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Five dimensional microstate geometries

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Five dimensional microstate geometries

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Content FIVE DIMENSIONAL MICROSTATE GEOMETRIES by Chih-Wei Wang A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) December 2007 Copyright 2007 Chih-Wei Wang Epigraph We put thirty spokes together and call it a wheel; But it is on the space where there is nothing that the usefulness of the wheel depends. We turn clay to make a vessel; But it is on the space where there is nothing that the usefulness of the vessel depends. We pierce doors and windows to make a house; And it is on these spaces where there is nothing that the usefulness of the house depends. Therefore just as we take advantage of what is, we should recognize the usefulness of what is not. – from “Dao De Jing” chap. 11, by Laozi (6th BC), tr. by Waley ii Dedication To my family. iii Acknowledgments I am very grateful to my advisor Nick Warner for sharing this interesting and rich subject with me, for his valuable guidance and for inspiring me by his enthusiasm about physics. I would also like to thank Iosif Bena for a lot of fruitful collaboration. Moreover, I thank the faculty of the physics department in USC for their interesting courses that further deepen my understanding of physics. Finally, I would like to thank the students in string theory group for many interesting discussions and for sharing their interesting ideas with me. iv Table of Contents Epigraph ii Dedication iii Acknowledgments iv List of Tables vii List of Figures viii Abstract x Chapter 1: Introduction 1 1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Asymptotic charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Building a black hole from string theory . . . . . . . . . . . . . . . . . 8 1.3.1 From string theory to supergravity . . . . . . . . . . . . . . . . 8 1.3.2 BPS branes and extremal black holes . . . . . . . . . . . . . . 11 1.3.3 Link p-branes to D-branes or vice versa . . . . . . . . . . . . . 16 1.4 Two-charge system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Chapter 2: Three-charge BPS Solutions and Classical Mergers 20 2.1 Build five-dimensional three-charge BPS supergravity solutions . . . . 22 2.1.1 Asymptotic charges and quantized charges . . . . . . . . . . . 28 2.2 A circular black ring with a black hole at the center . . . . . . . . . . . 32 2.2.1 Asymptotic, quantized charges and angular momenta . . . . . . 40 2.2.2 Horizon and Entropy . . . . . . . . . . . . . . . . . . . . . . . 43 2.3 A vertically shifted black hole . . . . . . . . . . . . . . . . . . . . . . 46 2.3.1 Asymptotic, quantized charges and angular momenta . . . . . . 52 2.3.2 Horizon and entropy . . . . . . . . . . . . . . . . . . . . . . . 54 2.4 The merger and the entropy . . . . . . . . . . . . . . . . . . . . . . . . 56 2.4.1 The merging process . . . . . . . . . . . . . . . . . . . . . . . 56 2.4.2 The entropy of mergers . . . . . . . . . . . . . . . . . . . . . . 59 2.5 conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 v Chapter 3: Bubbled Geometries 66 3.1 Gibbons-Hawking metrics . . . . . . . . . . . . . . . . . . . . . . . . 67 3.1.1 Asymptotic and local structure . . . . . . . . . . . . . . . . . . 68 3.1.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.1.3 Ambipolar generalization and nucleation . . . . . . . . . . . . 69 3.2 BPS solutions on ambipolar Gibbons-Hawking bases . . . . . . . . . . 72 3.2.1 Solving BPS equations . . . . . . . . . . . . . . . . . . . . . . 72 3.2.2 Regularity and asymptotic constraints . . . . . . . . . . . . . . 74 3.2.3 The fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3 Closed time-like curves and the bubble equations . . . . . . . . . . . . 78 3.4 Asymptotic charges and angular momenta . . . . . . . . . . . . . . . . 82 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Chapter 4: Bubble Mergers and Deep Microstates 87 4.1 Shallow microstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.1.1 A maximally spinning black hole blob . . . . . . . . . . . . . . 91 4.1.2 A maximally spinning black ring blob . . . . . . . . . . . . . . 95 4.2 Irreversible mergers and deep microstates of black holes . . . . . . . . 100 4.2.1 A zero-entropy black ring blob with a maximally-spinning black hole blob at the center . . . . . . . . . . . . . . . . . . . . . . 102 4.2.2 Irreversible mergers and scaling solutions . . . . . . . . . . . . 107 4.2.3 Numerical results for a simple merger . . . . . . . . . . . . . . 113 4.3 Irreversible merger of two zero-entropy black rings . . . . . . . . . . . 115 4.3.1 Two zero-entropy black ring blobs with an isolated GH point . . 117 4.3.2 Irreversible mergers and scaling solutions . . . . . . . . . . . . 121 4.3.3 Numerical results for a simple merger . . . . . . . . . . . . . . 127 4.4 Some general features about mergers and scaling solutions . . . . . . . 130 4.4.1 The necessary condition for general mergers . . . . . . . . . . 130 4.4.2 Scaling behaviors . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.4.3 Metric structure . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.5 Triangular solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.5.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.5.2 Abysses and the puzzle . . . . . . . . . . . . . . . . . . . . . . 143 4.6 A coarse-graining picture of microstate geometries . . . . . . . . . . . 144 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Chapter 5: Conclusion 150 References 154 vi List of Tables 2.1 The wrapping profile of M2/M5 branes . . . . . . . . . . . . . . . . . . 23 4.1 Distances between points in the scaling regime. . . . . . . . . . . . . . 115 4.2 Numerical results of a triangular solution . . . . . . . . . . . . . . . . 142 vii List of Figures 2.1 Coordinate system for black ring metric. The figure sketches a section at constantt and 2 . Surfaces of constanty are ring-shaped, whilex is a polar coordinate on theS 2 (roughlyx cos). Asymptotic infinity lies atx =y =1. (The credit of this graph belongs to [EEMR05]) . . . . 33 2.2 A vertically and a horizontally shifted black hole. The pictures show a section at constantt and 2 and the plane is (z; 1 ) plane. (1) The black hole is shifted vertically from the center of the black ring. (2) The black hole is shifted horizontally from the center of the black ring. . . . . . . 47 2.3 Three possible outcomes when we move the black hole from infinity to the center of the ring. In the first graph, the black hole is small and the radius of the ring decrease fromR 1 toR 0 . In the second graph, the size of the black hole is just big enough to catch the ring and their horizons touch at equator. In the third graph, the black hole is very big and the ring enter horizon at some particular angle cot = 0 . . . . . . . . . . 58 3.1 Topologically non-trivialS 2 . Any line connected two Gibbons-Hawking points has the fiber shrink at the both end-points and therefore, has two- sphere topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.1 AU(1)U(1) invariant merger. (1) A classicalU(1)U(1) merger on I R 4 that is shown on (z; 1 ) plane. By decreasing the black ring’s angular momentum,J T , one can decrease it’s radius and eventually merge with the black hole at the center. (2) A corresponding bubbled version of the same merger that is shown on thex-axis of GH base. By adjusting the flux parameter, one can decrease the distance between the black hole blob and the black ring blob and eventually they merge. . . . . . . . . . 101 4.2 The layout of GH points for the U(1)U(1) merger of a black ring microstate and a black hole microstate. Notice that every point here is just a point and there is no internal structure. . . . . . . . . . . . . . . . 114 viii 4.3 A U(1)U(1) invariant merger of two black rings. (1) A classical U(1)U(1) merger of two black rings on I R 4 that is shown on (z; 1 ) plane. By decreasing the interactive part of J L that comes from the fluxes , one can decrease the difference between two radii and eventu- ally the two black ring merge at particular radius. (2) A corresponding bubbled version of the same merger that is shown on thex-axis of GH base. By adjusting the flux parameters, one can decrease the distance between the two black ring blobs and eventually they merge. . . . . . . 116 4.4 The layout of the GH points for two shallow black rings . . . . . . . . . 128 4.5 Three branches of the bubble equations’ solutions. The plot shows the ring positions, r, and r 0 , along the horizontal axis withjJ int L j plotted on the vertical axis. The separations between two points in the blobs are too small to resolve. There are three branches: (i) The single, nearly vertical line in the center (in green) for which all four GH points remain extremely close together; (ii) The two outermost curves (in blue) where the two rings become progressive more widely spaced asjJ int L j! 0; (iii) The two curves (in red) that meet branch (ii) and show the scaling merger in which the two rings meet atjJ int L j = 0. . . . . . . . . . . . . 128 4.6 The AdS throat get longer and longer near the merger point and caps off with some finite size local structure. . . . . . . . . . . . . . . . . . . . 135 4.7 The layout of GH points for a triangular scaling solution. . . . . . . . . 139 4.8 TheX I gauge graph. Each point represent a gauge that make a particu- lar ring-blob has uniform flux parameter distribution. . . . . . . . . . . 147 ix Abstract In this thesis, we discuss the possibility of exploring the statistical mechanics descrip- tion of a black hole from the point view of supergravity. Specifically, we study five dimensional microstate geometries of a black hole or black ring. At first, we review the method to find the general three-charge BPS supergravity solutions proposed by Bena and Warner. By applying this method, we show the classical merger of a black ring and black hole on I R 4 base space in general are irreversible. On the other hand, we review the solutions on ambi-polar Gibbons-Hawking (GH) base which are bubbled geome- tries. There are many possible microstate geometries among the bubbled geometries. Particularly, we show that a generic blob of GH points that satisfy certain conditions can be either microstate geometry of a black hole or black ring without horizon. Fur- thermore, using the result of the entropy analysis in classical merger as a guide, we show that one can have a merger of a black-hole blob and a black-ring blob or two black-ring blobs that corresponds to a classical irreversible merger. From the irreversible mergers, we find the scaling solutions and deep microstates which are microstate geometries of a black hole/ring with macroscopic horizon. These solutions have the same AdS throats as classical black holes/rings but instead of having infinite throats, the throat is smoothly capped off at a very large depth with some local structure at the bottom. For solutions that produced fromU(1)U(1) invari- ant merger, the depth of the throat is limited by flux quantization. The mass gap is related x with the depth of this throat and we show the mass gap of these solutions roughly match with the mass gap of the typical conformal-field-theory (CFT) states. Therefore, based on AdS/CFT correspondence, they can be dual geometries of the typical CFT states that contribute to the entropy of a black hole/ring. On the other hand, we show that for the solutions produced from more general merger (without U(1)U(1) invariance), the throat can be arbitrarily deep. This presents a puzzle from the point view of AdS/CFT correspondence. We propose that this puzzle may be solved by some quantization of the angle or promoting the flux vectors to quantum spins. Finally, we suggest some future directions of further study including the puzzle of arbitrary long AdS throat and a general coarse-graining picture of microstate geometries. xi Chapter 1 Introduction 1.1 Motivations Einstein’s general relativity is one of the fundamental pillars of the theoretical physics. The theory has been proved to be quite successful in describing gravity. However, the incompatibility between general relativity and quantum mechanics is still an essential problem in physics. Particularly, the existence of black holes present several paradoxes that need to be solved in the correct quantum gravity theory. There are two most impor- tant puzzles that involved a black hole: Black holes entropy and no-hair theorem In 1972, Bekenstein [Bek72, Bek73] pro- posed the black hole should have a well-defined entropy associated with it’s hori- zon area: S Bek = A 4G 4 ; (1.1) where G 4 is the usual Newton constant in four-dimensional space time. On the other hand, black hole no-hair theorem stated that a black hole solution of Ein- stein’s general relativity is completely specified by three asymptotic observable charges: the mass (energy), electric charge and angular momentum. However, by the principle of statistical mechanics, the large macroscopic horizon area means huge number of microstates:e S Bek . The problem is where are those large number of the degrees of freedom that associated with these microstates? How can we 1 find such large number of microstates from the unique geometry of the black hole stated by the no-hair theorem? Information paradox It is well known that any object can not escape from the black hole after it crosses the event horizon. Any information about the object that falls into is completely locked inside the black hole forever. On the other hand, in 1975, Hawking showed [Haw75] that a virtual particle pair can be created near the horizon and one of them falls into the black hole while the other escape to infinity. The radiation from this effect is exactly thermal and can not carry any information. However, the energy of the black hole undergoes this process will be carried away by the radiation and eventually completely evaporated. Therefore, after the black hole evaporate away, the information stored inside is completely lost. In quantum theory, this corresponds to non-unitary evolution which violates one of the essential properties of the quantum physics, unitarity. A successful quantum gravity theory is expected to solve these two puzzles. String theory is one of the candidates of the quantum gravity theory. It has already provided the beginnings of the solution for the first puzzle. In [SV96], Strominger and Vafa showed that by counting the configurations of the branes and strings in the zero gravitational coupling limit, one can have the correct number of the microstates match Bekenstein’s entropy of a five-dimensional BPS three-charges black hole. Later, Johnson, Khuri and Myers [JKM96] generalized the argument to four-dimensional black holes. Even there are these successes, the geometric aspects of these microstates after the gravitational coupling is turned on are still unclear. Additionally, the information paradox still remain to be solved. To understand the geometric structure of the microstates of a black hole, we take the point of view that some of the microstates of a black hole can be understood as some geometries. These microstate geometries must be horizonless because the horizon 2 area corresponds to the entropy of a black hole and a microstate geometry should not have any entropy. Moreover, in order to avoid “naked” singularities, they should be smooth everywhere. Therefore, a microstate geometry is defined as a geometry which is horizonless and free of any singularity. From this point of view, the classical unique geometry of a black hole is a thermodynamic description of the ensembles of these microstate geometries with the same macroscopic properties as the black hole. Furthermore, in [Mat02], Mathur proposed a way to solve the information paradox which is later called “fuzzball conjecture”. He argued that quantum or semi-classical gravity effect can not be locked inside the planck-scale distance near the singularity and must be extended out to the horizon (which is usually macroscopic). This new picture is very different from the classical picture of a black hole we have (e.g. a point like singularity, empty space between the singularity and horizon). In Mathur’s conjecture, the microstate geometries with same macroscopic properties have similar geometries with each others outside of the “effective” horizon of the classical geometry. However, inside of the “effective” horizon, they start to deviate from each other and classical geometry and have their own non-trivial local structures. Therefore, the horizon of a classical black hole is actually an “effective” description which corresponds to the surface at which the microstate geometries start to deviate from each others. Moreover, Mathur’s conjecture has two forms. The weak form states that most of the microstates are the horizonal-sized ball with planck-scale curvature which can not be described in supergravity and must be addressed in string theory. On the other hand, the strong form suggest that majority of these microstates can be described in supergravity. Therefore, the number of supergravity solutions will be enough to account for the entropy of a black hole. In this thesis, we will explore the strong form and our main goal is looking for a family of supergravity solutions that correspond to the microstate geometries. The 3 supergravity solutions that we are looking for are asymptotical flat, horizonless and regular everywhere. Every such solution (geometry) will be a potential candidate for a microstate of a black hole. In this picture, the microstate geometries will give the correct statistical mechanics description of a black hole and the number of them will be large enough to account for the entropy of a black hole. Moreover, the information paradox will be solved because the microstate geometries do not have horizon and the light enter the black hole will eventually come out as Hawking radiation and carry the information that stored inside of the black hole. On the other hand, even if the weak form of Mathur’s conjecture is correct, one can still understand these microstate geometries from the point view of the holographic prin- ciple. Specifically, we will show in chapter 4 that there are some microstate geometries have very long AdS throats. From the point view of AdS/CFT correspondence, they can be the dual geometries of some typical conformal field states in [SV96]. Because the asymptotic charges play the roles of the only macroscopic properties that one can use to track the microstates geometries, they are very important for our main goal. Therefore, in the following section, we will explain how to read the asymptotic charges from the the asymptotic behavior of the metric. In the section 1.3, we describe schematically how to build a black hole with branes in string theory/supergravity. In sec- tion 1.4, we briefly mention some progresses in two-charge system. In the last section, we give the outline of this thesis. 1.2 Asymptotic charges In this section, we review the relation between the macroscopic properties of a black hole and the asymptotic behaviors of the metrics. We will closely follow the presentation in [Pee00]. 4 Since a classical black hole is a solution of general relativity, let us look at Einstein’s equation inD-dimensional space-time: R 1 2 g R = 8G D T ; (1.2) whereG D is Newton’s constant inD Dimensions,R is the Ricci tensor andT is the energy-momentum tensor. Far away from the black hole, the metric becomes flat. Then, the asymptotic behavior of the metric can be expressed as: g + h ; (1.3) where is the flat metric andh is a small deviation from it. If we plug this form of metric (1.3) into Einstein equation (1.2), for the leading order, we get 1 : R = 1 2 @ @ h + @ @ h @ @ h @ @ h = 8G D T 1 D 2 T : (1.4) However,h is not invariant under coordinate transformations and we can choose the coordinates to fixh in harmonic gauge: @ (h 1 2 h ) = 0 : (1.5) The name of this gauge choice comes from it’s similarity with the harmonic (Lorentz) gauge of the electromagnetism:@ A = 0. Under this gauge choice, (1.4) is simplified to: r 2 h = 16G D ~ T ; (1.6) 1 Notice that in the leading order, the indexes ofh are raised or lowered by the flat metric . 5 where ~ T is: ~ T = T 1 D 2 T : (1.7) To derive (1.6), we assume the system is non-relativistic and thus the time derivative of h can be neglected. The solution ofh can be constructed from Green functions in D 1 space dimensions: r 2 1 j~ x~ yj D3 = (D 3) D2 (j~ x~ yj); (1.8) and the result is: h (~ x) = 16G D (D 3) D2 Z d D1 ~ y ~ T j~ x~ yj D3 ; (1.9) where d is the solid angle ofS d and can be expressed as: d = 2 (d+1)=2 ( d+1 2 ) : (1.10) Moreover, we can expand (1.9) up to the dipole contribution: h (~ x) 16G D D2 1 (D 3)r D3 Z d D1 ~ y ~ T + x j r D1 Z d D1 ~ yy j ~ T ; (1.11) where ‘i’ is the space index andi = 1; 2:::D 1. In the rest frame of this system, the mass and the angular momentums is related with the energy momentum tensor by: M = Z d D1 ~ yT 00 ; J ij = Z d D1 ~ y (y i T j0 y j T i0 ) : (1.12) 6 Combine (1.12), (1.11) and (1.3), one can obtain the relation of the asymptotic behavior of the metric with the mass and the angular momenta: g 00 1 + 16G D (D 3) D2 r D3 D 3 D 2 M + 1 D 2 Z d D1 ~ yT i i 1 + 16G D (D 2) D2 M r D3 ; (1.13) g 0i 16G D D2 x j J ji r D1 ; (1.14) where we have used the fact that the system is non-relativistic and thusT 00 T i0 T ij . Besides of the mass and the angular momenta, the black hole can also have electric charges which can be observed from far away. To explain how to read electric charges, we use the familiar four dimensional electromagnetic system as the example. In four dimensional space time, we have one-form potential, A, and two-form field strength, F = dA, for the electromagnetic system. Maxwell’s equations with sources can be expressed concisely with differential forms: d? F = ? J; (1.15) where ? is the Hodge dual in the four-dimensional space-time and J is the one-form current. In the above equation, we ignore the constant coefficient beforeJ. The charge is the integral of the current and by Gauss’s (Stokes’) theorem, we have the total electric charge of a system: Q = Z S 2 ?F ; (1.16) 7 where S 2 is a two-sphere gauss surface that enclosed the system. The other half of Maxwell’s equations is simply Bianchi identity: dF = 0. The integral of this equation actually give the magnetic (topological) charge: P = Z S 2 F: (1.17) If the topology of the four-dimensional space-time is trivial, the magnetic charge is zero. However, in the four-dimensional space-time with nontrivial topology, we can have non- vanished magnetic charges. Moreover, the electric and magnetic charge should obey the Dirac quantization condition: QP = 2n; n2Z: (1.18) This imply if we have non-vanished magnetic (topological) charges, both of the electric and magnetic charges must be quantized. The exact quantized quantities will depend on the coefficient that we ignore in (1.15). Additionally, one can easily generalize the above electromagnetic story to a more general theory that contains various n-form potentials. Particularly, we will discuss about the 3-form potential of the eleven-dimensional super- gravity in the following chapter. 1.3 Building a black hole from string theory 1.3.1 From string theory to supergravity String theory is a theory about quantum relativistic strings and their interactions. The only two parameters in the theory are the string’s tension, 0 , and the coupling,g s . The 8 different values of 0 andg s define different “phases” of string theory. Roughly speak- ing, the string’s tension provide the only energy (or length) scale and the string coupling determine the strength of the gravitational force. Just like most of other quantum field theories, we only know how to do string theory perturbatively. However, in order to build a black hole from string theory, one needs to consider strong coupling. Fortu- nately, it was found out that the low energy limit of string theory is in fact supergravity. Therefore, for small 0 , the full string theory can be described by supergravity. In other words, if the curvature is small, one can use supergravity instead of string theory. It is well known that there are several types of superstring theory, namely, type I, type IIA, type IIB, E 8 E 8 heterotic and SO(32) heterotic string. They all have ten dimensional space-time and in low energy limit become type IIA or type IIB ten dimen- sional supergravity theories. However, it was shown that these theories are connected by several dualities and was proposed that they are actually different aspects of one unique theory, M-theory. Although the full story of M-theory is unclear so far, it was under- stood that the low energy limit of this theory is the eleven-dimensional supergravity. In the following, we use the eleven-dimensional supergravity to introduce the general idea about how to build a black hole in string theory. The reasons to use the eleven dimensional supergravity are both because it is the main focus of this thesis and it is the simplest. The eleven-dimensional supergravity has two bosonic fields, the metric and the 3- form potential and also fermionic fields. Because this thesis mainly concern the geom- etry, we will focus on the bosonic sector of the theory. The bosonic part of the eleven dimensional supergravity action [Joh02] is: S = 1 16G 11 Z d 11 x p g (R 1 48 jF 4 j 2 ) 1 96G 11 Z F 4 ^F 4 ^A 3 (1.19) 9 whereG 11 is the eleven-dimensional Newton constant, A 3 is the 3-form potential and F 4 is it’s corresponding field strength. From the action, one can see the bosonic part of the eleven-dimensional supergravity is very similar with general relativity in eleven- dimensional space-time. The only difference is it has an 3-form potential and the dynam- ics associated with this potential. The additional 3-form potential can be thought as the natural extension of one-form potential in four-dimensional electromagnetic system. For example, in four-dimensional electromagnetic system, a point electric charge with one dimensional world-line will source one-form potential. Correspondingly, the source of the 3-form potential in the eleven-dimensional supergravity is a two-dimensional object with three dimensional world-volume called M2-brane. On the other hand, just like a point magnetic charge 2 in four-dimensional space-time can source one-form potential “magnetically”, there is an object can source 3-form potential magnetically in eleven- dimensional supergravity. The dimensionality of this object can be easily calculated as the following: this object source the 3-form potential through the dual field strength which is 7-form in eleven dimensional space-time and therefore, this object should be five dimensional and is called M5-brane. There are many solutions in the eleven-dimensional supergravity that corresponds to the different distributions, wrappings and orientations of these branes. Since our goal here is to build a black hole from string theory, we are mostly interested in the solutions that look like a black hole in general relativity. Specifically, they should look like point objects in non-compact space. Therefore, the first thing we need to do is to compactify several spatial directions to some compact Calabi-Yau manifold in order to bring the eleven-dimensional space-time down to lower dimensional one. Then, if we wrap the M2/M5-branes around the non-trivial 2/5-cycles in the compact manifold, they will look 2 In order to have a point magnetic charge “theoretically”, one need to have non-trivial topology in the four-dimensional space-time 10 like point sources in the non-compact space. Consequently, the 3-form potential reduce to an 1-formU(1) gauge field in the non-compact space and the results will look like charged black holes in general relativity. 1.3.2 BPS branes and extremal black holes There exists a lot of solutions in the eleven-dimensional or type II supergravity that correspond to some branes. They are shown to be very useful to construct the black hole solutions. Among these branes, there is a special type of solutions called BPS branes which are very important for counting black hole microstates. In the following, we will explain the definition of “BPS”, the features of these BPS branes, why they are important and how they are connected to extremal black holes. At first, we start from the metric of a charged black hole to explain the meaning of “extremal”. The metric of four-dimensional Reissner-Nordstr¨ om black holes 3 is: ds 2 = 1 2M + Q 2 2 dt 2 + 1 2M + Q 2 2 1 d 2 + 2 d 2 2 =4 + ()4 ()dt 2 + 4 + ()4 () 1 d 2 + 2 d 2 2 ; 4 = 1 r r = M p M 2 Q 2 : (1.20) There are two horizons at r = r . However, if the charge larger than the mass, the horizon will not exist and the singularity at the center of the black hole become “naked”. In order to avoid the naked singularity, one should require the mass and charge satisfy the following bound: MjQj : (1.21) 3 Here, we use the particular units such that the Newton constant is normalized to1 11 A black hole is called extremal if the bound is saturated. The first thing one can notice isr + =r if the charge equal the mass. So, the two horizons coincide for the extremal case and the metric can be simplified to the following isotropic form by some coordinate change: ds 2 = 1 r 0 2 dt 2 + 1 r 0 2 d 2 + 2 d 2 2 = 1 + r 0 r 2 dt 2 + 1 + r 0 r 2 dr 2 + r 2 d 2 2 ; (1.22) where the two horizon coincide atr 0 and the new radial coordinate,r, is defined as: r r 0 : (1.23) One can see that this form of the metric can be naturally split into an isotropic space and time. Closer to the center of the source, the time is contracted while the space is isotrop- ically stretched. This can be thought as a typical behavior of zero dimensional BPS brane. The brane with tension squeeze the directions along it’s world line (volume) and stretch the transverse directions. One can naturally generalize this to a n-dimensional BPS brane and the metric look like: ds 2 = Z n (r) p k (dt 2 + n X i=1 dx 2 i ) + Z n (r) p ? (dx 2 ? ); (1.24) wherex i ;i = 1:::n are compact parallel directions andx ? are noncompact transverse directions.Z n (r) is the warp factor that controlled how much the space-time is stretched or contracted and it is a harmonic function in the noncompact space. The powers,p k and p ? are positive rational numbers and depend on the number of dimensions of the branes and the type of supergravity theory that we consider. 12 There are several reasons to consider BPS solutions. One of the reasons is the BPS solutions are usually simpler which can be seen very clearly by comparing (1.20) and (1.22). Furthermore, if one compute Hawking temperature of Reissner-Nordstr¨ om black holes, one have: T H = p M 2 Q 2 2 (M + p M 2 Q 2 ) 2 : (1.25) Therefore, the extremal solution in (1.22) has zero Hawking temperature and does not have Hawking radiation. This means the spectrum of the microstates will be stable and will not decay into something else by the radiation and therefore, it is easier to keep track of these microstates. On the other hand, an important feature of the BPS branes which may be used as the definition of “BPS” is that they preserve some fraction of supersymmetries in super- gravity. In order to see that, we need to look at the supersymmetry algebra. Consider the anti-commutator of the supersymmery generator: fQ ;Q g (C ) P + (C 1 :::p ) Z [ 1 :::p] ; (1.26) where ; are spinor indexes,C is the charge conjugation matrix, 1 :::p is the anti- symmetry product of gamma matrices,P is the momentum vector andZ is the charge associated with the p-dimensional brane. Notice that the indexes of gamma matrices in the second term run through the spatial directions that are parallel with the brane. In the rest frame of the system, the above anit-commutator becomes: fQ ;Q g (C 0 ) M + (C 1:::p ) Z [1:::p] ; (1.27) By requiring the supersymmetry transformed state,Qjphys >, has non-negative norm, we obtain the similar bound in (1.21). The state that saturated the bound must have 13 Q jphys>= 0 for some values of. This means the state preserve some supersymme- tries. The spinor parameters of the remain unbroken supersymmetries must satisfy the following projection condition: (1 +Sign(Z) 01:::p ) = 0 (1.28) This condition project out half of the supersymmetries. Therefore, the BPS branes with a single charge preserve only half of the supersymmetries. Another important feature of BPS branes is their superposition (linear) nature. It can be understood from the fact that their masses equal to their charges. Consider we add multiple “compatible” BPS branes, the result will be simply another BPS object. The reason is that the total charge is conserved and each mass of the branes equal to their charges individually and are added up to the total mass which again equal to total charge. Consequently, because the total mass is simply equal to the sum of all masses from every individual brane, there is no energy stored in the interaction between them. This means that those BPS branes will not ”feel” any force between each others and it can be thought as the result of the attraction of the gravity cancel the repulsion of the electricity. This feature of BPS branes make the multiple BPS branes behave like a linear system even thought the gravity is non-linear. There are several ways to put multiple branes together. The trivial way is to sim- ply consider many identical branes which are parallel with each other and located in different positions in non-compact space. Comparing this multiple-branes system and the single-brane system with the same total charge, one find out they both preserve half of the supersymmetries and have exactly the same asymptotic charge. Therefore, these system are considered as “one-charge” system or 1=2 BPS system because they preserve half of the supersymmetries. 14 The more general way to make multiple branes systems is to wrap two same branes on different spatial directions or consider two different branes with different spatial dimensions and let them intersect with each other by certain codimension. There are many possible ways to do that in different theories. However, not any possible com- bination will work because there is some compatibility issue. The intersection rules that determine the compatibility of two branes can be found in several different ways [PT96, Tse96, GKT96, AEH97]. For example, one can use dualities on the well-known intersection of two branes to obtain many other different intersection rules. Another way is to use the brane probe and the fact this probe should not feel force if it is mutually BPS with the background BPS branes. Nevertheless, in order for the multiple-branes system still be BPS, it must preserve some fraction of the supersymmetries and their asymptotic charges must be additive. This in general will require one to check the remaining unbro- ken supersymmetries carefully for each case. If two branes are shown to be compatible, the result metric has a simple form as one brane wrap on “top” of the other brane. For example, if we have a BPSp 1 -brane withp 1 spatial dimensions, the metric look like: ds 2 = Z p1 (r) p 1;k dt 2 + p 1 X i=1 dx 2 i + Z p 1 (r) p 1;? (dx 2 ? ): (1.29) Then let’s consider another BPS p 2 -brane which is compatible with p 1 -brane if they intersect with each other with codimension ‘p 1 ’ 4 . If we put thisp 2 -brane on the top of p 1 -brane, the resulting metric look like: ds 2 =Z p 2 (r 0 ) p 2;k Z p 1 (r 0 ) p 1;k dt 2 + p 1 X i=1 dx 2 i + Z p 1 (r 0 ) p 1;? p 2 X i=p 1 +1 dx 2 i + Z p 2 (r 0 ) p 2;? Z p 1 (r 0 ) p 1;? (dx 0 2 ? ); (1.30) 4 We assumep 2 >p 1 and this meansp 1 -brane should be completely inside of thep 2 -brane. 15 wherep 1;k ;p 1;? ;p 2;k ;p 2;? are all positive rational numbers,x ? ;x 0 ? are the overall trans- verse non-compact directions to the system andr;r 0 are the distance from the branes in the overall non-compact space which is the transverse space ofp 1 -brane in (1.29) and the transverse space ofp 2 -brane in (1.30). The above way to construct multiple branes solutions is based on harmonic ansatz [PT96, Tse96, GKT96]. The existence of thep 2 -brane in this multiple branes system further impose the sec- ond projection condition which is defined in (1.28) and project out half of the remain- ing unbroken supersymmetries that survive the first projection condition fromp 1 -brane. Therefore, the overall unbroken supersymmetries of this system are one fourth of the total supersymmetries. This kind of systems are called two-charge systems which have two charges associated with two branes or 1=4 BPS system because they preserve one fourth of supersymmetries. From the above very rough analysis, we can see how this goes in the simplest setup. For example, consider a space with a compact manifold without non-trivial intersection and if we wrapN compatible BPS branes around the cycles of this simple compact man- ifold, in general, we get N-charge system which preserve at least 1=2 N supersymme- tries. However, in more complicate situation, one need to analyze the supersymmetries that preserved by the system more carefully. Because from the unbroken supersymme- tries, we have associated killing spinors and they describe the isometric of the system. Therefore, a system with more charges preserve less supersymmetries and consequently the metric has less isometries and more complicated. 1.3.3 Link p-branes to D-branes or vice versa In string theory, we also have p-dimensional extended objects that preserved some super- symmetries and also couples to p-form potential called Dp-branes or simply D-branes. The simplest definition of Dp-branes is that they are p-dimensional extend objects on 16 which open strings can end. In other words, D-branes can be seen as the boundary con- ditions of the open strings. More generally, they are the coherent or solitonic states of closed strings that describe the collective behavior of open strings. The identification of BPS p-branes in type II supergravity as the low energy limit of Dp-branes in string the- ory is built through several remarkable development of Branes-physics [Pol96, Joh02]. Even though this success can not be extended to M-theory because the quantum theory of M-branes is unclear, one still can use the dualities to map M-branes to D-branes in order to utilize this identification. The impact of this identification is indeed profound. First of all, it gives us a way to quantize charges to some numbers with string units by identifying a BPS p-brane which has charge Q to N stacked Dp-branes. Secondly, as we have mentioned there is no energy stored in the interactions between BPS objects and consequently, the spectrum of the system should not change even we change the coupling strength,g s . This means the spectrum of Dp-branes that calculated at small g s should match exactly with the spectrum of BPS p-branes at large g s . This allow one to compute the entropy of the extremal (BPS) black hole by evaluating the number of the states on the conformal field theory that lives on the Dp-branes. The agreement of this computation [SV96] with Bekenstein’s entropy suggest that the large number of degrees of freedom of the black hole may come from the different states on Dp-branes. Moreover, this lead to a more direct conjecture, holography principle [Mal03], state that for every states of Dp- branes on the boundary corresponding to some solutions of string theory in the bulk. With the holography principle and Mathur’s conjecture, there is an expectation that one can find the dictionary that directly map the microstate geometries to the states on Dp- branes. Actually, there are already some success in two-charge system which will be explained in the next section. However, a two-charge supergravity solution does not have a macroscopic horizon and can not be used directly to verify if the counting match 17 with Bekenstein entropy. Thus, in order to prove or disprove strong form of Mathur’s conjecture, one need to extend the similar work to three-charge system. Therefore, the modern developments are trying to extent dictionary to the three-charge system. 1.4 Two-charge system Mathur’s conjecture has gone through several nontrivial test in two-charges systems. Specifically, Mathur, Lunin, Maldacena, and Maoz [LM02a, LM02b, LMM02, Mat05], showed that a state in the conformal field theory on a D1-D5 system can be mapped to a two-charge BPS supergravity solution that are asymptoticallyAdS 3 S 3 T 4 , and have no horizon and singularity. These solutions can be characterized by an arbitrary closed-curves in the space transverse to the D1 and D5 branes. However, as we have mentioned, two-charge supergravity solution does not have a macroscopic horizon and therefore, the microstates does not describe the true black hole. To extent the result to three-charge solutions in a D1-D5 system, one needs to add an additional momentum in one of the direction and thus become D1-D5-P system. Alternatively, one can consider three M2-branes in the eleven-dimensional supergravity which is our main focus in this thesis. 1.5 Outline In this thesis, we focus on five-dimensional three-charge supergravity solutions. The reason of studying five-dimensional system will be explained in the next chapter. Most of the original part in this thesis is based on the collaborated work done in [BWW06b], [BWW07a], [BWW06a] and [BWW07b]. The outline is the following: in the first chap- ter, we review a systematic method [BW05] for building general three-charges BPS supergravity solutions in five dimensional space time. Also, we describe the result about 18 the merger of the black ring and black hole in [BWW06b] which will be showed useful in probing the microstates of a black ring/hole. In the second chapter, we review the bub- ble geometries which will be our essential base to find the microstate geometries. In the third chapter, we discuss the merger of multiple Gibbons-Hawking points in the bubble geometries. In the last chapter, we give the conclusion and some future directions. 19 Chapter 2 Three-charge BPS Solutions and Classical Mergers In this chapter, we will review a very important technique upon which this thesis is based. In [BW05], Bena and Warner proposed a method to obtain a general supergrav- ity solution that corresponds to three-charge BPS system which preserve one eighth of supersymmetries. This method involves solving three coupled linear equations and the resulting metric in the non-compact space-time will look either like a five-dimensional black hole or black ring or any superposition of them. The first natural question to ask is why we consider a five dimensional rather then a four dimensional non-compact space-time. The main reason is actually more tech- nical rather than fundamental. When we build a black hole with branes, it is simpler to consider geometries with four transverse “spatial” directions and thus corresponding to a black hole/ring in five-dimensional space-time. In order to bring it down to the four-dimensional space-time, we need to do Kaluza-Klein reduction to compactify one spacial direction. During this procedure, a newU(1) gauge field and a charge associated to it will appear from the off-diagonal components of the metric involved the compacti- fied direction. This procedure will complicate the metric and field content of the system without significantly modifying the physics. Furthermore, the original entropy counting is done in five-dimensional [SV96] and it should be easier to connect the five dimen- sional solutions to the microstates in [SV96]. Therefore, to simplify our task, we will focus on five-dimensional solutions in this thesis. 20 As we have mentioned, the general three-charge five-dimensional solutions not only contain black holes but also black rings. One of the most significant feature of a black ring is that it has the horizon with topology S 2 S 1 rather than S 3 of a black hole. Another important difference between them is their angular momenta. Because, the rotation symmetry of the four dimensional space isSO(4) which can be decomposed to SU(2)SU(2), we have two independent angular momenta coming from two different SU(2). We can put the above statement in a more geometrical way, a four dimen- sional space can be decomposed into two orthogonal planes and thus we have two angular momenta associated to the rotation on these two planes. It was shown that a five-dimensional three-charge black hole must have these two angular momenta equal [BMPV97]. However, there is no such restriction on black ring solutions and in general, they have two different angular momenta. From the above statement, comes a conun- drum: what if we drop a small black ring into a large black hole? Is the final result still a black hole or it will become a black ring? After all, we are allowed to do so because they are all BPS and there should not be any force between them to prevent us from doing that. The reasonable guess is it will just be another bigger black hole. However, the problem is the two angular momenta will be different due to the black ring we drop in and the fact that the angular momentum should be conserved. Does that means it is possible to have a black hole with two different angular momenta? In [BWW06b], we use the technique that we are going to introduce to study the system contains a black hole and black ring and their merging process in order to understand this conundrum. Following the review of the technique, we will show the solution of this conundrum in the section 2.3. Even though the original motivation of investigating the classical merger of a black ring and a black hole is not related with the black hole/ring microstates, the result of this study gave us the important insight that lead us to discover the microstates of a black 21 hole/ring which has macroscopic horizon. The insight comes from studying the entropy of the result of the merger and the conditions for the reversible merge. These conditions will be shown in the section 2.4. 2.1 Build five-dimensional three-charge BPS supergrav- ity solutions In this section, we review the general method proposed by Bena and Warner in [BW05] for building a general three-charge BPS supergravity solution in five-dimensional space- time. Their starting point is M-theory and it’s corresponding low energy limit, the eleven-dimensional supergravity. As we have mentioned in the previous chapter, to build a three-charges BPS system, we need to wrap three compatible branes around sev- eral different directions. In the eleven-dimensional supergravity, we have M2 branes and it was understood that two orthogonal M2 branes are compatible. Therefore, the natural way to do is simply compactify six spacial directions to T 2 T 2 T 2 and wrap three M2 branes on these three two-torus. The result will be a point-like object in the five-dimensional space-time. However, it was shown that the general solutions can also have dipole charges coming from M5 branes. One may understand this from the following analogy: consider a electromagnetic system contained an electron circling around a closed path, there is no net magnetic charge in this system but only a mag- netic dipole charge. In the similar spirit, one can introduce dipole charges of M5 branes into the three-M2-brane system. However, we need to be careful how we wrap the M5 branes, in order not to break any supersymmetry further. Specifically, they must form electromagnetic pairs with three M2 branes. This means for each pair, the spatial part of the world volume of the M5-brane must be orthogonal to the M2-brane. Therefore, in 22 Branes 0 1 2 3 4 5 6 7 8 9 10 M2 1 ! ! ! M2 2 ! ! ! M2 3 ! ! ! M5 1 ! y () ! ! ! ! M5 2 ! y () ! ! ! ! M5 3 ! y () ! ! ! ! Table 2.1: The wrapping profile of M2/M5 branes Where, 0-4 are the directions in the non-compact space-time and 5-10 are the directions inT 2 T 2 T 2 . The arrow, !, means the branes extend in that direction. The wave symbol,, indicate the branes smear over in that direction. And,y () is the profile of closed loop that M5 branes extend in the four-dimensional space. The dot,, mean M2 branes appear as a point in the non-compact space-time and we are going to consider the case that smear M2 branes over the M5 branes profile. [BW05], they considered the wrapping profile of three pairs of M2/M5 branes as shown in Table 2.1. Notice that four spatial directions of the M5 brane wrap in the orthogonal com- pact directions of the corresponding M2 brane. The remain one spacial direction is extended in four dimensional space and thus become a closed curve which is describe by y (). And, the reason one should consider M2-branes smear over the transverse compact directions is to avoid to break the rotation symmetry along those directions. The resulting solution will look like a black ring with profile determined by y (). Moreover, it will preserve one eighth of the supersymmetries. The remain unbroken supersymmetries satisfy the following projection conditions: (1 + 056 ) = 0 (1 + 078 ) = 0 (1 + 09(10) ) = 0; (2.1) where is a spinor parameter that describe a supersymmetry transformation. 23 Form the wrapping profile, one can guess the full metric should have the following form: ds 2 11 = 1 Z 1 Z 2 Z 3 2=3 (dt +k) 2 + (Z 1 Z 2 Z 3 ) 1=3 h mn dx m dx n + Z 2 Z 3 Z 2 1 1=3 (dx 2 1 +dx 2 2 ) + Z 1 Z 3 Z 2 2 1=3 (dx 2 3 +dx 2 4 ) + Z 1 Z 2 Z 2 3 1=3 (dx 2 5 +dx 2 6 ); (2.2) A = A 1 ^dx 1 ^dx 2 +A 2 ^dx 3 ^dx 4 +A 3 ^dx 5 ^dx 6 ; (2.3) where Z I are warp factors associated with three different sets of M2/M5-branes, h mn is the four dimensional base space metric andA is the three-form potential sourced by M2/M5 branes. The A I are the three one-form potentials reduced fromA and they will couple to three charges respectively. Moreover, one can separateA I to two parts. One part is sourced by M2-branes electrically and the other part sourced by M5-branes magnetically. The corresponding field strengthdA I will be like: dA I = d dt +k Z I + I ; (2.4) where the first part comes from M2-branes in whichk is a one-form associated with the rotation of the black hole/ring in (2.2) and the second part is the dipole field strength from M5-branes. By utilizing the above metric ansatz and solving the killing spinors equations of the remain unbroken supersymmetries in (2.1), the following conclusions was shown in [BW05]: 24 1. The base space must be hyper-K¨ ahler This conclusion can be made from the pro- jection conditions. The supersymmetries,, that satisfied all the projection condi- tions in (2.1) must also satisfy the following conditions: (1 + 056 078 09(10) ) = (1 056789(10) ) = 0; (2.5) where, in the second equality, we use the Clifford algebra and take the mostly positive metric convention. From the above condition and the fact that 0123456789(10) =1 (we take the positive convention choice), we have: (1 1234 ) = 0: (2.6) This indicated the holonomy of the four dimensional space is completely inside one of theSU(2) in the rotation group,SO(4) =SU(2)SU(2). This means the four dimensional space is “half-flat” and thus the base space metric,h mn , must be hyper-K¨ ahler. 2. The full metric can be obtained by solving three linear equations In order to obtain the full metric, at first, one need to specify the base space metric, h mn . Then, one also need to specify the charges, the location of the M2 branes and the ring profile of M5 branes in fivedimensional space-time. Having done that, it was shown that one can obtain the corresponding warp factors,Z I , and one-form, 25 k, by solving the following three linear equations which can be thought as BPS equations: I = ? 4 I ; (2.7) r 2 Z I = 1 2 C IJK ? 4 ( J ^ K ); (2.8) dk +? 4 dk = Z I I ; (2.9) where? 4 is the Hodge dual on four dimensional non-compact space, and for M theory on (T 2 ) 3 ,C IJK =j IJK j. The first equation (2.7) indicate the dipole field strength must be self-dual. This implies the dipole field strength satisfy four dimensional Euclidean Maxwell equation. Therefore, one can obtain the dipole field strength by solving Maxwell equation for some particular source distribution and boundary condition at infinity and then extract the self-dual part of this solution. As we have mentioned, the dipole field strength is sourced by M5 branes and consequently determined by the dipole charges and the ring profile. The second step is to solve the warp factors,Z I . The homogeneous solutions of the second equation (2.8) are actually harmonic functions and as we have expected, they are sourced by the point-like M2 branes. The inhomogeneous solution comes from the “effective” M2-brane charges generated “magnetically” by the M5-branes. However, these “effective” M2-brane charges are not singular sources but continuously distributed and their distribution is exactly specified by the source term of the second equation. Therefore, once we obtained the dipole field strength from the ring profile, we can obtain the inhomogeneous part of Z I by solving the “electric potential” of this continuously distributed effective charges. 26 Finally, we need to solve one-form, k, which is basically related with the angular momenta of the system. Notice the left hand side of the equation (2.9) gives the self- dual part of dk. Thus, if we add anti-self-dual term to some solution, it will still be a solution. This corresponds to adding a homogeneous solution. Physical meaning of this homogenous solution can be thought of the spinning of the coordinate system or equivalently, the spinning of the system. However, spin a system very fast without adding the energy associated with this spinning can be very dangerous. Generally, it will lead to the appearance of the closed time-like curves. A closed time-like curve is a closed curve with time-like signature 1 and it will destroy the causality of the system. On the other hand, notice that onlydk appears in the third equation and this means that there is a gauge transformation that take a solution ofk to another solution like: k ! k + d(x); (2.10) where (x) is a function in four dimensional space. Again, doing the above gauge transformation is also equivalent to adding a homogeneous solution. However, this type of the homogeneous solution has a different physical meaning. The physical meaning of this gauge transformation ofk can be thought of doing a local time coordinate shift: t!t +(x). The reason for that can be seen clearly from the metric in (2.2) that doing this time coordinate change will generated fork. From the above discussion, one can see there are some subtleties in the final step involved the homogeneous solutions ofk. First of all, there are two types of homoge- neous solutions, one from the gauge transformation of k and the other from the anti- self-dual part of dk. The former type of the homogeneous solutions have no physical consequence and thus we are free to choose any gauge we want. However, choosing the 1 In our convention, the time-like signature is1 27 gauge wisely will simplify the solutions a lot. On the other hand, the homogenous solu- tion from anti-self-dual part ofdk will be dangerous and may have some physical effect on the final metric of the system. The choice of this type of the homogeneous solution should be made properly such that the full metric satisfy the following condition: There is no closed time-like curves in the space Asymptotic flat with non-rotating coordinate system at infinity The solution is free of singularity except at the positions of the sources The first condition we have discussed. The second condition is coming from the fact that we are looking for the asymptotic flat metric and it will requirek vanish at infinity. The third condition will require the regularity of k. These limitations will be helpful guidelines for us to constraint the possible form ofk. 2.1.1 Asymptotic charges and quantized charges In section 1.2, we have explain the general method to read the charges of the system from the asymptotic behavior of it’s metric. In this subsection, we apply this general method to study the asymptotic charges of the BPS system we have introduced. We also explain how to get the quantize charges with string units. As we have mentioned, the warp factors,Z I , behave much like the “electric poten- tial” of four dimensional Maxwell system. Therefore, we can expect the following asymptotic behavior: Z I 1 + Q I 2 + ::: ; (2.11) where is the radial direction of the four dimensional space andQ I can be treated as the total asymptotic charge of each pair of M2/M5 branes. Notice that we add a non-zero constant to make the metric become I R 4;1 asymptotically. We set this non-zero constant 28 to ‘1’ in order to fix the size of compact space at infinity. According to (1.13), we can extract the mass by examining the asymptotic behavior ofg 00 : g 00 =(Z 1 Z 2 Z 3 ) 2=3 1 + 2 (Q 1 +Q 2 +Q 3 ) 3 2 : (2.12) Compare the above behavior with (1.13), we have the following relation: G 5 M = 3 8 (Q 1 +Q 2 +Q 3 ) = 4 (Q 1 +Q 2 +Q 3 ) ; (2.13) whereG 5 is Newton constant in five-dimensional space-time. In the suitable unit, the above relation just indicated the system saturate the BPS bound and the summation of the three charges play the role of the central charge in (1.21). Moreover, we can read the angular momentum of the system from the asymptotic behavior ofg 0i as in (1.14): g 0i dx i = (Z 1 Z 2 Z 3 ) 2=3 k k 16G 5 3 x j J ji r 4 dx i ; (2.14) where J ji is a four dimensional antisymmetric generator of SO(4). One can choose the appropriate coordinate system to block diagonalize J ji to two 2 2 blocks. The two eigenvalues associate to these two blocks will be the two angular momenta of the system. Clearly, these two angular momenta will be related with the leading asymptotic behavior ofk. However, to further discuss the detail, we need to specify the coordinate system and therefore we postpone the discussion of the angular momentum until we mention some specific system. In the previous chapter, we have mentioned the connection between classical p- branes with quantized Dp-branes. Even thought the underlying quantum theory of the eleven-dimensional supergravity, M-theory, is unclear so far, one can utilize several 29 dualities to map M-brane story to D-brane one. Consequently, there are similar picture that connect classical BPS M-brane with charge Q with the N-stacked quantized M- branes. By this connection, one can express the classical charges with quanta of some units. Furthermore, these units must be able to be expressed with the two parameters in string theory, 0 andg s . First of all, the only fundamental scale in string theory is from the tension of the string: T = 1 2 0 1 2l 2 s ; (2.15) wherel s p 0 define the fundamental length scale of the string theory. On the other hand, we can define the plank length from the eleven-dimensional Newton constant and they are related withl s as the following [Joh02]: 16G 11 (2) 8 l 9 p = (2) 8 g 3 s l 9 s ; (2.16) where g s is the string coupling. The lower dimensional Newton constant are simply related toG 11 by: G D = G 11 V 11D ; (2.17) whereV 11D is the volume of the compact space. Therefore,G 5 can be expressed as: G 5 = (2) 8 l 9 p 16 (2L) 6 = l 9 p 4L 6 ; (2.18) whereL is the radius of the cycles inT 6 . One more ingredient we need is the tension of the quantized M-branes. It can be shown from the duality that the tensions of M2/M5- branes are [Joh02]: T 2 = 1 (2) 2 l 3 p ; T 5 = 1 (2) 5 l 6 p : (2.19) 30 From that, we can find out the masses of the quantized M2/M5-branes: m 2 = T 2 V T 2 = L 2 l 3 p ; m 5 = T 5 V T 4 S 1 = SL 4 (2)l 6 p ; (2.20) whereS is the length of the ring profile that M5-branes wrap on. Without losing the generality, we work on the system with only M2-branes. If we connect three classical M2-branes with asymptotic charges (Q 1 ;Q 2 ;Q 3 ) to the three sets of quantized M2-branes with number of (N 1 ;N 2 ;N 3 ), then we have: Q I = N I c 2 ; (2.21) where c 2 is the quantized string unit of the M2-brane charge that we are looking for. Rewrite (2.13) with the numbers of M-branes that we have introduce above, we have: G 5 (N 1 +N 2 +N 3 )m 2 = 4 (N 1 +N 2 +N 3 )c 2 : (2.22) From the expression ofm 2 andG 5 in (2.20) and (2.18), we obtainc 2 equal: c 2 = l 6 p L 4 : (2.23) How about the quantized unit for M5-branes? Recall that M5-branes generate the effec- tive M2-brane charges through wedge product of two dipole field strengths. Thus, in fact, the quantized unit of M5-branes is just the square root ofc 2 : c 5 = l 3 p L 2 : (2.24) 31 2.2 A circular black ring with a black hole at the center In this subsection, we apply the technique that we have reviewed to solve a particular example which hasU(1)U(1) invariance. The example we are going to consider is a circular black ring with a black hole at the center. To simplify the task, we choose the base space with the simplest possible hyper-K¨ ahler metric: I R 4 . This I R 4 can be naturally split into two orthogonal planes and the circular ring sit on one of the planes. Therefore, this system preserve the two rotation symmetries of the two planes. The natural coordinate system to use is the following: dS 2 IR 4 = (dz 2 + z 2 d 2 1 ) + (dr 2 + r 2 d 2 2 ) : (2.25) The first step is to solve the dipole field strength. For which, we need to fix our ring profile. In this system, we put the circular ring with radiusR at (z; 1 ) plane and it appear as a point in (r; 2 ) plane. The dipole field strength can be obtained by using Green function in four dimensional space and integrating the source vector over the ring: ~ a I (~ x) = q 0I Z 2 0 (cos 1 ~ e 1 + sin 1 ~ e z )Rd 1 R 2 +z 2 2Rz cos 1 +r 2 ; (2.26) I = (1 +? 4 )d(a I 1 zd 1 ) ; (2.27) whereq 0I are the dipole charges associated with three M5-branes anda I 1 is the 1 com- ponent of~ a I . One can work out I from the above integral. However, to simplify the following calculation, one should choose a coordinate system to make the dipole field 32 x =1 x = +1 x 1 y y =1 x = const Figure 2.1: Coordinate system for black ring metric. The figure sketches a section at constantt and 2 . Surfaces of constanty are ring-shaped, whilex is a polar coordinate on theS 2 (roughlyx cos). Asymptotic infinity lies atx = y =1. (The credit of this graph belongs to [EEMR05]) strength as simple as possible. Probably, the best coordinate system [EEMR04] for this purpose is the following: x = z 2 +r 2 R 2 p ((zR) 2 +r 2 )((z +R) 2 +r 2 ) ; (2.28) y = z 2 +r 2 +R 2 p ((zR) 2 +r 2 )((z +R) 2 +r 2 ) ; (2.29) in which one has1 x 1,1 < y1. A very nice picture of this coordinate system was shown in [EEMR05]. Here, we borrow it and show it in fig. 2.1. The metric on I R 4 in this coordinate system becomes: ds 2 IR 4 = R 2 (xy) 2 dy 2 y 2 1 + (y 2 1)d 2 1 + dx 2 1x 2 + (1x 2 )d 2 2 : (2.30) 33 And, the ring is located aty =1, while spatial infinity is atx!1;y!1. The dipole field strengths take a very simple form in this coordinate system 2 : I = 2q I (dx^d 2 dy^d 1 ); (2.31) whereq I are related withq 0I by a constant factor. In the second step, we need to solve (2.8) for Z I . The homogeneous solution is determined by the distribution of M2-branes. In this system, we put one set of three M2-branes with chargesY I at the center to make a black hole there and then distribute M2-branes uniformly along the ring with constant charge densities Q I . The part that comes from the black hole,H I , is just like the potential produced by a single point-like source: H I = Y I z 2 +r 2 = Y I R 2 xy x +y : (2.32) On the other hand, the part comes from M2-branes on the ring is: Z M2 I = Q I R 2R 2 p ((z +R) 2 +r 2 )((zR) 2 +r 2 ) = Q I R (xy) : (2.33) Moreover, we need to determine the inhomogeneous solution. The inhomogeneous solu- tion comes from the distribution of the “effective” charges generated by the dipole field strength. The charge distribution can be calculated from the source term: 1 2 C IJK ? 4 ( J ^ K ) = 4C IJK q J q K ? 4 (dy^dx^d 1 ^d 2 ) = 4C IJK q J q K xy R 4 : (2.34) 2 We take the convention, yx12 = 1. 34 The potential generated by the above charge distribution is: Z M5 I = 2C IJK q J q K (x 2 y 2 ) R 2 : (2.35) Now, we can combine (2.32), (2.33) and (2.35) together and we obtain the completeZ I : Z I = 1 + Q I R (xy) 2C IJK q J q K (x 2 y 2 ) R 2 Y I R 2 xy x +y : (2.36) In the final step, we need to solve (2.9) for one-form k. First of all, because the system hasU(1)U(1) symmetry, the components ofk can only depend on ‘x’ and ‘y’ and thus only few of the components in (2.9) contain non-trivial relations and they are: @ y k x @ x k y = 0; (y 2 1)@ y k 2 + (1x 2 )@ x k 1 = 0; @ x k 2 @ y k 1 = A + B (xy) + C (x 2 y 2 ) + D xy x +y ; (2.37) where (k y ;k x ;k 1 ;k 2 ) are the components ofk in (y;x; 1 ; 2 ) respectively andA,B,C andD are some parameters defined with the charges: A 2 X q I ; B 2 R (Q I q I ); C 4C IJK q I q J q K R 2 ; D 2Y I q I (2.38) 35 The natural gauge to choose for this system is to fix both ofk x andk y to zero. In this gauge choice, we only havek 1 andk 2 . To solve the equation 2.37, a natural guess of a particular solution is the following: k 1 = a 1 y + B 2 y 2 + C 3 y 3 D R 2 y 2 1 x +y ; k 2 = a 2 x + B 2 x 2 + C 3 x 3 D R 2 x 2 1 x +y ; (2.39) wherea 1 anda 2 are two constants that satisfya 2 a 1 =A. Now, we need to discuss the subtle issues related to the homogeneous solutions. As we have mentioned, we need to choose the homogeneous solution such that the metric is free of any CTC’s. Becausek has negative contribution to the metric, the obvious place to check is wherek becomes large. Sincek is stronger near the ring or the black hole, we are going to check this two places. The spatial part of the metric on ( 1 ; 2 ) plane is: ds 2 1 ; 2 =V 2 R 2 y 2 1 (xy) 2 d 2 1 + 1x 2 (xy) 2 d 2 2 V 4 (k 1 d 1 + k 2 d 2 ) 2 ; (2.40) where V (Z 1 Z 2 Z 3 ) 1=6 : (2.41) Since the ring is at y!1, we expand the above metric by y and check this limit order by order. First of all, the near ring behavior of the total warp factor, V , is the following: V 2 C 2 9R 2 1=3 y 2 + B C y x 2 B C x B 2 C 2 A C D CR 2 + 12E C 2 R 2 +O(y 1 ); (2.42) 36 where we have introduced another parameter: E 2 (q 1 q 2 Q 1 Q 2 +q 2 q 3 Q 2 Q 3 +q 1 q 3 Q 1 Q 3 ) (2.43) By using the near ring behavior ofV and using the particular solution ofk in (2.39), one can expand the metric on ( 1 ; 2 ) plane and the coefficients of the orders up toO(y) are the following : O(y 2 ) : ( C 2 9R 2 ) 1=3 R 2 d 2 1 ( 9R 2 C 2 ) 2=3 C 2 9 d 2 1 = 0; (2.44) O(y) : ( C 2 9R 2 ) 1=3 (2x)R 2 d 2 1 : (2.45) We see theO(y 2 ) term are completely canceled out. Therefore, we should not add any homogeneous solutions that contributed to this order or higher. Because onlyO(y 3 ) of k contribute to this order, it means we should not add anyO(y 3 ) or higher order term in the homogeneous solution to avoid add any addition term in this order. On the other hand, in theO(y) term, we already see the problem. Becausex take the value from1 to +1, this part is a potential dangerous term that can produce CTC’s around 1 direction if we look closer to the ring enough. To cancel this term, we need to add C 3 xy 2 tok 1 . However, this term alone is not the homogeneous solution, we need to take the following homogenous solution: k H 1 = C 3 x (y 2 1) ; k H 2 = C 3 y (x 2 1) : (2.46) One can check this indeed is the homogeneous solution of (2.37). After adding this homogenous solution to (2.39), theO(y 2 ) andO(y) parts of the metric in (2.40) will completely vanish. However, there is still a dangerous point atx =1. From the metric in (2.40), one can see ifx =1, the positive part ofd 2 2 is vanished. Therefore, ifk 2 37 does not vanish atx =1, there will be CTC’s around 2 direction. In order to make k 2 vanish atx =1, we need to seta 2 = C 3 and add a constant B 2 . Consequently,a 1 must be set toA C 3 . The remain issue that need to address is the asymptotic behavior ofk. Because we want the asymptotical flat metric, we would likek vanish at infinity (x!1;y!1). In order to achieve that, we simply add the appropriate constants tok 1 andk 2 to cancel the values ofk 1 andk 2 at infinity. The constant ofk 2 has been taken care of by adding B 2 and the constant ofk 1 isA B 2 . After solving these subtle issues, we can combine the homogenous and particular solutions and we have the following complete solution: k 1 = (y 2 1) C 3 (x +y) + B 2 D R 2 (x +y) A(y + 1); k 2 = (x 2 1) C 3 (x +y) + B 2 D R 2 (x +y) : (2.47) After dealing with the black ring, we change our focus to the black hole. The black hole is at the origin and in this coordinate system is at (x! 1;y!1) or equivalently, (x +y)! 0. The leading terms ofV 2 ,k 1 andk 2 at this limit are: V 2 2 (Y 1 Y 2 Y 3 ) 1=3 R 2 (x +y) k 1 D (y 2 1) R 2 (x +y) k 2 D (x 2 1) R 2 (x +y) (2.48) The leading behavior of the metric on ( 1 ; 2 ) plane at this limit is: ds 2 1 ; 2 (Y 1 Y 2 Y 3 ) 1=3 (x +y) (y + 1)d 2 1 (1x)d 2 2 (2.49) The above metric is completely coming from the positive part and the negative part from k vanishes at this limit. Notice that the metric near the black hole is independent of the black ring’s charges while the near-ring metric (2.40, 2.42) depends on the black hole’s charge explicitly. This fact is important for the entropy discussion and will be addressed 38 more carefully later. Moreover, the fact that positive part of the metric dominate near the black hole give us the possibility to add an homogeneous solutions ofk which diverge stronger thanO((x +y) 1 ). The reasonable guess is a term containO((x +y) 2 ) and one can show that the following is indeed a homogeneous solution: k BH 1 = K (y 2 1) R 2 (x +y) 2 k BH 2 = K (x 2 1) R 2 (x +y) 2 ; (2.50) whereK is a constant parameter which will be shown later corresponds to the angular momentum of the black hole. And adding this homogeneous solution is equivalent to adding an angular momentum to the black hole at the center. If we add this homogeneous solution, the leading term of the metric near the black hole becomes: ds 2 1 ; 2 (Y 1 Y 2 Y 3 ) 1=3 (x +y) (y + 1)d 2 1 (1x)d 2 2 1 (Y 1 Y 2 Y 3 ) 2=3 (x +y) 2 (K (y + 1)d 1 + K (1x)d 2 ) 2 : (2.51) From the above metric, one can see there is potential dangerous CTC’s can appear ifK is too large. Particularly, if one approach the black hole from it’s equator 3 and check the metric along 1 direction, one find the following condition must be satisfied in order to avoid CTC’s appear around the black hole’s equator: (Y 1 Y 2 Y 3 ) 1=3 K 2 (Y 1 Y 2 Y 3 ) 2=3 ) (Y 1 Y 2 Y 3 )K 2 (2.52) This is the upper bound of the black hole’s angular momentum coming from the chronol- ogy protection. 3 This approach direction corresponds to take the limitx! 1 first and then takey!1. 39 2.2.1 Asymptotic, quantized charges and angular momenta After one have solvedk, one obtains the full metric for this system. Two most important properties associated with the metric we just solved are the asymptotic charges and the horizon of the black ring and black hole. At first, we look at the asymptotic charges of this system. As we have mentioned, the three asymptotic charges can be read from the asymptotic behavior ofZ I as the (2.11). To study the asymptotic behavior, we need to take (x!1;y!1). One can use the following leading asymptotic limit of (xy) and (x +y): xy = 2R 2 p ((zR) 2 +r 2 )((z +R) 2 +r 2 ) 2R 2 z 2 +r 2 = 2R 2 2 x +y 2: (2.53) By using the above asymptotic limit, the asymptotic behaviors ofZ I for this system can be easily derived: Z I 1 + 1 2 2RQ I + 8C IJK q J q K + Y I + : (2.54) Therefore, the three asymptotic charges are: Q I = 2RQ I + 8C IJK q J q K + Y I : (2.55) As we have mentioned, the asymptotic charges related with the quantized charges by the units defined in (2.23) and from that we can obtain the relation between the localized classical charges or charge densities and the number of M2/M5 branes: Q I = N I 2R l 6 p L 4 ; q I = n I 4 l 3 p L 2 ; Y I = N BH I l 6 p L 4 ; (2.56) 40 whereN I is the total number of M2-branes on the ring,n I is the number of M5-branes and N BH I is the number of M2-branes on the black hole. Therefore, the quantized asymptotic charges are: N I = N I + 1 2 C IJK n J n K + N BH I : (2.57) Now, we turn our focus on the angular momenta. Since the space naturally split into two planes, there will be two angular momenta associated to the rotation of (z; 1 ) plane and (r; 2 ) plane. As we have mentioned, the angular momenta should be read from asymptotic behavior of g 0i in (2.14). However, the equation is expressed in Cartesian coordinate, we can convert it to the polar coordinate: g 0z 0; g 0 1 8G 5 3 z 2 J 1 4 ; g 0r 0; g 0 2 8G 5 3 r 2 J 2 4 ; (2.58) where p z 2 +r 2 andJ 1 andJ 2 are the eigenvalues ofJ ji on the two plane respec- tively and thus are the two angular momenta we are looking for. Therefore, to read this system’s angular momenta, we need to check asymptotic behaviors ofk 1 andk 2 as the following: k 1 c J J 1 z 2 (z 2 +r 2 ) 2 + ; k 2 c J J 2 r 2 (z 2 +r 2 ) 2 + ; z; r!1; (2.59) where c J 8G 5 3 = l 9 p L 6 ; (2.60) 41 in which we have used (2.18). To check the leading asymptotic behavior ofk 1 andk 2 , we can take (x!1;y!1) and use the following leading asymptotic limit: y + 1 2R 2 z 2 (z 2 +r 2 ) 2 ; x + 1 2R 2 r 2 (z 2 +r 2 ) 2 ; (2.61) and the asymptotic behaviors ofk 1 andk 2 are: k 1 z 2 (z 2 +r 2 ) 2 8R 2 3 C + 2R 2 B + 2R 2 A + 2D + K ; (2.62) k 2 r 2 (z 2 +r 2 ) 2 8R 2 3 C + 2R 2 B + 2D + K ; (2.63) where we have included the homogeneous solution in (2.50) that corresponding to adding angular momentum to the black hole. From the definition of c J and the rela- tion between various classical charges and quantized charges in (2.56), we have the two angular momenta expressed with the quantized charges: J 1 = J 4 + 1 6 C IJK n I n J n K + 1 2 n I N I + n I N BH I + J BMPV ;(2.64) J 2 = 1 6 C IJK n I n J n K + 1 2 n I N I + n I N BH I + J BMPV ; (2.65) whereJ BMPV is the quantized version ofK and defined as: K J BMPV l 9 p L 6 ; (2.66) andJ 4 is the difference ofjJ 1 j andjJ 2 j and come from the term containsA in (2.62): J 4 = R 2 L 4 l 6 p 3 X I=1 n I : (2.67) 42 2.2.2 Horizon and Entropy The other important aspect of the black holes/rings is their horizons. The geometry and topology of their horizons can be derived from studying the metric near the holes/rings. For example, in this system, the black hole is at the limitx! 1;y!1, while the black ring is aty!1. Take the corresponding limit of the spatial part of the metric, one can see the topology of the horizon is S 3 for the black hole and S 2 S 1 for the black ring. At first, we check the near ring metric. We have mentioned that theO(y 2 ) andO(y) term of the metric is vanished when we check the CTC’s near the ring. Therefore, the structure of the horizon comes from the constant order term of the metric when we take y!1 limit: ds 2 3 =V 2 R 2 dx 2 (xy) 2 (1x 2 ) + y 2 1 (xy) 2 d 2 1 + 1x 2 (xy) 2 d 2 2 V 4 (k 1 d 1 + k 2 d 2 ) 2 C 2 9R 2 2=3 4E R 2 B 2 4 + AC 3 + CD 3R 2 d 2 1 + C 2 9R 2 1=3 R 2 d 2 + sin 2 (d 1 +d 2 ) 2 +O(y 1 ); (2.68) where we have performed a coordinate transformationx! cos. The first part of the metric is a circle along the black ring and the second part is clearly a two-sphere with the radius equal to (( C 2 9R 2 )) 1=6 R) . The topology of the black ring’s horizon is thereforeS 2 S 1 . The horizon area (volume) is simply the length of the circle times the area of the two-sphere. Therefore, the horizon area (volume) of the black ring in the five-dimensional space-time is: A = 8 2 R p M (2.69) 43 whereM is defined by: M 4E B 2 4 R 2 + AC 3 R 2 + CD 3 = (2q 1 q 2 Q 1 Q 2 + 2q 1 q 3 Q 1 Q 3 + 2q 2 q 3 Q 2 Q 3 (q 1 Q 1 ) 2 (q 2 Q 2 ) 2 (q 3 Q 3 ) 2 ) 8q 1 q 2 q 3 2 (q 1 +q 2 +q 3 ) + 2 (q 1 Y 1 +q 2 Y 2 +q 3 Y 3 ) R 2 : (2.70) The entropy proposed by Bekenstein is related with the horizon area as the following: S BR = A 4G 5 = AL 6 l 9 p = p M; (2.71) whereM is the quantized version ofM: M 2n 1 n 2 N 1 N 2 + 2n 1 n 3 N 1 N 3 + 2n 2 n 3 N 2 N 3 (n 1 N 1 ) 2 (n 2 N 2 ) 2 (n 3 N 3 ) 2 4n 1 n 2 n 3 J 4 + n I N BH I (2.72) whereJ 4 is defined in (2.67). On the other hand, the near black hole metric can be obtained by taking the limit: x! 1 andy!1. However, it is awkward to express the black hole horizon metric with (y;x; 1 2 ) coordinate system and thus we convert it to (;; 1 ; 2 ) in which 2 z 2 +r 2 and (cos) 2 z 2 =(z 2 +r 2 ) by using the following near black hole limit: x +y 2 (z 2 +r 2 ) R 2 = 2 2 R 2 ; y 2 1 4 z 2 R 2 = 4 2 R 2 cos 2 ; 1x 2 4 r 2 R 2 = 4 2 R 2 sin 2 ; (2.73) 44 and the black hole horizon metric is: ds 2 3 2 (Y 1 Y 2 Y 3 ) 1=3 R 2 (x +y) 2 d 2 + cos 2 d 2 1 + sin 2 d 2 2 4 (Y 1 Y 2 Y 3 ) 2=3 (x +y) 2 K 2 R 2 cos 2 d 1 K 2 R 2 sin 2 d 2 2 (Y 1 Y 2 Y 3 ) 1=3 d 2 + cos 2 d 2 1 + sin 2 d 2 2 K (Y 1 Y 2 Y 3 ) 2=3 cos 2 d 1 sin 2 d 2 2 (2.74) The horizon of the black hole is actually a three-sphere deformed by the black hole’s angular momentum. The volume of this three-sphere is corresponding to the horizon area and thus the entropy of the black hole: S BH = 2 q N BH 1 N BH 2 N BH 3 (J BMPV ) 2 : (2.75) Comparing the entropy of the black hole (2.75) to the entropy of the black ring (2.71), one will notice the following important point of this section. The entropy/horizon of the black hole is completely independent of any black ring charge. On the other hand, the entropy/horizon of the black ring indeed depends on the charges of the black hole through the product of the dipole charge and black hole’s electric charge: n I N BH I . The black hole’s horizon can be understood as the manifestation of the fact the the black ring and black hole are mutually BPS and therefore, the black hole is completely ignorant of the existence of the black ring. However, from this point of view, the black ring case will be quite puzzling from this point of view. In fact, the unexpected product term in the black ring entropy can be understood as the angular momenta “difference” coming from the fluxes (“Poynting’s vector”) that generated by the cross product of the black 45 hole’s electric filed and the black ring’s magnetic field. Consider we define a intrinsic black ring angular momentum as the following: J T J 4 + n I N BH I : (2.76) This definition is certainly reasonable because when we takeN BH I = 0,J T is just the difference of two angular momenta, which should be treated as the special property of the black ring 4 . Therefore, the puzzle is more understandable, the “total” angular momenta difference, J 4 , equal to the sum of the angular momenta difference coming from the black ring itself plus the fluxes:J 4 = J T +(n I N BH I ). Using this definition, the entropy depends only onJ T which is the intrinsic property of the black ring. The next natural question to ask is: how does this cross product term depends on the relation between the black ring and the black hole? Particularly, if we move the black hole, will this term change? This question will be the focus of the next section and the answer will be one of the important point of this chapter. 2.3 A vertically shifted black hole In this section, we generalize the previous example by moving the black hole away from the center with some arbitrary distance. The most general situation is that you put the black hole in some arbitrary position. This in general will break all of the symmetry and the metric will be very complicated. The next easier to do is moving the black hole away from the center on one of the plane but keeping it at the origin on the other plane. This corresponding to a vertically or horizontally shifted black hole (see fig.2.2) and it break the originalU(1)U(1) to only oneU(1). Because of this lost of the symmetry, the 4 Recalled that a black hole must have the values of two angular momenta equal:jJ 1 j =jJ 2 j. 46 (1) (2) Figure 2.2: A vertically and a horizontally shifted black hole. The pictures show a section at constantt and 2 and the plane is (z; 1 ) plane. (1) The black hole is shifted vertically from the center of the black ring. (2) The black hole is shifted horizontally from the center of the black ring. metric will become more complicated and difficult to solve even though in principle it should be solvable. In this section, we will focus on a vertically shifted black hole. The horizontally shifted case is interesting and actually not much more difficult technically than a vertically shifted one but the interpretation of the physics is harder due to the symmetry associated to the circle of the black ring is broken and also it is not the main focus of this thesis. At first, let’s set up the system in the same coordinate system. To keep the dipole field strength simple, we leave the black ring at the same place (y =1.) By doing that, we automatically solve the first step, because the black hole does not source the dipole field strengths, I , and it should be the same as the previous example (2.31) even if we move the black hole. However, the black hole has electric charges which source Z I harmonic functions and by moving the location of it, we change Z I (not much different from the situation when you move a electric charge, you shift the electric potential produced by it.) In this section, we consider the system in which the black hole 47 is located at (z; 1 ;r; 2 ) = (0; 0;a; 0). For the convenience to express the solution, we introduce the following parameter: a R ; (2.77) and in (y;x; 1 ; 2 ) coordinate, the Euclidean distance, d 2 , from a generic point to the position of the black hole is: d 2 = z 2 + (r 2 +a 2 2ar cos 2 ) = R 2 (xy) ; (2.78) where is defined as: (1 + 2 )y + (1 2 )x + 2 p 1x 2 cos 2 (2.79) The contribution of the black hole toZ I ,H 0 I , is simply shifted version of (2.32): H 0 I = Y I d 2 = Y I (xy) R 2 : (2.80) Having done this, we solve the second step because the position of the black ring is fixed, the part ofZ I sourced by it remain the same and the totalZ I is: Z I = 1 + Q I R (xy) 2C IJK q J q K R 2 (x 2 y 2 ) + H 0 I : (2.81) The only difficulty of solving this system is in solving the equation ofk. However, we can take advantage of the linearity of (2.9) and isolate the source from the black hole 48 which is the only thing changed. Then, we focus on the part ofk, ^ k, which is generated by this source. The equation that we need to solve for it is: d ^ k +? 4 d ^ k = 2q I H 0 I (dx^d 2 dy^d 1 ): (2.82) In this system, we will not have the rotation symmetry of the (r; 2 ) plane but still have the symmetry of 1 and therefore, the components ofk should only depend onx,y and 2 . Rewrite (2.82) in the components, we have: (y 2 1) (1x 2 ) (@ y ^ k x @ x ^ k y ) @ 2 ^ k 1 = 0; (2.83) (y 2 1)(@ y ^ k 2 @ 2 ^ k y ) + (1x 2 )(@ x ^ k 1 ) = 0; (2.84) @ x ^ k 2 @ 2 ^ k x @ y ^ k 1 = D (xy) R 2 ; (2.85) where@ 1 @ @ 1 ,@ 2 @ @ 2 andD is defined in (2.38). To solve these equations, we need to fix the gauge freedom ofk first. The natural gauge choice is ^ k y = 0. In this gauge choice, we can take the derivative of (2.85) withy and use the first two equations to get the following ODE contains only ^ k 1 : (y 2 1)@ 2 y ^ k 1 + @ x (1x 2 )@ x ^ k 1 + 1 1x 2 @ 2 2 ^ k 1 = D R 2 @ y xy : (2.86) To obtain the particular solution of this equation require some guess work. Moreover, for any particular solution, one need to choose the corresponding appropriate homogeneous solution such that the full metric satisfy the conditions discussed in the section 2.1. The 49 checking of no-CTC’s condition proceed just like the section 2.2. The only constraint come from this condition are: k 1 D R 2 (1 + 2 ) y +O(y 0 ); y!1 k 2 ! 0; x!1 (2.87) On the other hand, the asymptotic condition require every components ofk should van- ish when we take the asymptotic limit: y!1. To find the correction combination of the homogenous and some particular solution that satisfy all these conditions require some trial-error approaches and the solution we got is: ^ k 1 = D 2R 2 (1 + 2 ) (y + 1) 1 1 (1 + 2 ) (xy) + 2 : (2.88) The other components can be obtained by integrating (2.83) and (2.84), but the result is very complicated. Instead of writing this result completely, one can choose the better gauge to simplify the solution first. Probably, the best gauge choice is the harmonic gauge:? 4 d? 4 k = 0. In this gauge choice, every components ofk is non-zero but much simpler: ^ k y = D R 2 1 + 2 p 1x 2 sin 2 (1y) (2.89) ^ k x = D R 2 1 + 2 p 1x 2 sin 2 (1x) (2.90) ^ k 1 = D 2R 2 (y + 1) 1 xy + 2 1 + 2 1 1 + 2 (2.91) ^ k 2 = D 2R 2 (x + 1) 1 xy 2 1 + 2 + 1 1 + 2 (2.92) First of all, we can clearly see that every components vanish at infinity (x!1;y! 1) and thus the asymptotic conditions are satisfied. The regularity is also manifested. 50 The only singularities are at the position of the black ring (y =1) and the black hole (^ = 0). Moreover, the no-CTC’s conditions in (2.87) are also satisfied. We will check their asymptotic and near ring/hole behaviors more carefully in the following subsection. Before we summarize the complete solution ofk, recall that one still have a freedom to add the homogeneous solution which is singular in the position of the black hole and that corresponds to adding an angular momentum to the black hole. In this system, we also have this freedom to add such homogeneous solution like (2.50). The physical form of this solution should be the same as (2.50), however because the position of the black hole is shifted in our coordinate system, we need to do the appropriate coordinate transformation onk BH in (2.50) to reflect this shifting. The result is: ^ k BH x = K(xy 1) sin 2 R 2 p 1x 2 2 ; ^ k BH y = K p 1x 2 sin 2 R 2 2 ; ^ k BMPV 1 = K (y 2 1) R 2 2 ; ^ k BMPV 2 = K x 2 1 +(xy) p 1x 2 cos 2 R 2 2 : (2.93) 51 The complete solution of one formk is obtained by combing the unchanged solution (the terms which containA,B andC), the changed solution, ^ k, that we just got in (2.89- 2.92) and finally, the homogeneous solution that related with the angular momentum of the black hole in (2.93): k y = D R 2 1 + 2 p 1x 2 sin (1y) ^ +k BMPV y ; (2.94) k x = D R 2 1 + 2 p 1x 2 sin (1x) ^ +k BMPV x ; (2.95) k = (y 2 1) C 3 (x +y) + B 2 A(y + 1) + D 2R 2 (1 +y) 1 ^ xy + 2 1 + 2 1 1 + 2 +k BMPV ; (2.96) k = (x 2 1) C 3 (x +y) + B 2 D 2R 2 (1 +x) 1 ^ xy 2 1 + 2 + 1 1 + 2 +k BMPV : (2.97) 2.3.1 Asymptotic, quantized charges and angular momenta The three asymptotic charges of this system can be obtained in the similar way as the central black hole case in the section 2.2.1. Actually, the charges are exactly the same with (2.55). It is not surprising because the total asymptotic charges will not “care” about the exact positions of the charges and therefore will not be effected if we move some of the charges. On the other hand, the angular momenta,J 1 andJ 2 , can be read from the asymptotic behaviors of k 1 and k 2 respectively as in (2.59). Using the similar procedure as the 52 section 2.2.1 we have the following leading asymptotic behavior of k 1 and k 2 for this system: k 1 z 2 (z 2 +r 2 ) 2 8R 2 3 C + 2R 2 B + 2R 2 A + 2D 1 + 2 + K ;(2.98) k 2 r 2 (z 2 +r 2 ) 2 8R 2 3 C + 2R 2 B + 2D 1 + 2 + K ; (2.99) again, from the definition ofc J and (2.56), we get the two angular momenta: J 1 = J 4 + 1 6 C IJK n I n J n K + 1 2 n I N I + n I N BH I 1 + 2 + J BMPV ; (2.100) J 2 = 1 6 C IJK n I n J n K + 1 2 n I N I + n I N BH I 1 + 2 + J BMPV : (2.101) Comparing the two angular momenta of this system in (2.100) and (2.101) with the two angular momenta of the central black hole system in (2.64) and (2.65), we find out the only difference comes from the term contains n I N BH I . Recall that we mentioned in the section 2.2.2 that this term is actually the momentum difference coming from the fluxes generated by the cross product of the electric field of the black hole and the magnetic field of the black ring. Since this term is different for the central black hole and vertical shifted black hole system, it shows that the momentum difference coming from the fluxes will change with the position change of the black hole. Consequently, the intrinsic angular momenta difference of the black ring defined in (2.76) should be modified accordingly: J T J 4 + n I N BH I 1 + 2 : (2.102) SinceJ T is the intrinsic property of the black ring, it should not depend on the position of the black hole. Therefore, one can notice thatJ 1 is actually independent of the position of the black hole. This statement actually fit our intuition quite nicely. Because theJ 1 is the angular momentum associate to the plane in which there is still aU(1) symmetry 53 and thus the angular momentum associated to this plane should be unchanged. We will discuss more detail in the section 2.4. 2.3.2 Horizon and entropy In this subsection, we look at the horizon and entropy of the black ring and black hole by the similar procedure introduced in the section 2.2.2. First, we look at the black ring’s horizon. First of all, the near ring behavior (y!1) ofV 2 andk are: V 2 C 2 9R 2 1=3 y 2 + B C y x 2 B C x B 2 C 2 A C D CR 2 (1 + 2 ) + 12E C 2 R 2 +O(y 1 ); (2.103) and, k y =O(y 1 ); k x =O(y 1 ); k = C 3 y 3 + C 3 x + B 2 y 2 A + C 3 + D (1 + 2 )R 2 y +O(y 0 ) k = C 3 (x 2 1)y +O(y 0 ): (2.104) Therefore, the near ring metric is: ds 2 3 =V 2 R 2 dx 2 (xy) 2 (1x 2 ) + y 2 1 (xy) 2 d 2 1 + 1x 2 (xy) 2 d 2 2 V 4 (k y dy + k x dx + k 1 d 1 + k 2 d 2 ) 2 C 2 9R 2 2=3 4E R 2 B 2 4 + AC 3 + CD 3R 2 (1 + 2 ) d 2 1 + C 2 9R 2 1=3 R 2 d 2 + sin 2 (d 1 +d 2 ) 2 +O(y 1 ); (2.105) 54 where k y and k x are the sub-leading terms near the ring and we take the coordinate transformation:x! cos. Therefore, the horizon area of the black ring is: A 0 = 8 2 R p M 0 ; (2.106) whereM 0 is defined by: M 0 4E B 2 4 R 2 + 1 3 ACR 2 + CD 3 (1 + 2 ) = (2q 1 q 2 Q 1 Q 2 + 2q 1 q 3 Q 1 Q 3 + 2q 2 q 3 Q 2 Q 3 (q 1 Q 1 ) 2 (q 2 Q 2 ) 2 (q 3 Q 3 ) 2 ) 8q 1 q 2 q 3 2 (q 1 +q 2 +q 3 ) + 2 (q 1 Y 1 +q 2 Y 2 +q 3 Y 3 ) (1 + 2 )R 2 : (2.107) The entropy is: S BR = A 0 4G 5 = p M 0 ; (2.108) whereM 0 is: M 0 = 2n 1 n 2 N 1 N 2 + 2n 1 n 3 N 1 N 3 + 2n 2 n 3 N 2 N 3 (n 1 N 1 ) 2 (n 2 N 2 ) 2 (n 3 N 3 ) 2 4n 1 n 2 n 3 J 4 + n I N BH I (1 + 2 ) : = 2n 1 n 2 N 1 N 2 + 2n 1 n 3 N 1 N 3 + 2n 2 n 3 N 2 N 3 (n 1 N 1 ) 2 (n 2 N 2 ) 2 (n 3 N 3 ) 2 4n 1 n 2 n 3 J T : (2.109) where J 4 is defined in (2.67) and in the second equality, we use the definition of J T defined in (2.102). In this form of the entropy, it becomes clearer that the entropy of the black ring indeed only depends on the intrinsic property of the black ring: J T and independent of the position of the black hole. 55 On the other hand, as we have mentioned, the black hole’s horizon is evidentally independent of the existence of the black ring. Therefore, the change of the relative distance of the black ring and black hole will not effect the horizon of the black hole in any way. Thus, beside of the position is shifted, the horizon and entropy of the black hole are exactly the same as (2.74) and (2.75). 2.4 The merger and the entropy 2.4.1 The merging process In this section, we use the results in previous sections to understand the merger of the black ring and the black hole. At first, imagine a process that we move a black hole from infinity (z = 0;r =1) along the axial direction to the center of the black ring (z = 0;r = 0.) If we start from a BPS solution and move the black hole adiabatically, each point in this continuous process should be also a BPS solution. Because there is no energy transfer to the system in this process and therefore the energy remain saturate the BPS bound. This implies that we can use the BPS solutions of the vertical shifted black hole system to describe this process if we change continuously from1 to 0. However, because an adiabatic process will not change the entropy of a system, the total horizon area should remain fixed during the merger process and therefore independent of. At first, the horizon of the black hole is not affected by the existence and the location of the black ring. Moreover, if we express the entropy of the black ring withJ T which is the intrinsicjJ 1 jjJ 2 j of a black ring defined in (2.102), the entropy of the black ring will also be fixed by it’s own intrinsic property and is independent of the position of the black hole. Therefore, the total horizon area is fixed during the merger process. 56 However, ifJ T is the intrinsic property of the black ring and is independent of,J 4 as the totaljJ 1 jjJ 2 j must change with: J 4 = J T n I N BH I (1 + 2 ) : (2.110) BecauseJ 4 defined in (2.67) is related with the embedded radius of the black ring, one have the following conclusion: R 2 ()L 4 l 6 p ( X n I ) = J T R 2 1 L 4 l 6 p ( X n I ) n I N BH I (1 + 2 ) ; (2.111) whereR 1 is a constant and can be understood as the intrinsic embedded radius of the black ring which can be considered as the embedded radius of the ring when the black hole and ring are widely separated. From the above equation, it is clearly that the exact embedded radius of the ring will change if one move the black hole. The behavior of R() is: R 2 () = R 2 1 l 6 p L 4 n I N BH I ( P n I )(1 + 2 ) : (2.112) Depending on the relative strengths ofJ T andn I N BH I , there are three possible out- comes when we move the black hole from infinity to the center of the ring: 1. J T R 2 1 L 4 l 6 p ( P n I ) > n I N BH I =) R(0) = R 0 , 2. J T R 2 1 L 4 l 6 p ( P n I ) = n I N BH I =) R(0) = 0 , 3. J T R 2 1 L 4 l 6 p ( P n I ) < n I N BH I =) R( 0 ) = 0 , in whichR 0 and 0 can be easily calculated from (2.112) for some particular charges. Also, one should remember thatR is the embedded radius not physical radius. There- fore, R = 0 in the second and third situation actually mean the ring touch the black 57 hole’s horizon which has non-zero physical size in general. We show these three out- comes at the figure 2.3. (1) (2) (3) R 0 δ Figure 2.3: Three possible outcomes when we move the black hole from infinity to the center of the ring. In the first graph, the black hole is small and the radius of the ring decrease from R 1 to R 0 . In the second graph, the size of the black hole is just big enough to catch the ring and their horizons touch at equator. In the third graph, the black hole is very big and the ring enter horizon at some particular angle cot = 0 Notice that in the third situation,R vanishes at some particular non-zero 0 . How- ever, we know that the vertical distance from the black hole to the center of the ring is a =R and whenR vanishes,a vanishes such that the black ring enter the horizon of the black hole at particular angle cot = 0 . From figure. 2.3, we see the merger of the black ring and black hole happen in second and third situation. So, we get the following merger condition: J T n I N BH I : (2.113) If a black ring and a black hole satisfies this condition, it will be possible to merge them through an adiabatic process. However, remember that the two angular momenta also depend on: J 1 = J T + 1 6 C IJK n I n J n K + 1 2 n I N I + J BMPV ; (2.114) J 2 = 1 6 C IJK n I n J n K + 1 2 n I N I + n I N BH I 1 + 2 + J BMPV ; (2.115) 58 in which we rewriteJ 1 by replacingJ 4 + n I N BH I 1+ 2 withJ T . In this form, we clearly seeJ 1 which is associate with the circle of the ring is remained constant butJ 2 is changed with the position of the black hole. This means in order to merge a black ring with a black hole, we need to apply torque in 2 direction somehow in order to changeJ 2 accordingly. This solve the conundrum we have mentioned in the beginning of this chapter. In order to drop a small black ring into a big black hole through adiabatic process, one needs to apply torque to “squeeze” the ring into the black hole. When they merged, the radius of the ringR! 0 and thusJ 4 jJ 1 jjJ 2 j = 0. This means the result of the merger has two angular momenta equal and it is exactly just a little bit bigger black hole. The angular momentum of the resulting black hole is: J BMPV final = J 1 = J T + 1 6 C IJK n I n J n K + 1 2 n I N I + J BMPV : (2.116) 2.4.2 The entropy of mergers We have mentioned the result of the merger of a black hole and a black ring is a big- ger black hole. But, how much bigger is it? We can put this question in a more for- mal way: Is the entropy (horizon area) of the black hole resulting from the merger larger, equal or smaller than total sum of the entropies of the ring and hole that merges? Another related question is if we merge a maximally spinning black hole (J = J max = p N BH 1 N BH 2 N BH 3 ) with a supertube (M = 0;J T = (J T ) max ), can we produce an over-spinning black hole such that chronology protection is no longer true and the CTC’s or singularities appear? To understand these questions, we need to study the entropy of the resulting black hole after merger. To simplify our task, we take all three sets of charges equal: n I = n; N I = N; N BH I = N; I = 1; 2; 3: (2.117) 59 Because the quantized asymptotic charges can not be changed during the merger and a black hole’s electric charges are equal to it’s asymptotic charges, the resulting black hole’s electric charges should be simply equal to the total asymptotic charges prior to the merger : N final = N + N + n 2 : (2.118) Take the angular momentum of the black hole before the merger asJ BMPV initial =J. Then, the angular momenta of the black hole after the merger is: J final = J T + n 3 + 3 2 nN + J: (2.119) Since the result of the merger is a black hole, it’s entropy is: S final = 2 q N 3 final J 2 final ; (2.120) therefore the entropy change is: S S final S initial = 2 q N 3 final J 2 final p M BR + 2 p M BH ; (2.121) whereM BR andM BH are: M BR = 3n 2 N 2 4n 3 J T ; M BH = N 3 J 2 : (2.122) In order to understand whether S is larger, equal or smaller than zero, it is equiv- alent to study the positiveness of the following function: G(N;n;N;J T ;J) = (N 3 final J 2 final ) 1 2 p M BR + p M BH 2 : (2.123) 60 Therefore, our task become studying whether the functionG(N;n;N;J T ;J) is positive, zero or negative under the following constraints: Non-negative black hole horizon area before merger =) J p N 3 Non-negative black ring horizon area before merger =) J T 3 4 (N 2 )=n The merger actually happen =) J T 3nN BecauseN;n;N are the number of branes and should be some arbitrary positive values, we fixN;n;N and varyJ andJ T under the three constraints to find the minimum ofG. At first, the first and second derivative ofG with respect toJ is: @G @J = J r M BR M BH 2(J T +n 3 + 3 2 nN); (2.124) @ 2 G @J 2 = N 3 M BH r M BR M BH : (2.125) We can see the second derivative is always positive and this means there are no maxi- mums but only minimums. Moreover, from the first derivative, we find the minimum is at: J min = N 3=2 1 q 1 +M BR = 2(J T +n 3 + 3 2 nN) 2 : (2.126) Clearly, this value is inside of the range of the first constraint and saturated only when M BR = 0. That is exactly when the ring is a supertube. Now if we plug thisJ min into G, we have the following function: G(N;n;N;J T ) =N 3 + 3 (n 4 N +N 2 N +NN 2 +n 2 N 2 + 2n 2 NN) J T (J T +n 3 + 3nN)N 3=2 r M BR + (2(J T +n 3 + 3 2 nN)) 2 ; (2.127) 61 remember thatM BR containsJ T . Take the derivative ofG with respect toJ T , we get: @G @J T = (2J T +n 3 + 3nN) 0 @ 1 + 2N 3=2 q M BR + (2(J T +n 3 + 3 2 nN)) 2 1 A : (2.128) Clearly, this is always negative. This means the minimum ofG is at the largest possible J T . However, we have two constrains onJ T , one is from requiring the black ring has positive horizon area and the other is from the merger actually happen. Which constraint is stronger depends on the relative values of N and N. If N 2n p N, the second constraint is equal or stronger than third one and we have the maximumJ T = 3N 2 4n . In this situation, by completing the square, one can show the minimum ofG is: G min1 = 1 16n 2 (2n p NN) 24 (n 3 +nN) p N (n p N) 2 +P 1 (n;N;N) ; (2.129) whereP 1 (n;N;N) is a strictly positive value and defined as: P 1 (n;N;N) 9N 3 + 18n p NN 2 + 20n 2 N 2 + 12n 4 N + 32n 4 N + 36n 2 NN + 16n 3 p NN: (2.130) From (2.129), we can see S is strictly positive ifN < 2n p N and only equal to zero whenN = 2n p N. The other possible situation is when N 2n p N and thus the third constraint is equal or stronger than the second one. The maximumJ T in this case is 3nN and the minimum ofG is: G min2 = P 2 q P 2 2 (N 2 4n 2 N) N 2 + 3NN + 3N 2 n 2 N 2 ; (2.131) 62 whereP 2 is: P 2 (n;N;N) (N 2 4n 2 N)(N + 3 2 N) + NN n 2 + 3N + 3 2 N : (2.132) BecauseN 2n p N,P 2 is a strictly positive value. Moreover, from (2.131), we can see S is strictly positive ifN > 2n p N and only equal to zero whenN = 2n p N. From the above analysis, we have seenG is non-negative and equal to zero only when the second and third constraints are saturated byJ T at the same time. Moreover, remember thatG isG evaluated atJ =J min and when the second constraint is saturated, J min = N 3=2 which saturate the first constraint. The conclusion is the entropy change after the merger is strictly positive and only equal to zero when all three constraints are saturated at the same time under the assumption that all three sets of charges are equal. What if the three sets of charges are different? The general problem is quite com- plicated and we expect the general behavior of S is similar and thus non-negative. However, we can look at the special situation when the three constraints are saturated. Consider a generic system defined by the following three charge “vectors” and each one has three components: (n 2 n 3 ;n 1 n 3 ;n 1 n 2 ); (N 1 ;N 2 ;N 3 ); (N BH 1 ;N BH 2 ;N BH 3 ): (2.133) Imposing the first and second constraints are saturated means requiring both the black ring and black hole is maximally spinning and thusJ andJ T are completely determined by the three charge vectors. Imposing the third constraint is saturated means the black hole is just big enough to catch the black ring or equivalently the black ring enter the black hole’s horizon from the equator and this condition put a constraint on the three charge vectors. The entropy change can be completely expressed with the three charge 63 vectors and in general non-zero. However, the entropy change become zero when the three charge vectors are parallel: (n 2 n 3 ;n 1 n 3 ;n 1 n 2 ) = p (N 1 ;N 2 ;N 3 ) = p BH (N BH 1 ;N BH 2 ;N BH 3 ); (2.134) wherep andp BH are some positive values. If we take the three charge vectors slightly away from this point, we find that the entropy change increase. For example, if we fix n I to some positive integers and start with all three charge vectors parallel and slightly change one ofN I orN BH I , we find out the first derivative vanishes but the second deriva- tive is strictly positive. This means if we consider S as a function in the charge vectors space, the points defined by (2.134) are actually local minimum which is exactly zero. Conversely, this means even if all three constraints are saturated, the merger is still ther- modynamically irreversible as long as the three charges vector are not parallel. This observation will be important for the discovery of the microstates of the black hole with non-zero horizon area. We will discuss this in the chapter 4. 2.5 conclusion In this chapter, we have introduced the tool to investigate the supergravity BPS solu- tions that correspond to a five-dimensional three-charges black hole, black ring or any arbitrary superposition of them. We also studied a particular system consisting of a black ring and a vertically shifted black hole. Using the solutions, we studied a adiabatic process which bring a black hole from infinity to the center of the black ring and the mergers. From the solutions, we realize the angular momenta coming from the fluxes change if we change the relative position of the black ring and black hole. Therefore, in order to dump a small black ring and a large black hole, we need to apply torque to change one of the angular momenta 64 and the result is indeed a larger black hole with two angular momenta equal. This solve the puzzle why one can not produce a black hole with two unequal momenta by dumping a small black ring in it. Moreover, we look at the entropy change after the black ring merge with the black hole. We found out in general this process is irreversible with one exception when all of the following conditions are satisfied: The black hole before the merger has zero horizon area The black ring has zero horizon area The black ring enter the black hole from the equator Three charge vectors ( 1 2 C IJK n J n K ,N I andN BH I ) are parallel 65 Chapter 3 Bubbled Geometries In chapter 2, we have found several supergravity solutions that corresponds to three- charge BPS systems in five-dimensional space-time by solving three BPS equations (2.7-2.9) in I R 4 base space. Particularly, we have a family of solutions called black rings which are specified by choosing an arbitrary closed curves. These solutions are possible candidates for the microstate geometries. However, as we have discussed in the introduction, we expect the microstate geometries are asymptotic flat, horizonless and regular everywhere. The asymptotic condition can be imposed on the solutions when we solve the equations. The horizonless condition can be achieved by maximizing the angular momentum of a black ring, J T , or a black hole, J BMPV . By doing that, one reduce their horizon area to zero and therefore, become horizonless. However, by rendering them horizonless this way, we expose some “mild” singularities at the position of the source. For example, by maximizingJ T of a black ring, the metric along the circle vanish on the ring and therefore, a null orbiford singularity appear. The black rings with zero horizon area are sometime referred to as supertubes. In [BW06, BGL06], it was proposed that one can resolve this singularity by a geom- etry transition that results in geometries without any branes or singularities but with non-trivial two-cycles. Due to the existence of large number of non-trivial two-cycles, these smooth geometries is called bubbled geometries. In order to describe bubbled geometries, one needs to replace the I R 4 base space with some more general hyper- K¨ ahler space. However, it was shown the only hyper-K¨ ahler metric which is regular, asymptotic to I R 4 and with the ‘+4’ signature is the globally flat metric on I R 4 . In 66 order to go around this, an ambipolar generalization of the hyper-K¨ ahler was proposed. This generalization allow the signature of the metric to flip from ‘+4’ to ‘4’. Particu- larly, in [BW06, BGL06], they introduced the ambipolar generalization of a multi-center Gibbons-Hawking space which is a hyper-K¨ aler space that preserve a triholomorphic U(1) isometry. They also used the nucleation processes (Bubbling) in an ambipolar gen- eralization of the multi-center Gibbons-Hawking spaces to generate the geometric tran- sitions that needed to resolve the singularities. There are very large number of smooth Lorentzian geometries that can be obtained from solving the three BPS equations (2.7- 2.9) on ambipolar Gibbons-Hawking bases. By checking carefully they satisfy all the conditions that needed for the microstate geometries, one obtain the all ingredients to build microstates geometries. Therefore, studying bubbled microstate geometries will be the main focus of this thesis. In the first section, we review the multi-center Gibbons-Hawking metrics and the ambipolar generalization of it. In the second section, we review the solutions of BPS equations on some Gibbons-Hawking space. 3.1 Gibbons-Hawking metrics Gibbons-Hawking metric is a particular sub-class of hyper-K¨ aler metric which has a tri- holomorphicU(1) isometry which means that it preserves all three complex structures. In four-dimensional space, thisU(1) isometry is realized as aU(1) fiberation over three dimensional flat space. The metric has the following form: ds 2 4 = 1 V (d +A) 2 + V (ds 2 IR 3); (3.1) 67 whereV is a harmonic function in I R 3 andA is a one form in I R 3 which is related with V by: ? 3 dA = dV =) ~ r ~ A = ~ rV ; (3.2) where? 3 is the Hodge dual in I R 3 andA ~ A:d~ y withy i ;i = 1; 2; 3, being the Carte- sian coordinates on I R 3 . To pick a metric from this family is equivalent to choose the harmonic functionV . Particularly, one can letV has a finite set of isolated sources with some arbitrary distribution on I R 3 . If there are N isolated sources located at~ y (i) , one hasV as the following: V (~ y) = " 0 + N X i=1 q i j~ y~ y (i) j ; (3.3) where q i are the charges associated with each sources and " 0 is some constant which determines the topology of the metric at infinity. 3.1.1 Asymptotic and local structure If we look at the metric in (3.1), there seems a local singularity at each source. However, the singularity can be just a coordinate one. For example, if we look at the metric near a particular source ‘j’, it has the following form: ds 2 4 q j d 2 + 2 4 ( d q j + cosd) 2 + (d 2 + sin 2 d 2 ) ; (3.4) where 2 p j~ y~ y (j) j, and are the standard spherical coordinates in I R 3 . Notice that, if we fix the periodicity of U(1) fiber, , to 4 for q j = 1, the angular part of the metric forq j = 1 is exactly aS 3 in I R 4 with (;; ) being the three Euler angles. Moreover, in order to avoid singularity appears at this local source,q j must be an integer. In general,q j can be any integer, the angular part therefore become aS 3 =jZ jq j j j. There 68 will be an orbifold singularity at the source ifq j 6= 1. That is because the periodicity of =q i is a fraction of 4 when one approaches the source which has q j 6= 1. How- ever, in string theory, if we let strings propagate in this space, they tend to resolve this singularity and smooth it out. So, usually, in string theory, we still consider this kind of metrics as smooth metrics. Finally, if we would also like to preserve the signature (+1; +1; +1; +1) for space globally, we need to require all of the charges are positive. The asymptotic structure of this metric is determined by " 0 and the total charges q 0 P q j . If" 0 6= 0,V !" 0 at infinity and the metric becomes flatR 3 S 1 because the size of the fiber remain small at infinity. This type of space we call it asymptotically locally flat (ALF) space. On the other hand, if " 0 = 0, we have V ! q 0 =r and the metric has the exactly same form as (3.4) with replaced by 2 p r andq j replaced byq 0 . Therefore, ifq 0 = 1, the space is asymptotic to I R 4 . In general, the metric is asymptotic to I R 4 =Z jq 0 j . This type of space is called asymptotically locally Euclidean (ALE) space. 3.1.2 Topology The multiple-centers Gibbons-Hawking space has some interesting topology. From the metric, one can see the fiber shrinks to zero size at any Gibbons-Hawking point with non-zero charge. Therefore, any line that connects one GH point to another GH point is a two-sphere topologically. That is because the fiber shrink at both end points of the line but remain the finite size at the middle of the line (see fig. 3.1). So, in the Gibbons-Hawking space withN Gibbons-Hawking (GH) points, there are 1 2 N (N 1) topologically non-trivialS 2 that connected any two points. 3.1.3 Ambipolar generalization and nucleation Recall that our main goal is the solutions which are smooth everywhere and asymptotic to I R 4 . Therefore, the base space must also satisfy both of the above conditions. The 69 Figure 3.1: Topologically non-trivial S 2 . Any line connected two Gibbons-Hawking points has the fiber shrink at the both end-points and therefore, has two-sphere topology. Gibbons-Hawking spaces that satisfy the above asymptotic condition are the special ALE spaces that has total charge equal to 1. On the other hand, in order for the space is regular everywhere (up to orbiford singularities), every Gibbons-Hawking charges in the space must be integers. If we want to impose the signature of space is (+1; +1; +1; +1) globally, we need to further require all charges are positive. However, the only possible configuration that satisfy all of the above conditions is actually the space with only one single point which has the charge equal to 1. The metric of this space take the form in (3.4) withq j = 1 and that is actually globally I R 4 . In [BW06, BGL06], it was proposed to relax this global positive signature condi- tion in order to have a more generalized space. In other words, the negative charges should be allowed. The resulting metrics are called ambipolar Gibbons-Hawking met- rics in [BW07]. However, allowing the existence of the negative charges has an immedi- ately dangerous consequence: the closed-time-like curves (CTC’s). In a pure Gibbons- Hawking space, this problem can not be solved. However, recall we only use Gibbons- Hawking space as a base and therefore, the full metric has warping factor in front of the base space metric (2.2). If the warp factor change sign at the same position where the Gibbons-Hawking potential,V , flip the sign, then the positive signature is still globally preserved. We will deal with this CTC problem more carefully in section 3.3. 70 By this ambipolar generalization, one can consider a Gibbons-Hawking space con- tains multiple points with both positive and negative charges such that they sum to 1. Moreover, one can use nucleation of the Gibbons-Hawking points to generate the geo- metric transition that is needed to resolve the null orbiford singularity of the supertubes. First of all, we explain the geometry transition. The null orbiford singularity is caused by the circle (S 1 ) of the M5-branes shrink to zero size. A geometric transition to resolve this singularity is a topological transition: The Gaussian surface, S 2 that enclosed the branes become topologically non-trivial and anotherS 2 appear due toS 1 shrink to zero size at the origin and the position of the ring. Then the branes and singularity disappear from the space and “dissolve” into the fluxes that go through theseS 2 . Therefore, after the geometry transition, there is no brane or singularity but only the non-trivialS 2 and the fluxes through it. In the ambipolar Gibbons-Hawking space, the similar geometry transition can be generated by a nucleation process. Consider a localized brane source which has no Gibbons-Hawking point. This source will be singular in space. The nucle- ation process made the source pair creating to two Gibbons-Hawking points with +Q andQ. The singularity will disappear and the branes dissolve into the fluxes through the non-trivialS 2 that connected these two Gibbons-Hawking points. More generally, one can nucleate a source to multiple Gibbons-Hawking points with total charges equal to 1. In this case, the branes will dissolve into multiple fluxes that go through the multi- ple no-trivialS 2 . This process indeed look like bubbling soap water. 71 3.2 BPS solutions on ambipolar Gibbons-Hawking bases In this section, we review the construction of BPS solutions on an ambipolar Gibbons- Hawking space which is basically parallel to the previous chapter but replacing I R 4 with an amipolar Gibbons-Hawking space. 3.2.1 Solving BPS equations Consider our base space is a ambipolar Gibbons-Hawking space with N Gibbons- Hawking points located at~ y (i) : ds 2 4 = 1 V (d +A) 2 + V ((dy 1 ) 2 + (dy 2 ) 2 + (dy 3 ) 2 ); (3.5) whereV andA is: V = N X i=1 q i r i ~ r ~ A = ~ rV ; (3.6) in whichr i j~ y~ y (i) j,q i 2 Z and P q i = 1. Recall the first step to solve three BPS equations (2.7-2.9) is to find the self-dual dipole field strengths I . In order to build I , it is natural to introduce the following set of frames: ^ e 0 = V 1 2 (d +A); ^ e a = V 1 2 dy a ; a = 1; 2; 3: (3.7) Under this set of frames, the natural bases for self-dual and anti-self-dual two forms are: (a) = ^ e 0 ^ ^ e a 1 2 abc ^ e b ^ ^ e c ; a = 1; 2; 3: (3.8) 72 (a) + are self-dual while (a) are anti-self-dual bases. Now, we can construct I by the self-dual bases as the following: I = 3 X a=1 (@ a ( K I V )) (a) + ; (3.9) where K I are some functions. Recall that the dipole field strength must be closed. However, I built from the above construction are closed if and only ifK I are harmonic in I R 3 . Therefore, we have three harmonic functions associated with three dipole field strengths and basically, these three harmonic functions are source by the dipole charges from M5-branes. The sources of these three harmonic functions in principle can be distributed arbitrarily. But, if the sources ofK I do not overlap with Gibbons-Hawking points, I will diverge at the isolated source ofK I at the location of isolated sources of K I . Therefore, in order to preserve the smoothness, one has to require that all sources ofK I are exactly at the location of Gibbons-Hawking points. Before we move on to the next step, notice that there is a “gauge transformation” ofK I which preserve I : K I ! K I + c I V ; (3.10) wherec I is some constant. In the second step, we need to solveZ I . By plugging I into (2.8), one can check the equation has the following solution [GG05, GGH + 03, BW06, BGL06]: Z I = 1 2 C IJK K J K K V + L I ; (3.11) whereL I are the homogenous solutions of (2.8) and thus they are the harmonic functions sourced by the M2-branes. In order to have smooth geometries, the sources ofL I must 73 also overlap with Gibbons-Hawking points. The final step is to solve the one formk. One can decomposek into to a one form alongU(1) and a one form,!, on I R 3 : k = (d +A) + ! = (d +A) + ~ !:d~ y; (3.12) , where is some function on I R 3 . By plugging this form ofk,Z I and I into (2.9), one can verify! and should be: = 1 6 C IJK K I K J K K V 2 + K I L I 2V + M; (3.13) ~ r~ ! = V ~ rMM ~ rV + 1 2 K I ~ rL I L I ~ rK I ; (3.14) where M is another harmonic function on I R 3 which comes from adding an anti-self dual part todk. From the above result, one can see the full metric of BPS solutions is completely determined by eight harmonic functions:V ,Z I ,L I andM up to three gauge transformations which are coming from the gauge transformation ofK I in (3.10): K I ! K I + c I V ; L I ! L I C IJK c J K K 1 2 c J c K V ; M ! M 1 2 c I L I + 1 12 C IJK (V c I c J c K + 3c I c J K K ); (3.15) where I = 1; 2; 3 and c I are three arbitrary constants associated with the three gauge transformations which leave the metric unchanged. 3.2.2 Regularity and asymptotic constraints As far as BPS equations are concerned, any set of these eight harmonic functions with any distributions of the sources specify a solution. However, we are looking for microstate geometries which are smooth, horizonless and asymptotic to I R 4 . For that 74 to be true, these eight harmonic functions can not be completely independent. First of all, all of the sources of these eight harmonic functions must be overlap on Gibbons- Hawking points. Thus they should take the following form: V = N X i=1 q i r i ; K I = N X i=1 k I i r i ; L I = l 0 + N X i=1 (l i ) I r i ; M = m 0 + N X i=1 m i r i ; (3.16) where q i , k i , l i and m i are the charges associated with the eight harmonic functions respectively,l 0 andm 0 are some constants. Notice that there is no constant term forV and that is because we choose" 0 = 0 in order to pick only the Gibbons-Hawking base which is asymptotic to I R 4 . Moreover, ifV vanish at infinity thenk must also vanish at infinity in order to have finite I at infinity. The constants,l 0 andm 0 , will also be fixed by requiring the metric is asymptotic to I R 4 and we will check them at the following discussion. Here we discuss the regularity and asymptotic constraints on charges and the con- stant terms separately: Regularity constraints In order to have smooth metric, one need to requireZ I regular everywhere. However, one can see there are singularities at the position of sources in general. For example, if one take a close look at the behavior of Z I near a Gibbons-Hawking point ‘i’: Z I 1 2 C IJK k J i k K i q i + (l i ) I 1 r i : (3.17) Thus, in order to maintain the regularities ofZ I , we need to require the coefficients ofO( 1 r i ) vanish and this means set the charges (l i ) I equal to the following values: (l i ) I = 1 2 C IJK k J i k K i q i : (3.18) 75 Moreover, we also need to requirek is regular. This means must be regular and again, we find in general there are singularities at the sources. The behavior of near a Gibbons-Hawking point ‘i’ is: 1 6 C IJK k I i k J i k K i q 2 i + k I i (l i ) I 2q i + m i 1 r i : (3.19) This means we need to set the chargesm i equal to: m i = 1 6 C IJK k I i k J i k K i q 2 i k I i (l i ) I 2q i = 1 12 C IJK k I i k J i k K i q 2 i ; (3.20) in which we have used the form of (l i ) I we just get in (3.18). Asymptotic constraints In order to have the metric asymptotic to I R 4 , we need to have Z I 6= 0 at infinity. The asymptotic behavior ofZ I is: Z I (l 0 ) I +O( 1 r ); (3.21) wherer is the radial coordinate on the base space and the coefficient of 1 r will be related with the asymptotic charges which will be discussed later. From the above asymptotic behavior, it is clear that (l 0 ) I should be some non-zero constant. In order to fix U(1) gauge couplings, we set it to 1. Also, just like what we have done in the previous chapter, one need to requirek vanish asymptotically in order to have non-rotating coordinate at infinity. Particularly, this means we should have vanish at infinity. The asymptotic behavior of is: m 0 + (l 0 ) I k I 0 2q 0 +O( 1 r ) = m 0 + 1 2 3 X I=1 k I 0 +O( 1 r ); (3.22) 76 where we have used (l 0 ) I = 1 andq 0 = 1 andk I 0 is defined as the sum ofk I i : k I 0 = N X i=1 k I i : (3.23) In order to have the constant term vanish, we need to havem 0 equal to the follow- ing: m 0 = 1 2 3 X I=1 k I 0 : (3.24) From the above analysis, we find out a smooth horizonless geometry which is also asymptotic to I R 4 is in fact specified by a distribution of a collection of N Gibbons- Hawking points with each one has GH charge q i and three flux parameters k I i . From here, we can see the moduli space of the solution is really huge and comes from the number,N, the positions of points,~ y (i) , the charges,q i and the flux parameters,k I i , up to the following gauge transformation coming from (3.15): k I i =) k I i + c I q i ; i = 1; 2;:::;N; (3.25) wherec I are three arbitrary constants. In principle, any combinations of these parame- ters will be a potential microstate geometry or superposition of some microstate geome- tries. However, before we can claim they are microstate geometries, we have to deal the trade-off of introducing ambipolar generalization: the closed-time-like curves (CTC’s). The no-CTC’s condition in general impose very non-trivial constraints on the possible charge distributions and we will discuss them in section 3.3. 3.2.3 The fluxes In section 3.1.3, we have mentioned a geometric transition in whichS 1 of the M5-branes shrink to zero size and dissolve into the flux through the gaussian surface,S 2 . This flux 77 prevent the two-sphere collapse and therefore make it become topologically non-trivial. Moreover, this flux through theS 2 encodes the charge of the M5-branes. To obtain the flux threading a particular two-sphere, we need to integrate the dipole field strength over it. For example, consider a line,~ y (ij) that connect point~ y (i) to~ y (j) and this line specify a topological non-trivial two sphere,4 ij . The integral of I over4 ij is: I ij 1 4 Z 4 ij I = 1 4 Z 4 ij 3 X a=1 (@ a ( K I V )) ^ e 0 ^ ^ e a = Z ~ y (ij) (@ a ( K I V )) ^ e a = K I V y (j) K I V y (i) = k I j q j k I i q i ; (3.26) where we include the normalization constant, 1=(4), to simplify the expression and for the later convenience. If this flux is not zero, it will hold up the two-sphere and therefore produce a non-trivialS 2 in space. 3.3 Closed time-like curves and the bubble equations Since we using ambipolar Gibbons-Hawking space as the base, checking the metric is positive definite and thus free of CTC’s is a very important issue and that is because we allow the signature of the base space to change from ‘+4’ to ‘4’. The first obvious dangerous closed cycles are those cycles in T 6 and to avoid CTC’s appear in these cycles, we need to require the warp factors associated with each cycles are non-negative (see (2.2)): Z 2 Z 3 0 ; Z 1 Z 3 0 ; Z 1 Z 2 0 : (3.27) 78 Then, we can shift our focus on the non-compact space. The full metric for the four- dimensional space at the constant time slice is: ds 2 4 = 1 W 4 ( (d +A) +!) 2 + W 2 1 V (d +A) 2 + V (dx 2 +dy 2 +dz 2 ) = Q W 4 V 2 (d +A) V 2 Q ! 2 + W 2 V (dx 2 +dy 2 +dz 2 ! 2 Q ) ; (3.28) whereW andQ are define by: W (Z 1 Z 2 Z 3 ) 1=6 ; Q Z 1 Z 2 Z 3 V 2 V 2 : (3.29) From the above metric, one can see in order to avoid CTC’s inU(1) fiber and ensure the metric in three dimensional space is positive definite, we need to require the following conditions: Q 0 ; W 2 V 0 ; (3.30) combine the second condition with the previous conditions (3.27), we have the following equivalent conditions: Q 0 ; V Z I 0 ; I = 1; 2; 3: (3.31) Having done that, there is still one more dangerous part related with !. Because Q 0, the contribution of! on three dimensional metric is non-positive and the metric has positive or negative signature depends on the competition between the regular I R 3 metric and ! 2 Q . One can show that the worst situation is in the cycle along! and to ensure the metric is positive in that direction, one need to check the following condition: 1 j~ !j 2 Q 0 ; (3.32) 79 where!~ !:d~ y and the length of the vector is evaluated by the flat I R 3 metric. Or, the following equivalent condition: Q j~ !j 2 ; (3.33) which is in fact a stronger condition thenQ 0. To verify all of these conditions above are satisfied globally in general is not an easy task and especially difficult for the condition 3.33. The reason is because these conditions not only depend on the charges but also the details of the distribution of Gibbons-Hawking points. However, there are several potentially dangerous regions that are relatively easier to check. One can test these regions to rule out the improper distribu- tions. After that, one can check other conditions more carefully. The obvious dangerous regions are at the positions where the Gibbons-Hawking points located. If one check the behavior ofQ near a particular point ’j’, the result is: Q Z 1 Z 2 Z 3 V 2 V 2 q i r i i 2 q i r i + ::: ; (3.34) where i is some constant. In the above equation, we have used the factZ I are regular 1 at the location of the points. From the above behavior ofQ, we know that in order to haveQ 0, we need to require vanish exactly at the positions of every Gibbons- Hawking points. However, recall that the divergent part is removed and the only remain task is to check the constant term of. For a space withN Gibbons-Hawking points, we will haveN conditions: (~ y =~ y (i) ) = 0 ; i = 1; 2; 3:::N; (3.35) 1 Recall that the singularities ofZ I are removed by adjusting the charges ofL I . 80 In order to evaluate the constant term of at charge ‘i’, we need to include the constant contribution from other charges. The algebra is kind of involving, let’s do it part by part as the following: I = 1 6 C IJK 3 k I i k J i q 2 i N X j6=i k K j r ij 2 k I i k J i k K i q 3 i N X j6=i q j r ij ! ; II = 1 4 C IJK k I i q i N X j6=i k J j k K j q j r ij + k I i k J i q 2 i N X j6=i k K i r ij k I i k J i k K i q 3 i N X j6=i q j r ij ! + 1 2q i 3 X I=1 k I i ; III = 1 2 3 X I=1 k I 0 + 1 12 C IJK N X j6=i k I j k J j k K j q 2 j r ij ; (3.36) wherer ij is defined as the distance between charge ‘i’ and ‘j’ andI is coming from the part involves three K’s,II is coming from K I L I 2V andIII is coming fromM. Combine all three parts above and after some algebra, one can show theN conditions,(~ y =~ y (i) ) = 0, are: 1 6 C IJK N X j6=i I ij J ij K ij q i q j r ij = 3 X I=1 ~ k I i ; i = 1; 2; 3:::N; (3.37) where I ij is the flux flow through the two sphere,4 ij , and is defined by (3.26) and ~ k I i is defined by: ~ k I i k I i q i k I 0 : (3.38) Notice that ~ k I i is invariant under the gauge transformation in (3.25). Also, one can check the summation of theseN conditions result to a trivial condition. So, we have (N 1) independent equations which are referred to by bubble equations in [BW06]. Naturally, the bubble equations are not sufficient to guarantee the space is free of CTC’s globally. 81 However, one can use them to rule out the improper charge distributions and in fact, in some simple cases, they alone are enough to ensure the space is free of CTC’s 3.4 Asymptotic charges and angular momenta As in the previous chapter, the asymptotic charges of this system can be read from the asymptotic behaviors ofZ I andk. Before we check that, it will be useful to look at the relation between the coordinate system in Gibbons-Hawking space and the two plane coordinate system we use in the previous chapter (2.25). As we have mentioned before, if we fix the total chargeq 0 = 1,V! 1 r and the space is asymptotic to I R 4 by the radial coordinate change,r = 2 4 : ds 2 4 d 2 + 2 4 (d 2 + sin 2 d 2 ) + 2 4 d + cosd 2 : (3.39) By comparing the above coordinate system with (2.25), one can find out the coordinate transformations that connect them are: z = cos( 2 ); r = sin( 2 ); 1 = 1 2 ( + ); 2 = 1 2 ( ): (3.40) At first, the asymptotic behaviors ofZ I in (3.11) is: Z I 1 + 1 2 C IJK k J 0 k K 0 N X i=1 k J i k K i q i ( 4 2 ); (3.41) wherek I 0 P k I i and we replacer by 2 4 . One can introduce the quantized flux param- eters as the following: k I i = k I i ( l 3 p L 2 ) ; (3.42) 82 where k I i are in fact half-integers which will become clear in the next chapter. The asymptotic quantized charges of this system are: N I = 2C IJK N X i=1 ~ k J i ~ k K i q i ; (3.43) where ~ k k I i q i k I 0 . By using the (2.59) and the coordinate relation (3.40), one can read the two angular momenta 2 from the asymptotic behavior ofk: k 1 4 2 ((J 1 +J 2 ) + (J 1 J 2 ) cos) d + ::: : (3.44) Recall that the coefficient ofk ind component is, so we need to check the asymptotic behavior of . If one look at the form of (3.13), one will find out that involves the dipole contribution ofK I =V . So, let’s check the asymptotic behavior ofK I =V first: K I V (~ y) k I 0 + ^ e y : N X i=1 k I i ~ y (i) k I 0 N X i=1 q i ~ y (i) ! 1 j~ yj + :::; k I 0 + ^ e y : N X i=1 ~ D I i ! 1 j~ yj + ::: ; (3.45) where ~ D I i is defined as: ~ D I i ~ k I i ~ y (i) : (3.46) 2 One may notice that the definition ofJ 2 has the opposite sign comparing to the previousJ 2 in (2.65). This is actually just a change of the convention. 83 The asymptotic behavior of is: (~ y) 1 6 C IJK k I 0 k J 0 k K 0 1 4 C IJK k I 0 N X i=1 k J i k K i q i + 1 12 C IJK N X i=1 k I i k J i k K i q 2 i ! 1 j~ yj + 1 2 ^ e y : 3 X I=1 N X i=1 ~ D i ! 1 j~ yj + ::: : (3.47) If we arrange our asymptotic coordinate such that P I P i ~ D I i point at the direction of = 0, then after some algebra, one can show has the following asymptotic behavior: 1 r 1 12 C IJK N X i=1 ~ k I i ~ k J i ~ k K i q 2 i + 1 2 j ~ Dj cos ! ; (3.48) whererj~ yj and ~ D is defined as: ~ D N X i=1 ~ D i ~ D i 3 X I=1 ~ D I i : (3.49) Usingr = 2 4 and (3.44), one can express the two angular momenta with the quantized flux parameters: J R J 1 +J 2 = 4 3 C IJK N X i=1 ~ k I i ~ k J i ~ k K i q 2 i ; (3.50) J L J 1 J 2 = 8j ~ Dj ; (3.51) where ~ D is: ~ D 3 X I=1 N X i=1 ~ k I i ~ y (i) ( L 4 l 6 p ) : (3.52) Moreover, one can derive the alternative form of J L to make the relation between J L and fluxes more transparent. Notice that if we multiply each bubble equation (3.37) by 84 ~ y (i) and the normalization constant, L 6 l 9 p , the right hand side becomes ~ D i . Then, if we further sum all of the equations and multiply the result by8, we get ~ J L 8 ~ D. The left hand side of it will become: ~ J L 8 ~ D = 4 3 C IJK N X i;j=1 i6=j I ij J ij K ij q i q j ~ y (i) r ij = 2 3 C IJK N X i;j=1 i6=j q i q j I ij J ij K ij ^ e ij ; (3.53) where ^ e ij is the unit vector point along the direction of ~ y (j) ~ y (i) , I ij are just the quantized version of I ij and in the second equality, we have used the anti-symmetry property of the fluxes, I ij = I ji . Also, one can define the “flux vector” associated with the two cycle that connect point ‘i’ and ‘j’: ~ J Lij 4 3 q i q j C IJK I ij J ij K ij ^ e ij ; (3.54) then ~ J L is simply the vector summation of all of the flux vectors: ~ J L = N X i;j=1 i<j ~ J Lij : (3.55) 3.5 Conclusion In this chapter, we have reviewed the Gibbons-Hawking metrics and the ambipolar gen- eralization. Also, we have discussed the nucleation in this space that resolve the null orbiford singularity of the supertube. Moreover, we review the procedure to solve the three BPS equations on an ambipolar Gibbons-Hawking base. The solutions are speci- fied by eight harmonic functions. However, by imposing the regularity and asymptotic 85 constraints, it was shown that the positions and charges of the sources of these harmonic functions can not be randomly chosen. The result is that the solutions are specified by the number, charges, and the positions of GH points and the three flux parameter k I associate to each points. However, there is potential dangerous CTC’s can appear in some general distribution of the charges. Therefore, we need to check the complete conditions that ensure the space is free of CTC’s. We also have reviewed the bubble equations which are necessary to rule out the improper distributions of the charges even though they alone can not guarantee the space is free of CTC’s. Finally, we explain how to read the asymptotic charges of a system with some arbitrary distribution of charges. 86 Chapter 4 Bubble Mergers and Deep Microstates In chapter 3, we have introduced the large number of the bubbled geometries that are possible microstate geometries of the black hole/ring. The natural question to ask is what the configurations of the positions, charges and flux parameters should be in order for them to be typical microstate geometries of a black ring or black hole. Moreover, what are the moduli of the microstate geometries that corresponding to some particular black hole/ring? Is this moduli space simply connected or disconnected? Can it be quantized? The answers of these questions may help us understand microscopic degrees of freedom that contribute to the entropy of a black hole/ring in statistical mechanics description and indeed worth to investigate. To understand these questions better, in [BWW07a], we show that a blob that con- tains large number of GH points with total GH charge equal to ‘+1’ and microscop- ically 1 fluctuated flux parameters is corresponding to a microstate geometry of the maximally spinning black hole (which has zero-horizon area). On the other hand, a microstate geometry of a zero-entropy black ring can be realized by a similar blob but with total GH charge equal to zero. To obtain a microstate geometry of a black hole with macroscopic horizon, we use the result of the classical mergers that we have studied in chapter 2. Specifically, in [BWW06a], we produced theU(1)U(1) invariant irreversible merger of a black hole blob and a black ring blob by violating the charge parallel condition. This condition is the last condition that listed in the conclusion of chapter 2. By adjusting the flux 1 The meaning of “microscopic” will be clear later 87 parameters, we managed to meet the merger condition, J T = n I N BH I but with non- parallel charge vectors. This corresponds to an irreversible merger of the microstates of a zero-entropy black ring and a maximally-spinning black hole. The resulting solution shows scaling behaviors on GH base. Specifically, the size of two blobs that participate in the merger decrease as they close with each other in such way the ratio of the size of the blobs and the distance between them is roughly fixed. Therefore, we called them “scaling solutions”. Furthermore, we checked the asymptotic charges of these scaling solutions and we found that they have the same asymptotic charges as a black hole with macroscopic horizon area. Therefore, we conclude these scaling solutions correspond to microstate geometries of a black hole with macroscopic horizon. When we reach the merger point, the distance between the two blobs and their sizes measured onR 3 are all scale to zero. However, the physical size is determined from the full metric which includes the warp factor. By checking the full metric, we found that physical sizes of these distances are fixed near the merger point. However, in the region away from these GH points, the metric is similar with a classical black hole and behaves like a AdS throat. Therefore, when these GH point move closer to each other on R 3 , this AdS region become larger and the AdS throat become longer. However, it will be shown that the length of this throat is controlled byJ L J 1 J 2 and since theJ L is quantized, the throat can not be arbitrary long and should have finite size. Therefore, the metric structure of the scaling solutions is a very long but finite AdS throat and smoothly caps off by the local structure of GH points. Moreover, in [BWW06a], it was shown that the mass gap from the length of these AdS throats match the mass gap of the typical CFT states on boundary. Therefore, by AdS/CFT correspondence, these deep microstate geometries can be the dual geometries of some typical CFT states. 88 Due to this long AdS throat, we called these microstate geometries “deep microstates”. Conversely, we will refer the microstate geometries of a zero-entropy black ring or a maximally spinning black hole, “shallow microstates”. In [BWW07b], by using the similarU(1)U(1) invariant irreversible merger of two black ring blobs, we found the scaling solutions that corresponding to deep microstates of a black ring with macroscopic horizon. Similarly, these solutions shows the scaling behavior and have very long but finite AdS throats. On the other hand, we also found the triangular scaling solution, which has no axial symmetry, by merging three GH points. For this solution, the length of AdS throat can be arbitrarily extended by continuously changing the shape of this triangle. This feature present a puzzle from AdS/CFT corre- spondence point of view. We will discuss this puzzle and possible solutions in section 4.5 The outline of this chapter is the following: In section 4.1, we introduce the shallow microstates of a maximally spinning black hole and zero-entropy black ring. In section 4.2, we use theU(1)U(1) irreversible merger of two shallow microstates to produce a deep microstate which corresponds to a black hole with macroscopic horizon. In section 4.3, we use the similar merger to produce a deep microstate of a black ring with macroscopic horizon. In section 4.4, we discuss some general features of these scaling solutions (deep microstates). Finally, in section 4.5, we introduce a more general merger withoutU(1)U(1) invariance. By this merger, we discover a deep microstate with an arbitrary long AdS throat. We show the numerical result of this solution and discuss the puzzle it present from the point view of AdS/CFT correspondence. And finally, we have the conclusion in section 4.7. 89 4.1 Shallow microstates In this section, we examine two configurations of bubble geometries. One is a blob that contains large number of GH points with total charge equal to ‘+1’ and uniformly distributed flux parameters with only microscopic variations. The other is a similar blob but with total charge equal to zero. By checking their macroscopic properties, we conclude the former is a microstate geometry of maximally spinning black hole blob and the later is a microstate geometry of a zero-entropy black ring. In order to verify that above picture, we check the asymptotic charges and angular momenta of the system with the general formula we derived in section 3.4 and com- pare them to classical quantities of a maximally-spinning black hole or a zero-entropy black ring. In other words, we connect microscopic values (parameters of the bubble geometries) to macroscopic quantities in classical geometries. In order to do this con- nection properly, one need to have more clear distinguish between a macroscopic value and microscopic value. Recall that the parameters to describe a bubble geometry are: the number of GH points,M, their charges, positions and flux parameters,k I i . The only parameter which has physical unit and is directly connected to macroscopic charges is the flux parameter. Therefore, just like a gas contains large number of molecules in which the macroscopic/microscopic scale is set by the total energy of the system and mean value of kinetic energy of the molecules, the macroscopic scale of a bubble geom- etry is set byO(Mk I ) while the microscopic scale is set byO(k I ), where k I is the mean value of the flux parameters. By including the unit ofk I i , ‘ l 3 p L 2 ’, and comparing it 90 to the unit of dipole, electric, angular momenta and horizon area, we have the following conclusion that the macroscopic values of these physical quantities should have: n I O(Mk I ); N I O(M 2 (k I ) 2 ); J R O(M 3 (k I ) 3 ); J L O(M 3 (k I ) 3 ); M O(M 6 (k I ) 6 ): (4.1) For any physical quantity has less order of ‘M’ than above list should be considered macroscopically zero. For convenience, in this chapter, we will set l 3 p L 2 = 1 and therefore, we will not distinguishk I i and k I i . 4.1.1 A maximally spinning black hole blob The first configuration we are going to consider is a blob of M GH points which has total GH charge equal to ‘+1’ and roughly uniformly distributed flux parameters with only microscopic variations. We will check the macroscopic properties of this blob and show that they matches with a classical quantities of a maximally-spinning black hole. The macroscopic properties that we are going to check include three asymptotic charges (which are equal to their electric charges), two identical angular momenta,J 1 andJ 2 (or equivalently,J R jJ 1 j +jJ 2 j andJ L jJ 1 jjJ 2 j withJ L = 0 for a black hole.) In order to simplify our task, we assume the total number of points, M to be odd number and M+1 2 GH points have +1 charges while the remaining GH points have1 charges. Then we take all of the flux parameters to be positive and have microscopic variations around the mean value: k I j = k I (1 +O(1)) ; (4.2) 91 wherek I is the mean value ofk I j : k I = 1 M M X j=1 k I j : (4.3) Notice thatk I are not invariant under the gauge transformation defined in (3.25). If we choose thec I as gauge transformation parameters, it will transform like: k I =) 1 M M X j=1 (k I j + q j c I ) = k I + c I M : (4.4) However, the value ofk I that we are going to use is actually a value in a particular gauge. The gauge is precisely the one that in which the flux parameters appears roughly uni- formly distributed in the blob. Therefore, it is a constant from the above consideration and become a trivial gauge invariant value. The three asymptotic charges can be read by using (3.43) and the definition of ~ k I i in (3.38): N BH I = 2C IJK M X j=1 q 1 j (k J j q j Mk J ) (k K j q j Mk K ) = 2C IJK M X j=1 q 1 j k J j k K j 2Mk J M X j=1 k K j + M 2 k J k K M X j=1 q j = 2C IJK M X j=1 q 1 j k J j k K j M 2 k J k K 2C IJK (M 2 +O(1))k J k K ; (4.5) 92 where we have used the fact that the sum of GH charges should equal to ‘+1’ and also the fact thatk I j are almost uniformly distributed. Therefore, in largeM limit, we have: N BH 1 4M 2 k 2 k 3 ; N BH 2 4M 2 k 1 k 3 ; N BH 3 4M 2 k 1 k 2 : (4.6) On the other hand, the angular momentum,J R , can be computed by using (3.50): J R = 4 3 C IJK M X j=1 q 2 j (k I j q j Mk I ) (k J j q j Mk J ) (k K j q j Mk K ) = 4 3 C IJK M X j=1 q 2 j k I j k J j k K j 3Mk I M X j=1 q 1 j k J j k K j + 3M 2 k I k J M X j=1 k K j M 3 k I k J k K M X j=1 q j 4 3 C IJK (2M 3 +O(M))k I k J k K = 16M 3 k 1 k 2 k 3 +O(M) (4.7) where we have also used the fact that fork I j k I (1 +O(1)) andjq j j = 1. And so one has: X j q 1 j k J j k K j = X j q j k J j k K j k J k K : X j k I j k J j k K j Mk I k J k K : (4.8) The other angular momentum, J L , not only depends on the flux parameters and charges but also on the distribution of GH points. To compute it, we can first evaluate the typical size of the flux vectors defined in (3.54): j ~ J L+ j = j ~ J L+ j 4 3 C IJK (2k I ) (2k J ) (2k K ) = 64k 1 k 2 k 3 ; j ~ J L++ j = j ~ J L j 0; (4.9) 93 where ~ J L+ is the flux vector between positive and negative charges and etc. The angu- lar momentum, J L , equal to the vector sum of all flux vectors between any two GH points (3.55) and therefore depends on their distribution. If we bias the distribution such that all negative charges at one side while all positive charges at the other side in order to maximizeJ L , the result is: J L = 16 (M 2 1)k 1 k 2 k 3 : (4.10) Therefore, in the most bias case, we haveO(M 2 ) term. However, in a generic random distribution, the vectors randomly point at different directions and there will be a lot of cancelations in the vector sum. Therefore, we should only haveO(M) term or even less. However, no matter in what distributions, J L will be macroscopically zero. Because according to the macroscopic scale listed in (4.1), the angular momentum needO(M 3 ) term in order to be macroscopic value. Therefore, one can say a blob that satisfied following conditions should haveJ L J 1 J 2 very small compared tojJ 1 j orjJ 2 j 2 and thereforejJ 1 j jJ 2 j: Large number of GH points with total charge equal to +1 Microscopically fluctuated flux parameters,k I j k I (1 +O(1)) Moreover, fromN I in (4.5) andJ R in (4.7), we have the following at the leading order ofM: J 2 1 J 2 2 1 4 J 2 R N BH 1 N BH 2 N BH 3 : (4.11) This indicates a blob that satisfy the above conditions has two equal angular momenta that saturate the boundJ 2 N BH 1 N BH 2 N BH 3 at the largeM limit. Therefore, it should be thought of as the microstate of a maximally spinning black hole. 2 Recall thatJ R J 1 + J 2 94 4.1.2 A maximally spinning black ring blob The next configuration that we are going to consider is a similar blob but with the total GH charge equal to zero. Recall that in order for the metric is asymptotic to I R 4 , the total GH charge of the system must be ‘+1. Therefore, we need to add one additional point of ‘+1’ GH charge. We will put this single isolated GH point far away from the blob. We will again check the macroscopic properties of this system included the dipole charges, electric charges, asymptotic charges and two angular momenta and show they match with the classical quantities of a zero-entropy black ring. Furthermore, we will show the macroscopic distance between the center of the blob with the isolated GH point corresponds to the radius of the ring. To evaluate the macroscopic properties of this system, we consider the similar sim- plified setup as the previous configuration. We take the total number of points in the blob,M 0 , to be even number with total GH charge equal to zero, with GH charges either ‘+1’ or ‘1’. The single isolated GH point is referred by the (M 0 + 1)-th point. We again consider uniformly distribution of k 0 I j with microscopic fluctuation in the blob. However, the flux parameter on the isolated GH point should not be the same as the mean value ofk 0 I j in the blob because otherwise, the flux will not be enough to maintain such large distance between the blob and the isolated point. In fact, we will show this flux parameter must be macroscopic size in order for this configuration to be microstate geometry of a black ring. Therefore for the flux parameters, we have: k 0 I j = k 0 I (1 +O(1)) ; j = 1; 2;:::M 0 ; k 0 I M 0 +1 a I 0 O(Mk 0 I ); (4.12) 95 wherek 0 I is the mean value ofk 0 I j : k 0 I 1 M 0 M 0 X j=1 k 0 I j : (4.13) Notice thatk 0I is a gauge independent value because P M 0 j=1 q j = 0. The three dipole charges are stored in the fluxes in the blob and are actually propor- tional to the total sum of the flux parameters in the blob: n I = 2 M 0 X j=1 k 0 I j = 2M 0 k 0I ; (4.14) which are clearly gauge independent values as it should be. The three asymptotic charges can be similarly obtained by (3.43): N I = 2C IJK M 0 X j=1 ~ k 0 J j ~ k 0 K j q j + M 02 k 0J k 0K = 2C IJK M 0 X j=1 k 0 J j k 0 K j q j 2(M 0 k 0 J +a J 0 )M 0 k 0 K + M 02 k 0J k 0K 2C IJK M 02 k 0 J k 0 K + 2a J 0 M 0 k 0 K = 1 2 C IJK n J n K + 2C IJK a J 0 n K ; (4.15) where ~ k is defined in (3.38). Comparing the above equation with (2.57), one can see the electric charge of the ring should be: N I = 2C IJK a J 0 n K : (4.16) 96 According to the macroscopic scale in (4.1), a macroscopic electric charge should have O(M 0 2 ) term. Therefore, in order for the system have macroscopic electric charges, one need to requirea I 0 are macroscopic. The angular momentum,J R , can be computed with (3.50) and it is: J R = 4 3 C IJK M 0 X j=1 ~ k 0I j ~ k 0J j ~ k 0K j M 03 k 0I k 0J k 0K = 4 3 C IJK M 0 X j=1 k 0I j k 0J j k 0K j 3 (M 0 k 0I +a I 0 ) M 0 X j=1 q j k 0 J j k 0 K j + 3 M 0 k 0I +a I 0 M 0 k 0J +a J 0 M 0 k 0K M 03 k 0I k 0J k 0K 4 3 C IJK 2M 03 k 0I k 0J k 0K + 6M 02 a I 0 k 0J k 0K + 3M 0 a I 0 a J 0 k 0K (4.17) In order to compute the other angular momentum,J L , we need to evaluate ~ J Lij and sum all of the flux vectors in the system. Similar with what we have seen in section 4.1.1, the flux vectors inside of the blob are too small to have macroscopic contribution. Therefore, the remaining part are those flux vectors that run between the isolated point and the points in the blob. These flux vectors have two typical size: j ~ J + j 4 3 C IJK (k 0I a I 0 ) (k 0J a J 0 ) (k 0K a K 0 ) ; j ~ J j 4 3 C IJK (k 0I +a I 0 ) (k 0J +a J 0 ) (k 0K +a K 0 ) ; (4.18) where ~ J + and ~ J are the flux vectors point from the isolated GH point to the positive and negative charge respectively. In principle, we need to do the vector sum of these vectors to obtainJ L . However, we consider the distance between the blob and the isolated GH point is macroscopic while the size of the blob is microscopic and therefore, we should have: r i;j r i;M 0 +1 ; r i;M 0 +1 r i;j = 1; 2:::M 0 ; (4.19) 97 wherer i;j is the distance between point ”i” and ”j" and r is the distance between isolated point and the center of the blob. Since the size of the blob is very small compared to r, every flux vectors run between the isolated point and the blob should roughly point at the same direction and the vector sum become the value sum: J L M 0 2 ~ J + + M 0 2 ~ J = 2M 0 3 C IJK (k 0I a I 0 ) (k 0J a J 0 ) (k 0K a K 0 ) + 2M 0 3 C IJK (k 0I +a I 0 ) (k 0J +a J 0 ) (k 0K +a K 0 ) = 4 3 M 0 C IJK 3a I 0 a J 0 k 0K + k 0I k 0J k 0K 4C IJK M 0 a I 0 a J 0 k 0K ; (4.20) where we drop the term M 0 k 0I k 0J k 0K , because it is not macroscopic value. There- fore, it is clear that in order to have macroscopicJ L , we needa I 0 M 0 k 0I . Moreover, recall that one can also computeJ L by finding dipole vector, ~ D, defined in (3.52): J L = 8j ~ Dj = 8 3 X I=1 M 0 X j=1 ~ k 0I j ~ y (j) 8 3 X I=1 M 0 X j=1 k 0I q j (M 0 k 0I + a I 0 ) r ^ e = 8M 0 3 X I=1 k 0I r = 4 3 X I=1 n I r ; (4.21) where~ y (j) is the position vector ofj-th point measured from the isolated GH point and we use the approximation in (4.19). If we compare the above result of J L with the definition ofJ 4 in (2.67), we find out the embedded radius of the ring is related with r by: R 2 4 = r: (4.22) 98 Consider the radial coordinate relation between Gibbons-Hawking space and I R 4 ,r = 2 4 , the distance between the blob and isolated GH point, r, in fact is exactly the embed- ded radius of the ring measured on GH base. WithJ R andJ L , one can easily deriveJ 1 andJ 2 . At the leading order ofM 0 , we have: jJ 1 j = 1 2 (J R +J L ) 2 3 C IJK 2M 03 k 0I k 0J k 0K + 6M 02 a I 0 k 0J k 0K + 6M 0 a I 0 a J 0 k 0K = 1 6 C IJK n I n J n K + 1 2 n I N I + J T ; jJ 2 j = 1 2 (J R J L ) 2 3 C IJK 2M 03 k 0I k 0J k 0K + 6M 02 a I 0 k 0J k 0K = 1 6 C IJK n I n J n K + 1 2 n I N I ; (4.23) whereJ T = J L . The two angular momenta indeed fit well with the classical quantities in (2.64) and (2.65) with settingN BH I = 0 and identifyingJ 4 withJ L . Moreover, one can compute the horizon area of the classical black ring that this microstate geometry corresponds to. The algebra is a little bit complicated. However, it can simplified by using the following formula: n 1 n 2 N 1 N 2 + n 2 n 3 N 2 N 3 + n 1 n 3 N 1 N 3 3n 1 n 2 n 3 J L + 32M 04 C IJK k 0I k 0J a K 0 2 ; (n 1 N 1 ) 2 + (n 2 N 2 ) 2 + (n 3 N 3 ) 2 2n 1 n 2 n 3 J L + 64M 04 C IJK k 0I k 0J a K 0 2 (4.24) 99 The approximate sign means we considera I 0 M 0 k 0I and only take macroscopic value. The horizon area is: M 2n 1 n 2 N 1 N 2 + 2n 1 n 3 N 1 N 3 + 2n 2 n 3 N 2 N 3 (n 1 N 1 ) 2 (n 2 N 2 ) 2 (n 3 N 3 ) 2 4n 1 n 2 n 3 J L 0; (4.25) where we have used the fact thatJ T =J L J 4 for a system with a single black ring. From the above analysis, one can clearly see that this configuration has exactly the same macroscopic properties as a maximally-spinning black ring at the largeM 0 limit and can be thought of a microstate geometry of it. 4.2 Irreversible mergers and deep microstates of black holes In section 4.1, we have introduced two shallow microstates which have the same macro- scopic properties of a zero-entropy black ring and maximally-spinning black hole. In this section, we will use the irreversible merger of them to produce a microstate of a black hole with macroscopic horizon area. To merge these two shallow microstates, the natural thing to do is to reproduce the corresponding microstate versions of the classical mergers we have introduced in section 2.4. However, we can not make exactly the same procedure on Gibbons-Hawking base. The reason is the space that contains a classical black ring with a vertically shifted black hole does not have a tri-holomorphicU(1) isometry as a Gibbons-Hawking space. Therefore, the best we can do isU(1)U(1) system which contains a black ring with a black hole at center. This system corresponding to haveJ T >n I N BH I and no merger happen in classical system. However, if one can further adjust the intrinsic angular momentum of the ring, J T , such that J T ! n I N BH I , the black ring will merge the 100 (1) (2) x bh br Figure 4.1: AU(1)U(1) invariant merger. (1) A classicalU(1)U(1) merger on I R 4 that is shown on (z; 1 ) plane. By decreasing the black ring’s angular momentum,J T , one can decrease it’s radius and eventually merge with the black hole at the center. (2) A corresponding bubbled version of the same merger that is shown on thex-axis of GH base. By adjusting the flux parameter, one can decrease the distance between the black hole blob and the black ring blob and eventually they merge. black hole through it’s equator (see fig. 4.1). Based on the analysis of the entropy in section 2.4.2, these kind of mergers can be either reversible or irreversible depending on the charge “vectors” (defined in (2.133)) of the black ring/hole are parallel or not. Therefore, one can use the corresponding bubble merger (see fig. 4.1) of a black ring blob and a black hole blob with non-parallel charges to produce aU(1)U(1) invariant microstate geometry of a black hole with macroscopic horizon. In section 4.2.1, we construct the bubble version of theU(1)U(1) classical sys- tem that was introduced in section 2.2. In section 4.2.2, we identify the meaning of the charge vectors parallel condition in bubble geometries. Then, we adjust the flux parameters to meet the bubble version of the merger condition (2.113) in irreversible way. From that, we show the solutions have scaling behaviors and they corresponding to deep microstate geometries of a black hole with macroscopic horizon. In section 4.2.3, we give a numerical example of a scaling solution. 101 4.2.1 A zero-entropy black ring blob with a maximally-spinning black hole blob at the center In this subsection, we will build a bubbled version of theU(1)U(1) classical system we have introduced in section 2.2. Specifically, the classical system that we will consider is a zero-entropy black ring with a maximally-spinning black hole at the center. The microstate geometries that corresponds to this two classical objects are the blobs with total GH charge equal to ‘+1’ and zero respectively. Therefore, the bubbled version of theU(1)U(1) classical system is simply a black hole blob at the origin with a black ring blob on the axis separated from it (see fig. 4.1). The two U(1)’s come from the U(1) fiber of GH base and the axial symmetry along the axis formed by the two blobs. To show that this picture is correct, we will check the macroscopic properties of the system and examine how they are decomposed to the macroscopic properties of two shallow microstates. Then, we show the results are match with the classical quantities in section 2.2. We consider the similar simplified setup we have used so far. We take the number of points in the black ring blob to beM 0 and in the black hole blob to beM. In which, we assumeM 0 to be even number andM to be odd number. Every GH charges are either ‘+1’ or ‘1’. Assigning the flux parameters is a little bit tricky. As we have mentioned, the flux parameters will be changed by the gauge transformation in (3.25). When we say the blob has roughly uniformly distributed flux parameters, the correct statement is we can find a particular gauge such that the distribution appears roughly uniform. However, in this system we have two blobs and the two gauges that these two blob required in order to have the uniformly distributed flux parameters, the black hole blob gauge and 102 the black ring blob gauge, are not necessarily equivalent 3 . Therefore, we need to pick one of them. Here, we fix the gauge such that the black hole blob has roughly uniformly distributed flux parameters,k I i k I (1 +O(1)). Then, the flux parameters in the black ring blob, k 0 I j , are related to the uniformly distributed flux parameters by some gauge transformation: k I i = k I (1 +O(1)) i = 1; 2;:::M k 0 I j = k 0I (1 +O(1))q 0 j X I j = 1; 2;:::M 0 ; (4.26) wherek I andk 0I are mean values ofk in the black hole and black ring blob respectively and X I specify the gauge transformation. Notice that X I also represent the relative different between the two gauge choices that we have mentioned above. Because every GH charge is either ‘+1’ or ‘1’, we have two typical values of the flux parameters associated to the positive and negative charge in the black ring blob: k 0 I + = k 0I (1 +O(1))X I ; k 0 I = k 0I (1 +O(1)) +X I : (4.27) Since the black hole blob has no dipole charge, the three dipole charges are coming from the black ring blob: n I = 2 M 0 X j=1 k 0I j = 2M 0 k 0I : (4.28) 3 Notice that although the global gauge “choice” is indeed unphysical, the relative difference between these two gauges is actually physical and the merger is reversible or irreversible depends on the size of this difference at the merger point. This will become clear later. 103 The three asymptotic charges can be computed similarly by (3.43): N I = 2C IJK M X i=1 q 1 i k J i q i (Mk J +M 0 k 0J ) k K i q i (Mk K +M 0 k 0K ) M 0 X j=1 q 0 1 j k 0 J j q 0 j (Mk J +M 0 k 0J ) k 0 K j q 0 j (Mk K +M 0 k 0K ) 2C IJK (Mk J +M 0 k 0J ) (Mk K +M 0 k 0K ) 2M 0 k 0J X K = 2C IJK M 2 k J k K + M 02 k 0J k 0K + 2M 0 k 0J (X K + Mk K ) =N BH I + 1 2 C IJK n J n K + 2C IJK n J (X K + Mk K ) ; (4.29) where we have used P M i=1 q i = 1 and P M 0 j=1 q j = 0. We only keep the leading term of M orM 0 but also keep any term containsX I because we don’t know yet how large it should be. The first term in the result of (4.29) is the electric charges of the black hole. In order to compare the rest part with the asymptotic charges of the black ring in (4.15), we need to understand the relation between the two parameters, X I and a I 0 in (4.15). Recall that a I 0 is the flux parameter of the isolate GH point in the gauge that the flux parameters in the black ring blob appear roughly uniform. Therefore, in order to know the relation between X I and a I 0 , we need to convert the gauge back to a I 0 ’s gauge by doing the opposite gauge transformation of the one in (4.26). The total flux parameter of the black hole blob will change accordingly: a I 0 = M X i=1 (k I i +q i X I ) = Mk I + X I : (4.30) By this identification, we can clearly see the total asymptotic charges in (4.29) are decomposed to: N I = N BH I + N I + 1 2 C IJK n J n K ; (4.31) 104 which indeed match the asymptotic charges of the U(1) U(1) classical system in (2.57). On the other hand, the angular momentum,J R , is: J R = 4 3 C IJK M X i=1 ~ k I i ~ k J i ~ k K i + M 0 X j=1 ~ k 0 I j ~ k 0 J j ~ k 0 K j 4 3 C IJK 2 (Mk I +M 0 k 0I ) (Mk J +M 0 k 0J ) (Mk K +M 0 k 0K ) + 6 (Mk I +M 0 k 0I )M 0 k 0J X K + 3M 0 k 0I X J X K 4 3 C IJK 2M 3 k I 0 k J 0 k K 0 + 3M 0 k 0I (M 2 k J k K ) + 2M 03 k 0I k 0J k 0K + 6M 02 k 0I k 0J (Mk K +X K ) + 3M 0 k 0I (Mk J +X J ) (Mk K +X K ) =J BH R + n I N BH I + j R : (4.32) In the last step, we identify the contribution from the black hole,J BH R , and from the ring, j R , by comparing the above equation to (4.7) and (4.17) and using (4.30). Moreover, one can see there is an extra term,n I N BH I , which is exactly the contribution from the fluxes that appear in the classical formula of the angular momenta ((2.64) and (2.65)). Now, we turn our focus on the other angular momentum,J L . To evaluate theJ L , we first evaluate the typical size of the flux vectors, ~ J Lij . The flux vectors in the ring/hole 105 blob are too small and therefore, the only flux vectors that have macroscopic contribu- tion toJ L are those flux vectors that run between two blobs. There are four typical flux vectors: ~ J L++ 0 4 3 C IJK (k 0I X I k I ) (k 0J X J k J ) (k 0K X K k K ) ^ e(4.33) ~ J L+ 0 4 3 C IJK (k 0I +X I +k I ) (k 0J +X J +k J ) (k 0K +X K +k K ) ^ e (4.34) ~ J L+ 0 4 3 C IJK (k 0I X I +k I )(k 0 J 0 X J +k J 0 )(k 0 K 0 X K +k K 0 )^ e (4.35) ~ J L 0 4 3 C IJK (k 0 I 0 +X I k I 0 )(k 0 J 0 +X J k J 0 )(k 0 K 0 +X K k K 0 )^ e(4.36) Where ^ e is the unit vector point from the black hole blob to the black ring blob and we use the approximation that the sizes of the two blobs are small compared to the distance between them and therefore every flux vectors between two blobs are roughly point at the same direction. Therefore, theJ L is: J L (M + 1)M 0 4 ( ~ J L++ 0 + ~ J L+ 0) + (M 1)M 0 4 ( ~ J L++ 0 + ~ J L+ 0) (M + 1)M 0 3 C IJK 6 (X I +k I ) (X J +k J )k 0K (M 1)M 0 3 C IJK 6 (X I k I ) (X J k J )k 0K 4C IJK (X I +Mk I ) (X J +Mk J )M 0 k 0K 4C IJK M 2 k I k J M 0 k 0K =J T n I N BH I ; (4.37) where we identify the contribution from the black ring asJ T and there is an additional term from the flux. Notice that if the three X I are all macroscopically zero, then J L is macroscopically zero. Moreover, if we identifyJ L J 4 , the above equation match exactly with classical angular momentum in (2.76). 106 Moreover, one can relateJ L to the embedded radius of the ring: J L = 8j ~ Dj 8 3 X I=1 M 0 X j=1 k 0I q j (Mk I + M 0 k 0I ) r ^ e = 8M 0 3 X I=1 k 0I r = 4 3 X I=1 n I r ; (4.38) where r is the distance between the centers of two blobs. One can see once we identify the position of the black hole blob and the position of the isolated GH point, the relation between the embedded radius of the ring andJ L is exactly the same as (4.21). However, J L in (4.37) is no longer equal toJ T due to the term coming from the fluxes. This is just like the classical system in which the embedded radius of the ring is changed by the existence of the black hole. From the above analysis, one can see this configuration indeed has the macroscopic properties that are decomposed to the properties of the black hole and ring exactly in the same way as theU(1)U(1) classical system that we have introduced in section 2.2. Therefore, it is the bubbled version of theU(1)U(1) classical system. 4.2.2 Irreversible mergers and scaling solutions In this subsection, we use the microstate system introduced by the previous subsection and the entropy analysis of classical mergers in section 2.4.2 to study the bubbled version of the irreversible mergers. We will show schematically that irreversible mergers in general leads to the scaling solutions. The more rigorous proof that specialized in the most biased black ring blob can be found in [BWW06a, BW07]. At first, we need to understand the condition for a merger to happen in the microstate system. Recall that in classical merger, there are three possible outcomes when we bring the black hole from infinity to the center of the black ring and they depends on the black 107 ring angular momentum,J T . ForJ T <n I N BH I , the black ring enter the black hole with an angle (fig. 2.3) and therefore, is notU(1)U(1) system and can not be described on Gibbons-Hawking space. On the other hand, forJ T >n I N BH I , the black hole sits at the center of the black ring and there is no merger happened. However, if we adjust the flux parameters to makeJ T !n I N BH I , the radisu of the ring will decrease and eventually, when J T = n I N BH I , the ring merge with the black hole. From (4.37), one can see that this merger condition is equivalent toJ L = 0. Therefore, the result of the merger is a microstate of a black hole. This merger condition can be expressed in microstate parameters by (4.37): 4C IJK X I X J M 0 k 0K + 8C IJK X I (Mk J ) (M 0 k 0K ) 0 ; (4.39) where the approximation means it should be macroscopically zero. The above merger of a black hole blob and a black ring blob can be reversible or irreversible based on the analysis in section 2.4.2. Recall that in section 2.5, we have concluded that there are four conditions in order for a reversible merger to happen. TheU(1)U(1) merger that we are trying to do in the microstate system has already satisfied the first three conditions, namely, a maximally spinning black hole, a zero- entropy black ring and the black ring merge with the black hole through it’s equator. However, there is a remaining condition: three charge vectors (C IJK n J n K ;N I and N BH I ) of the black ring and the black hole must be parallel in order for the merger to be reversible. Therefore, if we violate this charge parallel condition, the merger will 108 be irreversible. To understand this condition in microstate context, we can express this condition with the microstate parameters: C IJK n J n K == N BH I == N I =) C IJK k J k K == C IJK k 0J k 0K == C IJK k 0J (X K +Mk K ); =) (k 1 ; k 2 ; k 3 ) == (k 01 ; k 02 ; k 03 ) == (X 1 ; X 2 ; X 3 ): (4.40) Therefore, the merger condition in (4.39) and charge parallel condition in (4.40) define a reversible merger. To understand what happen during a reversible merger, we write the charge parallel condition as the following and plug it into the merger condition: X I = k I k 0I = k I (4.41) where are are two proportional constants. Before the merger, the black ring has macroscopic radius and thus the system has macroscopic size of J L (4.37). Conse- quently,X I must be macroscopic. Therefore, and should be: M M 0 ; O(1): (4.42) By plugging the charge parallel condition (4.41) into the merger condition in (4.39), we have: C IJK ( + 2M)k I k J k K = 6 ( + 2M)k 1 k 2 k 3 0 : (4.43) Since k I are all positive values, the only way that this condition can be satisfied is become macroscopically zero. Therefore, the only way to produce the reversible merger is to decrease the size ofX 1 ,X 2 andX 3 from macroscopic to microscopic. By doing that, every flux vectors run between the two blobs in (4.33)-(4.36) become microscopic. 109 Therefore, in reversible merger, the flux vectors run between two blobs become small and comparable to the typical flux vectors inside of the two blobs at the merger point. From the other point view, recall thatX I also represent the relative different between the black hole blob gauge and black ring blob gauge. By decreasingX I to macroscopically zero, one make these two gauges identical. Therefore, there will be a gauge choice such that every GH points have roughly the same flux parameter. The physical picture is the two blobs joint to form a bigger black hole blob with roughly uniformly distributed flux parameters and all of flux vectors in it are microscopic. This indicate the result is a bigger shallow microstate of a black hole. On the other hand, if the charge vectors are not parallel, in general, it is possible to maintain the macroscopic size of X I while meet the merger condition. Therefore, in the irreversible merger, one can merge the black ring blob and black hole blob while keeping the individual flux vectors between them macroscopic large. This seem strange because one expect the black ring blob will move very close to black hole blob at the merger point and then how can the flux vectors between them still maintain macroscopic size? In order to understand what happen during the irreversible merger, one needs to track the two scales, the size of two blobs and the distance between them. Given the number of GH points, GH charges and the flux parameters, the distribution of GH points can not be arbitrary and must be constrained by the no-CTC’s condition. At first, they need to satisfy the bubble equations of this system. Therefore, we are going to check the bubble equations in order to understand how these two scales behaves near the merger point. The bubble equations of the points in the black hole blob are: M X j6=i J ij r i;j + M 0 X j 0 =1 J ij 0 r i;j 0 = 3 X I=1 ~ k I i q i 3 X I=1 ( (Mk I + M 0 k 0I )); i = 1; 2:::M (4.44) 110 where we use unprimed ‘i’ and ‘j’ to label the points in the black hole blob and primed one for the black ring blob. And,J ij can be defined by the flux vectors: J ij 1 8 ~ J Lij :^ e ij = 1 6 C IJK q i q j I ij J ij K ij ; (4.45) where ^ e ij is the unit vector point fromi-th point toj-th point. The first terms of each bubble equation constrain the distribution of the points in the black hole blob while the second terms can be considered as the influences of the ring blob on the black hole blob. Before the merger, we have a black ring with macroscopic radius while the size of the two blobs are microscopic and therefore, we should have the following approximation: r i;j ;r i 0 ;j 0 r i;j 0 r 0 ; (4.46) where r 0 is the distance between two centers of the blobs. We can apply this approxima- tion and write the bubble equations before the merger in the following way: M X j6=i J ij r i;j q i 3 X I=1 (Mk I + M 0 k 0I ) r 01 M 0 X j 0 =1 J ij 0; i = 1; 2:::M: (4.47) The reason to write the bubble equations in this form is to estimate the effect of the ring on the distribution of the points in the blob during the irreversible merger. To reach the merger point, we need to change the flux parameters to decreaseJ L while keepX I large. One way to do that is by changing the “orientation” ofX I while keepingk I and k 0I fixed. If we produce the merger this way, the first term on the right hand side is a 111 constant during the merger. The second term has two typical values depending on the i-th point is positive or negative: i = + =) M 0 2 J ++ 0 + M 0 2 J + 0 r 01 = 3C IJK (X I +k I )(X J +k J )M 0 k 0K r 01 ; i = =) M 0 2 J + 0 + M 0 2 J 0 r 01 = 3C IJK (X I k I )(X J k J )M 0 k 0K r 01 : (4.48) For irreversible merger, we have macroscopicX I . Therefore, to the leading order ofM andM 0 , the bubble equations in (4.47) becomes: M X j6=i J ij r i;j q i 3 X I=1 (Mk I + M 0 k 0I ) 3M 0 C IJK X I X J k 0K r 01 ! : (4.49) Before the merger, the approximation in (4.46) is valid and therefore, the radius of the ring, r 0 , decrease linearly with J L (4.38). Since X I is macroscopic, the coefficient , C IJK X J X K M 0 k 0K , in general is macroscopic. Therefore, when one approach the merger point, the term that represent the effect of the ring begins to increase, dominate and become infinite large at the exact merger point. Moreover, at the leading order, this increasing of the black ring’s affect on the black hole blob is “uniform”. Therefore, in order to satisfy the bubble equations while the constants on the right hand side uniformly increase, every distances in the black hole blob should shrink uniformly. In other words, given any specific distribution of points in the black hole blob, the only effect of the ring blob on it is to decrease the overall size of the blob with slight deformation. The scale factor of the size of the blob depends on r 0 and therefore, onJ L : r 0 i;j r i;j P 3 I=1 (Mk I + M 0 k 0I ) 3M 0 C IJK X I X J k 0K r 1 P 3 I=1 (Mk I + M 0 k 0I ) 3M 0 C IJK X I X J k 0K r 01 r 0 r (4.50) 112 where we decrease r down to a very small value, r 0 . One can see the size of the blob decrease at the same rate as r. Therefore, the ratio of r i;j with r 0 is fixed when one approach the merger point. Therefore, these solutions that show scaling behaviors are called scaling solutions. Later, we will study more bubbled merger and at least for the case we have studied, the irreversible mergers in general lead to the scaling solutions. 4.2.3 Numerical results for a simple merger In this subsection, we show the above picture of scaling solutions by solving the bubble equations numerically. To simplify our task, we consider a black hole microstate that is made from three points, of GH chargesn, 2n + 1, andn and its merger with a black ring microstate that is made from two points, of GH chargesQ and +Q. This configuration can be considered as special simplified distributions of points in the black hole/ring blob. For example, putn points of charge +1 on top of each other to produce a point with charge +n. Moreover, we will put them all on a the axis and therefore, the system preservesU(1)U(1) symmetry (see fig. 4.2). We consider a configuration with 5 GH centers of charges: q 1 =12; q 2 = 25; q 3 =12; q 4 =20; q 5 = 20 (4.51) The first three centers give the black-hole blob and the last two centers give the black- ring blob. Therefore, corresponding total number of points in the black ring/hole blobs are: M = 49; M 0 = 40: (4.52) Thus, in this simplified model, the distinguish of macroscopic and microscopic is roughly like,O(100) withO(1). 113 x 1 x 2 x 3 x 4 x 5 BH BR Figure 4.2: The layout of GH points for the U(1) U(1) merger of a black ring microstate and a black hole microstate. Notice that every point here is just a point and there is no internal structure. The flux parameters of the black hole points are: k I 1 =jq 1 jk I ; k I 2 =jq 2 jk I ; k I 3 =jq 3 jk I ; (4.53) the mean values of the flux parameters are assigned by: k 1 = 5 2 ; k 2 = 3:08 3:18; k 3 = 1 3 ; (4.54) where we keepk 1 andk 3 fixed while changek 2 in order to meet the merger point. On the other hand, the parameters of the ring are: k 01 = 5 4 ; k 02 = 13 8 ; k 03 = 1; X 1 = 135 2 ; X 2 =103:4108:3; X 3 = 416 3 ; (4.55) where we changeX 2 withk 2 in such way the electric charges of the ring remained fixed. To solve the bubble equations (3.37) numerically, we put all of five points onx-axis and label their position byx i (fig. 4.2). During the irreversible merger, the positionsx i of the five points change as a function ofk 2 . The result is listed in the table 4.1. One can see the distance (x 4 x 3 ) r 0 decrease withJ L but the ratios of different distances are roughly fixed. This is the typical behavior of the scaling solution. 114 k 2 x 4 x 3 x 4 x 3 x 2 x 1 x 2 x 1 x 3 x 2 x 2 x 1 x 5 x 4 J L H 0 3.0833 175.5 2225 1.001 2.987 215983 .275 1 3.1667 23.8 2069 1.001 3.215 29316 .278 2 3.175 8.65 2054 1.001 3.239 10650 .279 3 3.1775 4.10 2049 1.001 3.246 5050 .279 4 3.178 3.19 2048 1.001 3.248 3930 .279 5 3.17833 2.59 2048 1.001 3.249 3183 .279 6 3.17867 1.98 2047 1.001 3.250 2437 .279 7 3.1795 .463 2046 1.001 3.252 570 .279 8 3.17967 .160 2045 1.001 3.253 197 .279 Table 4.1: Distances between points in the scaling regime. The parameter,H N 1 N 2 N 3 J 2 R =4 N 1 N 2 N 3 measures how far away the angular momentum of the resulting solution is from the angular momentum of the maximally-spinning black hole with identical charges. The value ofk 2 is varied to produce the merger. Both the total charges andJ R remain approximately constant, withJ R 3:53 10 7 . However, one should recall that the bubble equations alone are not enough to guar- antee the absence of CTC’s. Even in this simplified merger, checking the complete no- CTC’s conditions (3.31,3.33) analytically is quite difficult. Therefore, in [BWW06a], we check numerically that CTC’s are absent in this system. Specifically, we check the region besides of GH points 4 , namely, the regions near each points, insides of two blobs and the region between two blobs. This indicate that the scaling solutions are indeed microstate geometries of a black hole with macroscopic horizon. 4.3 Irreversible merger of two zero-entropy black rings Previously, we have built microstate geometries of a black hole with macroscopic hori- zon. The natural question is: Can we extend this success to a black ring? In classical geometry, one can image that by dumping a black ring on top of the other black ring, one 4 Recall that the no-CTC’s at every GH points are ensured by the bubble equations 115 q 0 = +1 1 st BR 2 nd BR (1) (2) x Figure 4.3: AU(1)U(1) invariant merger of two black rings. (1) A classicalU(1) U(1) merger of two black rings on I R 4 that is shown on (z; 1 ) plane. By decreasing the interactive part ofJ L that comes from the fluxes , one can decrease the difference between two radii and eventually the two black ring merge at particular radius. (2) A corresponding bubbled version of the same merger that is shown on thex-axis of GH base. By adjusting the flux parameters, one can decrease the distance between the two black ring blobs and eventually they merge. may merge these two black rings and become another bigger black ring. Although this classical system have not yet been studied, in this section, we show that by generalizing the previousU(1)U(1) merger, one can have the similarU(1)U(1) merger of two black ring blobs that result a microstate geometry of a black ring. The picture of this merger is shown in fig. 4.3. In this section, we first built the bubble version of the two concentric black rings. Then, we study theU(1)U(1) invariant merger of two shallow black ring microstates by generalizing the merger in the previous section. We again find the irreversible merger in general leads to the scaling solutions and the result is a microstate geometry of a black ring with macroscopic horizon. Furthermore, we show some numerical results of the scaling solutions. 116 4.3.1 Two zero-entropy black ring blobs with an isolated GH point In this subsection, we consider anotherU(1)U(1) configuration which contains two (shallow) black ring blobs sit on the common axis. Two U(1)’s are coming from the GH fiber and the axial symmetry along this axis. We will show this configuration is the bubble version of the two concentric black rings by checking the macroscopic properties and how they are decomposed to quantities of two individual rings. For the convenience, we will refer the black ring blob that is closer to the isolated GH point, the first blob, and the other one, the second blob and the isolated point by 0-th point (fig. 4.3). The setup will be similar as the previous configuration. We take the total number of points in the first black ring blob to beM and the second one to beM 0 . Both of them are even number. Every GH charges are either +1 or1. To setup the flux parameters for this system, one need to choose the gauge. There are three natural choice: The 0-th gauge: The gauge such that the flux parameter at the isolated GH point is zero The 1-st gauge: The gauge such that the flux parameters are uniformly distributed on the first blob The 2-nd gauge: The gauge such that the flux parameters are uniformly distributed on the second blob As the previous case, although the gauge choice is indeed unphysical, the relative differ- ences between these three gauges are actually physical. Particularly, we will show that the merger of the two rings is either reversible or irreversible depends on that the relative difference between 1-st gauge and 2-nd gauge is either macroscopic or microscopic. 117 Without losing the generality, we choose the 1-st gauge. In this gauge, the flux parameters are: k I 0 =a I 0 ; k I i =k I (1 +O(1)) i = 1; 2;:::M ; k 0I j =k 0I q 0 j X I +O(1) j = 1; 2;:::M 0 ; (4.56) where a I 0 are some unknown constants and X I define the gauge transformation from the 1-st gauge to the 2-nd gauge and therefore represent the relative difference between these two gauges. The two typical flux parameters in the second blob are: k 0I + = k 0I X I +O(1); k 0I = k 0I + X I +O(1): (4.57) The three dipole charges for the two blobs are: n I 1 = 2 M X j=1 k I j = 2Mk I ; n I 2 = 2 M 0 X j=1 k 0I j = 2M 0 k 0I : (4.58) The three asymptotic charges are: N I = 2C IJK ~ k J 0 ~ k K 0 + M X i=1 q 1 i ~ k J i ~ k K i + M 0 X i=1 q 01 i ~ k 0J i ~ k 0K i 2C IJK (Mk J +M 0 k 0J )(Mk K +M 0 k 0K ) 2 (Mk J +M 0 k 0J ) (Mk K +M 0 k 0K +a K 0 ) 2M 0 k 0J X K 2C IJK M 2 k J k K + M 02 k 0J k 0K + 2Mk J a K 0 + 2M 0 k 0J (X K +a K 0 ) + 2M 0 k 0J Mk K = 1 2 C IJK n J 1 n K 1 + N (1) I + 1 2 C IJK n J 2 n K 2 + N (2) I + C IJK n J 1 n K 2 ; (4.59) 118 whereN (1) I andN (2) I are the electric charges of the first and second rings respectively: N (1) I = 2C IJK n J 1 a K 0 ; N (2) I = 2C IJK n J 2 (X K +a K 0 ): (4.60) One can also write (4.59) in the following way: N I = N (1) I + N (2) I + 1 2 C IJK (n J 1 +n J 2 ) (n K 1 +n K 2 ): (4.61) Therefore, if one look at this combined system as a single black ring, it has the electric and dipole charges equal to the sum of the charges of the individual rings. The angular momentum,J R , is: J R = 4 3 C IJK ~ k I 0 ~ k J 0 ~ k K 0 + M X i=1 ~ k I i ~ k J i ~ k K i + M 0 X i=1 ~ k 0I i ~ k 0J i ~ k 0K i 4 3 C IJK 24 I 4 J 4 K + 64 I 4 J a K 0 + 34 I a J 0 a K 0 + 3M 0 k 0K 6M 0 k 0I (4 J +a J 0 )X K = 4 3 C IJK 2M 3 k I k J k K + 6M 2 k I k J a K 0 + 3Mk I a J 0 a K 0 + 2M 03 k 0I k 0J k 0K + 6M 02 k 0I k 0J (X K +a K 0 ) + 3M 0 k 0I (X J +a J 0 ) (X K +a K 0 ) + 6Mk I M 0 k 0J (X K +a K 0 ) + 6Mk I M 0 k 0J a K 0 + 6Mk I M 0 k 0J (Mk K +M 0 k 0K ) =J (1) R + J (2) R + n I 1 N (2) I + n I 2 N (1) I + C IJK n I 1 n J 2 (n K 1 +n K 2 ) ; (4.62) where4 I is defined as: 4 I = (Mk I + M 0 k 0I ) ; (4.63) andJ (1) R andJ (2) R are theJ R (4.17) for the first and second ring blob respectively. 119 To evaluateJ L , we assume the size of the two blobs are both small compared to the distances between them and their radii. Also, we ignore all of the flux vectors inside of the blobs because they are microscopic. Therefore, the angular momentum,J L , comes from the three kinds of the flux vectors that run between the isolated GH point, the first blob and the second blob: ~ J L0 4 3 C IJK (k I a I 0 )(k J a J 0 )(k K a K 0 )^ e (4.64) ~ J L0 0 4 3 C IJK (k 0I X I a I 0 )(k 0J X J a J 0 )(k 0K X K a K 0 )^ e (4.65) ~ J L 0 4 3 C IJK (k 0I X I k I )(k 0J X J k J )(k 0K X K k K )^ e (4.66) ~ J L 0 4 3 C IJK (k 0I +X I k I )(k 0J +X J k J )(k 0K +X K k K )^ e (4.67) where ^ e is the unit vector point along the common axis from the isolated point to the two blobs. Therefore,J L is: J L M 2 ( ~ J 0+ + ~ J 0 ) + M 0 2 ( ~ J 0+ 0 + ~ J 0 0) + MM 0 4 ( ~ J ++ 0 + ~ J + 0 + ~ J + 0 + ~ J 0) 4C IJK Mk I a J 0 a K 0 + 4C IJK M 0 k 0I (X J +a J 0 ) (X K +a K 0 ) + 8C IJK (Mk I ) (M 0 k 0J )X K = J (1) L + J (2) L + J int L ; (4.68) whereJ (1) L andJ (2) L are theJ L (4.20) for the first and second ring blob respectively and J int L comes from the sum of the all flux vectors between two blobs and it is: J int L = 8C IJK (Mk I ) (M 0 k 0J )X K = n I 1 N (2) I n I 2 N (1) I ; (4.69) where the sign of this value depends on the direction of the unit vector, ^ e. Notice that this value are either macroscopic or microscopic depends onX I . If all of the threeX I ’s 120 are microscopic,J int L will be macroscopically zero. Moreover, from (4.69), one can see this internal fluxes are actually corresponding to the Poynting vector that comes from the interaction of the electric fields of one ring with the magnetic fields of the other ring. We will show thisJ int L control the merger andX I control this merger is either reversible or irreversible. 4.3.2 Irreversible mergers and scaling solutions For the merger of the two classical black ring, the condition for irreversible merger has not be studied. However, one can generalize the previous story about the merger of a black ring and a black hole. Recall that for the merger of the ring and black hole blobs, the merger condition is J L = 0. For the two shallow blobs, J L mostly comes from the flux vectors between the blobs. Therefore the condition is equivalent to requiring the sum of the flux vectors between them become zero. Therefore, intuitively, one can generalize the merger condition for two shallow blobs to: J int L = 8C IJK (Mk I ) (M 0 k 0J )X K = 0 : (4.70) We will explain this condition more carefully in section 4.4.1. Moreover, we have observed that a important feature of the previous irreversible mergers is that they pre- serve the macroscopic sizes of the individual flux vectors between blobs near/at the merger point. We will show this story can be generalized to the two black ring merger. As we have mentioned, if allX I are microscopic, the interactive part,J int L (4.69) will be macroscopically zero and therefore the merger condition in (4.70) is trivially satisfied. However, if one reaches the merger by decreasing everyX I to zero, every individual flux vector between the two blobs ((4.66), (4.67)) becomes microscopic. This indicate it is a reversible merger. To verify the above picture, one can check the macroscopic properties 121 of the system at the merger point (X I ! 0). At first, notice that two electric charges of two rings become equal (4.60) if X I vanish. Therefore, the total electric charge is simply: N I = 2N (1) = 2N (2) = 2C IJK (n J 1 +n J 2 )a K 0 = 2C IJK n J a K 0 ; (4.71) wheren I is the total dipole charge and simply equal to the sum ofn I 1 andn I 2 . Moreover, ifX I vanish,J L in (4.68) equal to: J L 4C IJK (Mk I +M 0 k 0I )a J 0 a K 0 = 2C IJK n I a J 0 a K 0 : (4.72) By comparing the above electric charges and J L with the formulas of shallow ring microstate in (4.15) and (4.20) , one can see the result is indeed just a shallow microstate corresponds to a bigger zero-entropy black ring. It should not be a surprise after all. If we reduce the every flux vectors between two blobs down to microscopic size, all of the flux vectors inside of the combined blob will be microscopic and therefore is simply a shallow microstate. In other words, by decreasingX I to microscopic, there will be no macroscopic different between the 1-st gauge and 2-nd gauge. Therefore, there exist a gauge such the every GH points in the two black ring blobs are appears roughly uniform. On the other hand, if one maintain the macroscopic size ofX I but change it’s orien- tation such that: X I ?C IJK k J k 0K : (4.73) Then, the merger condition (4.70) is satisfied while the individual flux vectors between two blobs ((4.66) and (4.67)) remain macroscopic. This is an irreversible merger. We will show this irreversible merger in general leads to a scaling solution. 122 In the similar way to the merger of the black hole and black ring, we look for the scaling behavior of the irreversible merger of two black ring microstate geometries. We consider the following approximation: r i;j ;r i 0 ;j 0 r; r 0 ;r 0 ; < 0 (4.74) where we use unprimedi andj to label the points in the first ring blob and primed ones for the second ring blob. r, r 0 are the distance between the isolated GH point and the center of the two blobs respectively. r 0 is the distance between the center of the two blobs. Before we discuss the scaling solution, it is useful to understand the behavior of three macroscopic distances, r, r 0 and r 0 during the merger. To do that, we consider the “coarse grained” version of the bubble equations. The procedure of this coarse graining is simply replacing the two blobs by two points. Therefore, the originalM +M 0 independent bubble equations will reduce to two independent bubble equations. This two equations will constraint the behaviors of the three macroscopic distances. To obtain this two “coarse grained” bubble equations, we simply sumM andM 0 bubble equations in the two blobs respectively. The first one is: M X i=1 J i0 r i;0 + M X j6=i J ij r i;j + M 0 X j 0 =1 J ij 0 r i;j 0 = 3 X I=1 ~ k I i ! J (1) L 8 r + J int L 8r 0 = 3 X I=1 Mk I ; (4.75) 123 where J (1) L is the angular momentum, J L , of the first ring blob and J int L is defined in (4.69). The second one is: M 0 X i 0 =1 J i 0 0 r i 0 ;0 + M 0 X j 0 6=i 0 J i 0 j 0 r i 0 ;j 0 + M X j=1 J i 0 j r i 0 ;j = 3 X I=1 ~ k 0I i ! J (2) L 8 r 0 J int L 8r 0 = 3 X I=1 M 0 k 0I ; (4.76) whereJ (2) L is the angular momentum,J L , of the second ring blob. One can see the term that contains J int L basically represents the interaction between two blobs. And, as we have mentioned, the merger point is corresponding toJ int L = 0. As one approaches the merger point, there are two possible behaviors ofr 0 . The first one isr 0 remain finite as J int L ! 0. Consequently, the interaction part in the bubble equations will vanish and the two bubble equations become: J (1) L 8 r = 3 X I=1 Mk I =) J (1) L = 4 3 X I=1 n I 1 r ; J (2) L 8 r 0 = 3 X I=1 M 0 k 0I =) J (2) L = 4 3 X I=1 n I 2 r 0 : (4.77) From the above equations, one can see if the interaction part vanish, the distances between two blobs and the isolated GH point, r and r 0 , will simply correspond to their radii (4.21). Therefore, in this solution, there is no merger and it will look like superpo- sition of two concentric black rings in classical picture. The second possibility isr 0 shrink to zero asJ int L go to zero in such way thatJ int L =r 0 approach to some constant. The radius of the resulting ring, ~ r, can be obtained by adding (4.75) and (4.76): J (1) L +J (2) L 8 ~ r = 3 X I=1 (Mk I +M 0 k 0I ) =) ~ r = J (1) L +J (2) L 4 (n I 1 +n I 2 ) (4.78) 124 To see if this is a scaling solution, one need to look at the complete bubble equations. The bubble equations for the points in the first blob are: J i0 r i;0 + M X j6=i J ij r i;j + M 0 X j=1 J ij 0 r i;j 0 = 3 X I=1 ~ k I i : (4.79) For the situation that the approximation in (4.74) is valid, the above bubble equations come with two typical versions and can be written as: M X j6=i J ij r i;j 3 X I=1 Mk I +M 0 k 0I +a I 0 J +0 r M 0 2r 0 J ++ 0 + J + 0 ; M X j6=i J ij r i;j 3 X I=1 Mk I +M 0 k 0I +a I 0 J 0 r M 0 2r 0 J + 0 + J 0 ; (4.80) where the first one is for the positive charge while the second one is for the negative charge andJ +0 ,J 0 ,J ++ 0,J + 0,J + 0 andJ 0 are proportional to the typical sizes of the flux vectors in ((4.64)-(4.67)) and defined in (4.45). In order to reach the merger point, we need to change the flux parameters. One way to do that is to change the orientation ofX I while keep all of the other parameters fixed. Therefore, the first terms on the right hand sides of the above equations is a constant during the merger. Moreover, in order for the first black ring to have macroscopic radius, a I 0 must be macroscopic. Consequently,J +0 J 0 (4.64). Moreover, since we want to do irreversible merger, we need to keepX I macroscopic, therefore: J ++ 0 + J + 0 J + 0 + J 0 8C IJK X I X J k K : (4.81) 125 Putting the above information together, the bubble equations in (4.80) become: M X j6=i J ij r i;j q i 3 X I=1 Mk I +M 0 k 0I +a I 0 1 6 r C IJK a I 0 a J 0 a K 0 + 1 2r 0 C IJK X I X J M 0 k K i = 1; 2;:::M; (4.82) where r is the radius of the first ring at the present of the second ring and it approach to ~ r defined in 4.78 at the merger point. To estimate the effect of the second ring on the first ring, one can compare (4.82) with the bubble equations of the isolated 1-st ring blob: M X j6=i J ij r i;j q i 3 X I=1 Mk I +a I 0 1 6 r C IJK a I 0 a J 0 a K 0 i = 1; 2;:::M; (4.83) where r here is the “natural” radius of the first ring and defined in (4.77). Therefore, we see if the approximation in (4.74) is valid before the merger, when one approach the merger point, the effect of the second ring on the bubble equations of the first ring in general just increase the constant at the right hand side of (4.82) uniformly. Therefore, every distance between any two points in the first ring blob should scale to smaller size as the constant increase. Furthermore, to be more precise, one can compute the ratio of the two scales, the size of the first ring blob and the separation between two blobs. At first, recall that there are two typical microscopic flux vectors in the shallow blob and correspondingly we have: J + 8k 1 k 2 k 3 ; J + 8k 1 k 2 k 3 ; J ++ J 0: (4.84) 126 If we ignore the microscopic difference between J ij and consider r i;j are roughly the same, the bubble equations (4.82) can be further simplified. When the system is very close to the merger point,r 0 ! 0, we have: M X j6=i J ij r i;j M 2 q i 8k 1 k 2 k 3 r 1 q i 2r 0 C IJK X I X J M 0 k K i = 1; 2;:::M; =) r 0 r 1 C IJK X I X J M 0 k K 8Mk 1 k 2 k 3 ; (4.85) wherer 1 is the typical distance between two points in the first ring blob. This show that the ratio of the first ring blob’s size and the separation between two blobs is fixed to the value defined in (4.85) at the merger point. This ratio is roughly determined by the fluxes between two blobs and the fluxes inside of the first ring blob. Therefore, from the above analysis, we see there exist a similar scaling solution at the merger point through the irreversible merger of two black ring blobs. 4.3.3 Numerical results for a simple merger In order to see the scaling solutions explicitly and verify that there are no CTC’s globally, we constructed several numerical examples and we now discuss a representative case. We takeM =M 0 = 210 =Q and take the most biased distribution of points for the two blobs. That is we put all positive charges on a point and all negative charges on the other point for both of the blobs. Therefore, we have two points of charges, ‘+105’ and ‘105’, for each blob and we put them all on the common axis as in fig. 4.4. Therefore, it preservesU(1)U(1) symmetry. Then, we take the following flux parameters setup: n I 1 2Mk I = ( 50; 60; 40 ); a I 0 = ( 110; 560; 50 ); n I 2 2Mk 0I = ( 80; 50; 45 ); X I = (x;290; 230 ) ; (4.86) 127 +1 +Q +Q -Q -Q ρ σ Δ 1 Δ 2 Figure 4.4: The layout of the GH points for two shallow black rings 1800 2000 2200 2400 2600 2800 3000 3200 10000 20000 30000 40000 50000 60000 Figure 4.5: Three branches of the bubble equations’ solutions. The plot shows the ring positions, r, and r 0 , along the horizontal axis withjJ int L j plotted on the vertical axis. The separations between two points in the blobs are too small to resolve. There are three branches: (i) The single, nearly vertical line in the center (in green) for which all four GH points remain extremely close together; (ii) The two outermost curves (in blue) where the two rings become progressive more widely spaced asjJ int L j! 0; (iii) The two curves (in red) that meet branch (ii) and show the scaling merger in which the two rings meet atjJ int L j = 0. wherex is varied from about46 up to its merger value 5 ofx19:7. By solving the bubble equation numerically, we have the results of how the distances change withJ int L . The results are shown in fig. 4.5. As is evident from the graph, there are three sets of solutions to the bubble equations. Branch (i) exists for all values ofJ int L 5 Notice that due to the strong bias distribution, the merger condition is slightly modified by the internal fluxes in the two blobs 128 and has all four GH points in a very close cluster that scales asJ int L ! 0. This is appears as a very steep line at the center of fig. 4.5. The existence of the branch (i) is in fact outside of the scope of the analysis in previous subsection because the approximation in (4.74) is not valid before the merger. Branches (ii) and (iii) only appear at a bifurcation point when one hasjJ int L j 43; 500 orx38:3, and represent solutions in which the four GH points separate into two sets of very close pairs. On branch (ii) the two pairs move apart asJ int L ! 0 and the locations of the two bubbled rings is given by (4.77). Branch (iii) is the scaling merger solution that was described in the previous subsection. We have done extensive numerical searches for CTCs in all these solutions and we find that branch (i) is completely unphysical, with large regions of CTC’s, but that branches (ii) and (iii) are physical and have no CTC’s. Finally, one can compute the horizon area,M, of the corresponding classical black ring by pluggingn I (4.58),N I (4.60) andJ L (4.68) into the horizon area formula: M 2n 1 n 2 N 1 N 2 + 2n 1 n 3 N 1 N 3 + 2n 2 n 3 N 2 N 3 (n 1 N 1 ) 2 (n 2 N 2 ) 2 (n 3 N 3 ) 2 4n 1 n 2 n 3 J L : (4.87) The absolute number does not immediately convey useful information. On the other hand, we can compare this to the “maximal horizon area”,M 0 , of a black ring with the same values ofN I andn I , but withJ L = 0. For the configuration in (4.6) we find: R M M 0 0:14 ; (4.88) Thus the result of this merger of two shallow black rings is a microstate of a black ring with a macroscopic horizon area. We have studied several other such mergers with different values of flux parameters and found a number of solutions that are free of CTCs and have even higher values ofR. Indeed, one can arrange a very high value ofR if one 129 takes the outer ring to rotate in the opposite direction to the inner ring. One can achieve this in the foregoing example, (4.86), by taking, for example, X I = (410;290; 481:27 ) ; (4.89) while leaving all the other parameters unchanged. This configuration is very close to the merger point and hasR 0:638. As one would expect, one generates more entropy by merging states whose angular momenta are opposed to one another. 4.4 Some general features about mergers and scaling solutions So far, we have studied two kinds ofU(1)U(1) invariant irreversible mergers and the results are scaling solutions corresponds to the microstate geometries of a black hole and black ring with macroscopic horizon. In this section, we discuss several general features of the mergers and scaling solutions including the necessary general merger condition, their scaling behaviors and metric structures. 4.4.1 The necessary condition for general mergers We have seen two merger conditions:J L = 0 for the merger of a black hole and a black ring andJ int L = 0 for the merger of two black rings. The natural question to ask is: what is the merger condition in order to merge a collection of GH points in general? The sufficient conditions require further study. However, in this subsection, we will show the necessary condition in order to merge a collection of GH points. Consider a system contains M GH points with total charge equal to zero and an isolated GH point of ‘+1’ charge. Suppose each point has the flux parameter equal to 130 k I i and we merge all ofM GH points at~ r. We then have the followingJ L from (3.51) at the merger point: J L = 8j ~ Dj = 8 M X i=1 3 X I=1 ~ k I i ~ r = 8 M X i=1 3 X I=1 ~ k I i r; (4.90) where r is the distance between the merger point and the isolated GH point. The bubble equation for the isolated GH point is: 1 r M X i=1 J 0i = 3 X I=1 ~ k I 0 = M X i=1 3 X I=1 k I i = M X i=1 3 X I=1 ~ k I i : (4.91) From the above two equations, we have: J L = 8 M X i=1 J 0i = M X i=1 ~ J L0i ; (4.92) in the second equality, we use (4.45) and the fact that at the merger point, every unit vectors become identical. Recall thatJ L can be written as the vector sum of “all” flux vectors. However, in (4.92), we only have the vector sum of the partial flux vectors which run between the isolated GH point and the points in the merger. This imply that the vector sum of all flux vectors that connect the points inside of the merger must vanish at the merger point: J int L M X i;j=1 i<j ~ J Lij = 0 : (4.93) On the other hand, one can also merge all of the points included the isolated GH point by further decreasing the total flux vectors between them to zero. Therefore, the corresponding condition will be the vector sum of “all” flux vectors vanish and in other words, J L = 0. And, that is equivalent to require two angular momenta to be equal which is the feature of a classical black hole. 131 4.4.2 Scaling behaviors From the observation of the previous mergers, one can see that an interesting feature of the solutions near the merger is their scaling behaviors. For example, in the twoU(1) U(1) invariant merger solutions we have studied, every distance between the points decrease while keeping the ratios of them fixed. Moreover, these ratios are partially determined by the corresponding flux vectors. To illustrate the above picture more explicitly, a good example is the numerical result in section 4.3.3. There are two branches near the merger point, branch (i) and branch (iii). And both of them show the scaling behaviors but in “different ways”. For example, near the merger point (X 1 19:75), we have the following ratios of the flux vectors and distances for the branch (i): jJ 0j :jJ + 0j :jJ + 0j :jJ ++ 0j 1 : 1:077797 : 0:950461 : 1:028256 ; r 0 :r + 0 :r + 0 :r ++ 0 1 : 1:077783 : 0:950473 : 1:028255 : (4.94) There are six distances between 4 points and four of them have their ratios fixed by the size of the corresponding flux vectors. The other two distances in the blobs, r + and r 0 + 0, are stretched and consequently, the ratios of them with the other distances will be very different from the ratios of their corresponding flux vectors. On the other hand, for the branch (iii), we have: J + 0 +J ++ 0 J + 121855:6 ; J 0 +J + 0 J + 121858:1 ; r + 0 r + 121851:2 ; J 0 +J 0 + J 0 + 0 51730:8 ; J + 0 +J + 0 + J + 0 0 51732:5 ; r 0 + r 0 + 0 51737:7 ; (4.95) 132 therefore, the scaling distances of this branch are the sizes of the blobs and the distances between them. On the other hand, the differences between the individual flux vectors between blobs are ignored and the true scaling ratio is replaced by the combined flux vectors. This shows that this scaling solution is the “coarse grained” system in which the blobs are replaced by a point and the effect of this blob on other blobs or points are replace by the combined flux vectors. From the above observation, one can see there are probably multiple scaling solu- tions for a more complicated mergers by picking different distances to scale. It will be interesting to see how large is this multiplicity for a much more complicated merger within the constraint of no-CTC’s condition. On the other hand, we have mentioned the branch (i) is not physical because the presence of CTC’s. This may also imply the cooperated work of “proper” coarse graining with the bubble equations may be the key to avoid the CTC’s globally. However, the correct meaning of “proper” is not clear so far. We will discuss this “coarse grain” picture in more detail in section 4.6. 4.4.3 Metric structure In this subsection, we look at the metric structures of the scaling solutions near the merger point. The full metric on I R 3 including the warp factor is: ds 2 = (Z 1 Z 2 Z 3 ) 1=3 V (dr 2 + r 2 d 2 2 ); (4.96) where r is the radial coordinate on I R 3 . This metric is asymptotically flat due to the constant term we have added inZ I (3.21). And it become finite locally at GH points due 133 to the adding ofL I (3.18). Between these two extremes, in the region before the local structure of GH points become important, we have: (Z 1 Z 2 Z 3 ) 1=3 V 1 r 2 : (4.97) The coefficient is determined by the charges of GH points at the center. Recall that an extremal black hole (1.22) has this similar behavior that as one approach the singularity, a throat (called AdS throat) is opened up and go deeper and deeper while keeping the circumference fixed. In the classical black hole, this throat will be infinitely long. How- ever, in our case, this throat is “capping off” because Z I become finite at GH points. Therefore, we have the similar behavior of the metric with the classical black hole in the region where the effect of the “local structure” of GH points is unimportant (see fig. 4.6). And the metric begins to deviate from the classical black hole when one is close to the collection of GH points and the local structure becomes more and more significant. As we have seen from the previous mergers, we have some collection of GH points that become very close to each other near the merger point on GH base. When they are very close to each other, the region which is similar to the classical black hole become longer and longer. However, the physical distances between these points are actually determined by the full metric in 4.96. To estimate the physical distance between two very close points, first we can observe the dominate part of VZ I between these two points: V Z I (~ r) 1 2 C IJK q i q j J ij K ij 1 j~ r~ r i jj~ r~ r j j : (4.98) Therefore, the physical distance between these two points is: Z x j x i (Z 1 Z 2 Z 3 V 3 ) 1=6 dx q i q j (J ij ) 2 1=6 Z x j x i 1 p jxx i jjxx j j dx: (4.99) 134 Figure 4.6: The AdS throat get longer and longer near the merger point and caps off with some finite size local structure. where J ij is proportional to the size of the flux vector between point ‘i’ and ‘j’ and defined in (4.45). The integral in (4.99) is equal to independent ofx i andx j . Thus, the physical distance of two points is fixed to some constant proportional to the fluxes between them. From the above analysis, one can see the metric structure of a scaling solution near the merger point is: The AdS throat gets very long but then caps off with finite size local structure (see fig. 4.6). Due to this long throat, these scaling solutions are referred by “deep microstates”. The length of the throat is controlled by the size of the collection of GH points mea- sured on GH base. For the previousU(1)U(1) mergers, it is roughly determined by the distances between two black ring/hole blobs that we are trying to merge. If this dis- tance measured on GH base become zero, the throat become infinite long and thus, the metric become a classical black hole/ring. However, recall that this distance is propor- tional toJ int L which is quantized. Therefore, one can not have the throat continuously become infinite long. Since, for the microstate geometries, one should not consider the system with an infinitely long throat and therefore the one which has the longest throat is the system withJ int L = 1. For a black hole, this corresponds to a deep microstate with J L = 1. 135 From the point view of AdS/CFT correspondence, the feature of the long AdS throat make these deep microstates attractive candidates for dual geometries of typical bound- ary CFT states of black holes/rings. If these deep microstates are the dual geometries, the mass gap in this geometries should match with the mass gap of typical CFT states. To estimate the mass gap, one can consider the lightest excited mode at the bottom. The lightest excited mode is a wave with the longest wavelength determined by the circum- ference of the AdS throat. The circumference around certain angle can be estimated: Z 2 0 (Z 1 Z 2 Z 3 V 3 ) 1=6 rd 2 (Q 1 Q 2 Q 3 ) 1=6 ; (4.100) where Q I are asymptotic charges including physical units and in the approximation, we evaluate the integral at the region where the metric show the AdS throat and thus Z I V Q I =r 2 . The mass will be: m 1 (Q 1 Q 2 Q 3 ) 1=6 : (4.101) The energy will be red-shifted after this mode climb up the long AdS throat. Therefore, the energy observed at the boundary of AdS throat is: E r 0 = m p g 00 j r=r 0 = r 0 (Q 1 Q 2 Q 3 ) 1=2 = r 0 (Q 1 Q 5 Q p ) 1=2 ; (4.102) wherer 0 is the size of the collection of GH points on GH base and in the last step, we convert the charges in M2-M5 system to D1-D5-P system (which is basically one-to-one map) in order to use AdS/CFT correspondence. The size, r 0 , is controlled by J L in a black hole and byJ int L in a black ring. To be more specific, we use the black hole as an example: r 0 J L q 1 +q 2 +q 3 ; (4.103) 136 whereq I are the dipole charges of the black ring that merge with the black hole. We also need to considerQ 1 Q 5 > Q p in order to use AdS/CFT correspondence. In M2-M5 system, this corresponds toq 3 is dominated andq 3 q Q 1 Q 5 Qp . Therefore we have the mass gap for the scaling solution withJ L : E r 0 J L Q 1 Q 5 : (4.104) On the other hand, it was shown that D1-D5 CFT states can be characterized by various ways of breaking an long string of lengthN 1 N 5 into components strings. Each components of the string can carry one unit of J L . Therefore, the typical microstates that dual to the system with macroscopically largeJ L will be corresponds to many com- ponents string with roughly equal size. The length of each component string will be: l component = N 1 N 5 J L : (4.105) The mass gap is proportional to the inverse of this length: E CFT = J L N 1 N 5 : (4.106) One can see the mass gap evaluated at the boundary of AdS throat (4.104) match with the mass gap of the typical CFT states in (4.106). Therefore, the deep microstates are dual geometries of typical CFT states that contribute to the entropy of a black hole with macroscopic horizon. 137 4.5 Triangular solutions So far, all of the mergers we have considered have the axial symmetry on I R 3 . And, the merger is done by tuning the flux parameters to some critical values. However, this way to merge is considerably limited. In this section, we show by breaking the axial symmetry, one can have more possible mergers without tuning flux parameters and we also show the numerical results we have done for a triangle solution. Consider four GH points laid out as in Fig. 4.7 with GH charges: q 1 = + 1; q 2 = 2Q; q 3 = Q; q 4 = Q: (4.107) The bubble equations for this system take the form: J 12 r 12 + J 13 r 13 + J 14 r 14 = 3 X I=1 ~ k I 1 ; J 21 r 12 + J 23 r 23 + J 24 r 24 = 3 X I=1 ~ k I 2 ; J 31 r 13 + J 32 r 23 + J 34 r 34 = 3 X I=1 ~ k I 3 ; J 41 r 14 + J 42 r 24 + J 43 r 34 = 3 X I=1 ~ k I 4 ; (4.108) where only three of them are independent. To merge the three points 2,3 and 4, we need to decreaser 23 ,r 24 andr 34 such that: r 1i r; r ij r; i;j = 2; 3; 4: (4.109) 138 +1 +2Q -Q -Q r 12 r 23 r 34 θ r 42 Figure 4.7: The layout of GH points for a triangular scaling solution. Consequently, the bubble equations for the point 2,3 and 4 become: J 23 r 23 + J 24 r 24 0; J 32 r 23 + J 34 r 34 0; J 42 r 24 + J 43 r 34 0: (4.110) First of all, these equations require the orientation of flux vectors between these three points to form a closed loop. Moreover, they also require the distances between the three points are proportional to the size of the flux vectors: r ij = jJ ij j; 2i<j 4; (4.111) with! 0. Therefore, we have the scaling solution near the merger point. Moreover, (4.111) also shows that near the merger point, the triangle formed by the three GH points will have the same shape as the triangle formed by the three flux vectors between them. However, in order for these three flux vectors can form a closed triangular loop at merger point, they need to satisfy the following condition: Correct orientations:Sign(J 23 ) = Sign(J 34 ) = Sign(J 42 ) 139 Triangle inequality: The sizes of the three flux vectors must satisfy the triangle inequality: jJ 24 jjJ 34 j jJ 23 j jJ 24 j +jJ 34 j: (4.112) If these three flux vectors satisfy the above conditions, they can form a closed triangu- lar loop at merger point, the vector sum of them automatically vanishes and thus, the necessary condition in (4.93) is satisfied. For the layout in Fig. 4.7, there are four free parameters,r 12 ,r 23 ,r 34 and but only three independent bubble equations. Therefore, one can treat the angle, as a modulus of the solution. To meet the merger point, we only need to tune the angle such that the triangle has the shape determined by the flux vectors. In other words, for the system with the flux vectors have the correct orientations and satisfy triangle inequality, one do not need to fine-tune the flux parameters to meet the merger condition. Instead, one can continuously tune the angle,, to some critical value in order to merge the three points. 4.5.1 Numerical results Even though the bubble equations are not enough to guarantee the triangular solutions are free of CTC’s globally, we did find several examples are free of CTC’s under careful numerical checking. One such example has the following parameters: q i = (1; 210;105;105); k 1 i = (0; 525; 1200; 2210); k 2 i = (0;20000; 16000; 7887); k 3 i = (0; 6400; 1613; 7900); (4.113) where,i = 1;:::; 4. For the above flux parameters, we have the followingJ ij : J 23 = 8:0446 10 8 ; J 34 = 4:9063 10 8 ; J 24 = 1:1046 10 9 : (4.114) 140 Clearly, they satisfy the orientation requirement and triangular inequality. By solving the bubble equations numerically, we find the overall size of the ring blob depends on the shape of the triangle formed by the three charges. In Table 4.2 we show how the size of the triangle changes as we vary the angle,. One can see as the angle is changed toward a critical value, given by c 1:13638458710805705::: ; (4.115) the distances between three points (r 23 ,r 34 ,r 24 ) shrinks such that: r ij c : (4.116) Moreover, we have checked extensively the no-CTC’s conditions, (3.31) and (3.33) numerically for this solution and it is indeed free of CTC’s. Therefore, it satisfy the requirement of a microstate. As the system approach the merger point, it will look more and more like a classi- cal black ring. The horizon area of this classical black ring can be obtained from the macroscopic properties of the merger solution. The total electric and dipole charges of this system are fixed during the merger and can be obtained by plugging the parameters in (4.113) into the following formula: n I = 2 4 X i=2 k I i ; N I = 2C IJK 4 X i=1 ~ k J i ~ k K i q i 1 2 C IJK n J n K : (4.117) On the other hand, the angular momentum,J L , depends on the angle. At the critical angle c , we haveJ int L vanished andJ L is simply: J L 8jJ 12 +J 13 +J 14 j : (4.118) 141 r 12 r 23 r 34 r 24 r 34 =r 23 r 24 =r 23 0 580.889 28.601 19.150 47.751 .66954 1.6695 .2 532.623 27.820 18.577 46.175 .66776 1.6598 .4 537.742 25.439 16.851 41.482 .66238 1.6306 .6 546.005 21.341 13.943 33.779 .65333 1.5828 .8 557.025 15.323 9.8136 23.251 .64046 1.5174 1 570.302 7.0865 4.4196 10.178 .62366 1.4363 1.1 577.612 2.0172 1.2381 2.8049 .61375 1.3905 1.13 579.878 .36089 .22035 .49664 .61058 1.3762 1.136 580.335 .021823 .013311 .029969 .60993 1.3732 1.13635 580.361 1.96310 3 1.19710 3 2.69510 3 .60990 1.3731 1.136383 580.364 9.00810 5 5.49410 5 1.23710 5 .60989 1.3731 1.136384586 580.364 6.28910 8 3.83610 8 8.63510 8 .60989 1.3731 1.1363845871 580.364 4.57310 10 2.78910 10 6.27910 10 .60989 1.3731 1.136384587108 580.364 3.20710 12 1.95610 12 4.40310 12 .60989 1.3731 Table 4.2: Numerical results of a triangular solution This table shows the distances,r ij , between pointi and pointj in the triangle solution as a function of the modulus,. (See Fig. 3.) The angle,, is the angle between~ r 23 and~ r 34 and it is varied to produce the merger at = c (given below), while all the other parameters are kept fixed. One should also note that the ratios of the distances at merger are precisely the ratios of the fluxes:jJ 34 =J 23 j 0:609893 andjJ 24 =J 23 j 1:37308. 142 With the values of charges and angular momentum above, we found out the horizon area is non-zero. Moreover, we computed the horizon area ratio defined in (4.88) and we foundR 0:103. Therefore, the corresponding classical black ring has macroscopic horizon area and the scaling solution represent it’s microstate. 4.5.2 Abysses and the puzzle As with the scaling solutions from the U(1) U(1) invariant mergers, this triangle solution also has a very long AdS throat. However, the crucial difference is that the factor that control the length of the throat can be non-integer for triangle solutions. This factor is the vector sum of the flux vectors between the three points. Even though all of the three flux vectors are quantized, by continuously changing the angle between these vectors, one can have non-integer vector sum. Therefore, by adjusting the angle, one can have this factor, the vector sum of three flux vectors, arbitrarily close to zero and consequently, the AdS throat of the system becomes arbitrarily long. Therefore, this merger solution was referred by “abyss” in [BWW07b]. The existence of this abyss solution present a puzzle from the point of view of AdS/CFT correspondence. As we have mentioned in section 4.4.3, the length of the throat control the size of the mass gap. As the length of the throat become arbitrarily close to infinite, the red-shifted energy of the lightest excited mode climbing up the throat from the bottom will become arbitrarily small. Therefore, the mass gap will vanish for the abyss solution. However, just like the spectrum of quantum system in a box must be discrete, the mass gap of the states of CFT can not be zero. The most direct and easiest way to solve this puzzle is probably cutting off the throat by restoring the quantization of the vector sum of the flux vectors. It is not clear how to achieve this but one can have the following reasonable guesses: 143 The angles on I R 3 should be quantized in such way that the vector sum of the flux vectors is quantized. The individual ~ J Lij should be treated as quantum spins and therefore, the sum- mation of them obey the familiar quantum spin summation rule and therefore, the result should be quantized. 4.6 A coarse-graining picture of microstate geometries The essential problem of identifying a bubble geometry as a microstate geometry is checking that there are no CTC’s globally in the space. In section 4.3.3, we have found a unexpected merger solution with two points in each ring blob are relatively wide sep- arated than the scaling solution. This unexpected solution has a large region contains CTC’s around and between GH points. The reason that cause the appearance of CTC’s can be understood as the GH points in each ring blob are too widely separated. This may imply the system need to do some coarse graining by treating each ring blob as a small “dot” (element) in order to avoid CTC’s. Therefore, we have the following proposal to ensure there is no CTC’s globally. We believe it can be done by “properly” applying some coarse graining procedure and the bubble equations. The coarse graining proce- dure replace some subset of GH points by one “element”. Therefore, after the coarse graining procedure, the remaining objects in the space are several “elements” and the fluxes between them. These fluxes will depend on their internal structures. Moreover, the coarse-grained bubble equations constrain the distances between these elements. In order for this coarse graining procedure to be valid, one further requires the distances between the elements are much larger than the size of these elements. Additionally, one can imagine these elements can be isolated and therefore they must be well-behaved 144 blobs. This means that there will be no CTC’s when the system contains only one ele- ment. From the above observation, we have the proposal of the conditions that each element needs to satisfy: An element has total GH charge equal to zero. An element has positive dipole and electric charges The size of element is small compared to the distances with other elements If this picture is true, probing the moduli of the microstate geometries formed by a particular configuration of elements is equivalent to probing the moduli of the solution space of corresponding bubble equations. Even if there are constraints from the above conditions, there are still many possible elements. To begin, one should start from the simplest picture and the simplest element one can think of is a shallow ring-blob. It satisfy the first and second conditions obvi- ously. Moreover, if one use irreversible merger to merge multiple ring-blobs, the merger solutions are expected to be scaling solutions and consequently the third condition is protected by the scaling behaviors. Therefore, we have the following picture of the microstate geometries of a black ring/hole: By a more generalized irreversible merger, one can merge some number of shallow ring-blobs. The result should be a microstate of a black ring. If we merge multiple shallow ring-blobs with an GH point of ‘+1’ charge, the result should be a microstate of a black hole. In order to understand this picture better, we need to know the sufficient conditions to have more general merger solutions. From section 4.4.1, we know the necessary condition is that the vector sum of all of the flux vectors between these elements must vanish. Additionally, in section 4.5, we see this necessary condition is sufficient to merge three GH points to have triangle scal- ing solutions. Notice that, even we produce the triangle solutions of three GH points, it can be generalized to the triangle solutions of three shallow ring-blobs by replacing 145 each point with a ring-blob. As long as the flux vectors between the three ring-blobs form a closed triangle loop, we should have similar merger solution. Although the con- straints on the flux vectors for the more complicated merger require further study, one would expect for the merger of many elements (ring-blobs), the constraint may be even weaker. Also, as with the previous mergers we have studied, the irreversible merger require X I , the relative difference between any two particular gauges that make the distribution of the flux parameters in two different ring-blobs appears uniform, to be macroscopic. To generalize this story, one can define a three-dimensional gauge graph (see fig. 4.8). In the graph, one have three axes representX 1 ;X 2 andX 3 . With some reference point, one can pinpoint each ring-blob at the corresponding gauge that make the distribution of flux parameters in the blob appears uniform. Notice again that the absolute gauge is not physical while the relative differences between these gauges are all physical. Therefore, the gauge symmetry in (3.25) manifest itself as “global” translation symmetry in this graph 6 . And, fixing the gauge choice is equivalent to fixing the position of the origin and therefore all of the flux parameters in every ring blob are determined from their rela- tive position with the origin. The picture of the microstate geometry of a black ring/hole with macroscopic horizon in this graph will contain a lot of points widely separated and their distribution occupy large volume while the microstate geometries of zero entropy black hole/ring have their ring-blobs confined in microscopic box. To be more pre- cise, if the number of ring-blobs, N, and typical number of GH points in each blobs, M, both are large, there are three scales present: Macroscopic (O(NMk I )), middle scale (O(Mk I )) and microscopic scale (O(k I )). From this distinguish, the separations between ring-blobs in theX I gauge graph only need to be middle scale in order to let the corresponding classical black hole/ring has macroscopic horizon. 6 However, notice that there is no rotation symmetry in this graph 146 X 2 X X 1 3 Figure 4.8: TheX I gauge graph. Each point represent a gauge that make a particular ring-blob has uniform flux parameter distribution. In this picture, the counting of microstates that contributed to the entropy may come from several different ways. The most obvious and countable way is the number of arbitrary partitions of the total dipole and electric charge to each ring-blob. Recall that although the asymptotic charges are not additive, the dipole and electric charges are. Therefore, one can arbitrarily partition the total dipole and electric charges to all of the ring-blobs. For different partition, the configuration of the system in theX I gauge graph is different. And, for the microstates that correspond to a black hole/ring with macroscopic horizon, we need the distribution of the ring blobs is widely spread. By estimating the resulting J L for some configurations, one can show that we need the distribution of three electric charges must have uncorrelated fluctuation in order for the system to have J L less than maximal J L . The other possible way is the multiplicity of the merger solutions. In section 4.3.3, we found two branches of merger solutions 147 and they show different scaling behaviors near the merger point. Although one of them is shown to be not physical, it is still quite possible that there is large multiplicity of physical merger solutions for a more complicated merger that merger large number of ring-blobs. Nevertheless, all of these pictures and proposal require further investigate to verify. 4.7 Conclusion In this chapter, we have shown that a generic blob that contains large number of GH points with total GH charge equal to ‘+1’ and small varied flux parameters are corre- sponding to a microstate of a classical maximally spinning black hole. On the other hand, a similar generic blob with total GH charge equal to zero will be a microstate of a black ring with zero horizon area. Moreover, by arranging non-parallel charges, we produce an irreversible merger of two shallow microstates withU(1)U(1) invariance. The result is a scaling solution which corresponds to a deep microstate of a black hole/ring with non-vanishing horizon. These deep microstates have very long but finite AdS throats. The length of the throat is limited by the quantization of the fluxes. The long but finite AdS throat means the system has small but finite mass gap. It was further shown that this mass gap match the mass gap in dual CFT. On the other hand, we also showed the numerical result of a triangular scaling solu- tion produced from a more general merger withoutU(1)U(1) invariance. In which, one can reach the merger point by changing the angle of the triangle “continuously” to some critical angle which is predetermined by the flux vectors between the three points. Therefore, in triangular solution, one can come arbitrarily close to the merger point and consequently, the length of the AdS throat of the solution become arbitrarily long. This 148 present a puzzle from the point view of AdS/CFT correspondence. Because, the system with an infinite AdS throat will have a continuous spectrum while the spectrum of the quantum states on dual CFT in finite box should be discrete. One of the solutions is by quantizing the angle on I R 3 . Another is to treat the individual flux vectors, ~ J Lij , as some quantum spins. Both of these methods cut-off the AdS throat by limiting the ability of fine-tuning the angle. More of the other possible solutions of this puzzle are proposed in [BWW07b]. By studying merger and deep microstates, it leads us to several directions that we think are worth to pursue. First of all is to study the puzzle of arbitrarily long AdS throat and the possible solutions of it. The puzzle is certainly interesting and chal- lenge because the solution may give us some hints about quantum physics underline the eleven-dimensional supergravity. Furthermore, we propose a coarse-graining picture of microstate geometries. In the simplest picture, one can think the microstate geome- tries of a black ring/hole with macroscopic horizon as a gas of shallow ring-blobs. To understand and verify this picture require further study in future. 149 Chapter 5 Conclusion The two puzzles about black holes, namely black hole microstate problem and infor- mation paradox, provide great challenges that need to be solved in the correct quantum gravity theory. It is unclear at this moment what this quantum gravity theory should be. However, in this thesis, we took the point of view from string theory and supergravity. We ask if the classical geometry of a black hole is thermodynamic description, what is the microstate geometries of a black hole in classical statistical mechanics description? To understand this question, a good analogy [BW07] is the thermodynamic description of a gas and kinetic description of molecules in classical statistical mechanics. From this point of view, a black hole’s asymptotic charges correspond to thermodynamic quantities of a gas while the local structure of microstate geometries corresponding to details of the motions of molecules in particular microstates. Therefore, in this thesis, we search the microscopic parameters that describe these microstate geometries of a black hole in statistical mechanic sense. Furthermore, the microstate geometries we have found and analyzed are geometries that are horizonless, free of CTC’s and singularities, and asymptotically flat. From string theory point of view, some of these microstate geometries should be the solutions of supergravity. Therefore, we review the general method of obtaining three-charge BPS supergravity solutions. Based on this method, we study a particular solution on I R 4 base space that corresponding to a black ring with a black hole at center. Furthermore, we generalize the result to a black ring with a vertically shifted black hole and use this system to study an adiabatic process that bring a black hole from infinity to 150 the center of the black ring. We showed that if the ring is smaller that some critical size, the ring will merge with the black hole. By studying the entropy change before and after merger, we found out that in general this merger is irreversible. The reversible merger happen only when the black ring and black hole satisfy all of the four conditions listed in the conclusion of chapter 2. On the other hand, the three-charge BPS supergravity solutions on I R 4 base space can not be microstate geometries because they either have horizons or singularities. How- ever, it was shown by replacing I R 4 base with an ambipolar Gibbons-Hawking base, one can obtain geometries which are horizonless, smooth everywhere and asymptotically flat. The solutions on Gibbons-Hawking base have non-trivial two cycles and therefore called bubble geometries. The bubble geometries are specified by the positions of GH points and their charges and flux parameters. However, requiring the metric is free of CTC’s impose some constraints on these local parameters. Particularly, by requiring there is no CTC’s on every GH points, one obtain bubble equations which constrain the position of GH points when the charges and flux parameters are specified. However, the constraints from global absence of CTC’s are in general quite non-trivial and in most situation, are unclear. Based on the bubble geometries, we proposed a generic blob that contains large number of GH points with total GH charge equal to ‘+1’ and microscopically varied flux parameters is a microstate geometry of a maximally-spinning black hole while a similar blob with total GH charge equal to zero is a microstate geometry of a zero- entropy black ring. Furthermore, by arranging non-parallel charges, we produced the U(1) U(1) invariant irreversible merger of two generic blobs. By solving bubble equations and checking other no-CTC’s conditions numerically, we found the result of the merger is a scaling solution. Depending onJ L of the scaling solution, it corresponds to a microstate geometry of a black hole or a black ring with macroscopic horizon. These 151 microstate geometries produced from theU(1)U(1) irreversible mergers have very long but finite AdS throats and therefore are called “deep microstates”. Furthermore, it was shown that these microstate geometries have the same mass gaps as the typical CFT states on boundary. Therefore, from the point view of AdS/CFT correspondence, these microstate geometries can be the dual geometries of some typical CFT states. We also studied the more general mergers which have no axial symmetry on I R 3 . Particularly, we found the triangular scaling solution by merging three GH points. We found that as one arrange the angle of the triangle formed by the three GH points such that the shape is the same as the triangle formed by three flux vectors, the three GH points merge. This makes it possible to get arbitrarily close to the merger point by continuously adjusting the angle. The AdS throat becomes longer and longer as one approach the merger point. Therefore, for triangle solution, one can adjust the angle such that the solution has arbitrarily long AdS throat. This presents a puzzle from the point view of AdS/CFT because the mass gap of this solution will become arbitrarily small while the mass gap of the CFT states should be finite. Although the correct solution to this puzzle is unclear so far, we propose two directions. One is to quantize the angle on I R 3 and the other is to treat the flux vectors as some quantum spins. Both of them will limit the length of the throat. However, it is unclear how to perform the quantization of the angle or promote the flux vectors to quantum spins correctly and it is worth to do further investigation at these directions. Furthermore, we propose that by combing some coarse-graining procedures and the bubble equations, one can obtain bubble geometries which are free of CTC’s globally. The coarse-graining procedures replace several subsets of GH points by some “ele- ments” with the bubble equations constrain the distance between these elements. From this proposal, we have the following simplest picture for a microstate geometry of a 152 black hole/ring with macroscopic horizon: A gas of shallow ring blobs in which the dis- tances between ring-blobs are very large compared to their sizes (which can be thought of as the general feature of irreversible mergers or scaling solutions.) In this picture, the entropy may come from the many different ways to arbitrarily partition the total dipole and electric charges to these shallow ring-blobs and multiple ways to do the irreversible merger. One can further investigate how the density of microstate geometries in this picture changes with the total dipole and electric charges and compare their dependence with the classical horizon area’s dependence on the charges. This may give us the clues about what are the microscopic parameters to pinpoint a microstate of a black hole/ring in statistical mechanic description. The black hole is an essential object in general relativity. We only have limited understanding of it. In this thesis, we have initiated a study to understand the internal semi-classical structure, i.e. microstates of a black hole/ring. If we can have firmer grasp of these microstate geometries, it will greatly expand our understanding of a black hole. 153 References [AEH97] R. Argurio, F. Englert, and L. Houart. Intersection rules for p-branes. Phys. Lett., B398:61–68, 1997. [Bek72] J. D. Bekenstein. Black holes and the second law. Nuovo Cim. Lett., 4:737– 740, 1972. [Bek73] Jacob D. Bekenstein. Black holes and entropy. Phys. Rev., D7:2333–2346, 1973. [BGL06] Per Berglund, Eric G. Gimon, and Thomas S. Levi. Supergravity microstates for bps black holes and black rings. JHEP, 06:007, 2006. [BMPV97] J. C. Breckenridge, Robert C. Myers, A. W. Peet, and C. Vafa. D-branes and spinning black holes. Phys. Lett., B391:93–98, 1997. [BW05] Iosif Bena and Nicholas P. Warner. One ring to rule them all ... and in the darkness bind them? Adv. Theor. Math. Phys., 9:667–701, 2005. [BW06] Iosif Bena and Nicholas P. Warner. Bubbling supertubes and foaming black holes. Phys. Rev., D74:066001, 2006. [BW07] Iosif Bena and Nicholas P. Warner. Black holes, black rings and their microstates. 2007. [BWW06a] Iosif Bena, Chih-Wei Wang, and Nicholas P. Warner. Mergers and typical black hole microstates. JHEP, 11:042, 2006. [BWW06b] Iosif Bena, Chih-Wei Wang, and Nicholas P. Warner. Sliding rings and spinning holes. JHEP, 05:075, 2006. [BWW07a] Iosif Bena, Chih-Wei Wang, and Nicholas P. Warner. The foaming three- charge black hole. Phys. Rev., D75:124026, 2007. [BWW07b] Iosif Bena, Chih-Wei Wang, and Nicholas P. Warner. Plumbing the abyss: Black ring microstates. 2007. 154 [EEMR04] Henriette Elvang, Roberto Emparan, David Mateos, and Harvey S. Reall. A supersymmetric black ring. Phys. Rev. Lett., 93:211302, 2004. [EEMR05] Henriette Elvang, Roberto Emparan, David Mateos, and Harvey S. Reall. Supersymmetric black rings and three-charge supertubes. Phys. Rev., D71:024033, 2005. [GG05] Jerome P. Gauntlett and Jan B. Gutowski. General concentric black rings. Phys. Rev., D71:045002, 2005. [GGH + 03] Jerome P. Gauntlett, Jan B. Gutowski, Christopher M. Hull, Stathis Pakis, and Harvey S. Reall. All supersymmetric solutions of minimal supergravity in five dimensions. Class. Quant. 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Abstract (if available)

Abstract In this thesis, we discuss the possibility of exploring the statistical mechanics description of a black hole from the point view of supergravity. Specifically, we study five dimensional microstate geometries of a black hole or black ring. At first, we review the method to find the general three-charge BPS supergravity solutions proposed by Bena and Warner. By applying this method, we show the classical merger of a black ring and black hole in general are irreversible. On the other hand, we review the solutions on ambi-polar Gibbons-Hawking (GH) base which are bubbled geometries. There are many possible microstate geometries among the bubbled geometries. Particularly, we show that a generic blob of GH points that satisfy certain conditions can be either a microstate geometry of a black hole or black ring without horizon. Furthermore, using the result of the entropy analysis in classical merger as a guide, we show that one can have a merger of a black-hole blob and a black-ring blob or two black-ring blobs that corresponds to a classical irreversible merger. From the irreversible mergers, we find the scaling solutions and deep microstates which are microstate geometries of a black hole/ring with macroscopic horizon.

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Asset Metadata

Creator Wang, Chih-Wei (author)

Core Title Five dimensional microstate geometries

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 10/04/2007

Defense Date 09/12/2007

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag black hole's microstates,bubbled geometries,microstate geometries,OAI-PMH Harvest,supergravity

Language English

Advisor Warner, Nicholas P. (committee chair), Bonahon, Francis (committee member), Däppen, Werner (committee member), Johnson, Clifford (committee member), Pilch, Krzysztof (committee member)

Creator Email chihweiw@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m844

Unique identifier UC1491819

Identifier etd-Wang-20071004 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-558582 (legacy record id),usctheses-m844 (legacy record id)

Legacy Identifier etd-Wang-20071004.pdf

Dmrecord 558582

Document Type Dissertation

Rights Wang, Chih-Wei

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu