A NEW FORM OF THE C-METRIC KENNETH HONG CHONG MING (B.Sc. (Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2005/06 A New Form of the C-Metric A thesis submitted by Hong Chong Ming @ Kenneth (B.Sc. (Hons.), NUS) In partial fulfillment of the requirement for the Degree of Master of Science Supervisor A/P Edward Teo Ho Khoon Department of Physics National University of Singapore Singapore 119260 2005/06 A New Form of the C-metric Hong Chong Ming @ Kenneth June 2, 2006 Dedicated to my wife, Yivin Jou Yann Ting for her immense moral support and tolerance to my numerous shortcomings, and our son, Alpha Hong Yik Hang for the painstaking but joyful experiences he has brought on us. Acknowledgments There are many people I owe thanks to for the completion of this project. First and foremost, I am particularly indebted to my supervisor, A/P Edward Teo Ho Khoon, for the incredible opportunity to be his student. Without his constant support, patient guidance and invaluable encouragement over the years, the completion of this thesis would have been impossible. Being also my supervisor in my job as a teaching assistant here, his advice and help are indispensable. I have been greatly influenced by his attitudes and dedication in both research and teaching. I am also thankful to my seniors, Liang Yeong Cherng, for his advice on Sibgatullin’s integral method, and Brenda Chng Mei Yuen, for her discussion on the five-dimensional Cmetric. I am also grateful to Tan Hai Siong for his stimulating discussion on the generalized Weyl formalism. Special thanks also to Assistant Professor Sow Chorng Haur for granting me the flexibility in my work. The 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. I am deeply grateful to my wife, Yivin 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 could spend my time on this thesis. i Contents Acknowledgments i Summary vii Notations viii Overview § 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1.2 Historical review of the C-metric . . . . . . . . . . . . . . . . . . . . . . . . . § 1.3 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1.4 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uncharged C-Metric 11 § 2.1 New form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 § 2.2 Coordinate transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 § 2.3 Coordinate transformation to the Weyl form . . . . . . . . . . . . . . . . . . . 16 § 2.4 Weyl form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 § 2.5 Preliminary analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 § 2.5.1 Coordinate range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 § 2.5.2 Curvature singularities and asymptotic flatness . . . . . . . . . . . . . 23 ii CONTENTS iii § 2.5.3 Black hole and acceleration event horizons . . . . . . . . . . . . . . . . 24 § 2.5.4 Symmetric axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 § 2.5.5 Conical singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 § 2.5.6 Zero-acceleration limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 § 2.6 Weyl picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charged C-Metric 27 30 § 3.1 New form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 § 3.2 Weyl form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 § 3.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 § 3.4 Extremal charged C-metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 § 3.5 Ernst solution Rotating C-Metric 43 § 4.1 Pleba´ nski-Demia´ nski solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 § 4.2 Old form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 § 4.3 New form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 § 4.4 Weyl-Papapetrou form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 § 4.5 Physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 § 4.5.1 Coordinate range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 § 4.5.2 Curvature singularities and asymptotic flatness . . . . . . . . . . . . . 52 § 4.5.3 Black hole and acceleration event horizons . . . . . . . . . . . . . . . . 53 § 4.5.4 Symmetric axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 § 4.5.5 Conical singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 § 4.5.6 Torsion singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 § 4.5.7 Rod structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 CONTENTS iv § 4.5.8 Zero-acceleration limit . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotating Ernst Type Solution § 5.1 Rotating charged C-metric 57 59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 § 5.2 Stationary Harrison transformation . . . . . . . . . . . . . . . . . . . . . . . . 62 § 5.3 Rotating Ernst solution 63 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dilaton C-Metric 67 § 6.1 Dilaton charged black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 § 6.1.1 Einstein-Maxwell theory (α = 0) . . . . . . . . . . . . . . . . . . . . . . 69 § 6.1.2 Low energy string theory (α = 1) . . . . . . . . . . . . . . . . . . . . . 69 § 6.2 Emparan and Teo’s solution generating technique . . . . . . . . . . . . . . . . 70 § 6.3 Derivation of the dilaton C-metric . . . . . . . . . . . . . . . . . . . . . . . . 72 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 § 6.5 Physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 § 6.6 Dilatonic Harrison transformation . . . . . . . . . . . . . . . . . . . . . . . . 79 § 6.7 Dilaton Ernst solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 § 6.4 Coordinate transformation Five-dimensional “C-Metric” 84 § 7.1 Review of generalized Weyl solutions . . . . . . . . . . . . . . . . . . . . . . . 84 § 7.2 Five-dimensional “uncharged C-metric” . . . . . . . . . . . . . . . . . . . . . 87 § 7.3 Physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 § 7.4 Davidson and Gedalin’s solution generating technique . . . . . . . . . . . . . 94 § 7.5 Teo’s solution generating technique . . . . . . . . . . . . . . . . . . . . . . . . 96 § 7.6 Five-dimensional “dilaton C-metric” . . . . . . . . . . . . . . . . . . . . . . . 99 § 7.7 Five-dimensional “dilaton Ernst” solution . . . . . . . . . . . . . . . . . . . . 102 CONTENTS v Conclusion and Outlook § 8.1 Conclusion 104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 § 8.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 List of Figures 2.1 Graph of 1/B against mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 The structure of the uncharged C-metric in the x-y coordinate patch. . . . . . 23 2.3 The positions of the rods along the z-axis of the uncharged C-metric. . . . . . 29 3.1 The structure of the charged C-metric in the x-y coordinate patch. . . . . . . 35 3.2 The positions of the rods along the z-axis of the charged C-metric. . . . . . . 36 4.1 The structure of the rotating C-metric in the x-y coordinate patch. . . . . . . 51 4.2 The positions of the rods along the z-axis of the rotating C-metric. . . . . . . 57 5.1 The positions of the rods along the z-axis of the rotating charged C-metric. . 61 7.1 The positions of the rods of the five-dimensional “uncharged C-metric”. . . . 87 7.2 The positions of the rods along the z-axis in the massless limit. . . . . . . . . 92 vi Bibliography [1] § 1.2 A. 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[...]...Summary The C- metric describes a pair of black holes uniformly accelerating apart from each other We advocate a new form of the C- metric, which is related to the traditional one by a coordinate transformation It has the advantage that its properties become much simpler to analyze We explore the extension of this idea to the rotating (charged) C- metric However, it turns out that the new form of the rotating... in the vacuum If there exists an external electric field, some of these particle-antiparticle pairs may receive enough energy to materialize and become real particles These particles are then accelerated away by the external Lorentz force and describe an uniformly accelerated hyperbolic motion approaching asymptotically the speed of light These two charged particles approach each other until they come... et al found the dilatonic generalization of the charged C- metric for arbitrary dilaton coupling α [49] For each value of α, there exists a three-parameter family of black hole solutions labeled by mass m, the magnetic (or electric) charge q and the acceleration A This solution describes a pair of dilaton black holes uniformly accelerating apart Using the dilatonic generalization of the Ehlers-Harrison... coordinate transformation relating the static C- metric type solution to the Weyl form We will then cast the new form of the uncharged C- metric into the Weyl form explicitly This is then followed by a detailed analysis of the new form of the uncharged C- metric in both coordinate systems In Chapter 3, a new form of the charged C- metric [88] is presented together with the coordinate transformation relating... equations for a pair of charged black holes uniformly accelerating apart The geometrical properties of the C- metric were further investigated by Farhoosh and Zimmerman [68] and the asymptotic properties of the C- metric were analyzed by Ashtekar and Dray [1] In 1983, Bonnor [9] explored in detail the physical interpretation of the vacuum C- metric He transformed the vacuum C- metric into the Weyl form, ... [65] can also be constructed from the C- metric The C- metric and the Ernst solution describe two uniformly accelerated black holes in opposite directions under a source of acceleration The closest analogy is the pair of oppositely charged particles that are created in the Schwinger process Recall that in the Schwinger process virtual and short-lived particles are being created and rapidly annihilated... each of the black holes Furthermore, they pointed out that the vacuum C- metric is a member of the Weyl static axially symmetric class, whose mass sources are determined by the solutions of the Laplace equation The electromagnetic generalization of the vacuum C- metric, which is now known as the charged C- metric, was also discovered by them This can be interpreted as the solution of the Einstein-Maxwell... describes a pair of uniformly accelerated black holes It is then followed by the coordinate transformation relating the old and new forms of the uncharged C- metric After that, we will present a generic coordinate transformation between the static C- metric type solution and the Weyl form This will then be used to cast the uncharged C- metric into the Weyl form explicitly The properties of the uncharged C- metric. .. Each of these instantons corresponds a different way by which energy can be furnished to materialize the pair of black holes and then accelerate them apart It is therefore important to have a rather good understanding of the C- metric solutions at the level of classical general relativity For completeness, we now give a rather brief historical overview of the pair creation of § 1.4 Organization of the. .. Thambyahpillai [50] have found a solution describing an arbitrary numbers of collinear accelerating neutral black holes In 2003, Hong and Teo [88] rewrote the charged C- metric in a new form that simplifies the analysis on the charged C- metric It is to be noted that the C- metric is an important and explicit example of a general class of asymptotically flat radiative spacetimes with boost-rotation symmetry and . C- metric is a member of the Weyl static axially symmetric class, whose mass sources are determined by the solutions of the Laplace equation. The electromagnetic generalization of the vacuum C- metric, . performed the explicit calculation of the pair creation rate of non-extremal black holes. Later on, the problem of the pair creation of dilaton black holes in a background magnetic field was also. accelerating apart from each other. We advocate a new form of the C- metric, which is related to the traditional one by a coordinate transformation. It has the advantage that its properties become much simpler