˜ y “ ω2pV,˜ W˜ q

1 χpV,˜ W˜ q “ xV,˜ W˜ y ω 2 tr χ “ ω 1 2M χpV,˜ W˜ q “ p1 ´ ` |∇{ ω|2qxV,˜ W˜ y ´ 2HωpV,˜ W˜ q ω ω 2 2M tr χ “ 1 ´ ´ ω2∆{ log ω ω ω ´ ¯ ζpV˜ q “ ´V˜ log ω

2M ρ “ ω3 where V,˜ W˜ P ΓpT Σq and p¨, ¨q the round metric on S2.

Proof. The ﬁrst identity follows trivially from the metric gK upon restriction to Σ. From the Koszul formula and the fact that L is geodesic it is a straight for exercise

Lprq 1 ω to show that DV˜ L “ r V |Σ “ ω V . Denoting the Hessian of ω on Σ by H we

46 therefore have ˜ ˜ ˜ χpV, W q “ xDV˜ L, W y 1 “ xV,W y ω

v0 2 χpV,˜ W˜ q “ xD ˜ B , W˜ y ` |∇{ ω| xD L,W y ´ 2xD ˜ ∇{ ω, W˜ y 2M V v V V v “ 0 xD B ,W y ` V ωxD B ,W y ` W ωxD B ,Ly ` V ωW ωxD B ,Ly 2M V v L v V v L v ´ ¯ ` |∇{ ω|2χpV,W q ´ 2HωpV,˜ W˜ q

v r 1 “ 0 v xV,W y ` |∇{ ω|2xV,W y ´ 2HωpV,˜ W˜ q 2Mω ω where in the forth line we use the Koszul formula to evaluate the ﬁrst term and metric compatibility to show the last three terms vanish. We have

vr v u 1 F g 1 2M v “ “ “ p1 ´ q 2Mr 2Mr g1prq 2Mr 4M r r so the second and forth identities follow upon restriction to Σ. The third identity is simply a trace over Σ of the second. Similarly the ﬁfth follows our taking a trace of the forth and employing the fact that

1 ∆{ ω ´ |∇{ ω|2 “ ω∆{ log ω. ω

For ζ: 1 ζpV˜ q “ xD ˜ L,Ly 2 V 1 v “ xV, 0 B ` |∇{ ω|2L ´ 2∇{ ωy 2ω 2M v 1 “ ´ xV, ∇{ ωy ω 1 “ ´ xV,˜ ∇{ ωy ω 47 “ ´V˜ log ω.

1 From the ﬁrst identity we conclude that Σ has Gaussian curvature K “ ω2 ´ ∆{ log ω and therefore 2M xH,~ H~ y “ tr χ tr χ “ 4pK ´ q. ω3

Since ζ ´ d{ log tr χ “ ´d{ log ω ´ p´d{ log ωq “ 0 on Σ we have

1 2M ρ “ K ´ xH,~ H~ y “ . 4 ω3

It follows, in Schwarzschild spacetime, that all foliations to the past of a section of

the event horizon (r “ 2M) inside the standard nullcone (v “ v0) are (SP)-foliations since Σ “ tr “ ωu satisﬁes

1 1 1 2M xH,~ H~ y ´ ∆{ log ρ “ p1 ´ q ą 0 ðñ ω ą 2M. 4 3 ω2 ω

Moreover, equality is reached only at the horizon itself indicating physical signiﬁcance to our property (P). One of the motivating factors for our choice of mass functional comes from our ability, in this special case, to extract the exact mass content within

any Σ Ă Ω:

3 2 3 1 1 2M 2 2 1 1 p2Mq 3 2 3 2 2 mpΣq “ p 3 q dA “ 2 ω dS “ M. 2 4π Σ ω 2 4π Σ ω ´ ż ¯ ´ ż ¯ Lemma 3.2.2. Suppose Σ is a compact Riemannian manifold, then for any f P FpΣq

3 2 2 |f| f 3 dA “ inf ψ2dA dA ψą0 d ψ ´ ż ¯ ´ ż ż ¯

48 3 Proof. by choosing ψ “ |f| ` for some ą 0 it’s a simple veriﬁcation that

3 2 2 |f| |f| 3 2 2 p|f| ` q dA ě ψ dA dA ě inf ψ dA dA d ψ ψą0 d ψ ´ ż ¯ ż ż ż ż so by the Dominated Convergence Theorem

3 3 2 2 2 2 |f| f 3 dA “ lim p|f| ` q 3 dA ě inf ψ2dA dA. Ñ0 ψą0 d ψ ´ ż ¯ ´ ż ¯ ż ż

We show the inequality holds in the opposite direction from H¨older’s inequality

2 2 2 f 2 2 3 |f| 3 f 3 dA “ p q 3 ψ 3 dA ď ψ2dA dA ψ d ψ ż ż ´ ż ¯ ´ ż ¯

3 where the result follows from raising both sides to the 2 power and taking an inﬁmum over all ψ ą 0.

So given any 2-sphere with non-negative ﬂux ρ ě 0 in an arbitrary spacetime, deﬁning

ψ 1 ψ2dA ρ EH pΣq :“ 8π 4π ψ dA, we conclude that ş b ş ψ mpΣq “ inf EH pΣq ď EH pΣq ψą0 as desired. Recalling our use of H¨older’sinequality in the proof of Lemma 3.2.2, we see that mpΣq “ EH pΣq if and only if ρ is constant on Σ. So for Σ :“ tr “ ωu Ă Ω, where Ω is the standard nullcone in Schwarzschild spacetime, we see that mpΣq underestimates the Hawking energy EH pΣq with equality only if ρ hence ω is constant. Namely the round spheres within ‘time-symmetric’ slices given by t “ const ą 0 or

v equivalently u “ const ą 0 (so that v “ v0 implies r “ const) as expected from Sauters work ([25], Lemma 4.4).

49 3.3 Variation of EH

In this section we spend some time studying arbitrary normal variations of EH on admissible spheres following work of Bray, Jauregui and Mars ([7]). The authors of

~ ~K [7] consider ‘uniform area expanding flows’ according to the flow vector Bs “ I ` βI so we first spend some time extending their Plane Theorem to incorporate arbitrary

~ ~K normal ﬂows Bs “ αI ` βI . Subsequently, we show that an arbitrary null ﬂow is obstructed from monotonicity by a term with direct dependence upon ρ in analogy with the variation found by Christodoulou regarding the mass aspect function µ (see, for example, [25] Theorem 4.1). We hope that this points towards ρ being poten-

tially closer to an optimal choice of ﬂux for the Hawking Energy EH in capturing the ambient spacetime. The following proposition is known (see [5], Lemma 4), we provide proof to com- plement the Plane Theorem of [7] and to establish the result in the notation introduced in Deﬁnition 3.1.1.

Proposition 3.3.1 (Plane Derivation). Suppose Ω – I ˆ S2, for some interval I Ă R, is a hypersurface of M and α ‰ 0 is a smooth function on Ω. Assuming

the existence of a foliation of Ω by admissible spheres tΣsu according to the level set

~ H~ function s :Ω Ñ R whereby Bs|Σs “ αI “ ´α H2 then we have

1 dEH 1 2 “ pα¯ ´ αqp2Ks ´ H ´ 2∆{ log HqdA |Σs| ds Σs 2 p16πq3 ż b ˆ 2 ˆ 2 2 2 ` αp2Gpet, etq ` |IIr| ` |IIt| ` 2|αH | ` 2|∇{ log H| qdA ż ~ for IIrptq “ xII, erptqy where erptq is given in Deﬁnition 2.1.5.

Before proving Proposition 3.3.1 we will ﬁrst need to ﬁnd the second variation of area:

50 Lemma 3.3.1.

α xH,D~ H~ y “ ´α|II |2 ´ αRic pe , e q ´ H∆{ p q Bs r Ω r r H

Proof. In this lemma we temporarily denote the induced covariant derivative on Ω by ~ ~ D noticing that xH,DBs Hy calculates the same quantity as if the ambient connection was used. Taking a local basis tX1,X2u along the foliation we deﬁne γij :“ xXi,Xjy giving rise to the inverse metric γij. For any V P ΓpT Ωq parallel to the leaves of

the foliation (i.e. V |Σs “ ΓpT Σsq we have rBs,V ss “BspV psqq ´ V pBspsqq “ 0 giving rBs,V s|Σs P ΓpT Σsq. As such

~ ~ ~ ij ~ xH,DBs Hy “ xH,DBs pγ IIijqy

ik jl ~ ~ “ ´γ γ pxDBs Xk,Xly ` xDBs Xl,XkyqxIIij, Hy

ij ~ ` γ xDBs pDXi Xj ´ ∇{ Xi Xjq, Hy

ik jl ~ ~ ~ “ ´2γ γ pxrBs,Xks,Xly ´ xIIkl, BsyqxIIij, Hy

ij ~ ` γ px´RBsXi Xj ` DXi DBs Xj ` DrBs,XisXj ´ D∇{ X Bs, Hy Xi j

where we used the fact that rBs, ∇{ Xi Xjs|Σs P T Σs to get the last term. From the fact ~ that Bs “ αI it follows that

ik jl ~ ~ ~ ik ~ 2γ γ pxrBs,Xks,Xly ´ xIIkl, BsyqxIIij, Hy “ 2γ xDrBs,XksXi, Hy α ´ 2p´ qγikγjlxII~ , H~ yxII~ , H~ y H2 kl ij

ik ~ 2 “ 2γ xDrBs,XksXi, Hy ` 2α|IIr|

ij ~ ij ~ γ xDXi DBs Xj ` DrBs,XisXj, Hy “ γ xDXi rBs,Xjs ` DXi DXj Bs ` DrBs,XisXj, Hy

ij ~ “ γ xrXi, rBs,Xjss ` DrBs,Xj sXi ` DrBs,XisXj ` DXi DXj Bs, Hy

ij ~ ij ~ “ 2γ xDrBs,XisXj, Hy ` γ xDXi DXj Bs, Hy

having used the fact that rXi, rBs,Xjss P ΓpT Σsq to get the ﬁnal equality. This allows

51 us to simplify to

~ ~ 2 ~ ij ~ xH,DBs Hy “ ´2α|IIr| ` RicΩpBs, Hq ` γ xpDXi DXj Bs ´ D∇{ X Bsq, Hy. Xi j

α α Given also that Bs “ H er we see xBs,DX ery “ 2H Xxer, ery “ 0 for any X P ΓpT Σsq so we simplify the last two terms ~ xDXi DXj Bs, Hy “ ´HXiXjxBs, ery ` HXixBs,DXj ery ` HxDXj Bs,DXi ery α “ ´HX X p q ` αxD e ,D e y i j H Xj r Xi r α “ ´HX X p q ` αγklxD e ,X yxD e ,X y i j H Xj r k Xi r l α “ ´HX X p q ` α| II |2 i j H r ~ xD∇{ X Bs, Hy “ ´H∇{ X XjxBs, ery ` HxBs,D∇{ X ery Xi j i Xi j α “ ´H∇{ X p q Xi j H

and the result follows after we collect all the terms and take a trace over i, j.

Proof. (Proposition 3.3.1) The proof follows in parallel to the Plane Theorem of [7] (Theorem 2.1). From the ﬁrst variation of area formula:

~ dA9 s “ ´xH, BsydAs “ αdAs

9 ùñ |Σs| “ |Σs|α¯psq.

So variation of the Hawking Energy gives:

dEH d |Σs| 2 “ 16π ´ H dAs ds ds dp16πq3 ´ ´ ż ¯¯

|Σs| 1 2 ~ ~ 2 “ α¯ 16π ´ H dAs ´ 2 xH,DBs HydAs ´ αH dAs dp16πq3 2 ´ ´ ż ¯ ż ż ¯

|Σs| 1 2 2 α “ α¯p2Ks ´ H qdAs ` 2α|IIr| ` 2αRicΩper, erq ` 2H∆{ p qdA dp16πq3 2 H ´ ż ż 52 ´ αH2dA ż ¯ where we used the Gauss-Bonnet Theorem and Proposition 3.3.1 respectively to get the ﬁrst and second integrands of the last line. As in [7] we now trace the Gauss

equation for Σs in Ω twice over Σs to get

2 2 2RicΩper, erq “ S ´ 2Ks ` H ´ |IIr|

for S the scalar curvature of Ω. We then trace the Gauss equation for Ω in M twice over Ω to conclude

2 2 S “ 2Gpet, etq ` 2|αH | ` |IIt|

K since et P ΓpT Ωq. Substitution into our variation of EH therefore gives us after some algebraic manipulation that

dEH |Σs| 1 2 “ pα¯ ´ αqp2Ks ´ H qdAs ds dp16πq3 2 ´ ż 1 α ` 2Gpe , e q ` p|II |2 ´ H2q ` |II |2 ` 2|α |2 ` 2H∆{ p qdA . t t r 2 t H H s ż ¯ First performing an integration by parts on the last term

α α H∆{ p qdA “ p∆{ Hq dA H H ż ż

∆{ H 2 followed by the identity H “ ∆{ log H ` |∇{ log H| we obtain the ﬁrst line of the variation in Proposition 3.3.1. The second follows from the fact that 1 |II |2 “ |IIˆ |2 ` H2 r r 2 ˆ IIt “ IIt

We refer the reader to [7] (Theorem 2.2) for proof of the Cylinder Theorem:

53 ~K Proposition 3.3.2. Under the same hypotheses as in Proposition 3.3.1 with Bs “ βI

~K et for some smooth function β ‰ 0 on Ω and I “ H we have

1 dEH ˆ ˆ “ βp2Gpet, erq ` 2xIIr, IIty ` 4αH p∇{ log Hq ` 2∇{ ¨ αH qdAs. |Σs| ds Σs p16πq3 ż b

The full variation of EH is known from ([5], Lemma 3), we are now in a position to show it within our context:

Corollary 3.3.2.1. Under the same hypotheses as Proposition 3.3.1 and 3.3.2 with

~ ~K Bs “ αI ` βI

1 dEH 1 2 “ pα¯ ´ αqp2Ks ´ H ´ 2∆{ log HqdAs |Σs| ds Σs 2 p16πq3 ż b ˆ 2 ˆ 2 2 2 ` αp2Gpet, etq ` |IIr| ` |IIt| ` 2|αH | ` 2|∇{ log H| qdAs żΣs ˆ ˆ ` βp2Gpet, erq ` 2xIIr, IIty ` 4αH p∇{ log Hq ` 2∇{ ¨ αH qdAs żΣs

Proof. As in [7] (Theorem 1.13) variation of EH is achieved by summing the contributions from Propositions 3.3.1 and 3.3.2 since the variation of the area form and the mean curvature vector are known to be R-linear over the ﬂow vector decomposition.

Subsequently, we achieve an arbitrary past(future) directed null ﬂow by setting

~ ~K α “ ¯β ą 0 in Corollary 3.3.2.1 giving Bs “ αpI ¯ I q and

1 dEH 1 2 “ pα¯ ´ αqp2Ks ´ H ˘ 2∇{ ¨ αH ´ 2∆{ log HqdAs |Σs| ds Σs 2 p16πq3 ż b ˆ ˆ 2 2 ` αp2Gpet, et ¯ erq ` |IIr ¯ IIt| ` 2|αH ¯ ∇{ log H| qdAs. żΣs

54 It follows in an energy dominated spacetime that the only obstruction to a non- decreasing Hawking energy is the integrand

1 pα¯ ´ αqp2K ´ H2 ˘ 2∇{ ¨ α ´ 2∆{ log Hq “ 2pα¯ ´ αqρ . s 2 H ¯

dEH In particular ds ě 0 for any foliation where ρ is constant on each Σs, moreover, since m¯pΣsq “ EH pΣsq in this case (provided also ρ¯ ě 0) we have monotonicity of our quasi local mass as well. We extend beyond this case in the next section with the proof of Theorem 2.1.1.

55 4

Propagation of ρ

In this chapter we work towards proving Theorem 2.1.1 by finding the propagation of our flux function ρ along an arbitrary null flow.

4.1 Setup

We adopt the same setup as in [18] which we summarize here in order to introduce our notation:

Suppose Ω is a smooth connected, null hypersurface embedded in pM, x¨, ¨yq. Here we let L be a smooth, non-vanishing, null vector ﬁeld of Ω, L P ΓpT Ωq. It’s a well known fact (see, for example, [11]) that the integral curves of L are pre-geodesic so

we’re able to ﬁnd κ P FpΩq such that DLL “ κL. We assume the existence of an embedded sphere Σ in Ω such that any integral curve of L intersects Σ precisely once. As previously used, we will refer to such Σ

as cross-sections of Ω. This gives rise to a natural submersion π :Ω Ñ Σ sending

L L p P Ω to the intersection with Σ of the integral curve γp of L for which γp p0q “ p.

Given L and a constant s0 we may construct a function s P FpΩq from Lpsq “ 1

L and s|Σ “ s0. For q P Σ, if ps´pqq, s`pqqq represents the range of s along γq then 56 Ω

Σ L s1

L Σs0

Σs2

Σs3

Figure 4.1: Cross-sections: s3 ă s2 ă S´ ă s0 ă S` ă s1

letting S´ “ supΣ s´ and S` “ infΣ s` we notice that the interval pS´,S`q is non-

empty. Given that Lpsq “ 1 the Implicit Function Theorem gives for t P pS´,S`q

2 that Σt :“ tp P Ω|sppq “ tu is diﬀeomorphic to S through Σ. For s ă S´ or s ą S`, in the case that Σs is non-empty, although smooth it may no longer be connected.

We have that the collection tΣsu gives a foliation of Ω.

We construct another null vector ﬁeld L by assigning at every p P Ω L|p P TpM be the unique null vector satisfying xL,Ly “ 2 and xL, vy “ 0 for any v P TpΣsppq. As before each Σs is endowed with an induced metric γs, two null second fundamental forms χ “ ´xII~ ,Ly and χ “ ´xII~ ,Ly as well as the connection 1-form (or torsion)

1 ζpV q “ 2 xDV L,Ly. We will need the following known result ([25]):

Lemma 4.1.1. Given V P ΓpT Σsq,

• DV L “ ~χpV q ` ζpV qL

• DV L “ ~χpV q ´ ζpV qL

~ • DLL “ ´2ζ ´ κL

where, given V,W P ΓpT Σq, the vector ﬁelds ζ,~ ~χpV q are uniquely determined by xζ,~ V y “ ζpV q and x~χpV q,W y “ χpV,W q.

57 Proof. It suﬃces to check all identities agree by taking the metric inner product

with vectors L,L and an extension W satisfying W |Σs P ΓpT Σsq keeping in mind

that rL,W s|Σs P ΓpT Σsq. We leave this veriﬁcation to the reader.

For any cross-section Σ of Ω and v P TqpΣq we may extend v along the generator

L γq according to

V9 psq “ DV psqL

V p0q “ v.

Since x P TpΩ ðñ xL|p, xy “ 0 we see from the fact that

1 pxV ps9q,Lyq “ xD L,Ly ` κxV psq,Ly “ V psqxL,Ly ` κxV psq,Ly “ κxV psq,Ly, V psq 2 and xV p0q,Ly “ 0, that infact xV psq,Ly “ 0 for all s. As a result, any section W P ΓpT Σq is extended to all of Ω satisfying rL,W s “ 0. We also notice along each

generator 0 “ rL,W ss “ LpW sq “ W9 s such that W s|Σ “ 0 forces W s “ 0 on all

of Ω. We conclude that W |Σs P ΓpT Σsq and denote by EpΣq Ă ΓpT Ωq the set of such extensions oﬀ of Σ along L. We also note that linear independence is preserved along generators by standard uniqueness theorems allowing us to extend basis ﬁelds

tX1,X2u Ă ΓpT Σq oﬀ of Σ as well.

4.2 The Structure Equations

We will need to propagate the Christoﬀel symbols with the known result ([25]):

Lemma 4.2.1. Given U, V, W P EpΣq,

xrL, ∇{ V W s,Uy “ p∇{ V χqpW, Uq ` p∇{ W χqpV,Uq ´ p∇{ U χqpV,W q

where ∇{ the induced covariant derivative on each Σs.

58 Proof. Starting from the Koszul formula

2x∇{ V W, Uy “ V xW, Uy`W xV,Uy´UxV,W y´xV, rW, Usy`xW, rU, V sy`xU, rV,W sy we apply L to the left hand term to get

Lx∇{ V W, Uy “ xDL∇{ V W, Uy ` x∇{ V W, DLUy “ xrL, ∇{ V W s,Uy ` 2χp∇{ V W, Uq and to the right keeping in mind that rV,W s P EpΣq

L V xW, Uy ` W xV,Uy ´ UxV,W y ´ xV, rW, Usy ` xW, rU, V sy ` xU, rV,W sy ´ ¯ “ VLxW, Uy ` WLxV,Uy ´ ULxV,W y ´ 2χpV, rW, Usq

` 2χpW, rU, V sq ` 2χpU, rV,W sq

“ 2 V χpW, Uq ` W χpV,Uq ´ UχpV,W q ´ χpV, rW, Usq ´ ` χpW, rU, V sq ` χpU, rV,W sq ¯

“ 2 p∇{ V χqpW, Uq ` p∇{ W χqpV,Uq ´ p∇{ U χqpV,W q ` 2χp∇{ V W, Uq . ´ ¯ Equating terms according to the Koszul formula the result follows upon cancellation of the term χp∇{ V W, Uq.

Lemma 4.2.2. For S, U, V, W P EpΣq,

xrL, R{ VW Us,Sy “ p∇{ W ∇{ V χqpU, Sq ´ p∇{ V ∇{ W χqpU, Sq ` p∇{ W ∇{ U χqpV,Sq

´ p∇{ V ∇{ U χqpW, Sq ` p∇{ V ∇{ SχqpW, Uq ´ p∇{ W ∇{ SχqpV,Uq

where R{ the induced Riemann curvature tensor on Σs.

Proof. We notice any f P FpΣq can be extended to all of Ω by imposing Lpfq “ 0 along generators. As such fV P EpΣq and

rL, R{ fV W Us “ rL, R{ V fW Us “ rL, R{ VW fUs “ rL, fR{ VW Us “ frL, R{ VW Us.

59 Within EpΣq we conclude that both xrL, R{ VW Us,Sy and the right hand side of the

identity restricts to 4-tensors pointwise on each Σs. It therefore suﬃces to prove

the identity pointwise. In particular, for any v, w P TqΣs we extend to vector ﬁelds

V,W P EpΣq such that ∇{ V W |q “ 0. The Riemann tensor on Σs reads

xR{ VW U, Sy “ x∇{ rV,W sU, Sy ´ x∇{ V ∇{ W U, Sy ` x∇{ W ∇{ V U, Sy

“ x∇{ rV,W sU, Sy ´ V x∇{ W U, Sy ` x∇{ W U, ∇{ V Sy ` W x∇{ V U, Sy

´ x∇{ V U, ∇{ W Sy

so applying L to the terms on the right assuming restriction to q P Σs we have

L x∇{ rV,W sU, Sy ´ V x∇{ W U, Sy ` x∇{ W U, ∇{ V Sy ` W x∇{ V U, Sy ´ x∇{ V U, ∇{ W Sy ´ ¯ “ xrL, ∇{ rV,W sUs,Sy ´ VLx∇{ W U, Sy ` WLx∇{ V U, Sy

“ ´V xrL, ∇{ W Us,Sy ´ 2V χp∇{ W U, Sq ` W xrL, ∇{ V Us,Sy ` 2W χp∇{ V U, Sq

where the ﬁrst term in the second line vanishes as a result of Lemma 4.2.1 since

rV,W s P EpΣq and rV,W s|q “ 0. Using Lemma 4.2.1 on the ﬁrst and third terms of the third line we get

“ ´V p∇{ W χqpU, Sq ` p∇{ U χqpW, Sq ´ p∇{ SχqpW, Uq ´ 2V χp∇{ W U, Sq ´ ¯

` W p∇{ V χqpU, Sq ` p∇{ U χqpV,Sq ´ p∇{ SχpV,Uq ´ ¯ ` 2W χp∇{ V U, Sq

“ ´p∇{ V ∇{ W χqpU, Sq ´ p∇{ V ∇{ U χqpW, Sq ` p∇{ V ∇{ SχqpW, Uq ´ 2V χp∇{ W U, Sq

` p∇{ W ∇{ V χqpU, Sq ` p∇{ W ∇{ U χqpV,Sq ´ p∇{ W ∇{ SχqpV,Uq

` 2W χp∇{ V U, Sq.

We also note that restriction to q P Σs gives

0 “ p∇{ V χqp∇{ W U, Sq “ V χp∇{ W U, Sq ´ χp∇{ V ∇{ W U, Sq

60 allowing us to simplify the remaining terms above

´2V χp∇{ W U, Sq ` 2W χp∇{ V U, Sq “ 2 ´ χp∇{ V ∇{ W U, Sq ` χp∇{ W ∇{ V U, Sq ´ ¯ “ 2χpR{ VW U, Sq.

Since

LxR{ VW U, Sy “ xrL, R{ VW Us,Sy ` 2χpR{ VW U, Sq

the result follows upon cancellation of 2χpR{ VW U, Sq given that q was arbitrarily chosen.

Now we’re in a position to ﬁnd the structure equations that we’ll need to propa-

gate ρ. Recalling that the tensors γs, χ, χ and ζ are restrictions of associated tensors on Ω we measure their propagation with the Lie derivative along L. The following proposition is known ([25],[11]), we provide proof for completeness:

Proposition 4.2.1 (Structure Equations). 1 LK “ ´ tr χK ´ ∆{ tr χ ` ∇{ ¨ p∇{ ¨ χˆq (4.1) 2

LLγ “ 2χ (4.2)

1 1 L χ “ ´α ` |χˆ|2γ ` tr χχˆ ` ptr χq2γ ` κχ (4.3) L 2 4 1 L tr χ “ ´ ptr χq2 ´ |χˆ|2 ´ GpL,Lq ` κ tr χ (4.4) 2 1 1 1 L χ “ K ` χˆ ¨ χˆ ` GpL,Lq γ ` tr χχˆ ` tr χχˆ (4.5) L 2 2 2 ´ ¯ ´ Gˆ ´ 2Sp∇{ ζq ´ 2ζ b ζ ´ κχ

L tr χ “ GpL,Lq ` 2K ´ 2∇{ ¨ ζ ´ 2|ζ|2 ´ xH,~ H~ y ´ κ tr χ (4.6)

1 L ζ “ G ´ ∇{ ¨ χˆ ´ tr χζ ` d{ tr χ ` dκ{ (4.7) L L 2

where α is the symmetric 2-tensor given by αpV,W q “ xRLV L,W y, SpT q represents ˆ 1 the symmetric part of a 2-tensor T , GL “ GpL, ¨q|Σs and G “ G|Σs ´ 2 ptrγ Gqγ. 61 Proof. We prove each equation in turn, when used we will assume S, U, V, W P EpΣq:

1. Since Σ is of dimension two we have

KtxV,UyxW, Sy ´ xV,SyxW, Uyu “ xR{ VW U, Sy

Applying L to the left hand side of the equality we get

pLKqtxV,UyxW, Sy ´ xV,SyxW, Uyu ` 2KtxW, SyχpV,Uq ` xV,UyχpW, Sq

´ xW, UyχpV,Sq ´ xV,SyχpW, Uqu

so that a trace over V,U and then W, S gives

2LK ` 4 tr χK.

Applying L to the right hand side we have

LxR{ VW U, Sy “ xrL, R{ VW Us,Sy ` 2χpR{ VW U, Sq

allowing us to use Lemma 4.2.2. Taking a trace over V,U and W, S we get

2∇{ ¨ ∇{ ¨ χ ´ 2∆{ tr χ ` 2 tr χK

having used the fact that Ric{ “ Kγ in obtaining the last term. Equating terms we conclude that

1 LK “ ∇{ ¨ ∇{ ¨ χ ´ ∆{ tr χ ´ tr χK “ ∇{ ¨ ∇{ ¨ χˆ ´ ∆{ tr χ ´ tr χK 2

2. Coming from Lemma 4.1.1 we have already made extensive use of this identity:

pLLγqpV,W q “ LxV,W y “ xDLV,W y ` xV,DLW y

“ xDV L,W y ` xV,DW Ly

“ 2χpV,W q

62 3. pLLχqpV,W q “ LχpV,W q “ LxDV L,W y

“ xDLDV L,W y ` xDV L,DLW y

“ xRVLL ` DV DLL,W y ` x~χpV q ` ζpV qL, ~χpW q ` ζpW qLy

“ ´xRLV L,W y ` κχpV,W q ` x~χpV q, ~χpW qy

~ 1 having used Lemma 4.1.1 to get the third line. Since ~χpV q “ χˆpV q ` 2 tr χV we see that 1 x~χpV q, ~χpW qy “ x~χˆpV q, ~χˆpW qy ` tr χχˆpV,W q ` ptr χq2xV,W y 4 1 1 “ |χˆ|2xV,W y ` tr χχˆpV,W q ` ptr χq2xV,W y 2 4

using the fact that AB ` BA “ trpABqI for traceless symmetric 2 ˆ 2 matrices to get the second line. The result follows.

4. We will denote tensor contraction between the contravariant a-th and covariant

a b-th slots by Cb . Extending a local basis oﬀ of Σ and applying Gram-Schmidt

we get an orthonormal frame ﬁeld tE1,E2u allowing us to write

´1 ´1 g |Ω “ γ “ E1 b E1 ` E2 b E2

5 5 5 5 2 ´1 and γ “ E1 bE1 `E2 bE2. It’s an easy exercise to show C1 γ bγ “ δ ´Lbds whereby δpη, Xq “ ηpXq for any 1-form η and vector ﬁeld X. Since δ and Lbds are Lie constant along L

2 ´1 2 ´1 ´1 0 “ LLC1 γ b γ “ C1 pLLγ b γ ` γ b 2χq

giving

2 2 ´1 ´1 2 2 ´1 ´1 ´2C1 C1 γ b χ b γ “ ´C1 pC1 γ b 2χq b γ

2 2 ´1 ´1 “ C1 pC1 LLγ b γq b γ

2 ´1 1 ´1 “ C1 LLγ b pC2 γ b γ q 63 2 ´1 5 5 “ C1 LLγ b pE1 b E1 ` E2 b E2q

´1 “ LLγ .

As a result

1 2 ´1 L tr χ “ LLC1 C2 γ b χ

1 2 ´1 ´1 “ C1 C2 pLLγ b χ ` γ b LLχq

2 “ ´2|χ| ` tr LLχ

“ ´2|χ|2 ´ RicpL,Lq ` κ tr χ ` |χ|2

1 1 “ ´pχˆ ` tr χγq ¨ pχˆ ` tr χγq ´ GpL,Lq ` κ tr χ 2 2 1 “ ´ ptr χq2 ´ |χˆ|2 ´ GpL,Lq ` κ tr χ 2

5. pLLχqpV,W q “ LxDV L, W y “ xDLDV L, W y ` xDV L, DLW y

“ xRVLL ` DV DLL, W y ` x~χpV q ´ ζpV qL, ~χpW q ` ζpW qLy

~ ~ “ xRVLL, W y ` V x´2ζ ´ κL, W y ` x2ζ ` κL, DV W y

` x~χpV q, ~χpW qy ´ 2ζpV qζpW q

“ xRVLL, W y ´ 2p∇{ V ζqpW q ´ κχpV,W q ` x~χpV q, ~χpW qy

´ 2ζpV qζpW q.

Having used Lemma 4.1.1 to get the second and third lines.

Lemma 4.2.3. The first term satisfies the identity 1 1 1 xR W, Ly “ ´ K ´ xH,~ H~ y ` χˆ ¨ χˆ ` GpL,Lq xV,W y ` GˆpV,W q VL 4 2 2 ´ ¯ 1 ´ pcurl ζqpV,W q ` x~χpV q, ~χpW qy ´ x~χpV q, ~χpW qy 2 ´ ¯ Proof. From the first Bianchi identity followed by the Ricci equation ([21], pg.125)

64 xRVLW, Ly`xRLW V,Ly “ xRVW L,Ly

K ˜ ˜ ˜ ˜ “ xR{ VW L,Ly ` xIIpV,Lq, IIpW, Lqy ´ xIIpV,Lq, IIpW, Lqy where using Lemma 4.1.1

K K K K xR{ VW L,Ly :“ xDrV,W sL ´ rDV ,DW sL,Ly “ ´2p∇{ V ζqpW q ` 2p∇{ W ζqpV q

“ ´2pcurlζqpV,W q

˜ || IIpV,Lq :“ DV L “ ~χpV q

˜ || IIpV,Lq :“ DV L “ ~χpV q.

We conclude that the antisymmetric part satisﬁes 1 xR W, Ly ´ xR V,Ly “ ´pcurlζqpV,W q 2 VL WL ´ ¯ 1 ` x~χpV q, ~χpW qy ´ x~χpV q, ~χpW qy . 2 ´ ¯ Next we ﬁnd that 1 GpV,W q “ RicpV,W q ´ RxV,W y 2 1 “ xR L, W y ` xR L,W y ` tr xR p¨q,W y 2 LV LV γ p¨qV ´ ¯ 1 ` pGpL,Lq ` tr GqxV,W y. 2 γ

Since Σ is of dimension two we must have that trγxRp¨qV p¨q,W y9xV,W y with 1 ~ ~ 1 factor of proportionality K ´ 4 xH, Hy ` 2 χˆ ¨ χˆ coming from Proposition 3.0.1. We conclude therefore that the symmetric part satisﬁes

1 1 1 1 xR W, Ly`xR V,Ly “ GˆpV,W q´pK´ xH,~ H~ y` χˆ¨χˆ` GpL,LqqxV,W y 2 VL WL 4 2 2 ´ ¯ and the result follows as soon as we sum up the antisymmetric and symmetric contributions.

65 Combining the previous lemma with the propagation of χ we have

1 1 1 pL χqpV,W q “ K ´ xH,~ H~ y ` χˆ ¨ χˆ ` GpL,Lq xV,W y ´ GˆpV,W q L 4 2 2 ´ ¯ 1 ` pcurlζqpV,W q ´ x~χpV q, ~χpW qy ´ x~χpV q, ~χpW qy 2 ´ ¯ ´ 2p∇{ V ζqpW q ´ κχpV,W q ` x~χpV q, ~χpW qy ´ 2ζpV qζpW q 1 1 1 “ K ´ xH,~ H~ y ` χˆ ¨ χˆ ` GpL,Lq xV,W y ´ GˆpV,W q 4 2 2 ´ ¯ 1 ` x~χpV q, ~χpW qy ` x~χpV q, ~χpW qy ´ p∇{ ζqpW q ´ p∇{ ζqpV q 2 V W ´ ¯ ´ 2ζpV qζpW q ´ κχpV,W q.

Using again the fact that AB ` BA “ trpABqI for symmetric, traceless 2 ˆ 2 matrices it follows that

x~χpV q, ~χpW qy ` x~χpV q, ~χpW qy “ pχˆ ¨ χˆqxV,W y ` tr χχˆpV,W q ` tr χχˆpV,W q

1 ` xH,~ H~ yxV,W y 2

giving the result.

1 2 ´1 6. L tr χ “ LLpC1 C2 γ b χq

“ ´2χ ¨ χ ` trpLLχq

“ ´2χ ¨ χ ` 2K ` 2ˆχ ¨ χˆ ` GpL,Lq ´ 2∇{ ¨ ζ ´ 2|ζ|2 ´ κ tr χ

“ GpL,Lq ` 2K ´ 2∇{ ¨ ζ ´ 2|ζ|2 ´ xH,~ H~ y ´ κ tr χ

7. And ﬁnally, 1 pL ζqpV q “ LζpV q “ LxD L,Ly L 2 V 1 1 “ xD D L,Ly ` xD L,D Ly 2 L V 2 V L 1 1 “ xR L ` D D L,Ly ` x~χpV q ` ζpV qL, ´2ζ~ ´ κLy 2 VL V L 2 66 1 “ xR L,Ly ` V κ ` κζpV q ´ χpV, ζ~q ´ κζpV q 2 VL 1 “ ´∇{ ¨ χˆpV q ` V tr χ ´ tr χζpV q ` GpV,Lq ` V κ 2

having used Lemma 4.1.1 to obtain the third line and the Codazzi equation (3.2) to get the ﬁfth.

4.3 Propagation of ρ

From Proposition 4.2.1 we have the propagation of the ﬁrst two terms of ρ for the third and forth we’ll need

Corollary 4.3.0.1. Assuming tΣsu is expanding along L we have 1 Lp∇{ ¨ ζq “ ´2∇{ ¨ pχˆ ¨ ζq ´ 2 tr χ∇{ ¨ ζ ´ ∇{ ¨ ∇{ ¨ χˆ ` ∆{ tr χ ´ d{ tr χ ¨ ζ 2

` ∇{ ¨ GL ` ∆{ κ

3 1 L∆{ log tr χ “ ´2∇{ ¨ pχˆ ¨ d{ log tr χq ´ tr χ∆{ log tr χ ´ tr χ|d{ log tr χ|2 2 2 |χˆ|2 ` GpL,Lq ´ ∆{ ` ∆{ κ tr χ

Proof. When used we will assume V,W P EpΣq. For any 1-form η on Ω we have

LLp∇{ ηqpV,W q “ Lp∇{ V ηpW qq “ VLηpW q ´ Lηp∇{ V W q

“ V pLLηqpW q ´ pLLηqp∇{ V W q ´ ηprL, ∇{ V W sq

“ ∇{ V pLLηqpW q ´ ηprL, ∇{ V W sq from which we ﬁnd

1 2 ´1 ´1 Lp∇{ ¨ ηq “ C1 C2 pLLγ b ∇{ η ` γ b LLp∇{ ηqq 67 “ ´2χ ¨ ∇{ η ` trpLL∇{ ηq

1 “ ´2pχˆ ` tr χγq ¨ ∇{ η ` ∇{ ¨ pL ηq ´ ηp2∇{ ~¨ χˆq 2 L the last term coming from Lemma 4.2.1 after taking a trace over V,W . We conclude that

Lp∇{ ¨ ηq “ ´ tr χ∇{ ¨ η ´ 2∇{ ¨ pχˆ ¨ ηq ` ∇{ ¨ pLLηq.

The ﬁrst part of the corollary now straight forwardly follows from Proposition 4.2.1 for η “ ζ. For the second, since ∆{ log tr χ “ ∇{ ¨ d{ log tr χ we have

L∆{ log tr χ “ ´ tr χ∆{ log tr χ ´ 2∇{ ¨ pχˆ ¨ d{ log tr χq ` ∇{ ¨ pLLd{ log tr χq.

From the fact that

1 |χˆ|2 ` GpL,Lq ∇{ ¨ pL d{ log tr χq “ ∇{ ¨ pdL{ log tr χq “ ∆{ ´ tr χ ´ ` κ L 2 tr χ ´ ¯ the result follows as soon as we make the substitution

∆{ tr χ “ tr χ ∆{ log tr χ ` |d{ log tr χ|2 ´ ¯

Theorem 4.3.1 (Propagation of ρ). Assuming tΣsu is expanding along the ﬂow vector L “ σL´ we conclude that 3 σ 1 1 ρ9 ` σρ “ xH,~ H~ y |χˆ´|2 ` GpL´,L´q ` |τ|2 ´ GpL´,L`q 2 2 2 2 ´ ´ ¯ ¯ ´ 2 ´ ´ ´ ` ∆{ σp|χˆ | ` GpL ,L qq ´ 2∇{ ¨ pσχˆ ¨ τq ` ∇{ ¨ pσGL´ q ´ ¯ Proof. From Proposition 4.2.1 and Corollary 4.3.0.1 the proof reduces to an exercise in algebraic manipulation 1 1 L ρ “ L K ´ trχL tr χ ´ tr χL trχ ` L∇{ ¨ ζ ´ L∆{ log tr χ L L 4 L 4 L 68 1 1 1 “ ∇{ ¨ ∇{ ¨ χˆ ´ ∆{ tr χ ´ tr χK ´ tr χ ´ tr2 χ ´ |χˆ|2 ´ GpL,Lq ` κ tr χ 2 4 2 ´ ¯ ´ ¯ 1 ´ tr χ GpL,Lq ` 2K ´ 2∇{ ¨ ζ ´ 2|ζ|2 ´ xH,~ H~ y ´ κ tr χ 4 ´ ¯ 1 ´ 2∇{ ¨ pχˆ ¨ ζq ´ 2 tr χ∇{ ¨ ζ ´ ∇{ ¨ ∇{ ¨ χˆ ` ∆{ tr χ ´ d{ tr χ ¨ ζ ` ∇{ ¨ G ` ∆{ κ 2 L 3 1 ` 2∇{ ¨ pχˆ ¨ d{ log tr χq ` tr χ∆{ log tr χ ` tr χ|d{ log tr χ|2 2 2 |χˆ|2 ` GpL,Lq ` ∆{ ´ ∆{ κ tr χ

3 1 1 |χˆ|2 ` GpL,Lq 1 “ ´ tr χK ` tr χxH,~ H~ y ` xH,~ H~ y ´ tr χGpL,Lq 2 8 4 tr χ 4 ´ ¯ 3 1 3 ´ tr χ∇{ ¨ ζ ` tr χxH,~ H~ y ´ 2∇{ ¨ pχˆ ¨ pζ ´ d{ log tr χqq ` tr χ∆{ log tr χ 2 4 2 1 1 ` tr χ|ζ|2 ´ d{ tr χ ¨ ζ ` tr χ|d{ log tr χ|2 2 2 3 1 |χˆ|2 ` GpL,Lq 1 1 “ ´ tr χρ ` xH,~ H~ y ` tr χ|ζ ´ d{ log tr χ|2 ´ tr χGpL,Lq 2 4 tr χ 2 4 ´ ¯ |χˆ|2 ` GpL,Lq ` ∆{ ´ 2∇{ ¨ pχˆ ¨ pζ ´ d{ log tr χqq ` ∇{ ¨ G . tr χ L

The result therefore follows as soon as we express all terms according to the inﬂation

´ ` ´ basis tL ,L u where tΣsu is a ﬂow along L “ σL of speed σ “ tr χ.

´ Corollary 4.3.1.1. For tΣsu expanding along the ﬂow vector L “ σL and any u P FpΣsq 3 1 eu ρ9 ` σρ dA “ σeu |χˆ´|2 ` GpL´,L´q xH,~ H~ y ` ∆{ u Σs 2 Σs 4 ż ´ ¯ ż ´´ ¯´ ¯ 1 1 ` |2ˆχ´ ¨ du{ ` τ|2 ` GpL´, |∇{ u|2L´ ´ ∇{ u ´ L`q dA 2 4 ¯ Proof. We start by integrating by parts on the last three terms of Theorem 4.3.1

u ´ 2 ´ ´ ´ e ∆{ pσp|χˆ | ` GpL ,L qqq ´ 2∇{ ¨ pσχˆ ¨ τq ` ∇{ ¨ pσGL´ q dA ż ´ ¯ 69 “ σeu eúp∆{ euqp|χˆ´|2 ` GpL´,L´qq ` 2ˆχ´p∇{ u, ~τq ´ GpL´, ∇{ uq dA ż ´ ¯ “ σeu p∆{ u ` |∇{ u|2qp|χˆ´|2 ` GpL´,L´qq ` 2ˆχ´p∇{ u, ~τq ´ GpL´, ∇{ uq dA ż ´ ¯ “ σeu p|χˆ´|2 ` GpL´,L´qq∆{ u ` |χˆ´|2|∇{ u|2 ` 2ˆχp∇{ u, ~τq ż ´ ` GpL´, |∇{ u|2L´ ´ ∇{ uq dA. ¯ As a result 3 1 eu ρ9 ` σρ dA “ σeu p|χˆ´|2 ` GpL´,L´qq xH,~ H~ y ` ∆{ u 2 4 ż ´ ¯ ż ´ ´ ¯ 1 1 ` |χˆ´|2|∇{ u|2 ` 2ˆχ´p∇{ u, ~τq ` |τ|2 ` GpL´, |∇{ u|2L´ ´ ∇{ u ´ L`q dA. 2 4 ¯ ´ ´ 2 1 ´ 2 2 Sinceχ ˆ is symmetric and trace-free it follows that |χˆ ¨ du{ | “ 2 |χˆ | |∇{ u| from which the first three terms of the second line simplifies to give

1 1 |χˆ´|2|∇{ u|2 ` 2ˆχ´p∇{ u, ~τq ` |τ|2 “ |2ˆχ´ ¨ du{ ` τ|2 2 2

Remark 4.3.1. An interesting consequence of the above corollary in spacetimes sat-

isfying the dominant energy condition is the fact that any u P FpΣq gives

3 1 eu ρ9 ` σρ dA ě σeup|χˆ´|2 ` GpL´,L´qq xH,~ H~ y ` ∆{ u dA 2 4 ż ´ ¯ ż ´ ¯ The proof of Theorem 2.1.1 is a simple consequence of the following corollary:

´ Corollary 4.3.1.2. Assuming tΣsu is expanding along the ﬂow vector L “ σL with

each Σs of non-zero ﬂux (|ρpsq| ą 0) then

d 2 92 2 ρ 3 dA “ pρ 3 q ` σρ 3 dA ds żΣs żΣs 70 2 σ ´ 2 ´ ´ 1 ~ ~ 1 “ 1 |χˆ | ` GpL ,L q xH, Hy ´ ∆{ log |ρ| 3 Σ ρ 3 4 3 ż ´´ ¯´ ¯ 1 2 ` | χˆ´ ¨ d{ log |ρ| ´ τ|2 2 3 1 1 1 ` GpL´, |∇{ log |ρ||2L´ ` ∇{ log |ρ| ´ L`q dA 9 3 4 ¯ Proof. From the ﬁrst variation of Area formula

dA9 “ ´xH,L~ ydA “ ´σxH,L~ ´ydA “ σdA

2 1 u ´ 3 we get the ﬁrst equality. For the second we apply Corollary 4.3.1.1 with e “ 3 |ρ| , canceling the sign in the case that ρ ă 0.

4.3.1 Case of Equality

´ Lemma 4.3.1. For tΣsu expanding along L “ σL we have

´ ´ 2 ´ ´ LLτ ` στ ` ∇{ ¨ pσχˆ q “ σGL´ ` d{pσp|χˆ | ` GpL ,L qqq

Proof. By combining (8) and (11): 1 L pζ ´ d{ log tr χq “ G ´ ∇{ ¨ χˆ ´ tr χζ ` d{ tr χ ` dκ{ L L 2 1 |χˆ|2 ` GpL,Lq ´ d{ ´ tr χ ´ ` κ 2 tr χ ´ ¯ |χˆ|2 ` GpL,Lq “ ´ tr χpζ ´ d{ log tr χq ´ ∇{ ¨ χˆ ` G ` d{ . L tr χ

The result follows as soon as we switch to the inﬂation basis tL´,L`u.

Theorem 4.3.2. Let Ω be a null hypersurface in a spacetime satisfying the dominant energy condition with vector field L tangent to the null generators of Ω. Suppose tΣsu is an expanding (SP)-foliation defined as the level sets of a function s :Ω Ñ R satis- dm fying Lpsq “ 1 and achieves the case of equality ds “ 0. Then all foliations achieve 71 equality, moreover, we find an affine level set function r F Ω with r : r π P p q 0 “ |Σs0 ˝

such that any surface Σ:“ tr “ ω ˝ πu, for ω P FpΣs0 q, has data:

2 γ “ ω γ0

χ “ ωγ0

2 tr χ “ ω 2 r tr χ “ pK ´ 0 ´ ω2∆{ log ωq ω 0 ω ζ “ ´d{ log ω r ρ “ 0 ω3

2 where r0γ0 is the metric on Σs0 and K0 the Gaussian curvature associated to γ0. In the case that tr χ 0 our data corresponds with the the standard nullcone in |Σs0 “

r0 Schwarzschild spacetime of mass M “ 2 .

Proof. Without loss of generality we assume s0 “ 0. Immediately from Corollary 4.3.1.2 we conclude for this particular foliation that

|χˆ´|2 ` GpL´,L´q “ 0

2 | χˆ´ ¨ d{ log ρ ´ τ|2 “ 0 3 1 1 1 GpL´, |∇{ log ρ|2L´ ` ∇{ log ρ ´ L`q “ 0. 9 3 4

So from the ﬁrst equality we have bothχ ˆ´ “ 0 and GpL´,L´q “ 0. Combined with the second equality we conclude that τ “ 0 for this particular foliation and therefore

Lemma 4.3.1 ensures that GL´ “ 0 as well. Finally we may therefore utilize the ﬁnal equality to conclude also that GpL`,L´q = 0 so that, for any p P Ω and any

X P TpM, we have GpL´,Xq “ 0.

72 From this and Lemma 4.3.1 we have for any foliation oﬀ of Σ0 generated by some La (a ą 0) that

a a LLa τ ` aστ “ 0.

a a Given that τ |Σ0 “ τ|Σ0 “ 0 this enforces τ “ 0 by standard uniqueness theorems. We recognise this implies the case of equality for all foliations so without loss of generality we assume that L is geodesic. We are now in a position to show that

the ﬂux ρ P FpΩq is independent of the foliation from which it is constructed. In

particular, for any a ą 0, foliating oﬀ of Σ0 along the generator La will construct a

ρa which we would like to show agrees pointwise on Ω with ρ. From Theorem 4.3.1 we have 3 Lρ “ ´ tr χρ “ 3ρL log tr χ 2

L so for any p P Ω solving this ODE along the geodesic γπppqpsq gives

ρ ˝ sppq tr χppq 3 “ . ρp0q tr χp0q ´ ¯

For the generator La Theorem 4.3.1 gives 3 L ρ “ ´ tr χ ρ “ 3ρ pL log tr χ ´ κ q a a 2 a a a a a a 3ρ L log tr χ log a “ a ap a ´ q

“ 3ρaLaplog tr χq

where the penultimate line comes from the fact that

κaLa “ DLa La “ aLpaqL “ Laplog aqLa

and the ﬁnal line from the fact that tr χ a tr χ. Solving this ODE along the a “

La pregeodesic γπppqptq we have

ρ ˝ tppq tr χppq 3 ρ ˝ sppq a “ “ . ρap0q tr χp0q ρp0q ´ ¯ 73 Since we’re foliating oﬀ of Σ0 in both cases and ρ|Σ0 is independent of our choice of

null basis we have ρppq “ ρappq as desired.

We therefore deﬁne the functions r0 and r according to

1 r ˝ π “ ρ| , 0 “ ρ 2 Σs0 3 r0 r

(i.e. r|Σ0 “ r0) so that Theorem 4.3.1 gives

r 3 3 r ´3 0 L prq “ L pρq “ ´ tr χ ρ “ ´ tr χ 0 r4 a a 2 a 2 a r3

1 2 and therefore L prq “ tr χ r. It follows that if we scale L such that tr χ|Σ “ a 2 a 0 r0 1 2 1 2 then Lptr χrq “ ´ 2 ptr χq r ` tr χp 2 tr χrq “ 0 implies that tr χ “ r and Lprq “ 1. 2 So r is in fact our level set function. For r0γ0 the metric on Σ0, by Lie dragging γ0 along L to all of Ω we have

2 L pr2γ q “ 2rγ “ pr2γ q “ tr χpr2γ q. L 0 0 r 0 0

2 2 2 2 So from (6), LLpr γ0 ´γq “ tr χpr γ0 ´γq and r0γ0 ´γpr0q “ 0 giving γprq “ r γ0 by

uniqueness. We conclude that for any 0 ď ω P FpΣ0q the cross-section Σ :“ tr “ ω˝ πu

2 1 has metric γω “ γprq|Σ “ ω γ0 with Gaussian curvature Kω “ ω2 K0 ´∆{ log ω. More- over, r 1 0 “ ρ “ K ´ xH,~ H~ y ω3 ω ω 4 1 1 “ K ´ ∆{ log ω ´ tr χ ω2 0 2ω ω

having used the fact that ρω “ ρ|Σ (from independence of foliation) in the ﬁrst line and tr χ tr χ in the second. We conclude that, ω “ |Σ

2 r tr χ “ pK ´ 0 ´ ω2∆{ log ωq. ω ω 0 ω 74 1 In the case that tr χ|Σ0 “ 0 property (SP) forces 2 “ ρ|Σ0 to be constant by way of r0 the maximum principle. From our expression for tr χr0 we conclude that K0 “ 1 and

2 therefore γ0 is a round metric on S .

Remark 4.3.2. We bring to the attention of the reader that due the lack of information regarding the term Gˆ in (4.5) we are unable to conclude with any knowledge of the datum χ on Σ. In the case of vacuum this no longer poses a problem and one is able to correlate χ with χ as shown by Sauter ([25], Lemma 4.3). |Σ |Σr0

75 5

Foliation Comparison

In this chapter we show how the ﬂux function ρ of an arbitrary cross-section of Ω decomposes in terms of the ﬂux of the background foliation. With the appropriate asymptotic decay on Ω this allows us to prove Theorem 2.1.2.

5.1 Additional Setup

We follow once again the construction of [18] starting with a background foliation

as constructed in Section 4.1 oﬀ of an initial cross-section Σs0 . As before, each Σs allows a null basis tL, lu such that xL, ly “ 2. Also from Section 4.1 we have the diﬀeomorphism p ÞÑ pπppq, sppqq of Ω onto its image. Therefore any cross-section with associated embedding Φ : S2 Ñ Ω is equivalently realized with the map Φ˜ “ pπ, sq˝Φ. Expressing the component functions Ψ :“ π ˝ Φ and ω :“ s ˝ Φ we recognize that

2 2 Ψ: S Ñ Σs0 is a diﬀeomorphism and therefore the embedding Φ : S Ñ Ω is uniquely

´1 characterized as a graph over Σs0 with graph function ω ˝ Ψ . Without confusion we will simply denote the graph function by ω and it’s associated cross-section by

Σω. We wish to compare both the intrinsic and extrinsic geometry of Σω at a point q with the geometry of the surface Σspqq. We extend ω to all of Ω in the usual way by 76 Ω

Σs0 L L ω Σω

L q Σspqq

Figure 5.1:Σω as a graph over Σs0 imposing it be constant along generators of L, in other words, ωppq :“ pω ˝πqppq. For the extrinsic geometry of Σω we have the null-normal basis tL,Lu whereby L is given by the conditions xL,Ly “ 2 and xV,Ly “ 0 for any V P ΓpT Σωq. As before Σω has second fundamental form decomposing into the null components χ (associated to L) and χ (associated to L) with torsion ζ. For each Σs we equivalently decompose the second fundamental form into the components K (associated to L) and Q (associated to l) with torsion t. We will denote the induced covariant derivative on Σs by ∇ and on Σω by ∇{ . The following lemma is known ([18],[25]):

Lemma 5.1.1. Given q P Σω X Σspqq the map given by

Tω : TqΣspqq Ñ TqΣω

v Ñ v˜ :“ v ` vωL

is a well deﬁned isomorphism with natural extension EpΣs0 q Ñ EpΣωq. Moreover,

˜ ˜ • γωpV, W q “ γspV,W q

• χpV,˜ W˜ q “ KpV,W q

• ζpV˜ q “ tpV q ´ KpV, ∇ωq ` κxV, ∇ωy

77 • χpV,˜ W˜ q “ QpV,W q ´ 2tpV qxW, ∇ωy ´ 2tpW qxV, ∇ωy ´ |∇ω|2KpV,W q

´ 2HωpV,W q ` 2KpV, ∇ωqxW, ∇ωy ` 2KpW, ∇ωqxV, ∇ωy

´ 2κxV, ∇ωyxW, ∇ωy

• tr χ “ tr Q ´ 4tp∇ωq ´ 2p∆ω ´ 2Kˆ p∇ω, ∇ωqq ` tr K|∇ω|2 ´ 2κ|∇ω|2

ω for H the Hessian of ω on Σs.

Proof. For completeness we include a similar proof as in [18] (Proposition 1). Since

Tω : TqΣspqq Ñ TqΣω is clearly injective it suffices to showv ˜ P TqΣω. This follows from the fact thatv ˜ps ´ ωq “ vps ´ ωq ` vωLps ´ ωq “ ´vω ` vω “ 0 since ˜ Σω is locally characterised by s|Σω “ ω. For the extension V “ V ` V ωL we ˜ ˜ note that rL,V s “ 0 ùñ rL, V s “ 0 and it follows that V P EpΣωq (infact ˜ EpΣs0 q “ EpΣωq). From this and the fact that DLL “ κL the first two identities follow straight forwardly. For the third identity we find that L “ l ´ |∇ω|2L ´ 2∇ω since

xL, Ly “ xl, Ly “ 2

xL, V˜ y “ xl, V ωLy ´ 2x∇ω, V y “ 2V ω ´ 2V ω “ 0 giving 1 ζpV˜ q “ xD L, l ´ |∇ω|2L ´ 2∇ωy 2 V `V ωL 1 “ xD L ` κV ωL, l ´ |∇ω|2L ´ 2∇ωy 2 V 1 “ tpV q ´ |∇ω|2V xL,Ly ´ xD L, ∇ωy ` κxV, ∇ωy 4 V “ tpV q ´ KpV, ∇ωq ` κxV, ∇ωy.

For comparison between χ and Q we calculate

˜ ˜ 2 χpV, W q “ xDV `V ωLpl ´ |∇ω| L ´ 2∇ωq,W ` W ωLy 78 in three parts:

xDV `V ωLl, W ` W ωLy “ QpV,W q ` V ωxDLl, W y ` W ωxDV l, Ly

` V ωW ωxDLl, Ly

“ QpV,W q ´ V ωxl, DLW y ´ W ωxl, DV Ly

´ V ωW ωxl, DLLy

“ QpV,W q ´ 2V ωtpW q ´ 2W ωtpV q ´ 2κV ωW ω

2 2 ´|∇ω| xDV `V ωLL,W ` W ωLy “ ´|∇ω| xDV L,W ` W ωLy 1 “ ´|∇ω|2KpV,W q ´ |∇ω|2W ωV xL,Ly 2 “ ´|∇ω|2KpV,W q

´2xDV `V ωL∇ω, W ` W ωLy “ ´2xDV ∇ω, W y ´ 2W ωxDV ∇ω, Ly

´ 2V ωxDL∇ω, W y ´ 2V ωW ωxDL∇ω, Ly

“ ´2HωpV,W q ` 2W ωKp∇ω, V q ´ 2V ωLW ω

` 2V ωKp∇ω, W q ` 2V ωW ωx∇ω, DLLy

“ ´2HωpV,W q ` 2W ωKp∇ω, V q ` 2V ωKp∇ω, W q the third to last line coming from

xDL∇ω, W y “ Lx∇ω, W y ´ x∇ω, DLW y “ LW ω ´ KpW, ∇ωq

Collecting all the terms the result follows. The ﬁnal identity follows upon taking a trace.

5.2 Flux Comparison

We are now ready to prove our ﬁrst main result of this section. On Σω we will denote the ﬂux function (2.1) by ρ{ and on Σs by ρ the following theorem provides comparison between the two

79 Theorem 5.2.1 (Flux Comparison Theorem). At any q P Σω X Σs we have |Kˆ |2 ` GpL,Lq 1 ρ “ ρ ` ∇{ ¨ ∇{ ω ` |Kˆ |2 ` GpL,Lq |∇ω|2 { tr K 2 ´ ¯ ´ ¯ |Kˆ |2 ` GpL,Lq ` ∇ω ` GpL, ∇ωq ´ 2Kˆ p~t ´ ∇ log tr K, ∇ωq tr K

Remark 5.2.1. Revisiting Theorem 4.3.2 and the case that χˆ “ GpL, ¨q “ 0, Theo- rem 5.2.1 provides an alternative proof that ρ{ agrees with ρ point wise.

˜ ˜ ˜ Proof. When used, we assume V, W, U P EpΣs0 qp ùñ V, W, U P EpΣωq). We will need to know how to relate the covariant derivatives between the two surfaces so ﬁrst a lemma

~ ~ ˜ Lemma 5.2.1. Tω ∇V W ` V ωKpW q ` W ωKpV q ´ KpV,W q∇ω “ ∇{ V˜ W ´ ¯ ˜ Proof. Since ∇{ V˜ W |q “ pS ` SωLq|q “ TωpS|qq for some S P ΓpT Σspqqq it follows that ˜ x∇{ V˜ W,Uy “ xS,Uy for any U P EpΣs0 q. We ﬁnd 1 1 x∇{ ˜ W,U˜ y “ xD ˜ W˜ ` χpV,˜ W˜ qL ` χpV,˜ W˜ qL,Uy V V 2 2 1 “ xD ˜ W,U˜ y ` KpV,W qxL, Uy V 2

1 2 “ V˜ xW, Uy ´ xW,D˜ ˜ Uy ` KpV,W qxl ´ |∇ω| L ´ 2∇ω, Uy V 2

“ pV ` V ωLqxW, Uy ´ xW ` W ωL,DV `V ωLUy ´ KpV,W qUω

“ V xW, Uy ` 2V ωKpW, Uq ´ xW, ∇V Uy ` V ωKpW, Uq ´ W ωKpV,Uq ´ ¯ ´ KpV,W qUω

“ V xW, Uy ´ xW, ∇V Uy ` KpW, UqV ω ` KpV,UqW ω ´ KpV,W qUω ´ ¯ ~ ~ “ x∇V W ` V ωKpW q ` W ωKpV q ´ KpV,W q∇ω, Uy

~ ~ so S “ ∇V W ` V ωKpW q ` W ωKpV q ´ KpV,W q∇ω since EpΣs0 q|Σspqq “ ΓpT Σspqqq.

80 Now we proceed with the proof of Theorem 5.2.1 in 3 parts:

Step 1 Comparison between ∇{ ¨ ζ and ∇ ¨ t:

From Lemmas 5.1.1 and 5.2.1 we have ˜ ˜ ˜ ˜ p∇{ V˜ ζqpW q “ V pζpW qq ´ ζp∇{ V˜ W q

“ pV ` V ωLq tpW q ´ KpW, ∇ωq ` κxW, ∇ωy ´ ¯ ~ ~ ´ t ∇V W ` V ωKpW q ` W ωKpV q ´ KpV,W q∇ω ´ ¯ ~ ~ ` K ∇V W ` V ωKpW q ` W ωKpV q ´ KpV,W q∇ω, ∇ω ´ ¯ ~ ~ ´ κx∇V W ` V ωKpW q ` W ωKpV q ´ KpV,W q∇ω, ∇ωy.

Isolating the terms of the second line we get pV ` V ωLqptpW q ´ KpW, ∇ωq ` κW ωq

1 “ V tpW q ` V ω G pW q ´ ∇ ¨ Kˆ pW q ´ tr KtpW q ` W tr K ` W κ L 2 ´ ¯ ´ VKpW, ∇ωq ´ V ωpLLKqpW, ∇ωq ´ V ωKpW, rL, ∇ωsq

` V κW ω ` κV W ω ` V ωLκW ω

where (4.7) was used to give the ﬁrst line. To continue we’ll need an expression for

rL, ∇ωs and use (4.2) to get it:

2Kp∇ω, V q “ pLLγsqp∇ω, V q “ Lx∇ω, V y ´ xrL, ∇ωs,V y

“ LV ω ´ xrL, ∇ωs,V y

“ ´xrL, ∇ωs,V y

~ since rL, ∇ωs P ΓpT Σsq we conclude that rL, ∇ωs “ ´2Kp∇ωq. Substitution back into our calculation and using (4.3) in the form

~ ~ LLKpV,W q “ ´αpV,W q ` xKpV q, KpW qy ` κKpV,W q 81 gives pV ` V ωLqptpW q ´ KpW, ∇ωq ` κW ωq “ V tpW q ´ VKpW, ∇ωq

1 ` V ω G pW q ´ ∇ ¨ Kˆ pW q ´ tr KtpW q ` W tr K ` αpW, ∇ωq ` xK~ pW q, K~ p∇ωqy L 2 ´ ¯ ` V ωW κ ´ κV ωKpW, ∇ωq ` V κW ω ` κV W ω ` LκV ωW ω.

Collecting terms we get ˜ p∇{ V˜ ζqpW q “ V tpW q ´ tp∇V W q ` Kp∇V W, ∇ωq ´ VKpW, ∇ωq 1 ` V ω G pW q ´ ∇ ¨ Kˆ pW q ´ tr KtpW q ` W tr K ` αpW, ∇ωq ` xK~ pW q, K~ p∇ωqy L 2 ´ ¯ ´ V ωKpW,~tq ´ W ωKpV,~tq ` KpV,W qtp∇ωq

` V ωxK~ pW q, K~ p∇ωqy ` W ωxK~ pV q, K~ p∇ωqy ´ KpV,W qKp∇ω, ∇ωq

` V ωW κ ´ κV ωKpW, ∇ωq ` V κW ω ` κV W ω ` LκV ωW ω

2 ´ κ∇V W ω ´ κV ωKpW, ∇ωq ´ κW ωKpV, ∇ωq ` κKpV,W q|∇ω| .

So taking a trace over V and W

∇{ ¨ ζ “ ∇ ¨ t ´ ∇ ¨ pK~ p∇ωqq

1 ` G p∇ωq ´ p∇ ¨ Kˆ qp∇ωq ´ tr Ktp∇ωq ` ∇ω tr K ` αp∇ω, ∇ωq ` |K~ p∇ωq|2 L 2 ´ ¯ ´ 2Kp∇ω,~tq ` tr Ktp∇ωq ` 2|K~ p∇ωq|2 ´ tr KKp∇ω, ∇ωq

` 2∇ωκ ´ 3κKp∇ω, ∇ωq ` κ∆ω ` Lκ|∇ω|2 ` κ tr K|∇ω|2

1 1 “ ∇ ¨ t ´ ∇ ¨ pK~ p∇ωqq ` p∇ ¨ Kˆ qp∇ωq ´ ∇ω tr K ´ 2 Kp∇ω,~tq ´ tr Ktp∇ωq 2 2 ´ ¯ ´ ¯ ~ 2 ` 3|Kp∇ωq| ´ tr KKp∇ω, ∇ωq ` GLp∇ωq ´ tr Ktp∇ωq ` αp∇ω, ∇ωq

1 ` 2∇ωκ ´ 3κKˆ p∇ω, ∇ωq ` κ∆ω ` Lκ|∇ω|2 ´ κ tr K|∇ω|2 2

“ ∇ ¨ t ´ 2p∇ ¨ Kˆ qp∇ωq ` Hω ¨ K ´ 2Kˆ p∇ω,~tq ` 3|K~ p∇ωq|2 ´ tr KKp∇ω, ∇ωq ´ ¯ ` GLp∇ωq ´ tr Ktp∇ωq ` αp∇ω, ∇ωq 82 1 ` 2∇ωκ ´ 3κKˆ p∇ω, ∇ωq ` κ∆ω ` Lκ|∇ω|2 ´ κ tr K|∇ω|2 2 1 3 “ ∇ ¨ t ´ 2p∇ ¨ Kˆ qp∇ωq ´ Hω ¨ Kˆ ´ tr K∆ω ´ 2Kˆ p∇ω,~tq ` |Kˆ |2|∇ω|2 2 2 1 ` 2 tr KKˆ p∇ω, ∇ωq ` ptr Kq2|∇ω|2 ` G p∇ωq ´ tr Ktp∇ωq ` αp∇ω, ∇ωq 4 L 1 ` 2∇ωκ ´ 3κKˆ p∇ω, ∇ωq ` κ∆ω ` Lκ|∇ω|2 ´ κ tr K|∇ω|2. 2

Step 2 Comparison between ∇{ ¨ ζ ´ ∆{ log tr χ and ∇ ¨ t ´ ∆ log tr K:

Since tr χ “ tr K|Σω we start by comparing ∆{ log tr K with ∆ log tr K

log tr χ ˜ ˜ ˜ ˜ ˜ ˜ H pV, W q “ x∇{ V˜ ∇{ log tr K, W y “ V W log tr K ´ ∇{ V˜ W log tr K

So isolating the ﬁrst term we get

V˜ W˜ log tr K “ pV ` V ωLqpW ` W ωLq log tr K

“ VW log tr K ` pV W ω ` V ωW ` W ωV qL log tr K ` V ωW ωLL log tr K

and then the second ˜ ~ ~ ∇{ V˜ W log tr K “ p∇V W ` V ωKpW q ` W ωKpV q ´ KpV,W q∇ωq log tr K ~ ~ ` p∇V W ` V ωKpW q ` W ωKpV q ´ KpV,W q∇ωqωL log tr K

having used Lemma 5.2.1. Collecting terms log tr K ˜ ˜ H pV, W q “ VW log tr K ´ ∇V W log tr K

` pV W ω ´ ∇V W ωqL log tr K ` V ωW ωLL log tr K

´ V ωKpW, ∇ log tr Kq ` W ωKpV, ∇ log tr Kq ´ KpV,W qx∇ω, ∇ log tr Ky ´ ¯ ` KpV,W q|∇ω|2 ´ V ωKpW, ∇ωq ´ W ωKpV, ∇ωq ` V ωW ` W ωV L log tr K. ´ ¯ So that a trace over V and W yields

∆{ log tr K “ ∆ log tr K ` ∆ωL log tr K ` |∇ω|2LL log tr K ´ 2Kˆ p∇ω, ∇ log tr Kq

83 ´ 2Kˆ p∇ω, ∇ωqL log tr K ` 2∇ωL log tr K.

We take the opportunity at this point of the calculation to bring to the attention of the reader that we have not yet used any distinguishing characteristics of the function log tr K in comparison to an arbitrary f P FpΩq. In particular, we notice if f P FpΩq satisﬁes Lf “ 0, switching with log tr K above, yields

∆{ f “ ∆f ´ 2Kˆ p∇ω, ∇fq.

As a result,

Lemma 5.2.2.

∆{ g “ ∆g ` ∇{ ¨ pLg∇{ ωq ` ∇ωLg ´ 2Kˆ p∇ω, ∇gq

for any g P FpΩq.

Proof. We have

∆{ g “ ∆g ` ∆ωLg ` |∇ω|2LLg ´ 2Kˆ p∇ω, ∇gq ´ 2Kˆ p∇ω, ∇ωqLg ` 2∇ωLg

“ ∆g ` p∆ω ´ 2Kˆ p∇ω, ∇ωqqLg ` p∇ω ` |∇ω|2LqLg ` ∇ωLg ´ 2Kˆ p∇ω, ∇gq

“ ∆g ` ∆{ ωLg ` ∇{ ωLg ` ∇ωLg ´ 2Kˆ p∇ω, ∇gq

“ ∆g ` ∇{ ¨ pLg∇{ ωq ` ∇ωLg ´ 2Kˆ p∇ω, ∇gq

having used the fact that Lω “ 0 and the comment immediately preceding the statement of Lemma 5.2.2 to get the third equality.

Finishing up Step 2 we have

∇{ ¨ ζ ´ ∆{ log tr χ “ ∇ ¨ t ´ ∆ log tr K

´ 2p∇ ¨ Kˆ qp∇ωq ´ Hω ¨ Kˆ ´ 2Kˆ p∇ω,~t ´ ∇ log tr Kq

3 ´ tr Ktp∇ωq ` ∇ω tr K ` |Kˆ |2|∇ω|2 ` G p∇ωq ` αp∇ω, ∇ωq 2 L

84 1 1 ´ tr K∆ω ` ∆ωL log tr K ` ptr Kq2 ´ LL log tr K |∇ω|2 2 4 ´ ¯ ´ ¯ ´ ∇ω tr K ` 2∇ωL log tr K ` 2Kˆ p∇ω, ∇ωq tr K ` L log tr K ´ ¯ ´ ¯ 1 ` 2∇ωκ ´ 3κKˆ p∇ω, ∇ωq ` κ∆ω ` Lκ|∇ω|2 ´ κ tr K|∇ω|2 2 “ ∇ ¨ t ´ ∆ log tr K

´ 2p∇ ¨ Kˆ qp∇ωq ´ Hω ¨ Kˆ ´ 2Kˆ p∇ω,~t ´ ∇ log tr Kq

ˆ 2 2 ´ tr Ktp∇ωq ` ∇ω tr K ` |K| |∇ω| ` GLp∇ωq ` αˆp∇ω, ∇ωq

1 |Kˆ |2 ` GpL,Lq ` |Kˆ |2 ` GpL,Lq |∇ω|2 ` ∆ω ´ 2Kˆ p∇ω, ∇ωq 2 tr K ´ ¯ ´ ¯ 1 |Kˆ |2 ` GpL,Lq ` ´ p|Kˆ |2 ` GpL,Lq ´ κ tr Kq ` L |∇ω|2 2 tr K ´ ¯ |Kˆ |2 ` GpL,Lq 1 ` 2∇ω ` tr KKˆ p∇ω, ∇ωq ´ κKˆ p∇ω, ∇ωq ´ κ tr K|∇ω|2 tr K 2 “ ∇ ¨ t ´ ∆ log tr K

´ 2p∇ ¨ Kˆ qp∇ωq ´ Hω ¨ Kˆ ´ 2Kˆ p∇ω,~t ´ ∇ log tr Kq

ˆ 2 2 ´ tr Ktp∇ωq ` ∇ω tr K ` |K| |∇ω| ` GLp∇ωq ` αˆp∇ω, ∇ωq

|Kˆ |2 ` GpL,Lq |Kˆ |2 ` GpL,Lq |Kˆ |2 ` GpL,Lq ` ∆{ ω ` L |∇ω|2 ` 2∇ω tr K tr K tr K

` tr KKˆ p∇ω, ∇ωq ´ κKˆ p∇ω, ∇ωq

having used (8) to get the last two lines in the second equality, Lemma 5.2.2 to get

∆ω´2Kˆ p∇ω, ∇ωq “ ∆{ ω in the second equality followed by cancellation of the terms

1 ˆ 2 2 1 2 2 |K| ` GpL,Lq |∇ω| and 2 κ tr K|∇ω| . ´ ¯

Step 3 Comparison between ρ{ and ρ: ~ Denoting the Gauss curvature on Σs by C and the mean curvature vector h we have

85 from the Gauss equation (3.1) 1 1 1 1 K ´ xH,~ H~ y ` χˆ ¨ χˆ “ ´ R ´ GpL,Lq ´ xR L,Ly 4 2 2 4 LL

1 2 1 2 “ ´ R ´ GpL, l ´ |∇ω| L ´ 2∇ωq ´ xR 2 L, l ´ |∇ω| L ´ 2∇ωy 2 4 L l´|∇ω| L´2∇ω 1 1 “ C ´ x~h,~hy ` Kˆ ¨ Qˆ ` |∇ω|2GpL,Lq ` 2GpL, ∇ωq ´ xR l, Ly 4 2 L ∇ω

´ xRL ∇ωL, ∇ωy

1 1 1 “ C ´ x~h,~hy ` Kˆ ¨ Qˆ ` |∇ω|2GpL,Lq ` 2GpL, ∇ωq ´ xR l, Ly 4 2 2 L ∇ω ´ ¯ ´ αˆp∇ω, ∇ωq from this we conclude 1 1 K ´ xH,~ H~ y ` ∇{ ¨ ζ ´ ∆{ log tr χ ´ C ´ x~h,~hy ` ∇ ¨ t ´ ∆ log tr K 4 4 ´ ¯ ´ ¯ 1 “ Kˆ ¨ Qˆ ´ χˆ ¨ χˆ 2 ´ ¯ 1 ` |∇ω|2GpL,Lq ` 2GpL, ∇ωq ´ xR l, Ly ´ αˆp∇ω, ∇ωq 2 L ∇ω ´ ¯ ´ 2p∇ ¨ Kˆ qp∇ωq ´ Hω ¨ Kˆ ´ 2Kˆ p∇ω,~t ´ ∇ log tr Kq

ˆ 2 2 ´ tr Ktp∇ωq ` ∇ω tr K ` |K| |∇ω| ` GLp∇ωq ` αˆp∇ω, ∇ωq

|Kˆ |2 ` GpL,Lq |Kˆ |2 ` GpL,Lq |Kˆ |2 ` GpL,Lq ` ∆{ ω ` L |∇ω|2 ` 2∇ω tr K tr K tr K

` tr KKˆ p∇ω, ∇ωq ´ κKˆ p∇ω, ∇ωq

Isolating the ﬁrst two terms and using Lemma 5.1.1 we get Kˆ ¨ Qˆ ´ χˆ ¨ χˆ

“ Kˆ ¨ Qˆ ´ Kˆ ¨ Qˆ ´ |∇ω|2|Kˆ |2 ´ 4Kˆ p∇ω,~tq ` 2|Kˆ |2|∇ω|2 ´ ` 2 tr KKˆ p∇ω, ∇ωq ´ 2Kˆ ¨ Hω ´ 2κKˆ p∇ω, ∇ωq ¯ “ ´|Kˆ |2|∇ω|2 ´ 2 tr KKˆ p∇ω, ∇ωq ` 2Kˆ ¨ Hω ` 4Kˆ p∇ω,~tq ` 2κKˆ p∇ω, ∇ωq

86 and ﬁnally we have 1 1 K ´ xH,~ H~ y ` ∇{ ¨ ζ ´ ∆{ log tr χ ´ C ´ x~h,~hy ` ∇ ¨ t ´ ∆ log tr K 4 4 ´ ¯ ´ ¯ 1 “ |∇ω|2 |Kˆ |2 ` GpL,Lq ` G p∇ωq ´ 2Kˆ p∇ω,~t ´ ∇ log tr Kq 2 L ´ ¯ ˆ ˆ ` 2GLp∇ωq ´ xRL ∇ωl, Ly ´ 2p∇ ¨ Kqp∇ωq ` 2Kp∇ω,~tq ´ tr Ktp∇ωq ` ∇ω tr K ´ ¯ |Kˆ |2 ` GpL,Lq |Kˆ |2 ` GpL,Lq |Kˆ |2 ` GpL,Lq ` ∆{ ω ` L |∇ω|2 ` 2∇ω . tr K tr K tr K

Amazingly the third line vanishes by the Codazzi equation (3.2) as well as all terms with a factor κ giving 1 ρ ´ ρ “ |Kˆ |2 ` GpL,Lq |∇ω|2 ` G p∇ωq ´ 2Kˆ p∇ω,~t ´ ∇ log tr Kq { 2 L ´ ¯ |Kˆ |2 ` GpL,Lq |Kˆ |2 ` GpL,Lq |Kˆ |2 ` GpL,Lq ` ∆{ ω ` L |∇ω|2 ` 2∇ω tr K tr K tr K

and the result then follows from the fact that ∇{ ω “ ∇ω ` |∇ω|2L as well as

|Kˆ |2 ` GpL,Lq |Kˆ |2 ` GpL,Lq |Kˆ |2 ` GpL,Lq ∇{ ¨ ∇{ ω “ ∆{ ω ` ∇{ ω . tr K tr K tr K ´ ¯

5.3 Asymptotic ﬂatness

In this section we wish to study the limiting behaviour of our mass functional in the setting of asymptotic ﬂatness constructed by Mars and Soria [18]. Beyond the

assumption that we have a cross-section Σs0 of Ω we also assume for some (hence any) choice of past-directed geodesic null generator L (i.e. DLL “ 0) that S` “ 8.

L So all geodesics γq are ‘past complete’ with domain ps´pqq, 8q. We now take s0 “ 0

2 ignoring all points p satisfying sppq ď S´ and conclude that Ω – S ˆ pS´, 8q. Al-

though the value of S´ will depend on our choice of geodesic generator L our interest 87 2 lies only on the past of Σ0 (i.e. S ˆ p0, 8q) so we ignore this subtlety. A null hypersurface Ω with all the above properties is called extending to past null inﬁnity. In order to impose decay conditions of various transversal tensors (i.e. tensors satisfying T pL, ¨ ¨ ¨ q “ ¨ ¨ ¨ “ T p¨ ¨ ¨ ,Lq “ 0) we choose a local basis on Σ0 and extend it

to a basis ﬁeld tXiu Ă EpΣ0q. Given a transversal k-tensor T psq we say,

´m • T “ Op1q iﬀ Ti1...ik :“ T pXi1 , ..., Xik q is uniformly bounded and T “ Onps q iﬀ

m`j j s pLLq T psq “ Op1q p0 ď j ď nq

´m m ´m • T “ ops q iﬀ lim s T psqi1...i “ 0 and T “ onps q iﬀ sÑ8 k

m`j j s pLLq T psq “ op1q p0 ď j ď nq

X ´m • T “ on ps q iﬀ

smL ¨ ¨ ¨ L T psq “ op1q p0 ď j ď nq. Xi1 Xij

Now we’re ready to deﬁne asymptotic ﬂatness for Ω as given by the authors of [18]:

Deﬁnition 5.3.1. We say Ω is past asymptotically ﬂat if it extends to past null in-

ﬁnity and there exists a choice of cross-section Σ0 and null geodesic generator L with corresponding level set function s satisfying the following:

1. There exists two symmetric 2-covariant transversal and L Lie constant tensor

ﬁelds ˚γ and γ1 such that

2 X γ˜ :“ γ ´ s ˚γ ´ sγ1 “ o1psq X o2 psq

88 2. There exists a tansversal and L Lie constant one-form t1 such that

t t˜:“ t ´ 1 “ o ps´1q s 1

3. There exist L Lie constant functions θ0 and θ such that

θ θ θ˜ :“ tr Q ´ 0 ´ “ ops´2q s s2

1 4. The scalar xRXi Xi Xi3 ,Xi4 y along Ω is such that lim xRXi Xi Xi3 ,Xi4 y exists 1 2 sÑ8 s2 1 2 1 1 ´2 while its double trace satisﬁes ´ 2 R ´ GpL, lq ´ 4 xRLlL, ly “ ops q.

We will have the need to supplement the notion of asymptotic ﬂatness of Ω with a

´2 X stronger version of the energy ﬂux decay condition (GL “ ops q, LLγ˜ “ o1 p1q as given in [18]) with the following:

Definition 5.3.2. Suppose Ω is past asymptotically flat. We say Ω has strong flux decay if ´2 ˜ X ´1 j X 1´j GL “ ops q, t “ o1 ps q and LLγ˜ “ o3´jps q for 1 ď j ď 3 and strong decay if the condition on GL is dropped.

We will also need some results from [18] (Proposition 3, Lemma 2, Section 4) resulting directly from the asymptotically ﬂat restriction on Ω. One particularly valuable consequence is the ability to choose our geodesic generator L to give any conformal change on the ‘metric at null inﬁnity’, which turns out to be given by the 2-tensor, ˚γ. By the Uniformization Theorem we conclude that this covers all possible metrics on a Riemannian 2-sphere. We will denote the covariant derivative coming from ˚γ by ∇˚.

89 Proposition 5.3.1. Suppose Ω is past asymptotically ﬂat with a choice of aﬃnely

parametrized null generator L and corresponding level set function s. Letting γpsqij

denote the inverse of γpsqij, 1 1 γpsqij “ ˚γij ´ γ˚ij ` ops´3q (5.1) s2 s3 1 1 K “ s˚γ ` γ ` op1q (5.2) ij ij 2 1ij

K˚ K “ ` ops´2q (5.3) γpsq s2

2K˚ θ tr Q “ ` ` ops´2q (5.4) s s2 2 θ tr K “ ` ` ops´2q (5.5) s s2

ij where ˚γ is the inverse of ˚γij, tensors with ring˚ highlight the fact that indices have 1 ˚ been raised with ˚γ and θ “ ´ 2 trγ1. X It follows in case LLγ˜ “ o1 p1q that

1 t “ ∇˚ ¨ γ ` dθ{ ðñ G “ ops´2q 1 2 1 L

Proof. We refer the reader to [18] (Proposition 3) for proof.

As promised in Remark 2.1.1 we are now able to prove the following well known result:

Lemma 5.3.1. Suppose Ω extends to past null inﬁnity with null geodesic generator

L. Then any cross-section Σ ãÑ Ω satisﬁes tr K ě 0. If Ω is past asymptotically ﬂat then Σ is expanding along L.

Proof. For ω P FpΩq constructed by Lie dragging s|Σ along L we have Σ “ Σ1 for the geodesic foliation tΣλu given by s “ ωλ. So it suﬃces to prove the result along

90 an arbitrary geodesic foliation for Ω. From (8) we have, whenever tr Kps0q ă 0 for

some s0, that 1 1 1 L “ ` |χˆ´|2 ` GpL´,L´q ě tr K 2 2 ´ ¯ wherever it be deﬁned as well as

1 1 s ´ s psq ě ps q ` 0 tr K tr K 0 2

´ sÑs1 for any such s ě s0. So we can find an s1 ą s0 such that tr Kpsq ÝÝÝÑ´8. Since this contradicts smoothness we must have that tr K ě 0 on all of Ω. If Ω is past asymptotically flat it follows from Proposition 5.3.1 that tr Kpsq ą 0 for sufficiently large s. Since (4.4) gives

1 Lptr Kq “ ´ ptr Kq2 ´ |Kˆ |2 ´ GpL,Lq ď 0 2

we have tr Kps0q ě tr Kps1q for all s0 ď s1. So we must have that tr K ą 0 on all of Ω.

Lemma 5.3.2. On each Σs the diﬀerence tensor

˚ DpV,W q :“ ∇V W ´ ∇V W admits the decomposition

1 1 Dk “ p∇˚ γ˚k ` ∇˚ γ k ´ ∇˚kγ q ` Ops´2q. ij 2 i 1 j j 1 i 1ij s

Moreover, if f P FpΩq is Lie constant along L then

1 1 ∆f “ ∆˚f ` p´˚γij∇˚ ∇˚ f ´ p∇˚ ˚γijqf, `p∇˚iθqf, q ` ops´3q. s2 1 i j i 1 j i s3

Proof. The result follows from the well known fact (see, for example, [29]) that 1 xDpV,W q,Uy “ p∇˚ γpW, Uq ` ∇˚ γpV,Uq ´ ∇˚ γpV,W qq 2 V W U 91 s “ p∇˚ γ pW, Uq ` ∇˚ γ pV,Uq ´ ∇˚ γ pV,W qq 2 V 1 W 1 U 1 1 ` p∇˚ γ˜pW, Uq ` ∇˚ γ˜pV,Uq ´ ∇˚ γ˜pV,W qq. 2 V W U

The second is a simple consequence of the ﬁrst, we refer the reader to [18] (Lemma 2) for proof.

In the next Proposition we show that the decomposition of the metric given in

´4 Deﬁnition 5.3.1 part 1 allows us to ﬁnd Kγpsq up to Ops q:

2 Proposition 5.3.2. For a decomposition of the metric γpsq “ s ˚γ `sγ1 `γ˜ for some ﬁxed s we have:

K˚ 1 1 K “ ` pK˚θ ` ∇˚ ¨ ∇˚ ¨ γ ` ∆˚θq ` Ops´4q (5.6) γpsq s2 s3 2 1

˚ Proof. First we take the opportunity to show that V,W P EpΣ0q gives ∇V W P

EpΣ0q. Starting with the Koszul formula ˚ 2˚γp∇V W, Uq “ V ˚γpW, Uq ` W˚γpU, V q ´ U˚γpV,W q

´ ˚γpV, rW, Usq ` ˚γpW, rU, V sq ` ˚γpU, rV,W sq

˚ and the fact that ˚γ is Lie constant along L we conclude that L˚γp∇V W, Uq “ ˚ ˚γprL, ∇V W s,Uq on the left, applying L on the right we find everything vanishes ˚ since V,W P EpΣ0q ùñ rV,W s P EpΣ0q. Therefore ˚γprL, ∇V W s,Uq “ 0. Since ˚ ˚ rL, ∇V W s P ΓpT Σsq and ˚γ is positive definite it follows that rL, ∇V W s “ 0 and ˚ therefore ∇V W P EpΣ0q. To show the decomposition of Kγpsq we start by finding the decomposition of the Riemann curvature tensor on Σs:

Rs X ,X ∇ X ,X X ∇ X ,X ∇ X , ∇ X x XiXj k my “ x rXi,Xj s k my ´ ix Xj k my ` x Xj k Xi my

` Xjx∇Xi Xk,Xmy ´ x∇Xi Xk, ∇Xj Xmy ˚ ˚ ˚ ˚ “ x∇rXi,Xj sXk,Xmy ´ Xix∇Xj Xk,Xmy ` x∇Xj Xk, ∇Xi Xmy

92 ˚ ˚ ˚ ` Xjx∇Xi Xk,Xmy ´ x∇Xi Xk, ∇Xj Xmy ˚ ` xDprXi,Xjs,Xkq,Xmy ´ XixDpXj,Xkq,Xmy ` xDpXj,Xkq, ∇Xi Xmy ˚ ˚ ` x∇Xj Xk, DpXi,Xmqy ` XjxDpXi,Xkq,Xmy ´ xDpXi,Xkq, ∇Xj Xmy ˚ ´ x∇Xi Xk, DpXj,Xmqy

` xDpXj,Xkq, DpXi,Xmqy ´ xDpXi,Xkq, DpXj,Xmqy.

2 Using the decomposition γs “ s ˚γ ` Opsq we recognize the leading order term, com- 2 ˚ bining lines 3 and 4, is s ˚γpRXiXj Xk,Xmq. In order to ﬁnd the next to leading order term the fact that Rs X ,X s2˚γ R˚ X ,X deﬁnes a 4-tensor on each Σ x XiXj k my ´ p XiXj k mq s

allows us to search independently of our choice of basis tX1,X2u. In particular we ˚ L ˚ may assume that ∇Xi Xj “ 0 at q P Σs (hence on all of γq , since ∇Xi Xj P EpΣ0q). So assuming restriction to the generator through q and using Lemma 5.3.2 we have

Rs X ,X s2˚γ R˚ X ,X x XiXj k my ´ p XiXj k mq ˚ ˚ “ ´sXiγ1p∇Xj Xk,Xmq ` sXjγ1p∇Xi Xk,Xmq s ´ X p∇˚ γ pX ,X q ` ∇˚ γ pX ,X q ´ ∇˚ γ pX ,X qq 2 i Xj 1 k m Xk 1 j m Xm 1 j k s ` X p∇˚ γ pX ,X q ` ∇˚ γ pX ,X q ´ ∇˚ γ pX ,X qq 2 j Xi 1 k m Xk 1 i m Xm 1 i k ` Op1q

˚ ˚ “ ´sXiγ1p∇Xj Xk,Xmq ` sXjγ1p∇Xi Xk,Xmq s ∇˚ ∇˚ γ pX ,X q ` ∇˚ ∇˚ γ pX ,X q ´ ∇˚ ∇˚ γ pX ,X q 2 Xj Xi 1 k m Xj Xk 1 i m Xj Xm 1 i k ´ ˚ ˚ ˚ ˚ ˚ ˚ ´ ∇Xi ∇Xj γ1pXk,Xmq ´ ∇Xi ∇Xk γ1pXj,Xmq ` ∇Xi ∇Xm γ1pXj,Xkq ¯ ` Op1q.

It remains to simplify the two terms of the ﬁrst line in the second equality. Since

˚ ˚ ˚ ˚ ˚ Xiγ1p∇Xj Xk,Xmq “ ∇Xi γ1p∇Xj Xk,Xmq ` γ1p∇Xi ∇Xj Xk,Xmq 93 we conclude that

˚ ˚ ˚ ´Xiγ1p∇Xj Xk,Xmq ` Xjγ1p∇Xi Xk,Xmq “ γ1pRXiXj Xk,Xmq.

Moreover, it is easily shown using our choice of basis extension that 1 1 ∇˚ ∇˚ γ pX ,X q ´ ∇˚ ∇˚ γ pX ,X q ` γ pR˚ X ,X q 2 Xj Xi 1 k m 2 Xi Xj 1 k m 1 XiXj k m 1 “ pγ pR˚ X ,X q ´ γ pR˚ X ,X qq. 2 1 XiXj k m 1 XiXj m k

So we ﬁnally have from the fact that Σs is of dimension 2 that

Rs X ,X s2K˚ ˚γ ˚γ ˚γ ˚γ x XiXj k my “ p ik jm ´ im jkq s ` K˚p˚γ γ ´ ˚γ γ ` ˚γ γ ´ ˚γ γ q 2 ik 1jm im 1jk jm 1ik jk 1im s ` p∇˚ ∇˚ γ ´ ∇˚ ∇˚ γ ´ ∇˚ ∇˚ γ ` ∇˚ ∇˚ γ q ` Op1q. 2 j k 1im j m 1ik i k 1jm i m 1jk

Using (12) to take a trace over i, k: 1 pRicsq “ K˚˚γ ´ K˚θ˚γ jm jm s jm

1 K˚ ` p∇˚ p∇˚ ¨ γ q ` 2∇˚ ∇˚ θ ´ p∇˚2γ q ` p∇˚ ¨ p∇˚γ qq q ` p2θ˚γ ` γ q 2s j 1 m j m 1 jm 1 mj s jm 1jm ` Ops´2q

˚ “ K˚γjm

1 1 1 1 ` K˚θ˚γ ` K˚γ ` ∇˚ p∇˚ ¨ γ q ` p∇˚ ¨ p∇˚γ qq ` ∇˚ ∇˚ θ ´ p∇˚2γ q s jm 1jm 2 j 1 m 2 1 mj j m 2 1 jm ´ ¯ ` Ops´4q

and then over j, m:

2 1 2 2K “ K˚ ` 2K˚θ ´ 2K˚θ ` ∇˚ ¨ ∇˚ ¨ γ ` 2∆˚θ ` K˚θ ` Ops´4q γpsq s2 s3 1 s3 ´ ¯ giving the result.

94 Remark 5.3.1. Interestingly, in the case that Ω is asymptotically ﬂat satisfying the energy ﬂux decay condition we conclude that

K˚ 1 K “ ` pK˚θ ` ∇˚ ¨ t q ` Ops´4q γpsq s2 s3 1

according to Proposition 5.3.2.

Deﬁnition 5.3.3. For Ω past asymptotically ﬂat with background geodesic foliation

tΣsu we say a foliation tΣs‹ u is asymptotically geodesic provided

s “ φs‹ ` ξ

i X 1´i with scale factor φ ą 0 a Lie constant function along L and L ξ “ o2´ips q for

0 ď i ď 2. In addition (similarly to [18]), we will say tΣs‹ u approaches large spheres provided the class of geodesic foliations measuring φ “ 1 also induce ˚γ to be the

round metric on S2.

Remark 5.3.2. Given a basis extension tXiu Ă EpΣ0q (on tΣsu) and a foliation

tΣs‹ u as in Deﬁnition 5.3.3, Lie dragging s|Σs‹ along L to give ω P FpΩq we see at q P Σs‹ :

ωi “ φis‹ ` ξsωi ` ξi

ωij “ φijs‹ ` ξssωiωj ` ξsjωi ` ξsωij ` ξij

where ωi :“ Xiω, ωij :“ XjXiω, ξs :“ Lξ, ξss :“ LLξ, ξi :“ Xipξ|Σspqqq, ξsi “ Xipξs|Σs q and ξij “ XjXipξ|Σs q. The decay on ξ therefore gives us that:

φis‹ ` ξi ωi “ “ φis‹ ` ops‹q 1 ´ ξs

1 ξi ` φis‹ ξj ` φjs‹ ξi ` φis‹ ωij “ φijs‹ ` ξss ` ξsj ` ξij 1 ´ ξs 1 ´ ξs 1 ´ ξs 1 ´ ξs ´ ´ ¯´ ¯ ´ ¯ ¯ “ φijs‹ ` ops‹q.

95 From (12) and Lemma 5.3.2 we conclude that

2 ´1 ´1 ´1 dω|Σs‹ “ ps‹φq p dφ |Σs‹ ` ops‹ qq s‹ 1 ∆ω ∆˚φ o s´1 . |Σs‹ “ 2 ` p ‹ q φ s‹

5.3.1 Asymptotic Mass and Energy

Using Theorem 5.2.1 we prove a slightly weakened version of the beautiful result found by the authors of [18] (Theorem 1):

Proposition 5.3.3. Suppose Ω is past asymptotically ﬂat and tΣs‹ u is an asymptot-

X ically geodesic foliation with scale factor φ ą 0. Assuming LLγ˜ “ o1 p1q we have

2 ˚ 1 φ dA 1 ˚ ˚ ˚ ˚ lim EH pΣs‹ q “ Kθ ´ θ ´ ∆θ ` 4∇ ¨ t1 dA s‹Ñ8 16π d 4π φ ş ż ´ ¯ ˚ with ˚γ, K, θ, θ and t1 associated with the background geodesic foliation.

Proof. Given any ﬁxed s‹ we deﬁne ω P FpΩq by Lie dragging along L:

s|Σs‹ “ pφs‹ ` ξq|Σs‹

2 as before. From the decomposition γs “ s ˚γ ` sγ1 ` γ˜ and the standard identity for any invertible matrix M:

detpM ` sBq “ det Mp1 ` s trpM ´1Bq ` Ops2qq we have 1 detpγ q “ s2 detp˚γqp1 ´ θ ` ops´1qq. s s a a From the ﬁrst identity of Lemma 5.1.1 we therefore conclude that

2 2 ˚ dAs‹ “ dAs|Σs‹ “ s‹φ fdA

96 0 where f “ 1 ` ops‹q.

In Theorem 5.2.1, denoting the sum of all but the ﬁrst two terms by ηωp∇ωq we see

1 |Kˆ |2 ` GpL,Lq η p∇ωq “ p|Kˆ |2 ` GpL,Lqq|∇ω|2 ` GpL, ∇ωq ` ∇ω ω 2 tr K

´ 2Kˆ p~t ´ ∇ log tr K, ∇ωq

1 1 “ p|Kˆ |2 ` GpL,Lqq|∇ω|2 ` GpL, ∇ωq ´ ∇ωpL log tr K ` tr Kq 2 2

´ 2Kˆ p~t ´ ∇ log tr K, ∇ωq giving from Propositions 5.3.1 and 5.3.2

ˆ 2 EH pΣs‹ q |K| ` GpL,Lq 4π “ ρdA{ ω “ ρ ` ∇{ ¨ ∇{ ω ` ηωp∇ωqdAs‹ |Σs‹ | tr K 16π ż ż ´ ¯ b

“ ρ ` ηωp∇ωqdAs‹ ż K˚ 1 1 1 2 θ 2K˚ θ “ ` K˚θ ` ∇˚ ¨ ∇˚ ¨ γ ` ∆˚θ ´ p ` qp ` q ω2 ω3 2 1 4 ω ω2 ω ω2 ż ´ ´ ¯ ` ∇ ¨ t ` GLp∇ωq

1 ` p|Kˆ |2 ` GpL,Lqq|∇ω|2 2 1 ´ ∆ log tr K ´ ∇ωpL log tr K ` tr Kq ´ 2Kˆ p~t ´ ∇ log tr K, ∇ωq dA 2 s‹ ¯ ´1 ` ops‹ q

1 1 1 1 “ K˚θ ´ θ ` ∇˚ ¨ ∇˚ ¨ γ ` ∆˚θ ω3 2 2 2 1 ż ´ ´ ¯ 1 ` ∇ ¨ t ` G p∇ωq ´ ∇ω tr K ´ 2Kˆ p~t, ∇ωq L 2 1 ` p|Kˆ |2 ` GpL,Lqq|∇ω|2 2 ˆ ´ ∆ log tr K ´ ∇ωL log tr K ` 2Kp∇ log tr K, ∇ωq dAs‹ ¯ 97 ´1 ` ops‹ q having used the Divergence Theorem to get the second line. From Lemma 5.2.2 we have

´∆ log tr K ´∇ωL log tr K `2Kˆ p∇ log tr K, ∇ωq “ ´∆{ log tr K `∇{ ¨pL log tr K∇{ ωq

and therefore integrates to zero on Σs‹ by the Divergence Theorem. We also notice from the fact that L is geodesic and ∇{ ω “ ∇ω ` |∇ω|2L that

KpV, ∇ωq “ KpV,˜ ∇{ ωq “ χpV,˜ ∇{ ωq

for V P EpΣ0q. Lemma 5.1.1 and 5.2.1 therefore gives ˜ ˜ ˜ ˜ ˜ ˜ ˜ ∇{ V˜ ζpW q ` p∇{ V˜ pχ ¨ dω{ qqpW q “ V pζpW q ` χpW, ∇{ ωqq ´ ζp∇{ V˜ W q ´ χp∇{ V˜ W, ∇{ ωq

~ ~ “ pV ` V ωLqtpW q ´ tp∇V W ` V ωKpW q ` W ωKpV q ´ KpV,W q∇ωq ´ ~ ~ ˜ ´ Kp∇V W ` V ωKpW q ` W ωKpV q ´ KpV,W q∇ω, ∇ωq ´ χp∇{ V˜ W, ∇{ ωq ¯ “ ∇V tpW q ` V ωLLtpW q ´ V ωKp~t, ∇ωq ´ W ωKp~t, ∇ωq ` KpV,W qtp∇ωq where all terms in the penultimate line canceled from Lemma 5.2.1. Taking a trace over V,W ˆ ∇{ ¨ ζ ´ ∇{ ¨ p~χp∇{ ωqq “ ∇ ¨ t ` LLtp∇ωq ´ 2Kp~t, ∇ωq

1 “ ∇ ¨ t ` G p∇ωq ´ ∇ω tr K ´ 2Kˆ p~t, ∇ωq ´ ∇ ¨ Kˆ p∇ωq ` ∇ω tr K ´ tr Ktp∇ωq L 2 having used (11) to get the last line. We conclude that

1 ∇¨t`G p∇ωq´ ∇ω tr K´2Kˆ p~t, ∇ωqdA “ ∇¨Kˆ p∇ωq´∇ω tr K`tr Ktp∇ωqdA L 2 s‹ s‹ ż ż giving E Σ 1 1 1 1 H p s‹ q ˚ ˚ ˚ ˚ 4π “ 3 Kθ ´ θ ` ∇ ¨ ∇ ¨ γ1 ` ∆θ |Σs‹ | ω 2 2 2 16π ż ´ ´ ¯ b 98 1 ` ∇ ¨ Kˆ p∇ωq ´ ∇ω tr K ` tr Ktp∇ωq ` p|Kˆ |2 ` GpL,Lqq|∇ω|2 dA 2 s‹ ¯ ´1 ` ops‹ q.

Now we turn to simplifying the ﬁnal term in the integrand

1 p|Kˆ |2 ` GpL,Lqq|∇ω|2 “ p∇ω ´ ∇{ ωq tr K ´ ptr Kq2|∇ω|2. 2

2 θ X Denoting g :“ tr K ´ s ´ s2 we conclude from the hypothesis LLγ˜ “ o1 p1q that X ´2 g “ o1 ps q. So denoting gω :“ g|Σs‹ we have from Remark 5.3.2 (regarding the decay) that 1 ∇ω tr K| “ ∇ωg| ` ∇ωθ| Σs‹ Σs‹ s2 Σs‹ 1 “ ∇{ ωθ ` ops´3q ω2 ‹ 2 θ ∇{ ω tr K “ ∇{ ωg ` ∇{ ωp ` q ω ω ω2 2 θ “ ∇{ ¨ pg ∇{ ωq ´ ∆{ ωg ´ |∇{ ω|2 ` ∇{ ωp q ω ω ω2 ω2 2 θ “ ∇{ ¨ pg ∇{ ωq ´ p∆ω ´ 2Kˆ p∇ω, ∇ωqqg ´ |∇{ ω|2 ` ∇{ ωp q ω ω ω2 ω2 2 θ “ ∇{ ¨ pg ∇{ ωq ´ |∇{ ω|2 ` ∇{ ωp q ` ops´3q ω ω2 ω2 ‹ 1 1 2 θ ptr Kq2|∇ω|2| “ p ` q2|∇{ ω|2 ` ops´3q 2 Σs‹ 2 ω ω2 ‹ 2 2θ “ |∇{ ω|2 ` |∇{ ω|2 ` ops´3q. ω2 ω3 ‹

Combining terms we conclude

ˆ 2 2 ´3 p|K| ` GpL,Lqq|∇ω| “ ´∇{ ¨ pgω∇{ ωq ` ops‹ q. Σs‹ ˇ ˇ ˇ It’s a simple exercise to show Kˆ 1 γ θ˚γ oX 1 , so for dω “ ´ 2 p 1 ` q ` 1 p q |Σs‹ 99 2 2 1 ´1 ´1 “ ps φ qp´ dφ |Σ ` ops qq we have from Lemma 5.3.2 ‹ s‹ s‹ ‹ 1 1 ∇ Kˆ ∇ω ∇˚ γ θ˚γ ∇˚φ´1 o s´3 ¨ p q|Σs‹ “ 3 2 ¨ p 1 ` qp q ` p ‹ q s‹ 2φ 1 1 ∇ω tr K ∇˚φ´1θ o s´3 |Σs‹ “ ´ 3 2 ` p ‹ q s‹ φ 1 2 tr Kt ∇ω t ∇˚φ´1 o s´3 . p q|Σs‹ “ ´ 3 2 1p q ` p ‹ q s‹ φ

Therefore

1 φ2fdA˚ f 1 1 1 E pΣ q “ K˚θ ` ∇˚ ¨ ∇˚ ¨ γ ` ∆˚θ ´ θ H s‹ 8π d 4π φ 2 2 1 2 ş ż ´ ´ ¯ f ` ∇˚ ¨ pγ ` θ˚γqp∇˚φ´1q ` f∇˚φ´1θ ´ 2ft p∇˚φ´1q dA˚ ` ops0q 2 1 1 ‹ ¯ giving

2 ˚ 1 φ dA 1 1 ˚ 1 ˚ ˚ ˚ 1 lim EH pΣs‹ q “ Kθ ` ∇ ¨ ∇ ¨ γ1 ` ∆θ ´ θ s‹Ñ8 8π d 4π φ 2 2 2 ş ż ´ ´ ¯ 1 ` ∇˚ ¨ pγ ` θ˚γqp∇˚φ´1q ` ∇˚φ´1θ ´ 2t p∇˚φ´1q dA˚ 2 1 1 ¯ 1 φ2dA˚ 1 1 1 1 “ K˚θ ` ∇˚ ¨ ∇˚ ¨ γ ` ∆˚θ ´ θ 8π d 4π φ 2 2 1 2 ş ż ´ ´ ¯ 1 2 ´ φ´1∇˚ ¨ ∇˚ ¨ pγ ` θ˚γq ´ φ´1∆˚θ ` ∇˚ ¨ t dA˚ 2 1 φ 1 ¯

1 φ2dA˚ 1 “ K˚θ ´ θ ´ ∆˚θ ` 4∇˚ ¨ t dA˚ 16π d 4π φ 1 ş ż ´ ¯ having integrated by parts to get the second equality.

Remark 5.3.3. Suppose Ω is a past asymptotically flat null hypersurface with a background geodesic foliation tΣsu approaching large spheres (i.e ˚γ is the round metric 100 at infinity). Then for any other geodesic foliation of scale factor ψ it follows that the metric at infinity is ψ2˚γ (see [18], Section 4) approaching large spheres if and only if ψ solves the equation

1 ´ ψ2 “ ∆˚ log ψ. (5.7)

Proposition 5.3.3 shows all asymptotically geodesic foliations tΣs‹ u of the same scale factor φ share the limit

Epφq “ lim EH pΣs‹ q s‹Ñ8 which measures a Bondi energy EBpψq if ψ solves (5.7). The Bondi mass is therefore given by 2 ˚ mB “ inftEBpψq|1 ´ ψ “ ∆ log ψu.

Theorem 5.3.4. Suppose Ω is a past asymptotically ﬂat null hypersurface inside a spacetime satisfying the dominant energy condition. Then given the existence of an

asymptotically geodesic (P)-foliation tΣs‹ u approaching large spheres we have

mp0q ď EB

for EB the Bondi energy of Ω associated to tΣs‹ u. If equality is achieved on an

(SP)-foliation then EB “ mB the Bondi mass of Ω. In the case that tr χ|Σ0 “ 0 we conclude instead with the weak Null Penrose inequality

|Σ0| ď EB c 16π where equality along an (SP)-foliation enforces that any foliation of Ω shares its data (γ, χ, tr χ and ζ) with some foliation of the standard nullcone of Schwarzschild spacetime.

Proof. Since any asymptotically geodesic (P)-foliation has non-decreasing mass from

Theorem 2.1.1 and mpΣs‹ q ď EH pΣs‹ q from Lemma 3.2.2, it follows from [18] (The- 101 orem 1) that mpΣs‹ q converges since EH pΣs‹ q does. Moreover,

lim mpΣs‹ q ď lim EH pΣs‹ q s‹Ñ8 s‹Ñ8

and from [18] (Corollary 3) it follows that lims‹Ñ8 EH pΣs‹ q is the Bondi energy

associated to the abstract reference frame coupled to the foliation tΣs‹ u. Given

the case of equality, Theorem 2.1.1 enforces that mp0q “ mpΣs‹ q for all s‹. So

3 2 2 1 1 3 Theorem 4.3.2 applies and we conclude that mpΣq “ 2 4π r0 dA0 (for some

2 ´ ş ¯ positive function r0 on Σ0 of area form r0dA0) irrespective of the cross-section Σ Ă Ω. This gives, according to Remark 5.3.3 and Lemma 3.2.2,

3 2 2 1 1 3 lim mpΣs‹ q “ r0 dA0 “ EB ď inf Epφq ď mB. s‹Ñ8 2 4π φą0 ´ ż ¯

Since EB ď inf Epφq ď mB ď EB all must be equal.

If tr χ|Σ0 “ 0 property (P) gives

0 ě ∆{ log ρ{|Σ0

and the maximum principle implies ρ{|Σ0 “ K ` ∇{ ¨ τ is constant. From the Gauss- 4π Bonnet and Divergence Theorems we conclude that ρ|Σ “ from which it follows { 0 |Σ0|

|Σ0| that mp0q “ 16π . Under this restriction Theorem 4.3.2 enforces that any foliation b of Ω corresponds with a foliation of the standard nullcone in Schwarzschild with respect to the data γ, χ, tr χ and ζ.

From Proposition 5.3.3 and Lemma 3.2.2

3 1 1 ˚ ˚ 2 ˚ 2 inf Epφq “ pKθ ´ θ ´ ∆θ ` 4∇ ¨ t1q 3 dA φą0 4 4π ´ ż ¯ ˚ ˚ provided Kθ ´ θ ´ ∆θ ` 4∇ ¨ t1 ě 0. We show, given that Ω satisﬁes the strong

ﬂux decay condition, this quantity is infact lim mpΣs‹ q. We will need the following s‹Ñ8 proposition to do so:

102 Proposition 5.3.5. Suppose Ω is past asymptotically ﬂat with strong decay. Given a choice of aﬃnely parametrized null generator L and corresponding level set function, s, we have 1 ∇ ¨ ∇ ¨ K “ ´ ∇˚ ¨ ∇˚ ¨ γ ` ops´4q (5.8) 2s4 1

∆˚θ ∆ tr K “ ` ops´4q (5.9) s4 1 ∇ ¨ t “ ∇˚ ¨ t ` ops´3q (5.10) s3 1

Proof. From Lemma 5.3.2 and (5.2) 1 ∇ K “ ∇˚ ps˚γ ` γ q ´ Dk K ´ Dk K ` oX p1q i jm i jm 2 1jm ij km im jk 1 1 “ ∇˚ γ ´ ∇˚ γ ` oX p1q 2 i 1jm i 1jm 1 1 “ ´ ∇˚ γ ` oX p1q. 2 i 1jm 1 where the ﬁrst term of the second line comes from the fact that ∇˚˚γ “ 0. Next we compute

˚ k k k ∇i∇jKmn “ ∇i∇jKmn ´ Dij∇kKmn ´ Dim∇jKkn ´ Din∇jKmk 1 “ ´ ∇˚ ∇˚ γ ` op1q 2 i j 1mn

So contracting with (5.1) over j, m followed by i, n we get (5.8) and contracting instead over m, n and then i, j (5.9) follows. For (5.10)

˚ k ∇itj “ ∇itj ´ Dijtk 1 “ ∇˚ t ` ops´1q s i 1j

and the result follows as soon as we contract with (5.1) over i, j.

103 Remark 5.3.4. As soon as we impose that Ω has strong decay it follows from the

fact that rLX , LY s “ LrX,Y s that LXi γ˜, and LXi LXj γ˜ “ o1psq, since, for example,

LLLXi γ˜ “ LrL,Xisγ˜ ` LXi LLγ˜ “ op1q.

As a result its not hard to see early in the proof of Proposition 7 that

K˚ K K “ ` 1 ` O ps´4q γpsq s2 s3 1

for some Lie constant function K1. We may therefore provide a simpler proof using Proposition 5.3.5 and the propogation equation (4.1)

LK “ ´ tr KK ´ ∆ tr K ` ∇ ¨ ∇ ¨ K

in order to ﬁnd K1.

Theorem 5.3.6. Suppose Ω is past asymptotically ﬂat with strong ﬂux decay and

tΣsu is some background geodesic foliation. Then for any asymptotically geodesic

foliation tΣs‹ u with scale factor φ ą 0 we have

1 s3ρps q “ K˚θ ´ θ ´ ∆˚θ ` 4∇˚ ¨ t ` ops0q ‹{ ‹ 2φ3 1 ‹ ´ ¯ Proof. First let us remind ourselves of Theorem 5.2.1

|Kˆ |2 ` GpL,Lq 1 ρ “ ρ ` ∇{ ¨ ∇{ ω ` |Kˆ |2 ` GpL,Lq |∇ω|2 { tr K 2 ´ ¯ ´ ¯ |Kˆ |2 ` GpL,Lq ` ∇ω ` G p∇ωq ´ 2Kˆ p~t ´ ∇ log tr K, ∇ωq. tr K L

2 θ ´2 Denoting the exterior derivative on Σs by ds, since tr K “ s ` s2 `ops q, we conclude 1 ´1 that ds log tr K “ 2s dθ|Σs ` ops q giving

ˆ ~ ´3 Kpt ´ ∇ log tr K, ∇ωq|Σs‹ “ ops‹ q.

104 2 X ´1 ´1 Since LLγ˜ “ o1 ps q X o1ps q we also see that 1 2 2 1 2 θ |Kˆ |2 ` GpL,Lq “ ´L tr K ´ ptr Kq2 “ ´p´ ´ θq ´ p ` q2 2 s2 s3 2 s s2

X ´3 ´3 ` o1 ps q X o1ps q

X ´3 ´3 “ o1 ps q X o1ps q

and therefore, from Remark 5.3.2:

|Kˆ |2 ` GpL,Lq |Kˆ |2 ` GpL,Lq |Kˆ |2 ` GpL,Lq ∇{ ¨ ∇{ ω “ ∇{ ω ` ∆{ ω tr K tr K tr K ´ ¯ |Kˆ |2 ` GpL,Lq |Kˆ |2 ` GpL,Lq “ ∇ω ` |∇ω|2L tr K tr K ´ |Kˆ |2 ` GpL,Lq ` p∆ω ´ 2Kˆ p∇ω, ∇ωqq tr K Σs‹ ¯ˇ ´3 ˇ “ ops‹ q. ˇ

´3 From the strong ﬂux decay condition we have GLp∇ωq|Σs‹ “ ops‹ q also. From (5.9) we have ∆ tr K |∇ tr K|2 ∆ log tr K “ ´ tr K ptr Kq2

∆˚θ “ ` ops´3q 2s3

and combining this with Propositions 5.3.2 and 5.3.5:

´3 ρ{ “ ρ|Σs‹ ` ops‹ q 1 1 1 1 1 1 “ K˚θ ` ∇˚ ¨ ∇˚ ¨ γ ` ∆˚θ ´ θ ` ∇˚ ¨ t ´ ∆˚θ ` ops´3q ω3 2 2 1 2 ω3 1 2ω3 ‹ ´ ¯ 1 “ K˚θ ´ θ ´ ∆˚θ ` 4∇˚ ¨ t ` ops´3q 2ω3 1 ‹ ´ ¯ 1 ˚ ˚ ˚ ˚ having used Proposition 5.3.1 in the ﬁnal line to substitute 2 ∇ ¨ ∇ ¨ γ1 ` ∆θ “ ∇ ¨ t1 and the result follows.

105 Remark 5.3.5. We would like to bring to the attention of the reader our use of

(5.4) in the second to last equality in the proof of Theorem 5.3.6. Assuming tΣs‹ u is infact a geodesic foliation, running a parallel argument to decompose ρ{ as we did for

ρ allows us to conclude that (5.4) must also hold for tΣs‹ u. We refer the reader to [18] (Proposition 3) to observe that under the additional decay of Theorem 5.3.6, part 4 from Deﬁnition 5.3.1 is no longer necessary to give (5.4) for an arbitrary geodesic foliation provided it holds for at least one. We will exploit this fact in Section 6.2.

Corollary 5.3.6.1. With the same hypotheses as in Theorem 5.3.6 we have

3 1 1 2 2 ˚ ˚ ˚ 3 ˚ lim mpΣs‹ q “ pKθ ´ θ ´ ∆θ ` 4∇ ¨ t1q dA s‹Ñ8 4 4π ´ ż ¯ Proof. From Theorem 5.3.6 we directly conclude

2 2 2 1 3 2 4πp4mpΣ qq 3 “ p2ρq 3 dA “ K˚θ ´ θ ´ ∆˚θ ` 4∇˚ ¨ t ` op1q fω dA˚ s‹ { ω ω2 1 ż ż ´ ¯ giving

2 2 3 3 ˚ ˚ ˚ ˚ 4πp4 lim mpΣs‹ qq “ Kθ ´ θ ´ ∆θ ` 4∇ ¨ t1 dA s‹Ñ8 ż ´ ¯ by the Dominated Convergence Theorem.

Finally we’re ready to prove Theorem 2.1.2:

Proof. (Theorem 2.1.2) The ﬁrst claim of Theorem 2.1.2 is a simple consequence of Theorem 2.1.1. Property (P) and Theorem 5.3.6 enforces that

3 1 ˚ ˚ ˚ 0 ď lim s‹ρ{ “ 3 pKθ ´ θ ´ ∆θ ` 4∇ ¨ t1q s‹Ñ8 2φ

therefore, Theorem 2.1.1, Corollary 5.3.6.1, Lemma 3.2.2 and Proposition 5.3.3 gives

3 1 1 2 2 ˚ ˚ ˚ 3 ˚ mpΣ0q ď lim mpΣs‹ q “ pKθ ´ θ ´ ∆θ ` 4∇ ¨ t1q dA “ inf Epφq ď mB. s‹Ñ8 4 4π φą0 ´ ż ¯ The rest of the proof is settled identically as in Theorem 5.3.4.

106 6

Perturbing Spherical Symmetry

6.1 Spherical Symmetry

For the known null Penrose inequality in spherical symmetry (see [14]) we provide proof within our context in order to motivate a class of perturbations on the black hole exterior that maintain both the asymptotically ﬂat and strong ﬂux decay conditions. We also show the existence of an asymptotically geodesic (SP)-foliation for a subclass of these perturbations toward a proof of the null Penrose conjecture.

6.1.1 The metric

In polar areal coordinates [22] the metric takes the form

g “ ´apt, rq2dt b dt ` bpt, rq2dr b dr ` r2˚γ for ˚γ the standard round metric on S2. From which the change in coordinates pt, rq Ñ pv, rq given by b dv “ dt ` dr a produces the metric and metric inverse given by g “ ´he2βdv b dv ` eβpdv b dr ` dr b dvq ` r2˚γ

107 1 g´1 “ e´βpB b B ` B b B q ` hB b B ` ˚γ´1 v r r v r r r2

2Mpt,rq r 1 2 2β for h “ p1 ´ r q where Mpt, rq :“ 2 p1 ´ b2 q and apt, rq “ he .

It’s a well known fact that assigning Mpt, rq “ m0 ą 0 and βpt, rq “ 0 for m0 a constant the above metric covers the region given by v ą 0 in Kruskal spacetime or Schwarzschild geometry in an ‘Eddington-Finkelstein’ coordinate chart. We will therefore refer to the null hypersurfaces Ω :“ tv “ v0u as the standard nullcones (of spherically symmetric spacetime) as they agree with the similarly named hypersurfaces in the Schwarzschild case.

6.1.2 Calculating ρ

We approach the calculation similarly to the case of Schwarzschild. Denoting the

1 2 ´β gradient of v by Dv we use the identity DDvDv “ 2 D|Dv| to see L :“ Dv “ e Br

satisﬁes DLL “ 0 providing us our choice of geodesic generator for Ω and level set function s (as in Section 4.1). For convenience we will choose our background

foliation tΣru of Ω to be the level sets of the coordinate r. An arbitrary cross-section

Σ of Ω is therefore given as a graph over Σr0 (for some r0) which we Lie drag along

Br to the rest of Ω giving some ω P FpΩq. On Σ we therefore have the linearly independent normal vector ﬁelds

´β L “ e Br

´β Dpr ´ ωq “ e Bv ` hBr ´ ∇ω

where in this subsection (6.1.2) ∇ will temporarily denote the induced covariant

K derivative on Σr. We wish to ﬁnd the null section L P ΓpT Σq satisfying xL, Ly “ 2.

Since L “ c1L ` c2Dpr ´ ωq we have

´β 2 “ xL, Ly “ c2e Brpr ´ ωq

´β “ c2e

108 2 0 “ xL, Ly “ 2c1c2xL,Dpr ´ ωqy ` c2xDpr ´ ωq,Dpr ´ ωqy

´β 2 ´2β 2 ´β “ 2c1c2e ` c2pe xBv, Bvy ` |∇ω| ` 2e hxBv, Bryq

´β 2 2 “ 2c1c2e ` c2ph ` |∇ω| q

β 2β 2 giving c2 “ 2e and c1 “ ´e ph ` |∇ω| q so that β 2 β β L “ ´e ph ` |∇ω| qBr ` 2Bv ` 2he Br ´ 2e ∇ω

β 2 β “ 2Bv ` e ph ´ |∇ω| qBr ´ 2e ∇ω

β 2 β 2 “ 2Bv ` e ph ´ |∇{ ω| qBr ´ 2e p∇{ ω ´ |∇{ ω| Brq

β 2 β “ 2Bv ` e ph ` |∇{ ω| qBr ´ 2e ∇{ ω

2 having used the fact that ∇{ ω “ ∇ω`|∇ω| Br to get the third equality. We note from

2 the warped product structure (as for Kruskal spacetime) that EBr pΣr0 q “ LpS q|Ω where LpS2q is the set of lifted vector ﬁelds from the S2 factor of the spacetime product manifold. As a result we may globally extend V P EBr pΣr0 q to satisfy rBv,V s “ 0. The following facts are a direct application of the Koszul formula, we refer the reader to [21] (pg.206) for the details: 1 D B “ ´ B phe2βqe´βB (6.1) Br v 2 r r

DV Bv “ 0 (6.2)

DBr Br “BrβBr (6.3) 1 D B “ V. (6.4) V r r

Lemma 6.1.1. Suppose Ω “ tv “ v0u is the standard null cone in a spherically symmetric spacetime of metric

g “ ´he2βpv,rqdv b dv ` eβpv, rqpdv b dr ` dr b dvq ` r2˚γ

2Mpv,rq 2 where h “ p1´ r q and ˚γ is the round metric on S . Then for some cross-section

Σr0 Ă Ω and ω P FpΣr0 q, Σ:“ tr “ ω ˝ πu produces the data (writing ω ˝ π as ω): γ “ ω2˚γ

109 e´βpv0,ωq χ “ γ ω

2e´βpv0,ωq tr χ “ ω γ χ “ eβpv0,ωq ph ` |∇{ ω|2q ´ 2H˜ ω ´ 2β dω{ b dω{ ω r ´ ¯ 2eβpv , ωq tr χ “ 0 ph ´ ω2∆{ log ω ´ ωβ |∇{ ω|2q ω r ζ “ ´d{ log ω

2Mpv , ωq β ρ “ 0 ` ∆{ ω ` r |∇{ ω|2 ω3 ω

˜ Proof. For any V P EBr pΣr0 q we have from Lemma 5.1.1 that V :“ V `V ωBr|Σ P ΓpT Σq so that the ﬁrst identity follows directly from the metric restriction. From (25):

´β ´β β e D ˜ L “ e D pB q ` e V ωD L “ V V V r L r

so the second identity is given by ˜ ˜ ˜ χpV, W q “ xDV˜ L, W y

e´β “ xV,W y r

and a trace over V,W gives the third so that ∆{ log tr χ “ ´∆{ β ´ ∆{ log ω. For the forth identity:

˜ ˜ ˜ β 2 ˜ β ˜ β ˜ ˜ χpV, W q “ 2xDV˜ Bv, W y ` e ph ` |∇{ ω| qxDV˜ Br, W y ´ 2e xDV˜ ∇{ ω, W y ´ 2βre V ωW ω 1 “ eβph ` |∇{ ω|2q xV,˜ W˜ y ´ 2eβH˜ ωpV,˜ W˜ q ´ 2β eβpdω{ b dω{ qpV,˜ W˜ q ω r

˜ where xDV˜ Bv, W y “ 0 from (6.1) and (6.2) to give the second equality. Taking a trace over V,˜ W˜ we conclude with the ﬁfth identity:

2eβpv0,ωq tr χ| “ ph ` |∇{ ω|2 ´ ω∆{ ωq ´ 2β eβpv0,ωq|∇{ ω|2 Σ ω r 110 2eβ ∆{ ω |∇{ ω|2 “ ph ´ ω2p ´ qq ´ 2β eβ|∇{ ω|2 ω ω ω2 r 2eβ “ ph ´ ω2∆{ log ω ´ ωβ |∇{ ω|2q. ω r

As a result we have that

4 xH,~ H~ y “ tr χ tr χ “ ph ´ ω2∆{ log ω ´ ωβ |∇{ ω|2q. ω2 r

Since the metric on Σ is given by ω2˚γ we conclude that it has Gaussian curvature

1 1 K “ p1 ´ ∆˚ log ωq “ ´ ∆{ log ω ω2 ω2 and therefore 1 2Mpv , ωq β K ´ xH,~ H~ y “ 0 ` r |∇{ ω|2. 4 ω3 ω

Moreover, the torsion is given by 1 ζpV˜ q “ xD ˜ L,Ly 2 V e´β “ xV,Ly 2r 1 “ ´ V ω r 1 “ ´ V˜ ω r

˜ ˜ from which we conclude ζpV q|Σ “ ´V log ω and ∇{ ¨ ζ “ ´∆{ log ω, giving

2Mpv , ωq β ρ “ 0 ` ∆{ β ` r |∇{ ω|2. ω3 ω

Remark 6.1.1. We recover the data of Lemma 3.2.1 as soon as we set m0 “ M,

β “ 0 and r0 “ 2m0 as expected. 111 In comparison to the Schwarzschild spacetime, we have the two additional terms

βr 2 ∆{ β ` ω |∇{ ω| in the ﬂux function ρ. It turns out that a non-trivial GpL,Lq is 2 responsible. Since ∆{ β “ βrr|∇{ ω| ` βr∆{ ω and 1 GpL,Lq “ ´L tr K ´ ptr Kq2 ´ |Kˆ |2 2 2e´β 1 4e´2β “ ´e´βB p q ´ r r 2 r2 2β “ r e´2β r

βr 2 it follows, for arbitrary ω, that ∆{ βpωq ` ω |∇{ ω| “ 0 if and only if β is independent

of the r-coordinate and therefore GpL,Lq “ 0. For the function Mpv0, rq we look to

GpL,Lq along the foliation tΣru since:

2 ~ ~ GpL,Lq “ L tr χ ´ 2Ks ` 2∇ ¨ t ` 2|~t| ` xH, Hy

2eβ 2M 2 4 2M “ e´βB p p1 ´ qq ´ ` p1 ´ q r r r r2 r2 r 2β 2M 4M “ r p1 ´ q ´ r . r r r2

It follows from Lemma 4.3.1, on Σr, that

GL “ 0.

Since these components are all that contribute to the monotonicity of (2.2) for the

foliation tΣru we see that our need of the dominant energy condition reduces to

2M 0 ď hβ ď r r r

on th ě 0u X Ω. Next we show that tΣru is a re-parametrization of a geodesic (SP)-foliation:

112 6.1.3 Asymptotic ﬂatness

We now wish to choose the necessary decay on β and M in order to employ Theorem

´β 2.1.2. For L “ e Br the geodesic foliation tΣsu has level set function given by

r sprq “ eβptqdt żr0 for which ω “ const. ðñ s “ const. and therefore

2Mprpsqq ρpsq “ . rpsq3

1 ~ ~ 1 { h It follows from Lemma 6.1.1 that 4 xH, Hy ´ 3 ∆ log ρ “ rpsq2 ą 0 ðñ rpsq ą r0 “

2Mpv0, r0q as in Schwarzschild.

´1 0 Lemma 6.1.2. Choosing |βpv0, rq| “ o2pr q integrable and Mpv0, rq “ m0 ` opr q

for some constant m0, Ω is asymptotically ﬂat with strong ﬂux decay.

ds βprq eβ ´1 Proof. We’ve already veriﬁed that GL “ 0. Since dr “ e “ p1 ` β β q, |β| is

eβ ´1 ds ´1 integrable and β is bounded it follows that dr “ 1 ` f where |f| “ o2pr q is integrable. As a result

8 8 0 s “ r ´ r0 ` fptqdt ´ fptqdt “ r ´ c0 ` o3pr q żr0 żr

8 0 where β0 “ fptqdt and c0 “ r0 ´ β0. We conclude that rpsq “ s ` c0 ` o3ps q since r0 ş 8 0 our assumptions on β imply that rpsq fptqdt “ o3ps q. From the fact that ş c γ “ r2˚γ| “ ps ` c ` o p1qq2˚γ “ s2p1 ` 0 ` o ps´1qq2˚γ “ s2˚γ ` 2c s˚γ ` o psq˚γ s Σs 0 3 s 3 0 3

we seeγ ˜ “ o3psq˚γ ensuring condition 1 of Deﬁnition 5.3.1 holds up to strong decay given that all dependence on tangential derivatives falls on the L Lie constant tensor

113 ˚γ. Since ~t “ 0 for this foliation condition 2 follows trivially up to strong decay. If we

assume that Mpv0, rq “ m0 ` op1q for some constant m0 we see directly from Lemma 6.1.1

tr Q “ tr χ|Σs 2 2M “ p1 ´ q| ` ops´2q r r Σs 2 c 2m “ p1 ´ 0 qp1 ´ 0 q ` ops´2q s s s 2 c ` 2m “ ´ 2 0 0 ` ops´2q s s2

giving us the third condition of Deﬁnition 5.3.1. We refer the reader to [18] to observe the use of the forth condition of Deﬁnition 5.3.1 in proving (5.4) for an arbitrary geodesic foliation. As mentioned in Remark

2K˚ ´1 5.3.5, strong ﬂux decay bypasses our need of this condition since tr Q “ s ` ops q is veriﬁed above.

From Lemma 6.1.2, Theorem 2.1.2, Theorem 4.3.2 and the comments immediately proceeding Remark 6.1.1 we have the following proof of the known (see [14]) null Penrose conjecture in spherical symmetry:

Theorem 6.1.1. Suppose Ω:“ tv “ v0u is a standard null cone of a spherically symmetric spacetime of metric

2Mpv, rq ds2 “ ´ 1 ´ e2βpv,rqdv2 ` 2eβpv,rqdvdr ` r2 dϑ2 ` sin ϑ2dϕ2 r ´ ¯ ´ ¯ where

´1 1. |βpv0, rq| “ o2pr q is integrable

0 2. Mpv0, rq “ m0 ` opr q for some constant m0 ą 0

114 2Mr 3. 0 ď hβr ď r

Then,

|Σ| ď m0 c16π

for m0 the Bondi mass of Ω and Σ:“ tr0 “ 2Mpv0, r0qu. In the case of equality

we have β “ 0 and M “ m0 so that Ω is a standard null cone of Schwarzschild spacetime.

6.2 Perturbing Spherical Symmetry

We wish to study perturbations oﬀ of the spherically symmetric metric given in

Theorem 6.1.1 for the coordinate chart pv, r, ϑ, ϕq. We start by choosing a 1-form η

such that ηpBrpBvqq “ LBv η “ 0 and a 2-tensor γ satisfying γpBrpBvq, ¨q “ LBv γ “ 0

with restriction γ|pv,rqˆS2 positive definite. Finally we choose smooth functions M, β and α. Defining ~η to be the unique vector field satisfying γp~η, Xq “ ηpXq for arbitrary X P ΓpTMq and r2|~η|2 :“ γp~η, ~ηq the spacetime metric and its inverse are given by

g “ ´ph ` αqe2βdv b dv ` eβpdv b pdr ` ηq ` pdr ` ηq b dvq ` r2γ

1 g´1 “ e´βpB b B ` B b B q ` ph ` α ` |~η|2qB b B ´ p~η b B ` B b ~ηq ` γ´1. v r r v r r r r r2

´β We see that Ω :“ tv “ v0u remains a null hypersurface with Lp“ Dvq “ e Br P ΓpT ΩqXΓpT KΩq. Our metric resembles the perturbed metric used by Alexakis [1] to successfully verify the Penrose inequality for vacuum perturbations of the standard null cone of Schwarzschild spacetime. We’ll need the following to specify our decay conditions:

Deﬁnition 6.2.1. Suppose Ω extends to past null inﬁnity with level set function, s, for some null generator L. For a transversal k-tensor T

115 X ´m X ´m • We say T ps, δq “ δon ps q if T “ on ps q and

1 lim sup sup |smpL ...L T qps, δq| ă 8 for 0 ď j ď n Xi1 Xij δÑ0 Ω δ

• We deﬁne 2 2 ˚ 2 ˚m 2 |T |H˚m “ |T |˚γ ` |∇T |˚γ ` ¨ ¨ ¨ ` |∇ T |˚γ.

Decay Conditions on Ω:

2 2 1. r γ “ r ˚γ ` rδγ1 ` γ˜ where:

2 (a)˚γ is the Br-Lie constant, transversal standard round metric on S independent of δ

(b) γ1 is a Br-Lie constant, transversal 2-tensor independent of δ

i X 1´i (c)˜γ is a transversal 2-tensor satisfying pLBr q γ˜ “ δo5´ipr q for 0 ď i ď 3

α0 2. α “ δ r ` α˜ where α0 is a Br-Lie constant function independent of δ and ´1 |α˜|H˚2 ď δh1prq for h1 “ opr q

3. β satisﬁes:

´1 (a) |β| “ o2pr q is r-integrable

˚ ´1 (b) |∇β|H˚3 ď δh2prq for some integrable h2 “ opr q

˚ ´1 (c) |∇βr|H˚2 “ Opr q

4. M “ m0 ` m˜ where m0 ą 0 is constant independent of δ and |m˜ |H˚2 ď δh3prq

for h3 “ op1q

5. η is a transversal 1-form satisfying:

(a) η “ o2p1q

(b) |η|H˚3 ` r|LBr η˜|H˚3 ď δh4prq for h4 “ op1q. 116 6.2.1 A geodesic foliation

As in the spherically symmetric case we identify the null geodesic generator Dv “

´β e Br. We will again for convenience take the background foliation to be level sets of the coordinate r. We wish therefore to relate the given decay in r to the geodesic

foliation given by the generator L :“ Dv in order to show Ω is asymptotically ﬂat with strong ﬂux decay.

ds β eβ ´1 Once again dr “ e “ 1 ` f where f “ β β is r-integrable due to decay condition

3. Taking local coordinates pϑ, ϕq on Σr0 (for some r0) we have

s “ r ´ c0pϑ, ϕq ´ β1pr, ϑ, ϕq (6.5)

8 8 for β0pϑ, ϕq :“ fpt, ϑ, ϕqdt, c0 “ r0 ´ β0 and β1pr, ϑ, ϕq “ fpt, ϑ, ϕqdt. Since r0 r ş ş ˚ each Σr is compact, an m-th order partial derivative of f is bounded by C|∇f|H˚m´1 for some constant C independent of r (from decay condition 3). From decay condition

3, provided m ď 4, derivatives in ϑ, ϕ of β0 and β1 pass into the integral (for ﬁxed

r) onto f and are bounded. On any Σs (i.e ﬁxed s) it follows from (6.5) that

r r Bϑpϕqfpt, ϑ, ϕqdt B r “ ´ r0 “ ´e´β β eβdt ϑpϕq 1 ` f ϑpϕq ş żr0 with bounded derivatives up to third order. It’s a simple veriﬁcation in local coordinates, from

rps, ϑ, ϕq “ s ` c0pϑ, ϕq ` β1prps, ϑ, ϕq, ϑ, ϕq,

i X ´i ´β that Bsβ1 “ o3´ips q for 0 ď i ď 3. Coupled with the fact that LL “ e LBr on

i X 1´i transversal tensors we conclude that pLLq γ˜ “ o3´ips q for 0 ď i ď 3 and therefore

2 2 ˜ γs “ r γ|Σs “ s ˚γ ` sΓ1 ` Γ (6.6) where

Γ1 “ 2c0˚γ ` δγ1 117 ˜ 2 2 Γ “ γ˜ ` 2sβ1˚γ ` c0˚γ ` c0δγ1 ` β1˚γ ` 2c0β1˚γ ` β1δγ1

satisﬁes the requirements towards strong decay.

6.2.2 Calculating ρ

Since we will compare computations for the foliation tΣru with the geodesic foliation

of 5.4.1 we will revert back to denoting the covariant derivative on Σs by ∇ and

the covariant derivative on Σr by ∇{ . For the foliation tΣru we have the linearly independent normal vector ﬁelds

´β L “ e Br

´β 2 Dr “ e Bv ` ph ` α ` |~η| qBr ´ ~η from which similar calculations as in spherical symmetry yield the unique null normal satisfying xL,Ly “ 2 to be given by

β 2 β L “ 2Bv ` e ph ` α ` |~η| qBr ´ 2e ~η.

Lemma 6.2.1. We have δ 1 χ “ e´βpr˚γ ` γ ` pL γ˜qq 2 1 2 Br 2 δθ tr χ “ e´βp ` q ` δoX pr´2q r r2 4

1 ˚ for θ :“ ´ 2 trγ1. Moreover,

m δ ∇{ χ “ ´e´β ∇˚mγ ` δoX p1q, 0 ď m ď 4. 2 1 4´m

Proof. First we extend V,W P EBr pΣr0 q oﬀ of Ω such that rBv,V pW qs “ 0. Then for χ:

´β χpV,W q “ xDV pe Brq,W y

´β “ e xDV Br,W y 118 1 “ e´β B xV,W y 2 r δ 1 “ e´βpr˚γpV,W q ` γ pV,W q ` L γ˜pV,W qq 2 1 2 Br having used the Koszul formula to get the third line. So using a basis extension

tX1,X2u Ă EBr pΣr0 q Proposition 5.3.1 provides the inverse metric

1 δ γprqij “ ˚γij ´ γ˚ij ` δoX pr´3q r2 r3 1 5

and tr χ follows by contracting γprq´1 with χ. For the ﬁnal identity we note from

2 Lemma 5.3.2 we have for the decomposition γr “ r ˚γ ` rδγ1 ` γ˜ the diﬀerence tensor ˚ xDpV,W q,Uy “ x∇{ V W ´ ∇V W, Uy

rδ “ ∇˚ γ pW, Uq ` ∇˚ γ pV,Uq ´ ∇˚ γ pV,W q 2 V 1 W 1 U 1 ´ ¯ 1 ` ∇˚ γ˜pW, Uq ` ∇˚ γ˜pV,Uq ´ ∇˚ γ˜pV,W q 2 V W U ´ ¯

for V, W, U P EpΣr0 q. So proceeding as in Proposition 5.3.5

∇{ χ ∇˚ χ Dmχ Dmχ i jk “ i jk ´ ij mk ´ ik jm δ 1 “ ∇˚ pre´β˚γ ` e´β γ ` e´β pL γ˜q q ´ e´βδ∇˚ pγ q ` δoX p1q i jk 2 1jk 2 Br jk i 1jk 4 δ δ “ r˚γ ∇˚ pe´βq ´ e´β ∇˚ γ ` γ ∇˚ pe´βq ` δoX p1q jk i 2 i 1jk 2 1jk i 3 δ “ ´e´β ∇˚ γ ` δoX p1q. 2 i 1jk 3

Iteration provides our result

m δ ∇{ χ “ ´e´β ∇˚mγ ` δoX p1q, 1 ď m ď 4 2 1jk 4´m from decay condition 3.

119 For χ we have

β 2 β χpV,W q “ 2xDV Bv,W y ` e ph ` α ` |~η| qxDV Br,W y ´ 2xDV e ~η, W y

2β 2 β “ 2xDV Bv,W y ` e ph ` α ` |~η| qχpV,W q ´ 2∇{ V pe ηqpW q and using the Koszul formula on the ﬁrst term we see

2xDV Bv,W y

β β “ V pe ηpW qq ` BvxV,W y ´ W pe ηpV qq ´ xV, rBv,W sy ` xBv, rW, V sy ` xW, rV, Bvsy

β β “ ∇{ V pe ηqpW q ´ ∇{ W pe ηqpV q

“ curlpeβηqpV,W q so that a trace over V,W yields tr χ “ e2βph`α ` |~η|2q tr χ´2∇{ ¨ peβηq and therefore

xH,~ H~ y “ e2βph ` α ` |~η|2qptr χq2 ´ 2∇{ ¨ peβηq tr χ

2M α 2 δθ 2 “ 1 ´ ` δ 0 ` ` δoX pr´3q r r r r2 2 ´ ¯´ ¯ 2M α 4 4δθ “ 1 ´ ` δ 0 ` ` δoX pr´3q r r r2 r3 2 ´ ¯´ ¯ 4 2m θ α “ 1 ´ 0 ` δ ` δ 0 ` δoX pr´3q r2 r r r 2 ´ ¯ from decay conditions 2-5. For ζ we have

´β β ´β ζpV q “ xDV pe Brq, Bvy ´ e xDV pe Brq, ~ηy

´β “ ´V β ` e xDV Br, Bvy ´ xDV Br, ~ηy

´β β “ ´V β ` e xDV Br, Bvy ´ e χpV, ~ηq.

From the Koszul formula

2xDV Br, Bvy

“ V xBr, Bvy ` BrxV, Bvy ´ BvxV, Bry ´ xV, rBr, Bvsy ` xBr, rBv,V sy ` xBv, rV, Brsy

β β “ e V β ` Brpe ηpV qq 120 β β “ e V β ` LBr pe ηqpV q

1 e´β β β from which we conclude that ζpV q “ ´ 2 V pβq ` 2 LBr pe ηqpV q ´ e χpV, ~ηq and 1 1 ∇{ ¨ ζ “ ´ ∆{ β ` ∇{ ¨ pe´βL peβηqq ´ ∇{ ¨ peβχp~ηqq 2 2 Br 1 1 1 “ ´ ∆{ β ` ∇{ ¨ pβ ηq ` ∇{ ¨ pL ηq ´ eβχp∇{ β, ~ηq ´ eβ∇{ ¨ pχp~ηqq 2 2 r 2 Br

X ´3 “ δo2 pr q

having used decay conditions 3, 5 and Lemma 6.2.1 for the ﬁnal line.

Lemma 6.2.2. Ω satisfies conditions 1, 2 and 3 of Definition 5.3.1. Ω additionally satisfies strong flux decay if and only if

1 ∇˚ ¨ γ ` dθ “ 0 2 1

1 ˚ for θ “ ´ 2 trγ1 and is subsequently past asymptotically ﬂat.

Proof. Having already veriﬁed condition 1 up to strong decay for γs of our geodesic

foliation tΣsu we continue to show conditions 2 and 3.

Given V P EBr pΣr0 q Lemma 5.1.1 ensures V ´ V sL|Σs P ΓpT Σsq and we see that rV ´ V sL,Ls “ rV,Ls ` LV sL

β ´β ´β “ e V pe qL ` e V pBrsqL

“ peβV pe´βq ` e´βV peβqqL

“ 0.

So V ´ V sL P EpΣ0q and Lemma 5.1.1 gives

tpV ´ V sLq “ tpV q “ ζpV q ` χpV, ∇{ sq

1 1 1 “ ´ V pβq ` β ηpV q ` pL ηqpV q ´ eβχpV, ~ηq ` χpV, ∇{ sq 2 2 r 2 Br

X ´1 ´1 “ pδo3 pr q X o1pr qqpV q ` χpV, ∇{ sq 121 1 “ pδoX pr´1q X o pr´1qqpV q ` re´β˚γpV, ∇˚sq 3 1 r2 e´β “ V β ` pδoX pr´1q X o pr´1qqpV q r 0 3 1

having used decay conditions 3 and 5 to get the second line, Lemma 6.2.1 for the third and (6.5) for the last. Moreover,

pLV ´V sLtqpW ´ W sLq “ pV ´ V sLqptpW ´ W sLqq ´ tprV,W sq

“ pLV tqpW q ´ V sLptpW qq

dβ “ pL ´ e´βV sL qp 0 qpW q ` opr´1q V Br r 1 “ L pdβ qpW q ` opr´1q r V 0 1 “ pL dβ qpW ´ W sLq ` opr´1q r V ´V sL 0

where the last line follows since β0 is L-Lie constant. With a basis extension

1 ´1 tXiu Ă EpΣ0q we therefore conclude that LXi t “ s LXi dβ0 ` ops q so that condition 2 for asymptotic ﬂatness is satisﬁed up to strong decay with t1 “ dβ0. From Proposition 5.3.1 and (6.6): 2 1 tr K “ ´ trΓ˚ ` ops´2q s 2s2 1 2 1 “ ´ tr˚p2c ˚γ ` δγ q ` ops´2q s 2s2 0 1 2 δθ ´ 2c “ ` 0 ` ops´2q s s2

and 1 Kˆ “ K ´ tr Kγ 2 s 1 1 2 δθ ´ 2c “ s˚γ ` Γ ´ ` 0 ` ops´2q γ ` op1q 2 1 2 s s2 s ´ ¯ 1 1 2 δθ ´ 2c “ s˚γ ` p2c ˚γ ` δγ q ´ p ` 0 qps2˚γ ` sp2c ˚γ ` δγ qq ` op1q 2 0 1 2 s s2 0 1 122 δ “ ´ pγ ` θ˚γq ` op1q. 2 1

For condition 3 we take r|Σs P FpΣsq and Lie drag it to the the rest of Ω along

Br (hence L) to give rs P FpΩq. Using Lemma 5.1.1 from the vantage point of the cross-section Σs amongst the background foliation tΣru:

´β ´β β 2 β 2 e tr Q “ e tr χ ´ 4pζ ` d{ log e qp∇{ rsq ´ 2∆rs ` |∇{ rs| e tr χ ´ 2βr|∇{ rs|

From the expression of rpsq in 5.4.1, recalling Remark 5.3.2, we see drs “ ´dβ0 `op1q

1 ˚ ´2 from which Lemma 5.3.2 implies that ∆rs “ ´ s2 ∆β0`ops q. From decay conditions 3, 5 and Lemma 6.2.1 we have

∆˚β tr Q “ tr χ| ` 2 0 ` ops´2q Σs s2

∆˚β “ e2βph ` α ` |~η|2q tr χ ´ 2∇{ ¨ peβηq | ` 2 0 ` ops´2q Σs s2 ´ ¯ 2 δθ ´ 2c 2M α ∆˚β “ p ` 0 qp1 ´ ` δ 0 q ` 2 0 ` ops´2q s s2 s s s2

2 δθ ´ 2c ´ 4M ` 2δα ∆˚β “ ` 0 0 ` 2 0 ` ops´2q s s2 s2 2 c ` 2M 1 “ ´ 2 0 ` pδθ ` 2∆˚β ` 2δα q ` ops´2q s s2 s2 0 0

X and condition 3 follows as soon as we set M “ m0 ` δo2 p1q. As in the spherically 2K˚ ˚ symmetric case the highest order term for tr Q agrees with s where K “ 1 is the Gaussian curvature of ˚γ. We recall that our use of condition 4 depends on whether Ω has strong ﬂux decay (Remark 5.3.5). From Proposition 5.3.1 and (6.6) we will have strong ﬂux decay if and only if 1 1 dβ “ t “ ∇˚ ¨ Γ ´ d{trΓ˚ 0 1 2 1 2 1 1 “ ∇˚ ¨ p2c ˚γ ` δγ q ` d{pδθ ´ 2c q 2 0 1 0 123 1 δ “ dp´2β q ` ∇˚ ¨ γ ` δdθ ` 2dβ 2 0 2 1 0 1 “ dβ ` δp ∇˚ ¨ γ ` dθq 0 2 1

1 ˚ which in turn holds if and only if 2 ∇ ¨ γ1 ` dθ “ 0.

Henceforth we will adopt the conditions of Lemma 6.2.2 for Ω. From Proposition 5.3.2

1 δ 1 X ´3 K 2 “ ` θ ` ∇˚ ¨ ∇˚ ¨ γ ` ∆˚θ ` δo pr q r γ r2 r3 2 1 4 ´ ¯ 1 δ “ ` θ ` δoX pr´3q. r2 r3 4

From Lemma 6.2.1 we have

δ ∇{ ∇{ χ “ ´ ∇˚ ∇˚ γ ` δoX p1q i j mn 2 i j 1mn 2

so that contraction with γprq´1 ﬁrst in mn then ij gives

δ ∆{ tr χ “ ∆˚θ ` δoX pr´4q r4 2

2 { ∆{ tr χ |∇{ tr χ| which we use in ∆ log tr χ “ tr χ ´ ptr χq2 to conclude

δ ∆{ log tr χ “ ∆˚θ ` δoX pr´3q. 2r3 2

Finally we have ρ 1 ρ “ K 2 ´ xH,~ H~ y ` ∇{ ¨ ζ ´ ∆{ log tr χ r γ 4 1 δ 1 2m θ α δ “ ` θ ´ ` 0 ´ δ ´ δ 0 ´ ∆˚θ ` δoX pr´3q r2 r3 r2 r3 r3 r3 2r3 2 2m δ 1 “ 0 ´ p ∆˚θ ` α q ` δoX pr´3q r3 r3 2 0 2

124 2m δ 1 “ 0 ´ p ∆˚θ ` α q ` δoX pr´3q r3 r3 2 0 2

and 1 1 1 2m θ α xH,~ H~ y ´ ∆{ log ρ “ 1 ´ 0 ` δ ` δ 0 4 3 r2 r r r ´ ¯ 1 2m δ 1 ´ ∆{ log 0 ´ p ∆˚θ ` α q ` δoX pr´3q ` δoX pr´3q. 3 r3 r3 2 0 2 2 ´ ¯ We may now use Lemma 5.3.2 to decompose the last term 2m δ 5 ∆{ log 0 ´ p ∆˚θ ` α q ` δoX pr´3q r3 r3 2 0 2 ´ ¯ 1 ˚ δ 1 ˚ X ´2 “ 2 ∆ log 1 ´ p ∆θ ` α0q ` δo2 p1q ` δopr q r 2m0 2 ´ ¯ 1 ˚ δ 1 ˚ ´2 “ 2 ∆ log 1 ´ p ∆θ ` α0q ` δopr q r 2m0 2 ´ ¯ giving

1 1 1 2m 1 δ 1 ~ ~ { 0 ˚ ˚ ´2 xH, Hy ´ ∆ log ρ “ 2 1 ´ ´ ∆ log 1 ´ p ∆θ ` α0q ` δopr q. 4 3 r r 3 2m0 2 ´ ´ ¯¯

Since m0 ą 0 we notice for suﬃciently small δ our perturbation ensures ρ ą 0 for all r ą 0. However, from our construction so far it’s not yet possible to conclude 1 ~ ~ 1 that some δ ą 0 will enforce 4 xH, Hy ě 3 ∆{ log ρ along the foliation. Moreover, the existence of a horizon (tr χ “ 0) is equally questionable.

6.2.3 Smoothing to Spherical Symmetry

We will solve this diﬃculty by ‘smoothing’ away all perturbations in a neighborhood of the (desired) horizon in order to obtain spherical symmetry on r ă r1 for some r1 ą 0 yet to be chosen. The resulting spherical symmetry will uncover the horizon 1 ~ ~ 1 at r “ r0 ă r1 and will also provide a choice of δ ą 0 so that 4 xH, Hy ą 3 ∆{ log ρ away from it, causing the foliation tΣru to be an (SP)-foliation.

125 We will use a smooth step function 0 ď Sδprq ď 1 such that Sδprq “ 0 for

1 r ă r1 and Sδprq “ 1 for r ą r2 for some ﬁnite r2pδq chosen to ensure |Sδprq| ď δ. By ﬁrst choosing parameter functions for the desired spherically symmetric region; ˜ ˜ ˜ ˜ βpv, rq and 0 ă Mpv, rq “ m0 ` op1q such that r0 “ 2Mpv0, r0q and 2Mpv0, rq ă r

for r ą r0 we induce spherical symmetry on r ă r1 with the following substitutions:

γ˜ Ñ δrpSδprq ´ 1qγ1 ` Sδprqγ˜

˜ βpr, ϑ, ϕq Ñ Sδprqβpr, ϑ, ϕq ` p1 ´ Sδprqqβpv0, rq

˜ Mpr, ϑ, ϕq Ñ SδprqMpr, ϑ, ϕq ` p1 ´ SδprqqMpv0, rq

δα α˜ Ñ S prqα˜ ´ p1 ´ S prqq 0 δ δ r

η Ñ Sδprqη.

We leave the reader the simple veriﬁcation that these changes to our perturbation

tensorsγ ˜, β, M,α ˜ and η maintain the decay conditions 1-5. Clearly for r ą r2 our substitutions leave the metric unchanged while inducing spherical symmetry on ˜ ˜ r ă r1 with the spherical parameter functions β , M: ˜ r0 “ 2Mpv0, r0q

r “ r1

r ą r1

Figure 6.1: Perturbing Spherical Symmetry

An example Sδprq is given by the function

0 r ď r1 k e r1´r Sδprq “ $ k k r1 ă r ă r2 ’ e r1´r `e r´r2 &’ 1 r2 ď r ’ %’ 126 4e4 1 1 ekr where k “ and r2pδq “ r1 ` k. Since Sδprq “ P p ` q for P prq “ kr δ r1´r r2´r 1`e satisfying the logistic equation

P 1prq “ kP p1 ´ P q

we have

1 1 1 Sδprq “ kSδprqp1 ´ Sδprqqp 2 ` 2 q pr ´ r1q pr ´ r2q

Sδprq 1 ´ Sδprq “ k 2 p1 ´ Sδprqq ` kSδprq 2 pr ´ r1q pr ´ r2q

Sδprq 1 ´ Sδprq ď k 2 ` 2 . pr ´ r1q pr ´ r2q ´ ¯

Elementary analysis reveals on the interval r1 ă r ă r2 that

k 2 e r1´r 4e 0 ď 2 ď 2 pr ´ r1q k

1 1 2 0 ď k k ď e r r r r 2 e 1´ ` e ´ 2

4 Sδprq 1´Sδprq 2e yielding from simple symmetry arguments that both 2 , 2 ď 2 and there- pr´r1q pr´r2q k fore 4e4 0 ď S1 prq ď k “ δ δ k2 ˜ as desired. Denoting mpr, δq :“ Sδprqm0 ` p1 ´ SδprqqMprq the new metric gives

2M˜ pv0,rq r3 , r ă r1 ρ “ 2mpr,δq δ 1 ˚ X ´3 $ r3 ´ r3 p 2 ∆θ ` α0q ` δo2 pr q, r1 ď r ď r2 ’ 2m0 δ 1 ˚ X ´3 & r3 ´ r3 p 2 ∆θ ` α0q ` δo2 pr q, r2 ă r ’ 1 ~ ~ 1 % and 4 xH, Hy ´ 3 ∆{ log ρ “

1 2M˜ pv0,rq r2 p1 ´ r q, r ă r1 1 2mpr,δq 1 ˚ δ 1 ˚ ´2 $ r2 1 ´ r ´ 3 ∆ log 1 ´ 2mpr,δq p 2 ∆θ ` α0q ` δopr q, r1 ď r ď r2 ’ ’ 1 2m0 1 ˚ δ 1 ˚ ´2 & 2 ´1 ´ ´ ∆ log 1´´ p ∆θ ` α0q `¯¯δopr q, r2 ă r. r r 3 2m0 2 ’ ´ ´ ¯¯ %’ 127 ˜ Since Cpr1q ě mpr, δq ě m0 for Cpr1q :“ supr0ărăr1 M we see for any choice of ˜ r1 ą Cpr1q (which is possible since M “ m0 ` op1q) and suﬃciently small δ the foliation tΣru satisﬁes property (SP). If we therefore restrict to perturbations satisfying the dominant energy condition on Ω then Theorem 2.1.2 implies the following:

Theorem 6.2.1. Let gδ be a metric perturbation oﬀ of spherical symmetry given by

2β β 2 gδ “ ´ph ` αqe dv b dv ` e pdv b pdr ` ηq ` pdr ` ηq b dvq ` r γ where

2 2 1. r γ “ r ˚γ ` rδγ1 ` γ˜ is trasversal with ˚γ the transversal Br-Lie constant round

2 metric on S independent of δ, γ1 a transversal Br-Lie constant 2-tensor inde- ˚ ˚ i X 1´i pendent of δ satisfying ∇¨γ1 “ dptrγ1q and pLBr q γ˜ “ δo5´ipr q for 0 ď i ď 3.

α0 2. α “ δ r ` α˜ where α0 is Br-constant, independent of δ and |α˜|H˚2 ď δh1prq for ´1 h1 “ opr q

3. β satisﬁes:

´1 (a) |β| “ o2pr q is r-integrable

˚ ´1 (b) |∇β|H˚3 ď δh2prq for some integrable h2 “ opr q

˚ ´1 (c) |∇βr|H˚2 “ Opr q

4. M “ m0 ` m˜ where m0 ą 0 is constant, independent of δ and |m˜ |H˚2 ď δh3prq

for h3 “ op1q

5. η is a transversal 1-form satisfying:

(a) η “ o2p1q

(b) |η|H˚3 ` r|LBr η˜|H˚3 ď δh4prq for h4 “ op1q.

128 Then for sufficiently small δ, Ω:“ tv “ v0u is past asymptotically flat with strong flux decay. In addition, for any choice of spherical parameters β˜pv, rq and M˜ pv, rq ˜ ˜ ˜ such that 0 ă Mpv0, rq “ m0 ` op1q, r0 “ 2Mpv0, r0q and 2Mpv0, rq ă r for r ą r0, smoothing to spherical symmetry with the step function Sδprq (as above) according to:

γ˜ Ñ δrpSδprq ´ 1qγ1 ` Sδprqγ˜

˜ βpr, ϑ, ϕq Ñ Sδprqβpr, ϑ, ϕq ` p1 ´ Sδprqqβprq

˜ Mpr, ϑ, ϕq Ñ SδprqMpr, ϑ, ϕq ` p1 ´ SδprqqMprq

δα α˜ Ñ S prqα˜ ´ p1 ´ S prqq 0 δ δ r

η Ñ Sδprqη

˜ we have that Σ:“ tr0 “ 2Mpv0, r0qu is marginally outer trapped and the coordinate

spheres tΣrurěr0 form an (SP)-foliation. Moreover, if gδ respects the dominant energy condition on Ω we have the Penrose inequality:

|Σ| ď mB c16π where mB is the Bondi mass of Ω.

129 Appendix A

Bartnik data of coordinate spheres in boosted Schwarzschild

The Schwarzschild metric in isotropic coordinates is given by,

α 2 ds2 “ ´ dt2 ` β4pdx2 ` dy2 ` dz2q β ´ ¯

M M 2 2 2 where α “ 1 ´ 2R and β “ 1 ` 2R for R “ x ` y ` z . Alternatively making a change to sphericala coordinates,

x “ R sin ϑ cos ϕ

y “ R sin ϑ sin ϕ

z “ R cos ϑ

transforms the metric to

α 2 ds2 “ ´ dt2 ` β4 dR2 ` R2pdϑ2 ` psin ϑq2dϕ2q . β ´ ¯ ´ ¯

Deﬁnition A.0.1. For semi-Riemannian manifolds pB, gBq and pF, gF q, given f ą 0

a smooth function on B the warped product M “ B ˆf F is the product manifold

130 B ˆ F furnished with the metric tensor,

‹ 2 ‹ g “ π1pgBq ` pf ˝ π1q π2pgF q where π1 and π2 represent the canonical projections onto B and F respectively.

For Warped Products we have a pointwise decomposition of each tangent space into orthogonal compliments TpM “ kerpdpπ2qpq ‘ kerpdpπ1qpq known as normal and tangent vectors (i.e. TpM “ norpTpMq ‘ tanpTpMq) respectively. We also in the case of product manifolds recall the submodules LpBq,LpF q Ă ΓpT Mq of lifted vector ﬁelds, whereby X P LpBq Ă kerpdπ2q is the canonical representative of some ˜ ˜ X P ΓpTBq such that dπ1pXq “ X. For every pp, qq P B ˆf F we refer to the semi- Riemanian submanifold B ˆ q as a leaf and p ˆ F as a ﬁbre. Therefore, returning to Schwarzschild geometry, we recognize a Warped Product

2 M α 2 2 4 2 structure P ˆRβ2 S with leaves isometric to P “ R ˆ p 2 , 8q, β dt ` β dR and

´ ` ˘ 2 ¯ ﬁbres homothetic to the standard round sphere via the function Rβ on P. Introducing the boosted coordinates,

cosh ψ ´ sinh ψ t t¯ “ ´ sinh ψ cosh ψ z z¯ ˆ ˙ ˆ ˙ ˆ ˙ it is easily shown that the standard boosted slice tt¯ “ 0u (or tt ´ tanhpψqz “ 0u) inherits the induced metric:

α 2 ds2 “ β4 dx¯2 ` dy¯2 ` pcosh2 ψ ´ sinh2 ψqdz¯2 β3 ´ ´ ¯ ¯ forr ¯ :“ x¯2 ` y¯2 ` z¯2 “ R p1 ´ tanh2 ψ cos2 ϑq. It is also easy to show that b the boosteda time slice is asymptotically ﬂat with the metric clearly displaying the required asymptotic behaviour for larger ¯. Using the ambient Warped Product structure our goal is to study the geometry

2 2 2 2 of the coordinate spheres, Sr¯, given by constantr ¯ “ x¯ ` y¯ ` z¯ that foliates this 131 a 2 boosted slice. We start by realizing Sr¯ as the intersection of the slices: t “ tanh ψR cos ϑ (A.2a)

R2 ´ t2 “ r¯2 (A.2b) yielding the equations r¯tanh ψ cos ϑ tpϑq “ (A.3a) 1 ´ tanh2 ψ cos2 ϑ a r¯ Rpϑq “ (A.3b) 1 ´ tanh2 ψ cos2 ϑ a t or equivalently, for tanh η :“ tanh ψ cos ϑ “ R , t “ r¯sinh η (A.4a)

R “ r¯cosh η (A.4b)

(for constantr ¯).

2 2 From these equations we recognize a family of embeddings, jr¯ : S Ñ Sr¯:

2 fr¯ 2 S /PˆRβ2 S σ id $ 2 S

fr¯ σ pϑ, ϕq /ptpϑq,Rpϑq, ϑ, ϕq /pϑ, ϕq 2 ¯ As such, given any V P ηpT S q, by denoting djr¯pV q as V , we have (assuming restric- 2 tion to Sr¯ throughout): ¯ V “ V ptqBt ` V pRqBR ` VS2

“ V ηX ` VS2

2 for X “ RBt ` tBR and VS2 P LpS q.

Henceforth we will drop the subscript VS2 and will refer interchangably between elements of LpS2q and ΓpT S2q as the meaning should remain clear from the context. Our use of the Warped Product structure will be extensively due to, ([21],pg.206)

132 Proposition A.0.1. On a Warped Product M “ B ˆf F , if X,Y P LpBq and V,W P LpF q then

1. DX Y P LpBq is the lift of DX Y on B.

Xf 2. DX V “ DV X “ f V

xV,W y 3. norDV W “ ´ f Df

4. tanDV W P LpF q is the lift of ∇V W on F .

Armed with this Proposition we extract some necessary results from the ambient spacetime

2 Corollary A.0.1.1. For X “ RBt ` tBR P LpPq and V,W P LpS q,

M 1 1. (a) D B “ ´ B BR R R2 β R

M 1 (b) D B “ D B “ B BR t Bt R R2 αβ t

α M 1 (c) D B “ p q B Bt t β3 R2 β4 R

M α M t 2M t 2. D X “ pR ` ´ p q2qB ` pt ` qB X β4 β3 β R R αβ R t

xV,W y α 3. D W “ ∇ W ´ DpRβ2q “ ∇ W ´ R pV,W q V V Rβ2 V β

Where x¨, ¨y denotes the ambient metric tensor, p¨, ¨q and ∇ the standard round

metric and corresponding covariant derivative on the sphere S2.

4 α 2 Proof. We recall that, xBR, BRy “ β and xBt, Bty “ ´p β q :

1 3 M Mβ3 1. • xDBR BR, BRy “ 2 BRxBR, BRy “ 2β p´ 2R2 q “ ´ R2

133 1 • xDBR BR, Bty “ ´xBR,DBR Bty “ ´ 2 BtxBR, BRy “ 0

1 • xDBR Bt, BRy “ 2 BtxBR, BRy “ 0

1 α 2 α M • xDBR Bt, Bty “ 2 BRp´p β q q “ ´ β3 R2

1 α M • xDBt Bt, BRy “ ´ 2 BRxBt, Bty “ β3 R2

1 • xDBt Bt, Bty “ 2 BtxBt, Bty “ 0

D , D , x BR BR BRy x BR BR Bty M Giving D BR “ BR ` Bt “ ´ 2 BR, b) and c) follow simi- BR xBR,BRy xBt,Bty R β larly.

2 2 2. Since DX X “ t DBR BR ` tBt ` 2tRDBR Bt ` RBR ` R DBt Bt the result follows from 1.

3. Given DV W “ tanDV W `norDV W , this is a simple application of Proposition A.0.1. and the fact that

2 4 xV,W y “ R β pdπ2V, dπ2W q ˝ π2

2 With all these tools in place we ﬁnally direct our attention towards Sr¯.

2 A.1 The Fundamental Forms of Sr¯

Proposition A.1.1.

M α M t 2 α D ¯ W¯ “ rV ηW ηtR ` ´ p q u ` tV W η ´ R pV,W qsB V β4 β3 β R β R 2M t ` rV ηW ηtt ` u ` RV W ηsB αβ R t α t ` ∇ W ` tV ηW ` W ηV u V β R

134 Proof. ¯ DV¯ W “ DV ηX`V pW ηX ` W q

“ V ηW ηDX X ` V ηDX W ` V W ηX ` W ηDV X ` DV W

XpRβ2q “ V ηW ηD X ` pV ηW ` W ηV q X Rβ2

xV,W y ` V W ηX ´ DpRβ2q ` ∇ W Rβ2 V the ﬁnal equality following from Proposition A.0.1. Now using Corollary A.0.1.1 we have, M α M t 2M t “ V ηW ηtpR ` ´ p q2qB ` pt ` qB u β4 β3 β R R αβ R t

tB pRβ2q xV,W y ` R pV ηW ` W ηV q ` V W ηttB ` RB u ´ DpRβ2q ` ∇ W Rβ2 R t Rβ2 V M α M t 2M t α t “ V ηW ηtpR ` ´ qB ` pt ` qB u ` pV ηW ` W ηV q β4 β3 β R R αβ R t β R α ` V W ηttB ` RB u ´ R pV,W q ` ∇ W R t β V collecting up all the terms the result follows.

2 In order to extract the extrinsic geometry of Sr¯ from Proposition A.1.1 we’ll be needing an orthonormal basis tν, ν‹u for the normal bundle. Recalling equations (A.4a) and (A.4b) we obtain a normal frame ﬁeld,

1 ∇η β ∇η tDpR ´ r¯cosh ηq,Dpt ´ r¯sinh ηqu “ t B ´ r¯sinh η , ´p q2B ´ r¯cosh η u β4 R R2β4 α t R2β4

1 B 1 t ∇η β β 1 ∇η “ t p R q ´ , ´ p B q ´ u β2 β2 β2 R Rβ2 α α t β2 Rβ2

where ∇η :“ gradS2 η. Thus, we may construct a normal vector ﬁeld in LpPq from the linear combination

135 t β α BR t β given by DpR ´ r¯cosh ηq ´ R Dpt ´ r¯sinh ηq “ α p β3 p β2 q ` R p α Btqq. By denoting

α 2 t 2 2 Γ “ p β3 q ´ p R q and Λ “ 1 ´ Γ|∇η| we ﬁnd our orthonormal frame ﬁeld from

2 Lemma A.1.1. For suﬃciently large r¯, Sr¯ inherits an orthonormal frame ﬁeld tν, ν‹u given by:

1 α BR t β Γ 2 ν “ p q ` p B q β3 β2 R α t

1 1 ‹ t BR α β ∇η Γ 2 Λ 2 ν “ p q ` p B q ` Γ R β2 β3 α t Rβ2

Proof. It’s clear that xν‹, ν‹y “ ´1 and xν, ν‹y “ 0 so it suﬃces to show that

xν‹, V¯ y “ 0 for any V P ηpT S2q. So we calculate,

1 1 ‹ t BR α β ∇η Γ 2 Λ 2 xν , V¯ y “ V ηx p q ` p B q, tB ` RB y ` Γx ,V y R β2 β3 α t R t Rβ2

t2 α α “ V ηt β2 ´ R u ` Rβ2Γp∇η, V q R β3 β t α “ Rβ2V ηtp q2 ´ p q2u ` Rβ2ΓV η R β3

“ 0

2 Remark A.1.1. Here ‘suﬃciently large r¯’ serves to ensure that Γ ą 0 on Sr¯ and as a consequence Λ ą 0.

2 Denoting the second fundamental form of Sr¯ by IIr¯ we have,

Proposition A.1.2.

1 α t 2 M α 2 t 2 1. Γ 2 xν, II pV,W qy “ V ηW ηrR t1 ´ p q u ` tp q ´ pα ` 2qp q us r¯ β R β2 β3 R α ´Rp q2pV,W q β

136 1 1 ‹ 2 α 2 t t 2 α 2 1 ` 2α 2. Γ 2 Λ 2 xν , II pV,W qy “ V ηW ηrtβ t1 ´ p q u ` Mβ tp q ´ p q us r¯ β3 R R β3 α R α ` tβ2tp q2η ´ upV,W q r¯ β

Proof.

¯ 1. Since xDV¯ W , νy “ xIIr¯pV,W q, νy the result follows from Proposition A.1.1 and Lemma A.1.1 following a tedious yet straight forward calculation.

2. As in the proof of 1., after the use of Proposition A.1.1 and Lemma A.1.1 the expression eventually simpliﬁes to,

1 1 ‹ 2 α 2 t 3 β α 2 t Γ 2 Λ 2 xν , II pV,W qy “ V ηW ηrtβ t1 ´ p q u ´ Mβp q ´ M p q s r¯ β3 R α β3 R

2 2 ´ Rβ ΓV W η ` Rβ Γp∇V W, ∇ηq ´ tαβpV,W q

` 2tαβΓV ηW η

at which point the third term admits the substitution p∇V W, ∇ηq “ V pW, ∇ηq´

pW, ∇V ∇ηq “ V W η ´ pW, ∇V ∇ηq. This in turn removes V W η from the ex-

pression and pW, ∇V ∇ηq satisﬁes the following identity:

Lemma A.1.2. As a function on round S2, the Hessian of η satisﬁes

η tR t H 2 pV,W q “ ´ ˝ f pV,W q ` 2 ˝ f V ηW η. S r¯2 r¯ R r¯ ´ ¯ ´ ¯

R 2 t R 2 t Proof. From the identity ∇η “ p r¯ q ∇p R q we have V η “ p r¯ q V p R q for any V P ΓpT S2q as well as t r¯ R t pW, ∇ ∇ηq “ W p qV pp q´2q ` p q2pW, ∇ ∇p qq V R R r¯ V R t t R t “ W p qV pp1 ´ p q2q´1q ` p q2pW, ∇ ∇p qq R R r¯ V R t t t t R t “ 2 p1 ´ p q2q´2W p qV p q ` p q2pW, ∇ ∇p qq R R R R r¯ V R 137 t R t t R t “ 2 p q4W p qV p q ` p q2pW, ∇ ∇p qq. R r¯ R R r¯ V R

So proving the Lemma is equivalent to showing that

t t pW, ∇ ∇p qq “ ´ pV,W q V R R

t and since R “ tanh ψ cos ϑ this is, in turn, equivalent to

cos ϑ H 2 cos ϑ˚γ 0 S ` ”

for ˚γ the standard round metric. This is easily veriﬁed for the basis tBϑ, Bϕu.

After substituting the identity of Lemma A.1.2 the expression simpliﬁes to the result.

~ 2 In order to ﬁnd the mean curvature H of Sr¯ we will need the induced metric and its inverse. The metric we ﬁnd from α xV,¯ W¯ y “ V ηW ηxX,Xy ` xV,W y “ V ηW ηtt2β4 ´ R2p q2u ` xV,W y β

“ R2β4tpV,W q ´ ΓV ηW ηu

So in a coordinate basis the induced metric and it’s inverse take the form

2 4 gij “ R β t˚γij ´ Γηiηju

1 Γ gij “ t˚γij ` ηiηju R2β4 Λ

i ij for η “ ˚γ ηj.

138 Corollary A.1.2.1.

2 4 1 α 2 β t 2 2 1. R β Γ 2 Λxν, H~ y “ ´Rp q r2 ´ tΓ ` p1 ´ p q u|∇η| s β α R M α t ` tp q2 ´ p2 ` αqp q2u|∇η|2 β2 β3 R

2 4 1 3 ‹ 2 2 α 2 t t 2 α 2 1 ` 2α 2. R β Γ 2 Λ 2 xν , H~ y “ |∇η| rtβ t1 ´ p q u ` Mβ tp q ´ p q us β3 R R β3 α R α ` tβ2tp q2Γ ´ up2 ´ Γ|∇η|2q r¯ β

Proof. We make the following observations:

1 Γ |∇η|2 gijη η “ p|∇η|2 ` |∇η|4q “ i j R2β4 Λ ΛR2β4

1 Γ 2 ´ Γ|∇η|2 gij˚γ “ p2 ` |∇η|2q “ ij R2β4 Λ ΛR2β4

In either expression of Proposition A.1.1 taking a trace with gij yields the above factors. A straight forward simpliﬁcation yields the result.

4 1 1 r¯ Remark A.1.2. We notice that lim pRβ Γ 2 xν, H~ yq “ ´2, lim Γ 2 “ and lim β “ 1. r¯Ñ8 r¯Ñ8 R r¯Ñ8 2 1 1 As a result we deduce that xν, H~ y “ ´ ` Op q and similarly that xν‹, H~ y “ Op q. r¯ r¯2 r¯2 This is to be expected since, for large r¯, the spacelike normal Rν approaches the po-

4 sition vector ﬁeld P “ tBt ` RBR in Minkowski (or R1) and it’s a well known fact 3 4 that the coordinate sphere of radius r¯ in any boosted R Ă R1 has mean curvature ~ 2 P H “ ´ r¯ r¯ .

A.2 Curvature and Energy

Letting λ “ lim Λpr¯q we reﬁne our estimate of xν, H~ y r¯Ñ8

139 Corollary A.2.0.1.

2 f 1 1. xν, H~ y “ ´ ` ` Op q, where r¯ r¯2 r¯3

R r¯ f “ 2M tp1 ` λ´1qp2p q2 ` 1q ´ 4u r¯ R

g 1 2. xν‹, H~ y “ ` Op q, where r¯2 r¯3

t ´ 1 ´1 r¯ 2 g “ ´2M λ 2 t3 ` λ ´ p q u r¯ R

Proof. 1. We start by approximating Γ, α t Γ “ p q2 ´ p q2 β3 R α t “ p q2 ´ 1 ` 1 ´ p q2 β3 R r¯ R α “ p q2t1 ´ p q2p1 ´ p q2qu R r¯ β3 r¯ R 1 1 “ p q2t1 ´ 4M u ` Op q R r¯ r¯ r¯2

giving,

1 r¯ R 1 1 Γ 2 “ t1 ´ 2M u ` Op q R r¯ r¯ r¯2

and,

1 R 1 1 RΓ 2 “ r¯t1 ´ 2M u ` Op q. r¯ r¯ r¯

From this we conclude that,

1 R 1 f 1 RΓ 2 xν, H~ y “ p1 ´ 2M qp´2 ` q ` Op q r¯ r¯ r¯ r¯2 R 1 1 “ ´2 ` p4M ` fq ` Op q. r¯ r¯ r¯2

140 Now,

1 α 2 β t 2 2 M α 2 Λp2 ` RΓ 2 xν, H~ yq “ 2Λ ´ p q r2 ´ pη ` p1 ´ p q q|∇η| s ` pp q β3 α R Rβ4 β3 t ´ p2 ` αqp q2q|∇η|2 R α α α β “ 2 1 ´ p q2 ` p q2|∇η|2t p q2 ´ 1 ` ´ 1 u β3 β3 β3 α ´ ¯ ´ ¯ ´ ¯ t α β α ` p q2|∇η|2t 1 ´ p q2 ` 1 ´ p q2 u R β3 α β3 ´ ¯ ´ ¯ M α t ` pp q2 ´ p2 ` αqp q2q|∇η|2 Rβ4 β3 R

1 where we’ve isolated with ¨ all terms of magnitude Op r¯q. Asr ¯ Ñ 8, ´ ¯ R 1 λp4M ` fq “ lim Λ¯rp2 ` RΓ 2 xν, H~ yq r¯ r¯Ñ8 r¯ r¯ r¯ t r¯ r¯ “ 8M ` |∇η|2t´4M ` M u ` |∇η|2p q2t4M ` 3M u R R R R R R r¯ t ` M t1 ´ 3p q2u|∇η|2 R R r¯ r¯ t “ 8M ´ 2M p1 ´ 2p q2q|∇η|2 R R R

2 R 2 t 2 r¯ 2 making the substitutions |∇η| “ p r¯ q p1 ´ λq and p R q “ 1 ´ p R q the result follows.

2. From our approximation of Γ in 1. we see

R R 1 1 p q2Γ ´ 1 “ ´4M ` Op q r¯ r¯ r¯ r¯2

so that R α R β ´ α p q2Γ ´ “ p q2Γ ´ 1 ` r¯ β r¯ β r¯ R 1 1 “ Mp ´ 4 q ` Op q R r¯ r¯ r¯2 141 R 1 1 “ Mp1 ´ 4p q2q ` Op q. r¯ R r¯2

Using this we see directly from Corollary A.1.2.1

R 3 2 ‹ 2 4 1 3 ‹ λ 2 lim r¯ xν , H~ y “ lim R β Γ 2 Λ 2 xν , H~ y r¯ r¯Ñ8 r¯Ñ8 t t t “ |∇η|2t4M ` M pp q2 ´ 3qu R R R t R r¯ ` M p1 ´ 4p q2qp2 ´ p q2|∇η|2q R r¯ R t R “ M t6|∇η|2 ` 2λ ´ 8p q2u R r¯ t R R “ ´2M t3p q2λ ` p q2 ´ λu R r¯ r¯ tR r¯ “ ´2Mλ t3 ` λ´1 ´ p q2u r¯2 R

Deﬁnition A.2.1. The Hawking Energy of a closed surface Σ is given by;

|Σ| 1 ~ ~ EH “ 1 ´ xH, HydA c16π 16π Σ ´ ż ¯ It’s a well known fact that the Hawking Energy for coordinate spheres in an asymptotically ﬂat hypersurface approach the ADM energy asr ¯ Ñ 8. In Schwarzschild we see therefore that coordinate spheres in the cononical time slice tt “ 0u are round (of radiusr ¯ “ Rβ2) and satisfy t “ η “ 0 so that Corollary 2 gives us

~ ~ ~ 2 4 α 2 xH, Hy “ xν, Hy “ R2 p β3 q . Therefore

2 4 2 4πR β 1 2 4 4 α 2 Rβ α 2 R EH “ p1´ 4πR β 2 p 3 q q “ p1´p q q “ pα`βqpα´βq “ M. c 16π 16π R β 2 β 2

As a result of boosting this slice to the rapidity ψ we expect the ADM energy to boost to M cosh ψ. We are now in a position to verify this expectation.

142 Proposition A.2.1. For the boosted time slice tt¯ “ 0u the Hawking Energy EH pr¯q

2 of the coordinate sphere Sr¯ satisﬁes

lim EH pr¯q “ M cosh ψ r¯Ñ8

Proof. From the change ϑ Ñ ϑ¯ given by,

sin ϑ sin ϑ¯ “ 1 ´ tanh2 ψ cos2 ϑ a sechψ cos ϑ cos ϑ¯ “ 1 ´ tanh2 ψ cos2 ϑ it’s easily shown that a

¯2 ¯ 2 2 R 2 2 2 2 1 2 dϑ ` psin ϑq dϕ “ p q tλdϑ ` psin ϑq dϕ u “ lim ds 2 . r¯ r¯Ñ8 r¯2 Sr¯

2 2 We conclude for larger ¯, Sr¯ approaches round S and therefore, 2 ? |Sr¯| detg 1 R 2 4 r¯Ñ8 1 R 2 “ dϑdϕ “ Λ 2 p q β sin ϑdϑdϕ / λ 2 p q sin ϑdϑdϕ “ 4π. r¯2 r¯2 r¯ r¯ ż ż ż From our reﬁned decomposition of H~

2 f 2 1 4 f 1 xH,~ H~ y “ ´ ` ` Op q “ ´ 4 ` Op q r¯ r¯2 r¯4 r¯2 r¯3 r¯4 ´ ¯ we are able to approximate EH pr¯q,

2 1 | r¯| R 1 1 1 1 S 2 2 2 4 2 4 EH pr¯q “ 2 p q rr¯pλ ´ Λ β q ` fΛ β s sin ϑdϑdϕ ` Op q 4π c16πr¯ r¯ r¯ ´ż ¯ giving,

1 1 R 2 1 1 4 1 lim EH pr¯q “ p q r lim pr¯pλ 2 ´ Λ 2 β qq ` fλ 2 s sin ϑλϑdϕ. r¯Ñ8 4π 2 r¯ r¯Ñ8 ż

143 So in order to calculate the limit, we will also need next to leading order information ? about detg. We ﬁnd this next order term by noting that since

α 4M 1 Λ “ 1 ´ Γ|∇η|2 “ λ ` p1 ´ p q2q|∇η|2 “ λ ` |∇η|2 ` Op q β3 R r¯2

we have that

2 1 1 2M M 1 1 2M 1 |∇η| 1 2 4 2 2 4 2 2 Λ β “ λ p1 ` |∇η| qp1 ` q ` Op 2 q “ λ ` pλ ` 1 q ` Op 2 q λR 2R r¯ R λ 2 r¯

giving

2 1 1 4 r¯ 1 |∇η| lim pr¯pλ 2 ´ Λ 2 β qq “ ´2M λ 2 p1 ` q. r¯Ñ8 R λ so that 1 r¯ |∇η|2 lim EH pr¯q “ f ´ 2M p1 ` qdA. r¯Ñ8 8π 2 R λ żS Using Corollary A.2.0.1 it’s an easy calculation to show that 2 r¯ |∇η| 1 R 2 R 1 r¯ R R 2 2 2 2 4 tf ´ 2M p1 ` quλ p q “ 2Mt 1 ` λ pp q ´ 2p q qu R λ r¯ λ 2 r¯ R r¯ r¯ R R “ 2Mt2 cosh ψ ` sechψpp q2 ´ 2p q4qu r¯ r¯

so that ﬁnally we calculate π 2π M R 2 R 4 lim EH pr¯q “ t2 cosh ψ ` sechψpp q ´ 2p q qu sin ϑdϑdϕ r¯Ñ8 4π r¯ r¯ ż0 ż0 M 1 1 1 “ 2M cosh ψ ` sechψ ´ 2 dx 2 1 ´ tanh2 ψx2 p1 ´ tanh2 ψx2q2 ż´1 M B 1 1 “ 2M cosh ψ ` sechψ ´ p1 ` c q 2 2 dx 2 Bc ´1 1 ´ c x c“tanh ψ ” ż ıˇ ˇ M 2 2 ˇ 2 1 “ 2M cosh ψ ` sechψ ´ tanh´1 c ´ cp´ tanh´1 c ` q 2 c c2 c 1 ´ c2 c“tanh ψ ” ıˇ ˇ “ M cosh ψ. ˇ

144 We will now work towards a stronger result from which we can deduce Proposition

1 A.2.1. Speciﬁcally, we calculate K 2 H,~ H~ from the Gauss equation: Sr¯ ´ 4 x y

2 Lemma A.2.1. In the case of Schwarzschild, P ˆRβ2 S , we have,

2M 1 α 2 1. xRBRBt BR, Bty “ ´ R2 Rβ2 p β q

Rβ2 M 2. (a) H pBR, BRq “ R2

Rβ2 (b) H pBR, Btq “ 0

Rβ2 M α 2 (c) H pBt, Btq “ ´ R2 p β3 q

Proof. Using Corollary A.0.1.1 thoughout:

1. xRBRBt BR,Bty “ xDBt DBR BR, Bty ´ xDBR DBt BR, Bty

“BtxDBR BR, Bty ´ xDBR BR,DBt Bty ´ BRxDBt BR, Bty ` xDBt BR,DBR Bty M α 1 M M M “ 0 ` xB , B y ´ B p xB , B yq ` p q2xB , B y R2β β3 β4 R2 R R R αβR2 t t αβR2 t t

Mα M 2α M 2 “B p q ` ´ R R2β3 R4β4 R4β4

2Mα M α 1 ` α 2M 2α “ ´ ` pB p q ´ M q ` R3β3 R2 R β3 R2β4 R4β4 2Mα M “ ´ p1 ´ q R3β3 Rβ

2Mα2 “ ´ R3β4

α M 1 ` α where we used the fact that B p q “ in the sixth line. R β3 R2 β4

145 2 2 2 2 2. (a) xBR,DBR DpRβ qy “ BRpRβ q ´ xDBR BR,DpRβ qy M “B2 pRβ2q ` B pRβ2q R R2β R M M M Mα “B pβ2 ´ β q ` pβ2 ´ β q “ B pαβq ` R R R2β R R R2

Mα M β ´ α “ αB β ` βB α ` “ p ` αq R R R2 R2 2 M “ R2

2 2 BRpRβ q 1 (b) xBR,DBt DpRβ qy “ β4 2 BtxBR, BRy “ 0

B pRβ2q αβ 1 (c) xB ,D DpRβ2qy “ R xB ,D B y “ B xB , B y t Bt β4 t Bt R β4 2 R t t α α m “ p´ q. β3 β3 R2

Proposition A.2.2. Let M “ B ˆf F be a warped product with Riemannian curvature tensor R. If X,Y,Z P LpBq and U, V, W P LpF q, then

B 1. RXY Z P LpBq is the lift of RXY Z on B.

Hf pX,Y q f 2. RVX Y “ f V , where H is the Hessian of f.

3. RXY V “ RVW X “ 0

xV,W y 4. RXV W “ f DX Df

F xDf,Dfy 5. RVW U “ RVW U ´ f 2 txV,UyW ´ xW, UyV u

Proof. see [21] pg.210.

2 Proposition A.2.3. For Σ “ Sr¯

1 1 M K ´ xH,~ H~ y ` xIIˆ, IIˆy “ p2 ` Γ|∇η|2q Σ 4 2 ΛpRβ2q3 146 Proof. Knowing that Gp¨, ¨q vanishes identically in Schwarzschild our task is to cal-

‹ 1 ‹ ‹ culate the quantity xRνν‹ ν, ν y “ 4 xRLLL,Ly (for L “ ν ´ν and L “ ν `ν ) so that the result follows from the Gauss equation (Proposition 3.0.1). We use our standard

‹ ‹ 1 ‹ ‹ choice for tν, ν u and break ν into the components Λ 2 ν “ ν1 ` ν2 where:

t BR α β r 2 s ` 3 r Bts ‹ R β β α ν1 “ 1 Γ 2

and

1 ∇η ν “ Γ 2 2 Rβ2

(we temporarily denote also ν1 “ ν). We will need to ﬁnd the determinant of the linear map

t 1 α β ‹ 2 3 Bt 1 ν R β β α 2 1 α 1 t β “ Γ BR ν1 ˜ β3 β2 R α ¸ ˆ ˙ ˆ ˙

which is easily seen to give

t 2 1 α 2 1 p q ´ p 3 q 1 R αβ β αβ “ ´ . Γ αβ

Thus Proposition A.2.2 gives

‹ ‹ ‹ Λ R ‹ ν, ν R ‹ ν , ν ν R ‹ ν , ν R ν , ν x νν y “ x ν1 ν1 `ν2 1 1 ` 2y “ x ν1 ν1 1 1 y ´ x ν2 ν1 1 2y

1 xν , ν y 2 “ xR B , B y ´ 2 2 HRβ pν , ν q α2β2 BRBt R t Rβ2 1 1

2 2M Γ|∇η| 1 α 1 2 t β 2 “ ´ ´ r tp q2 HRβ pB , B q ` p q2p q2HRβ pB , B qus pRβ2q3 Rβ2 Γ β3 β4 R R R α t t

2M |∇η|2 α 1 M t β M α “ ´ ´ rp q2 ´ p q2p q2 p q2s pRβ2q3 Rβ2 β3 β4 R2 R α R2 β3

2M |∇η|2 α M β “ ´ ´ p q2 p q2Γ pRβ2q3 Rβ2 β3 R2 α 147 M “ ´ p2 ` Γ|∇η|2q pRβ2q3

Having used Lemma A.2.1 in the forth.

2 Lemma A.2.2. Sr¯ satisﬁes 1 xIIˆ, IIˆy “ Op q. r¯4

Proof. From Proposition A.1.2 and Corollary A.1.2.1 we conclude that the quantities 1 xIIˆpV,W q, νy “ xIIpV,W q, νy ´ xV,W yxH,~ νy 2 1 xIIˆpV,W q, ν‹y “ xIIpV,W q, ν‹y ´ xV,W yxH,~ ν‹y 2

converge asr ¯ Ñ 8 for any V,W P ΓpT S2q. Therefore, from the choice of orthonormal frame

Bϑ Bϕ te1 “ 1 , e2 “ 2 u Λ 2 Rβ2 sin ϑRβ we conclude that

1 xIIˆ, IIˆy “ xIIˆpe , e q, νy2 ´ xIIˆpe , e q, ν‹y2 “ Op q i j i j r¯4 i,j ÿ ´ ¯

Proposition A.2.2 therefore allows us to directly conclude that

3 1 ~ ~ 1 ˆ ˆ M r¯ 3 r¯ 2 2 lim r¯ tKΣ ´ xH, Hy ` xII, IIyu “ p q p2 ` p q |∇η| q. r¯Ñ8 4 2 λ R R

Moreover, from Lemma A.2.2

2 1 |Sr¯| 1 1 lim E r¯ lim K 2 H,~ H~ IIˆ, IIˆ dA H p q “ Sr¯ ´ x y ` x y r¯Ñ8 r¯Ñ8 4π 16π 2 4 2 c żSr¯ π 1 M r¯ r¯ 2 2 “ 1 p2 ` p q |∇η| q sin ϑdϑ 4 0 λ 2 R R ż 148 1 π tanh2 ψ sin2 ϑ “ M cosh ψ p1 ´ tanh2 ψ cos2 ϑqp2 ` q sin ϑdϑ 4 1 ´ tanh2 ψ cos2 ϑ ż0 1 1 “ M cosh ψ 2 ´ 2 tanh2 ψx2 ` tanh2 ψp1 ´ x2qdx 4 ż´1 1 4 2 “ M cosh ψp4 ´ tanh2 ψ ` tanh2 ψp2 ´ qq 4 3 3 “ M cosh ψ as expected.

A.3 Normal Connection and Momentum

2 Finally, we ﬁnish our study of Sr¯ with its connection 1-form. From this we are able to deduce the ADM-momentum of our boosted slice.

‹ 2 2 Deﬁnition A.3.1. Given a normal frame ﬁeld tν, ν u for Sr¯ and V P ΓpT S q, the connection 1-form associated to this frame is given by:

‹ ανpV q “ xDV¯ ν, ν y.

Proposition A.3.1. In our standard normal frame tν, ν‹u we have,

1 α α α 2 M 1 t 2 α 2 ΓΛ 2 α pV q “ V ηr t p q ´ 1u ` t2p1 ` αqp q ´ p q us ν β3 β β3 R β4 R β3

Proof.

1 1 1 DV¯ pΓ 2 νq “ V ηDX pΓ 2 νq ` DV pΓ 2 νq

1 2 1 1 Γ 2 νpRβ q “ V ηttD pΓ 2 νq ` RD pΓ 2 νqu ` V BR Bt Rβ2

Now

1 t M 1 ` α t β 1. tD pΓ 2 νq “ t B ´ B u : BR R R β6 R R α t

1 α 1 α 1 t β t β D pΓ 2 νq “ B p qB ` D B ` B p qB ` D B BR R β3 β2 R β3 β2 BR R R R α t R α BR t 149 M 1 ` α 1 α M 1 α 1 M 1 “ t ` uB ´ B R2 β4 β2 β3 R2 β3 R β3 β2 R2 β R t β t M 1 t β M 1 ` t´ ´ uB ` B R2 α R R2 α2 t R α R2 αβ t

M 1 ` α t β “ B ´ B R2 β6 R R2 α t

1 β M 1 M t 1 2. RD pΓ 2 νq “ p ` qB ` B : Bt α R β6 t R R β6 R

1 α 1 1 β t β D pΓ 2 νq “ D B ` B ` D B Bt β3 β2 Bt R R α t R α Bt t α 1 M 1 1 β t β α 1 M “ t ` uB ` B β3 β2 R2 αβ R α t R α β3 β4 R2 R

1 2 Γ 2 νpRβ q 1 α 3. “ p q2 : Rβ2 R β3

1 2 2 Γ 2 νpRβ q α 1 B pRβ q α 1 1 M 1 “ R “ p ´ q Rβ2 β3 β2 Rβ2 β3 β2 R R2 β

Giving

1 β t 2 M 1 t M 2 ` α 1 α 2 D ¯ pΓ 2 νq “ V ηrt p1 ´ p q q ` uB ` B s ` p q V. V α R R β6 t R R β6 R R β3

As a result,

1 1 1 ‹ 1 ‹ xDV¯ pΓ 2 νq,Γ 2 Λ 2 ν y “ ΓΛ 2 xDV¯ ν, ν y

β t M 1 t M 2 ` α t B α β “ V ηxt p1 ´ p q2q ` uB ` B , p R q ` p B qy α R R β6 t R R β6 R R β2 β3 α t

α2 ` V η Γ β4

α α t M t 2 ` α α 1 “ V ηr t Γ ´ p1 ´ p q2qu ` tp q2 ´ p q2 us β3 β R R R β4 β3 β4 α α α M 1 t α “ V ηr t p q2 ´ 1u ` t2p1 ` αqp q2 ´ p q2us β3 β β3 R β4 R β3 150 It now easily follows that

h 1 Corollary A.3.1.1. ανpV q “ V ηt r¯ ` Op r¯2 qu for,

R ´ 1 r¯ 2 h “ ´2M λ 2 t1 ` 2p q u r¯ R

Proof.

In order to get hold of the second fundamental from K of our slice tt¯ “ 0u we

will need to ﬁnd the unit timelike normal N~ 9Dt¯ which we now work towards.

Lemma A.3.1. t Dt¯“ ´ cosh ψt DpR ´ r¯cosh ηq ´ Dpt ´ r¯sinh ηqu R t 1 B β β 1 t ∇η “ ´ cosh ψt p R q ` p B q ` p1 ´ p q2q u R β2 β2 α α t β2 R Rβ2

Proof. We recall that t¯“ cosh ψt ´ sinh ψz “ cosh ψt ´ sinh ψR cos ϑ so that,

β 1 R Dt¯“ cosh ψt´p q2B ´ tanh ψp cos ϑB ´ sin ϑB qu. α t β4 R R2β4 ϑ

´ tanh ψ sin ϑ We also recall that ∇η “ Bϑ allowing for the above expression to 1 ´ tanh2 ψ cos2 ϑ be written as

β 1 t R t Dt¯“ ´ cosh ψtp q2B ` B ` p1 ´ p q2q∇ηu α t β4 R R R2β4 R

and the result follows.

¯ ¯ 2 t 2 1 β 2 1 t 2 2 Remark A.3.1. We notice that xDt, Dty “ cosh ψtp R q β4 ´p α q ` β4 p1´p R q q|∇η| u has the same limit for large r¯ as ´ cosh2 ψΓΛ.

151 Since Dt¯ is the timelike normal to our boosted slice tt¯“ 0u and therefore normal

2 2 to Sr¯ we must have some function ϕn P FpS q such that,

Dt¯ N~ :“ ´ “ cosh φ ν‹ ` sinh φ ν. |Dt¯| n n

It’s an easy calculation to show that

1 α 2 ¯ 2 p q ´ 1 ~ xDt, Γ νy t β β3 sinh φn “ xN, νy “ ´ 1 “ 1 |Dt¯|Γ 2 R α |Dt¯|Γ 2

1 ~ ‹ which has magnitude Op r¯q indicating that N approaches ν for larger ¯. Since sinh φ lim n “ 1: r¯Ñ8 φn

α 2 r¯ t r¯p1 ´ p 3 q q t 4M lim rϕ¯ “ ´ lim β “ ´ R n 1 r¯ 1 r¯Ñ8 R r¯Ñ8 ¯ 2 R 2 2 |Dt|η p R q λ

or equivalently,

2 ~ ‹ Lemma A.3.2. Given φn P FpS q such that N “ cosh φnν ` sinh φnν we have, k 1 ϕn “ r¯ ` Op r¯2 q for,

t ´ 1 k “ ´4m d 2 . r¯

Proof.

Corollary A.3.1.2. For the orthonormal frame ﬁeld tN,~ ~nu in the normal bundle

2 of Sr¯ such that xBr¯, ~ny ą 0:

~ ~ l 1 1. xN, Hy “ r¯2 ` Op r¯3 q for,

t ´ 1 ´1 r¯ 2 l “ 2M λ 2 t1 ´ λ ` p q u r¯ R

152 s 1 2. α~npV q “ V ηt r¯ ` Op r¯2 qu for,

R ´ 1 s “ ´2m λ 2 r¯

Proof. 1. Since ~ ~ ‹ ~ ~ xN, Hy “ cosh φnxν , Hy ` sinh φnxν, Hy

according to Corollary A.2.0.1 and Lemma A.3.2,

2 t ´ 1 ´1 r¯ 2 t ´ 1 lim r¯ xN,~ H~ y “ ´2M λ 2 t3 ` λ ´ p q u ´ 4M λ 2 p´2q r¯Ñ8 r¯ R r¯

t ´ 1 ´1 r¯ 2 “ 2M λ 2 t1 ´ λ ` p q u. r¯ R

2. From

‹ ~n “ cosh φnν ` sinh φnν

it’s an easy exercise to show that α~n “ αν ´ dφn giving

R ´ 1 r¯ 2 t ´ 1 lim rα¯ ~npV q “ V ηr´2M λ 2 t1 ` 2p q us ´ V p´4M λ 2 q r¯Ñ8 r¯ R r¯

R 2 t R ´ 1 r¯ 2 t R ´ 1 “ p q V p qr´2M λ 2 t1 ` 2p q us ` 4MV p λ 2 q r¯ R r¯ R R r¯

t R 3 ´ 1 R 2 R ´ 1 “ V p qr´2Mp q λ 2 t1 ` 2p q u ` 4M λ 2 s R r¯ r¯ r¯

t R ´ 1 R 2 “ V p qp2M λ 2 qr2 ´ tp q ` 2us R r¯ r¯

R ´ 1 “ ´2M λ 2 V η r¯

R 1 ´ 2 having used the fact that cosh ψ “ r¯ λ in the second line.

We are now in a position to verify that the ADM-momentum of our boosted slice is given by P~ “ p0, 0,M sinh ψq,

153 Proposition A.3.2. In the boosted isotropic coordinates px1, x2, x3q “ px,¯ y,¯ z¯q,

1 j Pi :“ lim tKij ´ Kg¯iju~n dA “ M sinh ψδi3 r¯Ñ8 8π 2 żSr¯

~ where Kp¨, ¨q “ ´xIIt¯p¨, ¨q, Ny is the scalar second fundamental form of our boosted

ij 2 slice (K “ g¯ Kij) and ~n is the outward unit normal to Sr¯.

Proof. From the induced metric on our boosted slice we read oﬀ,

4 Bx¯ “ β Dx¯

4 By¯ “ β Dy¯ α B “ β4tcosh2 ψ ´ p q2 sinh2 ψuDz¯ z¯ β3

giving

|Dpr¯2q|~n “ Dpr¯2q

“ 2¯xDx¯ ` 2¯yDy¯ ` 2¯zDz¯

2¯x 2¯y 2¯z “ B ` B ` B . β4 x¯ β4 y¯ 2 α 2 2 z¯ cosh ψ ´ p β3 q sinh ψ

2 2 Thus, under restriction to Sr¯ we see |Dpr¯ q| “ 2¯r ` Op1q so that, x¯ 1 R 1 1 x~n, B y “ ` Op q “ sin ϑ cos ϕ ` Op q “ sin ϑ¯cos ϕ ` Op q x¯ r¯ r¯ r¯ r¯ r¯ y¯ 1 R 1 1 x~n, B y “ ` Op q “ sin ϑ sin ϕ ` Op q “ sin ϑ¯sin ϕ ` Op q y¯ r¯ r¯ r¯ r¯ r¯ z¯ 1 R 1 1 x~n, B y “ ` Op q “ sechψ cos ϑ ` Op q “ cos ϑ¯ ` Op q. z¯ r¯ r¯ r¯ r¯ r¯

2 Bϑ Bϕ Dr¯ For the orthonormal frame ﬁeld te1 “ 1 , e2 “ 2 , ~n “ 2 u we have Rβ2D 2 Rβ sin ϑ |Dr¯ |

4 2 r¯ ´ 1 x¯ 2 r¯ ´ 1 R xe , B y “ β xe ,Dx¯y “ β Λ 2 B p q “ β Λ 2 B p sin ϑq cos ϕ 1 x¯ 1 R ϑ r¯ R ϑ r¯

2 R λ 1 1 “ sechψβ p q 2 cos ϑ cos ϕ “ cos ϑ¯cos ϕ ` Op q r¯ Λ r¯ 154 1 xe , B y “ β2 sin´1 ϑp´ sin ϑ sin ϕq “ ´β2 sin ϕ “ ´ sin ϕ ` Op q, 2 x¯ r¯

4 2 r¯ ´ 1 y¯ 2 r¯ ´ 1 R xe , B y “ β xe ,Dy¯y “ β Λ 2 B p q “ β Λ 2 B p sin ϑq sin ϕ 1 y¯ 1 R ϑ r¯ R ϑ r¯

2 R λ 1 1 “ sechψβ p q 2 cos ϑ sin ϕ “ cos ϑ¯sin ϕ ` Op q r¯ Λ r¯ 1 xe , B y “ β2 sin´1 ϑpsin ϑ cos ϕq “ β2 cos ϕ “ cos ϕ ` Op q 2 y¯ r¯

and ﬁnally

2 r¯ ´ 1 2 α 2 2 z¯ xe , B y “ β Λ 2 tcosh ψ ´ p q sinh ψuB p q 1 z¯ R β3 ϑ r¯

2 r¯ ´ 1 2 α 2 2 R “ β Λ 2 tcosh ψ ´ p q sinh ψuB psechψ cos ϑq R β3 ϑ r¯

2 R λ 1 2 α 2 2 “ ´β p q 2 tcosh ψ ´ p q sinh ψu sin ϑ r¯ Λ β3 1 “ ´ sin ϑ¯ ` Op q r¯

xe2, Bz¯y “ 0.

2 Now given any ambient vector ﬁeld X restricted to Sr¯,

KpX, ~nq ´ KxX, ~ny “ xX, ~nypKp~n,~nq ´ Kq ` xX, eiyKpei, ~nq i ÿ ~ ~ “ xX, ~nyxH, Ny ´ xX, eiyα~npeiq i ÿ ~ ~ “ xX, ~nyxH, Ny ´ xX, e1yα~npe1q

~ where we used xIIr¯pei, eiq, Ny “ ´K ` Kp~n,~nq in the second line and α~n9dϑ in i ÿ the third line (following from Proposition A.3.1 given that φn is independent of ϕ).

From our calculations so far we recognize that KpBx¯, ~nq ´ KxBx¯, ~ny “ F1psin ϑq cos ϕ and K , ~n K , ~n F sin ϑ sin ϕ for some F and F . So given that detg 2 pBy¯ q´ xBy¯ y “ 2p q 1 2 Sr¯

is independent of ϕ we have P1 “ P2 “ 0. For P3 we see a

155 ~ ~ xBz¯, ~nyxH, Ny ´ xe1, Bz¯yα~npe1q

t ´ 1 ´1 r¯ 2 R ´ 1 r¯ ´ 1 1 1 “ r2M λ 2 t1 ´ λ ` p q u cos ϑ¯ ´ 2M λ 2 sin ϑ¯ λ 2 η s ` Op q r¯ R r¯ R ϑ r¯2 r¯3

sinh ψ R 2 ´ 1 2 2 2 2 “ 2M p q λ 2 rcos ϑt1 ´ sinh ψp1 ´ tanh ψ cos ϑqu cosh2 ψ r¯ 1 1 ` cosh2 ψ sin2 ϑs ` Op q r¯2 r¯3

sinh ψ R 2 ´ 1 2 2 2 2 2 4 1 “ 2M p q λ 2 rcosh ψ ´ 2 sinh ψ cos ϑ ` sinh ψ tanh ψ cos ϑs cosh2 ψ r¯ r¯2 1 ` Op q r¯3

r¯ 2 ´ 1 1 1 “ 2M sinh ψp q λ 2 ` Op q R r¯2 r¯3

giving

1 ~ ~ xBz¯, ~nyxH, Ny ´ xe1, Bz¯yα~npe1qdA 8π 2 żSr¯ π M r¯ 2 ´ 1 R 2 1 1 “ sinh ψ p q λ 2 p q Λ 2 sin ϑdϑ ` Op q 2 R r¯ r¯ ż0 π M Λ 1 1 “ sinh ψ p q 2 sin ϑdϑ ` Op q 2 λ r¯ ż0 1 “ M sinh ψ ` Op q r¯

the result follows.

Remark A.3.2. (see [31]) In Proposition A.3.2 we recall for X “Bz¯ that xBz¯, ~ny “

¯ 1 ¯Bϑ¯ 1 ¯ ¯ cos ϑ ` Op r¯q giving for large r¯, xBz¯, e1ye1 « ´ sin ϑ r¯ « r¯∇ cos ϑ. So for any α‹ “ 0 1 0 2 α~n ´ dφ‹ where φ‹ “ φ‹ ` Op r¯q and φ‹ P FpS q

156 ~ ~ lim xBz¯, ~nyxH, Ny ´ α~npxBz¯, e1ye1qdA r¯Ñ8 2 żSr¯

¯ 2 ~ ~ ¯ ¯ “ cos ϑ limpr¯ xH, Nyq ´ limprα¯ ~nqp∇ cos ϑqdσ ¯2 żS ¯ 2 ~ ~ ¯ ¯ “ cos ϑ limpr¯ xH, Nyq ` ∇ ¨ plimprα¯ ~nqq cos ϑdσ ¯2 żS

2 3 cos ϑ¯ lim r¯ H,~ N~ lim r¯ ∇ 2 α dσ “ p p x yq ` p Sr¯ ¨ ~nqq ¯2 żS

2 3 2 cos ϑ¯ lim r¯ H,~ N~ lim r¯ ∇ 2 α lim r¯ ∆ 2 rφ¯ dσ “ p p x yq ` p Sr¯ ¨ ‹q ` p Sr¯ ‹qq ¯2 żS ¯ 2 ~ ~ 3 “ cos ϑplimpr¯ xH, Nyq ` limpr¯ ∇ ¨ α‹q ´ 2 limprφ¯ ‹qqdσ ¯2 żS ¯ ¯ ¯ having used ∆S2 cos ϑ “ ´2 cos ϑ after an integration by parts twice to obtain the ﬁnal term of the ﬁnal line.

From this remark we see that the momentum (at least in the limit) is completely encoded within the connection 1-form provided we choose an orthonormal frame that r¯2 ~ ~ ~ ~ ~ ‹ ‹ yields lim rϕ¯ ‹ “ lim xH, Ny. Noticing that H “ xH, νyν´xH, ν yν is dominated r¯Ñ8 r¯Ñ8 2 ~ 2 ‹ by xH, νyν « ´ r¯ν for larger ¯ we may choose the orthonormal frame ﬁeld tνH , νH u whereby,

H~ ν :“ ´ “ cosh ϕ ~n ` sinh ϕ N~ H H H H

‹ ~ νH :“ sinh ϕH~n ` cosh ϕH N

with associated connection 1-form given by αH “ αn ´ dϕH .

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160 Biography

Henri Petrus Roesch was born in Pretoria, South Africa on May 12, 1989 to parents Elize and Henning Roesch. Most recently, Roesch recieved Ph.D and M.A. degrees in mathematics from Duke University in Durham, North Carolina. Prior to that, he recieved a B.Sc. degree in mathematics with theoretical physics from University College London and a M.A.St degree in theoretical physics from Cambridge Univer- sity in the United Kingdom. In July 2017, Roesch will be joining the faculty of the University of California, Irvine Department of Mathematics as a National Science Foundation Postdoctoral Fellow. In July 2018, Roesch will be joining the faculty of Columbia University Department of Mathematics as a J.F. Ritt Assistant Professor.

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