BLACK RINGS IN FIVE DIMENSIONS KENNETH HONG CHONG MING (B.Sc. (Hons.), M.Sc. NUS) NATIONAL UNIVERSITY OF SINGAPORE 2013 BLACK RINGS IN FIVE DIMENSIONS KENNETH HONG CHONG MING (B.Sc. (Hons.), M.Sc. NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2013 Dedicated to my wife, Jou Yann Ting for her immense moral support and tolerance to my numerous shortcomings, and our son, Alpha Hong Yik Hang for his shares on taking care of his another two much younger siblings, our daughter, Beta Hong Yik Tong our son, Gamma Hong Yik Zhe for the painstaking but joyful experiences they have brought on us, and (planned but yet-to-be-born) our daughter (??), Zeta Hong Yik Yan for the painstaking but joyful experiences you will bring on us although you not have chance to be with us when this work is being carried out. Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Kenneth Hong Chong Ming 14 August 2013 i Acknowledgments First and foremost, I am particularly indebted to my supervisor, A/P Edward Teo Ho Khoon, for the incredible opportunity to be his student since I was an undergraduate student in 1999. Under his patient supervision and guidance over the years, I have completed various research projects including two UROPS projects (1999–2001), Honours year project (2001–2002), Master degree project (2002–2006), and lastly PhD degree project (2007–2013). It is not an exaggeration that the totality of my knowledge on general relativity, black holes, Maple, LATEX, etc, are all credited to him. Being also my academic supervisor during my initial stage (2002–2005) as a teaching staff here, his advice and help are much indispensable. I have been greatly influenced by his attitudes and dedication in both research and teaching. I am also thankful to Chen Yu. I benefited a lot from him on inverse scattering method and rod structure in our endless conversations and stimulating discussions. Our collaborations have produced two publications which essentially form the main content of this thesis. Special thanks also to department for granting me the flexibility in my work so that research work can be carried out. Certainly, encouragement and help from other colleagues and friends are much appreciated too. I would also like to express my sincere gratitude to my family members in Malaysia. Last but not least, I am deeply grateful to my wife, Jou Yann Ting, for her valuable cooperation in my life and for sharing a major part of the responsibility on family affairs, so that I can spend my time on research work. This thesis is also dedicated to my dearest children, Yik Hang (Alpha), Yik Tong (Beta) and Yik Ze (Gamma). I hope that one day in the future, they will also learn to appreciate the beauty of physics as what I had. iii Contents Declaration i Acknowledgments iii Summary vii List of Figures viii Introduction 1.1 Motivations of higher-dimensional gravity . . . . . . . . . . . . . . . . . . 1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . Review of some known black hole solutions 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Four-dimensional Kerr black hole . . . . . . . . . . . . . . . . . . . . . . 2.3 Five-dimensional Myers–Perry black hole . . . . . . . . . . . . . . . . . . 11 2.4 Emparan–Reall black ring . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Figueras black ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.6 Pomeransky–Sen’kov black ring . . . . . . . . . . . . . . . . . . . . . . . 21 Stationary and Axisymmetric Solutions in Vacuum 29 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Canonical form of the metric . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3 Rod structure of static solutions . . . . . . . . . . . . . . . . . . . . . . . 33 iv § 7.3. Four-dimensional Kaluza–Klein C-metric in Fig. 7.10. The explicit solution is given below: µ1 µ5 ρ µ3 µ2 µ4 , , , µ3 µ2 µ4 µ1 µ5 µ1 µ5 R12 R13 R14 R23 R25 R34 R35 R45 = k2 . 2 µ3 R15 R24 R11 R22 R33 R44 R55 G0 = diag − 2γ0 e (7.8) Using the inverse scattering method, we then perform the following 3-soliton transformation on this seed: (1) 1. Remove a soliton at each of z1 , z2 and z4 , with trivial BZ vectors m0 = (0, 0, 1), (2) (3) m0 = (0, 1, 0) and m0 = (0, 1, 0), respectively; (1) 2. Add back a soliton at each of z1 , z2 , and z4 , with non-trivial BZ vectors m ˜0 = (2) (3) (C1 , 0, 1), m ˜ = (0, 1, C2 ) and m ˜ = (0, 1, C4 ), respectively. Here, C1 , C2 and C4 are the BZ parameters. We note that if C2 = C4 = 0, we recover the ISM procedure of the static electric Kaluza– Klein C-metric above. On the other hand, if C1 = C2 = and z1 = z2 , we effectively recover the ISM procedure of the static magnetic Kaluza–Klein C-metric above. We need to fix some choices on the values of BZ parameters so that the desired solution can be obtained. Counting the rods from the left, we need to join up Rods and 2, as well as, Rods and 5, by requiring that Rod has the same normalised direction as Rod 2, and Rod has the same normalised direction as Rod 5. These two conditions will fix the values of the BZ parameters C1 and C4 . We also need to impose a condition on the BZ parameter C2 so that the normalised direction of Rod does not have a time component. Some preliminary analyses have been carried on the resulting solution and it does seem to have the interpretation as the four-dimensional static dyonic Kaluza–Klein C-metric. t φ w z1 z2 z3 z4 Figure 7.11: The rod sources of the seed solution for rotating electric Kaluza–Klein C-metric when lifted to five dimensions. 126 § 7.3. Four-dimensional Kaluza–Klein C-metric We also expect that it is possible to generate a four-dimensional rotating electric Kaluza–Klein C-metric from a singular diagonal seed given by the rod structure given in Fig. 7.11. The explicit solution is given in (7.6). Using the inverse scattering method, we then perform the following 3-soliton transformation on this seed: (1) 1. Remove a soliton at each of z1 , z2 and z3 , with trivial BZ vectors m0 = (0, 0, 1), (3) (2) m0 = (0, 1, 0) and m0 = (1, 0, 0), respectively; (1) 2. Add back a soliton at each of z1 , z2 , and z3 , with non-trivial BZ vectors m ˜0 = (2) (3) (C1 , 0, 1), m ˜ = (0, 1, C2 ) and m ˜ = (1, C3 , 0), respectively. Here, C1 , C2 and C3 are the BZ parameters. We note that if C2 = C3 = 0, we recover the ISM procedure of the static electric Kaluza–Klein C-metric above. The construction above is also very similar to the case of unbalanced Pomeransky–Sen’kov doubly rotating black ring discussed in Chapter 5. Again, the BZ parameters C1 , C2 and C3 are to be fixed at some specific values so that the resulting solution can be interpreted as the four-dimensional rotating electric Kalaza–Klein C-metric upon dimensional reduction along the w-direction. t φ w z1 z2 z3 z4 Figure 7.12: The rod sources of the seed solution for rotating magnetic Kaluza-Klein C-metric when lifted to five dimensions. A four-dimensional rotating magnetic Kaluza–Klein C-metric is expected to be generated from a similar ISM procedure from a diagonal seed given by the rod structure given in Fig. 7.12. The explicit solution is given in (7.7). Using the inverse scattering method, we then perform the following 3-soliton transformation on this seed: (1) 1. Remove a soliton at each of z1 , z2 and z3 , with trivial BZ vectors m0 = (0, 1, 0), (2) (3) m0 = (1, 0, 0) and m0 = (0, 1, 0), respectively; (1) 2. Add back a soliton at each of z1 , z2 , and z3 , with non-trivial BZ vectors m ˜0 = (2) (3) (C1 , 1, 0), m ˜ = (1, C2 , 0) and m ˜ = (0, 1, C3 ), respectively. Here, C1 , C2 and C3 are the BZ parameters. 127 § 7.3. Four-dimensional Kaluza–Klein C-metric We note that if C1 = C2 = 0, we recover the ISM procedure of the static magnetic Kaluza–Klein C-metric above. Again, the BZ parameters C1 , C2 and C3 are to be fixed at some specific values so that the resulting solution can be interpreted as the four-dimensional rotating magnetic Kalaza–Klein C-metric upon dimensional reduction along the w-direction. t φ w z1 z2 z3 z4 z5 Figure 7.13: The rod sources of the seed solution for rotating dyonic Kaluza–Klein C-metric when lifted to five dimensions. The most general four-dimensional Kaluza–Klein C-metric is the rotating dyonic solution. We conjecture that such a solution can be generated from a singular diagonal seed in five dimensions (7.8) with the rod structure given in Fig. 7.13. Using the inverse scattering method, we then perform the following 4-soliton transformation on this seed: (1) 1. Remove a soliton at each of z1 , z2 , z3 and z4 , with trivial BZ vectors m0 = (0, 0, 1), (4) (3) (2) m0 = (0, 1, 0), m0 = (1, 0, 0) and m0 = (0, 1, 0), respectively; (1) 2. Add back a soliton at each of z1 , z2 , z3 and z4 , with non-trivial BZ vectors m ˜0 = (2) (3) (4) (C1 , 0, 1), m ˜ = (0, 1, C2 ), m ˜ = (1, C3 , 0) and m ˜ = (0, 1, C4 ), respectively. Here, C1 , C2 , C3 and C4 are the BZ parameters. Again, the BZ parameters C1 , C2 , C3 and C4 are to be fixed at some specific values so that the resulting solution can be interpreted as the four-dimensional rotating dyonic Kalaza–Klein C-metric upon dimensional reduction along the w-direction. The ISM construction of the rotating dyonic solution above has made apparent the role played by each solitonic transformation. The solitonic transformations at z = z1 and z = z4 are to generate the electric and magnetic charges respectively. Roughly speaking, the solitonic transformations at z = z2 and z = z3 are to turn on the rotation and provide an additional parameter which can be tuned to eliminate undesired physical properties, for instance, Dirac–Misner singularities, from the resulting solution. 128 § 7.3. Four-dimensional Kaluza–Klein C-metric To the best of our knowledge, the above class of four-dimensional Kaluza–Klein Cmetric solutions (except for the static electric and static magnetic cases) are not known in the current literature. Furthermore, the ISM constructions of the static electric and static magnetic Kaluza–Klein C-metric have not been reported anywhere in the literature. Thus, it is definitely worth to study these problems. Some works are in progress along this direction which we hope to report it elsewhere in the future. 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D2 (1970), 2141–2160. 141 [...]... non-extremal minimally rotating black ring at jψ = 27/32 : an upper branch of thin black rings and a lower branch of fat black rings Fat black rings always have smaller area than the Myers–Perry black hole Their curves meet at the same zero-area naked singularity at jψ = 1 The curves (2.34) and (2.36) are plotted in Fig 2.2 Contrary to what happens for rotating black hole in four dimensions and for the singly-rotating... Emparan and Reall in [44] The new solution that does not exist in four dimensions is a black ring, i.e., a black hole with horizon topology S 2 × S 1 For a certain range of mass and angular momentum, there exists three different stationary black hole solutions, a thin black ring, a fat black ring and a Myers–Perry black hole 13 § 2.4 Emparan–Reall black ring The black ring solution brings forth the violation... doubly rotating black ring that was remarkably simple, considering the generality of the solution The properties of this solution were further studied in [40] It is of obvious interest to generalise these vacuum black ring solutions to include charge This would allow the embedding and study of black rings in string theory, among other possibilities Like five-dimensional black holes, black rings can carry... the generation of the doubly rotating black ring using the ISM By combining the techniques used to generate the S 1 -rotating and S 2 rotating black rings, this solution was first obtained by Pomeransky and Sen’kov [109] in 2006 Although they mentioned that they had obtained the most general doubly rotating black ring solution, only the balanced case was presented in their paper Furthermore, Pomeransky... (2.36) 17 § 2.4 Emparan–Reall black ring aH √ 2 2 Myers-Perry black hole 1 thin black ring fat black ring √ 0 27 32 1 jψ Figure 2.2: (jψ , aH ) phase diagram for five-dimensional (regular) black ring and Myers– Perry black hole rotating along their ψ-directions The dashed curve corresponds to five-dimensional singly rotating Myers–Perry black hole The solid curve for black rings has two branches that meet... for a few well-known solutions including the five-dimensional doubly rotating Myers–Perry solution, the Emparan–Reall singly rotating black ring solution, the Figueras singly rotating black ring solution and the Pomeransky–Sen’kov doubly rotating ring solution The Pomeransky–Sen’kov solution is well known to describe an asymptotically flat doubly rotating black ring in five dimensions, whose self-gravity... known black hole solutions 2.1 Introduction In this chapter, we will review some well-known asymptotically flat black hole solutions in four and five space-time dimensions These include the four-dimensional Kerr black hole, the five-dimensional doubly rotating Myers–Perry black hole, the Emparan–Reall S 1 -rotating black ring, the Figueras S 2 -rotating black ring and the Pomeransky–Sen’kov doubly rotating... This branch of solution meets the singular Myers–Perry black hole The fat black rings always have smaller area than the thin black rings Moreover, as µ → 1, the fat black ring will meet the singular singly rotating Myers–Perry solution, i.e., a naked singularity, at (jψ = 1, aH = 0) For the spherical Myers–Perry solution with rotation in the ψ direction, the corresponding result is 2 aH = 2 2(1 − jψ )... include the four-dimensional Kerr black hole, the five-dimensional doubly rotating Myers–Perry black hole, the Emparan–Reall S 1 -rotating black ring, the Figueras S 2 -rotating black ring and the Pomeransky–Sen’kov doubly rotating black ring We will mainly focus on the physical properties of these solutions In Chapter 3, we will review the formalism and notation used in studying stationary axisymmetric vacuum... Emparan–Reall singly rotating black ring 61 Rod sources of the alternative seed solution for the Emparan–Reall singly rotating black ring 64 4.5 Rod sources of the seed solution for the Figueras singly rotating black ring 65 4.6 Rod sources of the seed solution for the Pomeransky–Sen’kov doubly rotating black ring . BLACK RINGS IN FIVE DIMENSIONS KENNETH HONG CHONG MING (B.Sc. (Hons.), M.Sc. NUS) NATIONAL UNIVERSITY OF SINGAPORE 2013 BLACK RINGS IN FIVE DIMENSIONS KENNETH HONG CHONG MING (B.Sc an asymptotically flat doubly rotating black ring in five dimensions, whose self-gravity is exactly balanced by the centrifugal force arising from the rotation in the ring direction. In this thesis, we generalise. escape. In the past decade, there has been an increasing attention on black holes in higher dimensions. There are a number of reasons to be interested in such studies. Unification of fundamental interactions It