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Bookbinding Co University of South Wales Bound by Abbey Bookbinding Co. 105 Cathays Terrace. Cardiff CF24 4HU, U.K. 2060331 Tel: +44 (0)29 2039 5882 Email: [email protected] www.bookbindersuk.com Rotating perfect fluid bodies in Einstein's general theory of relativity Paul Henry Messenger April 2005 A submission presented in partial fulfilment of the requirements of the University of Glamorgan for the degree of Doctor of Philosophy Abstract The study of rotating astrophysical bodies is of great importance in understanding the structure and development of the Universe. Rotating bodies, are not only of great interest in their own right, for example pulsars, but they have also been targeted as prime possible sources of gravitational waves, currently a topic of great interest. The ability of general relativity to describe the laws and phenomena of the Universe is unparalleled, but however there has been little success in the description of rotating astrophysical bodies. This is not due to a lack of interest, but rather the sheer complexity of the mathematics. The problem of the complexity may be eased by the adoption of a perturbation technique, in that a spherically symmetric non-rotating fluid sphere described by Einstein's equations is endowed with rotation, albeit slowly, and the result is expressed and analysed using Taylor's series. A further consideration is that of the exterior gravitational field, which must be asymptotically flat. It has been shown from experiment that, in line with the prediction of general relativity, a rotating body does indeed drag space-time around with it. This leads to the conclusion that the exterior gravity field must not only be asymptotically flat, but must also rotate. The only vacuum solution to satisfy these conditions is the Kerr metric. This work seeks to show that an internal rotating perfect fluid source may be matched to the rotating exterior Kerr metric using a perturbation technique up to and including second order parameters in angular velocity. The equations derived, are used as a starting point in the construction of such a perfect fluid solution, and it is shown how the method may be adapted for computer implementation. Contents Preface IX 1 Introduction 1 1.1 Background ............................ 1 1.2 General relativity ......................... 2 1.3 Rotation and general relativity .................. 3 1.4 Overview of the work presented ................. 6 2 General relativity - an introduction 8 2.1 An overview of general relativity ................. 8 2.1.1 Bianchi identity ...................... 9 2.2 Physical explanation of energy conservation .......... 11 2.3 Einstein tensor and Einstein's equations ............ 12 2.3.1 Einstein tensor ...................... 12 2.3.2 Einstein's equations .................... 12 2.4 Solutions of Einstein's equations ................. 13 2.4.1 Internal solutions ..................... 14 2.5 Extrinsic curvature ........................ 18 3 Boundary conditions 21 3.1 Introduction ............................ 21 3.2 The junction conditions ..................... 22 3.2.1 Darmois conditions .................... 23 CONTENTS VI 3.2.2 Lichnerowicz conditions ................. 24 3.2.3 O'Brien and Synge conditions .............. 24 3.3 Equivalence of sets of conditions ................. 24 3.3.1 An example ........................ 26 3.3.2 Concluding remarks.................... 29 4 Rotation 31 4.1 Introduction ............................ 31 4.2 Metrics for rotating systems ................... 31 4.3 Kerr metric ............................ 35 4.4 A missing result ?......................... 37 4.5 Evidence of ergospheres ..................... 39 4.6 Conclusion and discussion .................... 39 4.7 A final point of interest ...................... 40 5 Robertson Walker source endowed with first order rotation 42 5.1 Introduction ............................ 42 5.2 The non-rotating case ...................... 43 5.2.1 Continuity of the gab ................... 45 5.2.2 Continuity of the extrinsic curvature .......... 45 5.3 The slow rotation case ...................... 48 5.3.1 Continuity of the g^ ................... 49 5.3.2 Continuity of the Kab ................... 49 6 A general first order rotating perfect fluid source 53 6.1 The non-rotating case ...................... 53 6.2 Slowly rotating systems ...................... 58 6.2.1 Einstein's equations .................... 60 6.3 Darmois boundary conditions .................. 62 6.3.1 Continuity of the extrinsic curvature .......... 64 CONTENTS vii 7 Sources of second order rotation 73 7.1 Introduction ............................ 73 7.2 The source metric ......................... 74 7.3 The boundary ........................... 74 7.4 Einstein's equations ........................ 74 8 Perfect fluid sources for the Kerr metric with second order rotation 77 8.1 Introduction ............................ 77 8.1.1 The metrics ........................ 77 8.2 Continuity of the extrinsic curvature .............. 80 8.3 Einstein's equations ........................ 83 8.4 Pressure and density ....................... 85 8.5 Boundary descriptions ...................... 86 8.6 Generating a solution ....................... 87 8.7 Generation of solutions ...................... 91 8.7.1 Construction of a solution when X^ + ^ j^ 0 ...... 92 9 The Schwarzschild interior solution endowed with rotation 98 9.1 Introduction ............................ 98 9.2 Boundary calculations ...................... 99 9.3 Metric components ........................ 99 9.4 Density at the boundary ..................... 100 9.4.1 Finding the values of /o and C in the limits of K = 0 and K = 1 ......................... 102 9.5 Behaviour of the boundary .................... 106 9.6 Angular velocity .......................... 106 10 General second order metrics 108 10.1 Introduction ............................ 108 10.2 Metrics and coordinates ..................... 108 CONTENTS Vlll 10.3 Continuity of the metric tensor ................. 112 10.4 Einstein's equations ........................ 115 10.5 Continuity of the extrinsic curvature ..............115 10.5.1 The boundary ....................... 115 10.5.2 Simplification and matching of the Kab ......... 116 10.6 Continuity conditions summary ................. 125 11 Robertson Walker fluid sources endowed "with second order rotational parameters - analytic solutions 128 11.1 Introduction ............................ 128 11.2 Metrics ............................... 129 11.3 Einstein's equations ........................ 129 11.3.1 Simplification of Einstein's equations .......... 132 11.4 The search for analytic solutions ................. 136 11.5 Result development ........................ 139 11.6 A subclass of solutions ...................... 142 11.7 Particle radial velocity ...................... 142 11.8 A subclass of solutions continued ................ 145 11.9 A specific example ........................ 148 12 Results, conclusions and future work 150 Expansions of terms 154 Preface I would like to thank Professor Ron Wiltshire for his eternal enthusiasm and optimism, regardless of how often I had a query and also the rest of the staff from the Division of Mathematics for their support and advice. In addition I should like to extend my thanks to the LRC staff for speedily obtaining all my requests for books and papers. On a family note, thanks to my mother for reading various versions of the thesis and tracking down spelling and grammatical errors, and also to my cats for their never failing companionship on those long dark nights of writing up and worry. I would also like to thank my wife for her continual "Are you ever going to finish your Ph.D. thing ?" - a highly motivational factor in completing this work. And not forgetting The Clangers (Major Clanger, Mother Clanger, Small Clanger, Tiny Clanger and numerous Aunts and Uncles) and their friends (The Soup Dragon, The Froglets, The Glow Buzzers, The Skymoos, The Cloud, The Iron Chicken and The Music Trees) who first inspired me to look through a telescope. IX Chapter 1 Introduction 1.1 Background Almost every object in the Universe displays some degree of rotation. Locally, the Earth rotates about its own axis, the Moon rotates about the Earth while the planets rotate about the Sun. These facts may be established today by anyone using time lapsed photography (see figure 1.1). In historical times for example, the eighteenth century models of the Solar System called orreries were built to establish the relative motions of the planets (see figure 1.2). The rotation of the Sun about its own axis may be established by observing the motion of sunspots (see figure 1.3). On a universal scale the Sun is located in the outer arm of the Milky Way spiral galaxy (see figure 1.4). The galaxy rotates about its galactic centre. Hence the Sun and the Solar System also rotate about the galactic centre. The Solar System currently completes one rotation about the centre in approximately 220 million years, at a velocity of about 250 km/s. As explained in the narrative of figure 1.4, it would be impossible to photograph the structure of the Milky Way galaxy, but photographs of other spiral arm galaxies and whirlpool galaxies immediately give the sense of rotation about the centre (see figures 1.5 and 1.6). On a still larger scale the Universe itself 1.2 GENERAL RELATIVITY 2 may be rotating. The idea of a rotating universe is at present a topic of discussion, with interesting research papers being produced. 1.2 General relativity Towards
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