The phases of supersymmetric black holes in five dimensions

The supersymmetric solutions of single black hole also called BMPV blackhole because it was first discovered by Breckenridge, Myers, Peet and Vafa andsingle black ring with horizon topol

THE PHASES OF SUPERSYMMETRIC BLACK HOLES

IN FIVE DIMENSIONS

JIANG YUN (B.Sc., ZJU)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE

2010

of California, Davis Under his invaluable encouragement, patient guidance andunwearied revision from the initial to the final level, I am able to develop anunderstanding of the subject and complete this thesis at the end, additionally toget a general introduction to supersymmetry Besides the academic aspects, I wasdeeply benefited from his sincere treatment to students and his punctilious attitudetoward research For example, he carefully read through my thesis and correctedalmost every error, even a nonsignificant misusage of punctuation.

Much of this thesis I understood in the early preparation is from the manyinteractions with my senior, Chen Yu, so I must acknowledge my special intellectualdebt to him He also provided essential suggestions on the order of chapters.Meanwhile, I am greatly indebted to my girlfriend, Qu Yuanyuan, for her persistentsupport during the completion of my Master study For the thesis, she madeimportant contributions in programming to figure out the phase structures forvarious systems and drawing the Figure (4.5) and Figure (A.3) She also generouslytook the time to read and comment on the early drafts

i

There are many teachers at National University of Singapore (NUS) whohelped me learn high energy physics, to whom I am indebted Especially, I wouldlike to thank Prof Belal E Baaquie for his useful course and his strong recommen-dation without hesitation as soon as he learnt that I planned to apply for PhD in

US schools I would like to show my gratitude to Dr Wang Qinghai who gave me

my comprehensive exposure to quantum field theory in a superb course requiredsitting in the whole Saturday afternoon this semester As long as I had a ques-tion, he always spared the time to help me in every way possible in order for someconcept or derivation to really sink in even if he is busy

In addition, I owe my deepest gratitude to Prof Feng Yuanping, the head

of Physics department In the last three years he has made available his help in anumber of ways from academic consultations to the aspect of personal matters I

am grateful to all the faculty, staff and administration of NUS for their considerateservices and the Physics department particularly for providing the equipment.Thanks are also due to my parents, and my friends, far too many to namehere, who always care and encourage me to go ahead with a bold head Luckily,

I do not idle time away and accomplished some achievements during two-yearpostgraduate study at NUS More importantly, I am still on my way to pursue mylongtime dream after graduation

Jiang Yun

NUS, Singapore

July, 2010

Acknowledgement i

Contents iii

Summary vii

List of Figures ix

List of Symbols xi

1 Introduction · 1 · 1.1 Black holes in higher dimensional space-time · 1 ·

1.1.1 Why we are interested in higher dimensional black holes · 2 ·

1.1.2 New features of D = 5 black holes · 3 ·

1.1.3 Phases of D = 5 vacuum black holes · 5 ·

1.2 Black holes of D = 5 supergravity · 7 ·

1.2.1 Nonextremal solution of D = 5 supergravity · 8 ·

1.2.2 Supersymmetric solutions of minimal D = 5 supergravity · 8 ·

1.2.3 Why minimal D = 5 supergravity · 9 ·

1.3 Objective and organization · 10 ·

iii

2 Elements of General Relativity in Higher Dimensions · 13 ·

2.1 Conserved charges · 13 ·

2.2 Dimensionless scale · 16 ·

3 Review of Supersymmetric Solutions in Five Dimensions · 19 · 3.1 What is a supersymmetric black hole · 19 ·

3.1.1 N = 2, D = 4 supergravity · 20 ·

3.1.2 Minimal N = 1, D = 5 supergravity · 21 ·

3.2 BMPV black hole · 27 ·

3.3 Supersymmetric black ring · 32 ·

4 Generating SUSY Solutions via Harmonic Functions · 45 · 4.1 Harmonic function method formalism · 46 ·

4.2 The single supersymmetric black ring · 48 ·

4.3 The single BMPV black hole · 51 ·

4.4 Supersymmetric multiple concentric black rings · 54 ·

4.4.1 General multi-bicycles (tandems) solution · 55 ·

4.4.2 Symmetric bicycling rings solution · 58 ·

4.4.3 Multiple di-ring solution · 61 ·

4.5 Supersymmetric bi-ring Saturn · 62 ·

4.5.1 General supersymmetric bi-ring Saturn solution · 63 ·

4.5.2 Bi-ring Saturn with a central static BMPV black hole · 66 ·

CONTENTS v

5 Phases of Supersymmetric Black Systems in Five Dimensions · 69 ·

5.1 Full phase of extremal black systems in five dimensions · 70 ·

5.2 Singularities and causality on supersymmetric black holes · 71 ·

5.2.1 Absence of closed timelike curves (CTCs) · 71 ·

5.2.2 Absence of Dirac-Misner strings · 74 ·

5.3 Phases of D = 5 SUSY black bi-ring Saturn with equal spins · 77 ·

5.3.1 Black bi-ring Saturn phase and its boundaries · 78 ·

5.3.2 Non-uniqueness of the phase for black bi-ring Saturns · 81 ·

5.4 Phases of general D = 5 SUSY black bi-ring Saturn · 82 ·

6 Conclusion and Outlook · 85 · 6.1 Overall Conclusion and Discussion · 85 ·

6.2 Future Studies · 87 ·

Bibliography · 91 · A Coordinate Systems · 99 · A.1 Hyperspherical coordinates · 100 ·

A.2 Gibbons-Hawking coordinates · 100 ·

A.3 Ring coordinates · 102 ·

A.4 Near-horizon spherical coordinates · 104 ·

B Angular Momenta in the Ring Planes · 107 · C Absence of Dirac-Misner Strings for the SUSY Solutions · 111 · C.1 General multiple concentric black rings · 112 ·

C.2 General black bi-ring Saturn · 112 ·

In contrast to four dimensions, in five dimensions black hole solutions canhave event horizons with nonspherical topology and violate the uniqueness theo-rems The supersymmetric solutions of single black hole (also called BMPV blackhole because it was first discovered by Breckenridge, Myers, Peet and Vafa) andsingle black ring (with horizon topology S1× S2) have been discovered in minimal

D = 5 supergravity The first part of the thesis is devoted to the recent opments in five-dimensional supersymmetric black holes: first I briefly describeminimal N = 1, D = 5 supergravity theory, next I review the well-known super-symmetric black hole solutions in five dimensions and study the physical properties

devel-of the BMPV black hole and the supersymmetric black ring However, the BPS(Bogomol’nyi-Prasad-Sommerfield) equations solved by the black ring appear to

be nonlinear, hence this obscures the construction of multiple ring solutions viasimple superpositions

In order to construct solutions describing multiple supersymmetric black rings

or superpositions of supersymmetric black rings with BMPV black holes, in thesecond part I review an alternative approach — the harmonic function method Ifirst introduce four harmonic functions to characterize the single supersymmetricblack ring solution Via simple superpositions of harmonic functions for each ring,supersymmetric multiple concentric black rings are then constructed, in which therings have a common center, and can lie either in the same plane or in orthogonal

vii

planes In addition, this solution-generating method can also be applied to symmetric solutions with a black hole by taking the limit R → 0 As a result, Iconstruct the most general supersymmetric solution — black multiple bi-rings Sat-urn, which consists of multiple concentric black rings sitting in orthogonal planeswith a BMPV black hole at the center In the thesis I focus particularly on thebi-rings Saturn solution.

super-In the last part, I present the phase diagram of the established metric bi-ring Saturn in five dimensions and show that its structure is similar tothose for extremal vacuum ones: a semi-infinite open strip, whose upper bound onthe entropy is equal to the entropy of a static BMPV black hole of the same totalmass for any value of the angular momentum Following this, I provide a detailedanalysis of the configurations that approach its three boundaries Remarkably, Iargue that for any j ≥ 0 the phase with highest entropy is a black bi-ring Saturnconfiguration with a central, close to static, S3 BMPV black hole (accounting forthe high entropy) surrounded by a pair of very large and thin orthogonal blackrings (carrying the angular momentum) Moreover, I also study the outstandingfeature of non-uniqueness arising from this exotic configuration Possible general-izations to more supersymmetric black hole solutions including black Saturn with

supersym-an off-center hole, non-orthogonal ring configurations supersym-and rings on Eguchi-Hsupersym-ansonspace are discussed at the end of the thesis

List of Figures

1.1 Phases of five dimensional vacuum black holes · 6 ·

3.1 Phase of BMPV black hole · 31 ·

3.2 Phase of single supersymmetric black ring · 41 ·

4.1 Phase of BMPV black hole and SUSY black ring at R → 0 · 54 ·

4.2 Multiple bicycling black rings · 55 ·

4.3 Phase of symmetric bicycling black rings · 60 ·

4.4 Di-ring in the same plane · 61 ·

4.5 Supersymmetric bi-ring Saturn configuration · 63 ·

4.6 Dimensionless area aH and angular momentum j functions for the BMPV black hole · 66 ·

5.1 Full phase of the extremal black system in five dimensions · 70 ·

5.2 Phases of the supersymmetric black bi-ring Saturn · 79 ·

5.3 Phase of general supersymmetric black bi-ring Saturn · 83 ·

A.1 Gibbons-Hawking coordinates · 102 ·

A.2 Ring coordinates for flat four-dimensional space · 104 ·

A.3 Near-horizon spherical coordinates on R3 · 105 ·

ix

List of Symbols

Variables

auppH Upper boundary of horizon area

A, Ai 1-form gauge field

f , fi Scalar field of the solution of minimal D = 5 supergravity

2f (dω + ?4dω)

h Harmonic function (hr for black ring, hh for black hole)

hµν Weakly perturbative gravitation field

¯

hµν h¯µν ≡ hµν− 1

2hρρηµν

H Harmonic function on D = 4 hyper-K¨ahler base space

xi

Symbol Definition

l, li S1 radius in the topology of the horizon for the black ring

L5 Lagrangian for minimal D = 5 supergravity

N Number of supercharges in the supergravity theory

q Dipole parameter (qh for black hole, qi for black ring)

Q Electric charge parameter (Qh for black hole, Qi for black ring)

R Scalar curvature (Chapter 2); Radius of black ring (Other chapters)

S5 Action for minimal D = 5 supergravity

SGR Hilbert action for general relativity

γ Parameter related to the spin for BMPV black hole

ωL, ˆωQ Charge-independent and charge-dependent component of ˆω

ΩH Angular velocity of the horizon

Λ Constants (Λh for black hole, Λi for black ring)

LIST OF SYMBOLS xiii

Conventional Notations

GD Newton’s constant in D dimensional space-time

ηµν Minkowski metric for D dimensional flat space-time

ηµν=diag (−1, +1, +1, )

Ωd−1 Area of a unit (d − 1)-sphere, Ωd−1 = 2πd/2/Γ d2

∇α, Dα Covariant derivative operator on flat-space Rn

∆ Laplacian on D = 4 hyper-K¨ahler base space

 d’Alembertian operator on flat-space Rn,  ≡ ∂µ∂µ = −∂t2+ ∇2

?4 Hodge dual on D = 4 hyper-K¨ahler base space

iYA p-form contraction with vector Y , (iYA)α1 αp−1 ≡ YβAβα1 αp−1

Coordinate Systems(r, θ, ϕ, ψ) Gibbons-Hawking coordinates

(ρ, Θ, φ2, φ2) Transformed Gibbons-Hawking coordinates

Introduction

The attempt at generalizing the well established theory of standard dimensional) Einstein gravity, that is, classical general relativity, to higher dimen-sions has been the subject of increasing attention in recent years Even though

(four-it has not found any direct observational and experimental support so far, thedevelopment of higher dimensional gravity has been strongly influenced in recentdecades by string theory

According to the general theory of relativity, a black hole is a region of spacefrom which nothing, not even light, can escape It is the result of the deformation

of space-time caused by a very compact mass Each black hole has no materialsurface; all of its matter has collapsed into a singularity that is surrounded by

a spherical boundary called its event horizon The event horizon is a one-waysurface: particles and light rays can enter the black hole from outside but nothingcan escape from within the horizon of the hole into the external universe So the

· 1 ·

importance of black holes for gravitational physics is clear: their existence is atest of our understanding of strong gravitational fields, beyond the point of smallcorrections to Newtonian physics, and a test of our understanding of astrophysics,particularly of stellar evolution [1] In addition, black holes have offered a potentialinsight into non-perturbative effects in the territory of quantum gravity since blackhole radiation involves a mixture of gravity and quantum physics [2, 3] Thereforehigher dimensional black holes will play a crucial role in our finding of the missinglink in a complete picture of the fundamental forces to the ‘theory of everything’,which unifies gravity with the other forces in nature.

1.1.1 Why we are interested in higher dimensional black

as a solution of low-energy effective field theory [4–8]

• The AdS/CFT correspondence is the conjectured equivalence between a stringtheory defined on one space, and a quantum field theory without gravity de-fined on the conformal boundary of this space It relates the dynamics of

a D-dimensional black hole with those of a quantum field theory in D − 1dimensions [9, 10]

§ 1.1 BLACK HOLES IN HIGHER DIMENSIONAL SPACE-TIME · 3 ·

• This theoretical work has led to the possibility of proving the existence ofextra dimensions If higher-dimensional black holes could be produced in

a particle accelerator such as the Large Hadron Collider (LHC), this wouldprovide a conceivable evidence that large extra dimensions exist [11, 12]

On the other hand, higher-dimensional gravity is also of intrinsic interest

We believe that the space-time dimensionality D should be endowed as a tunableparameter in general relativity, particularly in its most basic solution: black holes.Four-dimensional black holes are known to have a number of remarkable features,such as uniqueness, spherical topology, dynamical stability, and the satisfaction of

a set of simple laws — the laws of black hole mechanics [13] No-hair theoremspostulate that a stationary, asymptotically flat, vacuum black hole is completelycharacterized by its mass and spin [14–16], and event horizons of nonsphericaltopology have been proved to be forbidden by the Gauss-Bonnet theorem [17].One would like to know which of these are peculiar to four-dimensions, and whichhold more generally

1.1.2 New features of D = 5 black holes

The natural change is that there is the possibility of rotation in several dependent rotation planes when space-time has more than four dimensions Thefirst higher-dimensional generalization of the static, asymptotically flat, vacuumsolution — Schwarzschild black hole was discovered in 1963 [18] It was not until

in-1986 that the higher-dimensional extension of the Kerr black hole, rotating either

in a single plane or arbitrarily in each of the N ≡ D−12  = d

2 independent tation planes, was found by Myers and Perry They also showed that stationaryasymptotically flat black hole solutions of the Einstein-Maxwell equations exist for

ro-all D ≥ 4 space-time dimensions [19] All these black holes with spherical horizontopology are characterized by their mass and N independent angular momenta.Recent discoveries have shown that five dimensional black holes exhibit quali-tatively new properties not shared by their four-dimensional siblings A remarkablefeature of asymptotically flat black holes in five space-time dimensions, in contrast

to four dimensions where the event horizon in a stationary black hole must havetopology S2, is that they can have event horizons with nonspherical topology Thefirst explicit example of such a black hole was the discovery of the ‘black-ring’ so-lution of the vacuum Einstein equations by Emparan and Reall in 2002 [20], whoseevent horizon has topology S1 × S2 The black ring is required to be rotating tobalance the self-gravitational attraction More strikingly, there is a small range ofspin within which it is possible to find a black hole and two black rings with thesame mass and angular momentum Hence, the existence of this solution impliesthat the uniqueness theorems valid in four dimensions cannot simply extend tofive-dimensional solutions except for the static case It is also suggested that as

a five-dimensional black hole is spun up, a phase transition occurs from the blackhole to a black ring, which can have an arbitrarily large angular momentum for agiven mass In addition to the S1 rotating black ring discovered in [20], a blackring with rotation in the azimuthal direction of the S2 was found in [21], and thedoubly-spinning black ring solution, which is allowed to rotate freely in both the

S2 and S1 directions was successfully constructed by a smart implementation ofthe inverse scattering method (ISM) by Pomeransky and Sen’kov in 2006 [22]

In 3+1 dimensions, configurations of multiple black holes can only be kept inequilibrium by adding enough electric charge to each black hole, for example, thewell-known multi-Reissner-Nordstr¨om black hole [23, 24] Multi-Kerr black holespace-times [25] cannot be in equilibrium because the spin-spin interaction [26] is

§ 1.1 BLACK HOLES IN HIGHER DIMENSIONAL SPACE-TIME · 5 ·

not sufficiently strong to balance the gravitational attraction of black holes withregular horizons [27–29] However, for 4+1-dimensional stationary vacuum solu-tions, angular momentum does provide sufficient force to keep two black objectsapart Using the ISM, recently we can construct more exact asymptotically flatbalanced solutions containing a number of black objects such as black Saturn [30]:

a black ring surrounding a concentric spherical black hole, black di-ring [31, 32]where the two concentric rings lie in the same plane and bicycling black bi-rings,

or simply called black bi-rings most recently [33, 34], which is a balanced uration of two singly spinning concentric black rings placed in orthogonal planes.Obviously, the most general system is constructed from two doubly spinning blackrings in orthogonal planes and even more exotic generalizations, which includemulti-bicycles (tandems) and bi-ring Saturn

config-1.1.3 Phases of D = 5 vacuum black holes

A scatter-plot sampling of the parameter space of the exact solutions in[30] shows regions of the phase diagram where black Saturns are the entropicallydominating solutions, which has been proved to be true in [35] throughout the entirephase diagram: for any value of the total mass M > 0 and angular momentum

J > 0, the phase with highest entropy is a black Saturn The total entropy ofthis black Saturn configuration approaches asymptotically, in the limit of infinitelythin ring, to the entropy of a static black hole with the same total mass [35] alsoargues that in fact for any given value of the total angular momentum J , thereexist black Saturns spanning the entire range of areas

0 < AH < AmaxH = 32

3

r2π

3 (GM )

Figure 1.1: Phases of five dimensional vacuum black holes: dimensionless area aH vs.angular momentum j for fixed total mass M = 1 The solid curves correspond to asingle Myers-Perry black hole and black ring The semi-infinite shaded strip, spanning

0 ≤ j < ∞, 0 < aH < 2√2 is covered by black Saturns

Here Amax is the horizon area of the static Myers-Perry black hole So the fullphase of vacuum black holes illustrated by the gray strip in the region of the plane(j, aH) in Figure 1.1 is anticipated to be a semi-infinite open strip

0 < aH < 2√

2

0 ≤ j < ∞

(1.2)

plus an additional point (j, aH) = (0, 2√

2) for the static Myers-Perry black hole

As stated in [35], each point in the gray strip actually corresponds to a parameter family of black Saturn solutions The top end aH = 2√

one-2 with j 6= 0 isreached only asymptotically for black Saturns with infinitely long rings Solutions

at the bottom aH = 0 are naked singularities For a fixed value of j we can movefrom the top of the strip to the bottom by varying the spin of the central Myers-Perry black hole jhole from 0 to 1 For a fixed area we can move horizontally by

§ 1.2 BLACK HOLES OF D = 5 SUPERGRAVITY · 7 ·

having jhole< 0 and varying the spin of the ring between | jhole| and ∞

The other remarkable gravitational phenomenon of the five-dimensional uum black hole phase space is continuous non-uniqueness The balanced black Sat-urn solution with a single black ring exhibits two-fold continuous non-uniquenessfor fixed mass and angular momentum [30] Adding n more rings into this simplebalanced configuration results in 2n-fold continuous non-uniqueness since the totalmass and angular momentum can be distributed continuously between the centralblack hole and the n surrounding black rings in such a space-time Including thesecond angular momentum gives doubly spinning multi-rings black saturns with3n-fold continuous non-uniqueness, also for the J1 = J2 = 0 configurations [30].Moreover, a ring can have an effect as small as desired on the total black Saturn

vac-So for generic values of M , J and AH ∈ (0, Amax

H ) there are black Saturns with anarbitrarily large number of rings The phase space of the five-dimensional blackholes therefore shows a striking infinite intricacy In chapter 5, I will provide adetailed analysis of a more complicated configuration — the extremal bi-ring Sat-urn and then discuss the phase of the non-supersymmetric zero-temperature blackholes since the extremal ones are similar to the cases of the supersymmetric blackhole and black ring [33]

Black holes of D = 5 supergravity, which is non-singular on and outside theevent horizon are just solutions of five-dimensional supergravity theory [36], whichhas been shown to be equivalent to dimensional reductions of 10-dimensional and11-dimensional supergravity, the low-energy limit of superstring theories and Mtheory respectively due to string dualities [37]

1.2.1 Nonextremal solution of D = 5 supergravity

The charges and dipoles of black rings actually provide the basis to interpretthem as objects in string/M theory Black holes can only carry electric chargewith respect to a two-form The construction of stationary, charged, topologically-spherical, black-hole solutions (‘charged Myers-Perry’) in maximal supergravitytheories was described in [38] However, black rings can not only carry conservedelectric charges coupled to a two-form field or to a dual magnetic one-form butalso support the nonconserved dipole charges that are independent of all conservedcharges, in fact they can be present even in the absence of any gauge charge Thefirst regular example of a charged rotating black ring was obtained in [39], whichcarries a U(1) electric charge and local fundamental string charge (see also [40]).And the solutions describing the rotating black rings with these dipoles, called

‘dipole rings’ were first found in [41] Most recently, the charged doubly spinningblack ring was constructed by adding fundamental and momentum charges to aneutral five-dimensional solution of Einstein’s vacuum equations [42]

1.2.2 Supersymmetric solutions of minimal D = 5

super-gravity

Supersymmetric solutions of supergravity theories, saturating a Prasad-Sommerfield (BPS) inequality [43], are of particular importance in stringtheory because such solutions often have certain stability and non-renormalizationproperties that are not possessed by non-supersymmetric solutions [44] For exam-ple, it has been possible to give a microscopic description of Bekenstein-Hawkingentropy of the supersymmetric black holes studied in [45] This is also the sim-plest example, as it carries the minimum number of net charges (D1-brane and

Bogomolnyi-§ 1.2 BLACK HOLES OF D = 5 SUPERGRAVITY · 9 ·

D5-brane charges and linear momentum) necessary to have a finite area regularhorizon Then the successful extension to rotating black holes with a single in-dependent rotation parameter was taken by Breckenridge, Myers, Peet, and Vafa(BMPV) in 1997 [46] (see also [47]) This solution has the same three charges andequal angular momenta in two orthogonal planes, for which the mass is fixed bythe BPS relation Alternatively, the electrically charged BMPV black hole [46] can

be obtained as a solution of N = 1,D = 5 minimal supergravity [36]

The first example of a supersymmetric black ring solution was obtained forminimal D = 5 supergravity in [48] using a canonical form for supersymmetricsolutions of this theory [44] This was then generalized to a supersymmetric blackring solution of the U(1)3 supergravity in [49] The latter solution has 7 indepen-dent parameters, which can be taken to be the 3 charges, 3 dipoles and J1 Themass is fixed by the BPS relation and J2 is determined by the charges and dipoles.Remarkably, the only known asymptotically flat supersymmetric (BPS) blackhole solutions, which can be naturally viewed as the solutions of minimal super-gravity theory, exist only for D = 4 and D = 5 [36] For D = 5, one can ob-tain supersymmetric black-hole solutions as an extremal limit of the nonextremalcharged rotating solutions But we note that a supersymmetric (BPS) black hole

is necessarily extremal, but the converse is not true because the BPS limit and theextremal limit are distinct for rotating black holes [50]

1.2.3 Why minimal D = 5 supergravity

One motivation for studying D = 5 supergravity is that it is equivalent to11-dimensional supergravity, which may generate considerable excitement as thefirst potential candidate for the theory of everything

Firstly, in order to saturate the BPS inequality M ≥| Q |, angular tum must vanish for supersymmetric D = 4 black holes But in five dimensionsthere exist rotating supersymmetric black holes, which have non-singular and non-rotating horizons The effect of rotation on the horizon is not to make it rotate but

momen-to deform it from a round 3-sphere momen-to a squashed 3-sphere [36] Thus the phase

of supersymmetric black holes in D = 5 will be significantly richer than those in

D = 4

Secondly, although we have constructed the static solution in D > 4 gous with multi-Reissner-Nordstr¨om black hole [24] in D = 4 and a more com-plicated solution describing a Myers-Perry black hole with a concentric dipolering [51], the exact solutions for non-supersymmetric black holes tend to be harder

analo-to construct than their supersymmetric cousins, particularly true for multi-centeredsolutions because for which we have to solve the full second-order Einstein’s equa-tions [30] The first-order nature of the supersymmetry conditions [44] makes iteasy to write down stationary superpositions corresponding to multiple BMPVblack holes [36] and multiple concentric black rings [52, 53] I will review the su-persymmetric solutions and make further analysis in detail in chapter 3

Among the many different supergravity theories in various numbers of time dimensions D and number (N ) of supersymmetry charges, minimal super-gravity (mSUGRA), i.e N = 1, D = 5 supergravity, is the easiest for us to studysince it contains the least N , which in the sense that it is the smallest possiblesupersymmetric extension of Einstein’s theory of general relativity Therefore, inthe thesis we only consider the black hole solutions of minimal N = 1, D = 5

space-§ 1.3 OBJECTIVE AND ORGANIZATION · 11 ·

supergravity The purpose of this thesis is, using the harmonic function method,

to construct explicit solutions for the supersymmetric bi-ring Saturn configurationconsisting of two orthogonal black rings with a BMPV black hole at the commoncenter and find the full phase diagram of supersymmetric black objects in fivedimensions

The organization of this thesis is as follows It begins with some basic cepts and definitions of higher-dimensional general relativity and some technicaltricks in chapter 2 that are important for later use Then chapter 3 is devoted tothe recent progress in five-dimensional supersymmetric black holes: first I brieflydescribe minimal D = 4 and D = 5 supergravity, next I review all known super-symmetric black hole solutions in five dimensions, in particular the supersymmetricblack ring In chapter 4, I introduce an alternative approach to recover all the su-persymmetric solutions; in this case all the solutions are expressed in terms of theharmonic functions Using this method I can via simple superpositions constructmore supersymmetric configurations including the multiple concentric black ringsand bi-ring Saturn listed at the end of this chapter In chapter 5, I discuss thesingularities and causality problems of supersymmetric black hole solutions Fol-lowing this, I present the phases of supersymmetric bi-ring Saturn of three types:(i) symmetric bi-ring Saturn, (ii) nonsymmetric bi-ring Saturn and (iii) generalbi-ring Saturn, ending with a detailed analysis of their phase structures Finally,chapter 6 concludes my finding of minimal supersymmetric bi-ring Saturn solutionsconsisting of a central BMPV black hole surrounded by two concentric black rings,

con-or even multiple black rings sitting in con-orthogonal planes with a discussion of thefull phases of supersymmetric black holes in five dimensions Possible directions forfuture studies to find more supersymmetric solutions of new families are also men-tioned Four useful appendices are contained at the end of the thesis Appendix A

distinguishes all coordinate systems applied in the thesis, appendix B contains thederivation of angular momenta in orthogonal ring planes and appendix C studiesthe condition for the absence of Dirac-Misner strings in the supersymmetric bi-ring Saturn In appendix D source codes for plotting the phase diagram of variousconfigurations are provided for reference.

where GD is the D dimensional Newton’s constant and has dimensions of [L]N −1

to guarantee the dimensionlessness of the action And Lmatter is the Lagrangiandensity for other matter fields

Extremizing the action (2.1) yields the Einstein equation in the conventionalform

Gµν ≡ Rµν − 1

where Tµν = 2(δLmatter/δgµν)

The mass, angular momenta, and other conserved charges of a black hole

or any isolated system are defined through comparison to the field created nearasymptotic infinity by a weakly gravitating system This will be the D dimensionalgeneralization of the ADM mass and angular momentum

In the linear (weakly gravitating field) approximation

h00 = 16πGD(d − 1)Ωd−1

dimen-is mass M since classical general relativity in vacuum dimen-is scale invariant Thus mensionless quantities for the angular momenta ja and the horizon area aH can bedefined

di-jaD−3= cJ J

D−3 a

aD−3H = cA A

D−3 H

(2.17)

cA= ΩD−32(16π)D−3(D − 2)D−2 D − 4

D − 3

D−32

(2.18)

§ 2.2 DIMENSIONLESS SCALE · 17 ·

For the case D = 5,

ja = 34

r 3π2G5

Ja

M3/2

aH = 316

r3π

AH(G5M )3/2

(2.19)

In the thesis, I will seek for the function aH (ja) for each supersymmetricsystem, which is equivalent to the entropy, or the area AH, as a function of Ja forfixed mass

Review of Supersymmetric Solutions in Five Dimensions

In this chapter I first briefly describe two representative supergravity theories

in four and five dimensions, next I review all known supersymmetric black holesolutions in five dimensions including BMPV black hole and supersymmetric blackring solutions, where we would determine various properties of the ADM mass, ofthe angular momentum and of the horizon structure In addition, I will use analternative approach to recover these black hole solutions of minimal supergravity

Supersymmetric black holes, which we are interested principally in the thesis,are naturally viewed as BPS-saturated solutions of D-dimensional supergravitytheories, which are the dimensional reduction of D = 10, 11 supergravity, the low-energy effective theory for superstring and M-theory, respectively

· 19 ·

3.1.1 N = 2, D = 4 supergravity

The D = 4 case is already well-understood The bosonic sector of N = 2,

D = 4 supergravity is Einstein-Maxwell theory, in which all the supersymmetricsolutions (admitting super-covariantly constant spinors) were presented in [55] andsome simple generalizations to SU(4) supergravity with axion and dilaton as well[56] For any asymptotically flat solution of Einstein-Maxwell theory that is non-singular on and outside the event horizon, the mass M is bounded below by thecharges [43] In geometrized units of charge the bound is

supergrav-ds2 = −



1 − G4Mr

2

dt2+



1 −G4Mr

§ 3.1 WHAT IS A SUPERSYMMETRIC BLACK HOLE · 21 ·

which is equivalent to the saturation of a BPS bound (3.1)2 in the absence ofmagnetic charge Its inner and outer horizon coincide r+ = r−= G4M and it has

a naked singularity r = 0 for the extremal Reissner-Nordstr¨om black hole Thecalculation of the Bekenstein-Hawking entropy for the D = 4 extremal Reissner-Nordstr¨om black hole was subsequently performed in type II string theory [61, 62].Recently, a supersymmetric D = 4 rotating black hole was obtained by Kaluza-Klein reduction of five-dimensional supersymmetric black rings wrapped on thefiber of a Taub-NUT space [63]

3.1.2 Minimal N = 1, D = 5 supergravity

Minimal five-dimensional supergravity, a theory with eight supercharges, wasconstructed in [64] The action3 of its bosonic sector including Einstein-Maxwelltheory and an additional ‘AFF’ Chern-Simons (CS) term with a particular coeffi-cient is

S5 = 14πG5

3 Here the metric has mostly positive signature (−, +, +, +, +).

Gµν = 8πTµν (3.6)with the Maxwell stress-energy tensor

numer-5 Symplectic Majorana spinors are defined as a = ab b where ab is antisymmetric with

12= 1 It is also convenient to introduce  ab such that  12 = 1.

The symplectic Majorana condition is ¯  a ≡  a† γ 0 =  aT C, where the charge conjugation matrix C is real and antisymmetric and satisfies Cγ T C−1 = γ

§ 3.1 WHAT IS A SUPERSYMMETRIC BLACK HOLE · 23 ·

be obtained, for example

Equation (3.19) implies D(αVβ)= 0 and hence V is a Killing vector field [66] thatpreserves the field strength (i.e LVF = 06 where LV denotes the Lie derivative).

It further implies that V generates a symmetry of the full solution In addition,equation (3.19) can be written as