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Matter Models in General Relativity with a Decreasing Number of Symmetries

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Matter Models in General Relativity with a Decreasing Number of Symmetries Charles University in Prague Faculty of Mathematics and Physics DOCTORAL THESIS Norman Gürlebeck Matter Models in General Relativity with a Decreasing Number of Symmetries Institute of Theoretical Physics Supervisor of the doctoral thesis: Prof. RNDr. Jiří Bičák, DrSc., dr.h.c. Study program: Physics Specialization: Theoretical physics, astronomy and astrophysics Prague, 2011 Acknowledgements First, I am much obliged to my supervisor Jiří Bičák. With his deep understanding of physics, he was a constant help. Additionally, I always enjoyed and benefited from his crash courses in Czech arts and culture. I am greatly indebted to David Petroff. He is a splendid colleague and friend. With his deliberate and candid manner, discussions could take surprising turns providing insights beyond physics. I also thank his family, which received me several times so warmly in their home. I thank Jörg Frauendiener and Tomáš Ledvinka for agreeing to be the reviewers of my thesis. Furthermore, I thank the colleagues at the institute for creating a nice working atmosphere. All of them and especially Tomáš Ledvinka, Otakar Svítek and Martin Žofka had always an open ear for my question and were never short of helpful remarks. I sincerely appreciate the constant work of Jiří Horáček and Oldřich Semerak that kept my funding going like the Grant No. GAUK 22708. I was also financially supported by the Grant No. GAČR 205/09/H033. The friendship of Ivan Pshenichnyuk, Martin Scholtz and of the spaceship crew Otakar Svítek and Martin Žofka was a big support in the last years (even though we did not meet with success in all space missions). I owe gratitude to Martin Scholtz and Martin Žofka, in particular, for all the translations they did for me and all the help with public affairs. Let me also thank several institutes that invited me during the last years. These are the Albert Einstein Institute in Golm, the Institute of Theoretical Physics of the Friedrich Schiller University in Jena and the University of Wisconsin in Milwaukee. Among those who made these stays a memorable experience are Marcus Ansorg, John Friedman and Reinhard Meinel. I much appreciated the discussions with them on Dedekind ellipsoids. I am obliged to my family not just for their support in the last years. My father showed me the love for mathematics and my mother that there is also something apart and beyond. My sister was always a role model for me and is greatly responsible for me turning out the way I did. At last, let me thank Julia Häuberer. She helped me a lot in the vicissitudes of my PhD student’s life. She is the best reason imaginable to leave the Dedekind ellipsoids and dipole shells to their own fate for another night and go home. I declare that I carried out this doctoral thesis independently, and only with the cited sources, literature and other professional sources. I understand that my work relates to the rights and obligations under the Act No. 121/2000 Coll., the Copyright Act, as amended, in particular the fact that the Charles University in Prague has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 paragraph 1 of the Copyright Act. In Prague, July 15, 2011 Norman Gürlebeck Title: Matter Models in General Relativity with a Decreasing Number of Symmetries Author: Norman Gürlebeck Institute: Institute of theoretical physics Supervisor: Prof. RNDr. Jiří Bičák, DrSc., dr.h.c. Abstract: We investigate matter models with different symmetries in general relativity. Among these are thin (massive and massless) shells endowed with charge or dipole densities, dust distributions and rotating perfect fluid solutions. The electromagnetic sources we study are gravitating spherical symmetric condensers (including the implications of the energy conditions) and arbitrary gravi- tating shells endowed with a general test dipole distribution. For the latter the Israel formalism is extended to cover also general discontinuous tangential components of the electromagnetic test field, i.e., surface dipole densities. The formalism is applied to two examples and used to prove some general properties of dipole distributions. This is followed by a discussion of axially symmet- ric, stationary rigidly rotating dust with non-vanishing proper volume. The metric in the interior of such a configuration can be determined completely in terms of the mass density along the axis of rotation. The last matter models we consider are non-axially symmetric, stationary and rotating perfect fluid solutions. This is done with a first order post-Newtonian (PN) approximation to the Dedekind ellipsoids. We investigate thoroughly two limits of this 1-PN sequence, where the 1-PN Dedekind ellipsoids become axially symmetric 1-PN Maclaurin spheroids or degenerate to a rod-like singularity. Keywords: Dipole layers in curved backgrounds, rigidly rotating dust, Dedekind ellipsoids, post- Newtonian approximation Název práce: Modely hmoty v obecně relativitě s klesajícím počtem symetrií Autor: Norman Gürlebeck Ústav: Ústav teoretické fyziky Školitel: Prof. RNDr. Jiří Bičák, DrSc., dr.h.c. Abstrakt: V práci zkoumáme modely hmoty s různými symetriemi v obecné relativitě. Mezi nimi jsou tenké (hmotné a nehmotné) slupky s nábojovou či dipólovou hustotou, řešení s prachem či rotující ideální tekutinou. Elektromagnetické zdroje, které studujeme, jsou gravitující sféricky symetrické kondenzátory (zohledňující důsledky energetických podmínek) a libovolné gravitující slupky s obecným testovacím rozložením dipólů. Pro ty jsme zobecnili Israelův formalizmus na případ obecných nespojitých tečných složek testovacího elektromagnetického pole, tj. plošné hus- toty dipólů. Formalizmus je aplikován na dva příklady a použít k dokázání některých obecných vlastností rozložení dipólů. Potom následuje diskuze axiálně symetrického, stacionárního rozložení prachu, který rotuje jako tuhé těleso a má nenulový vlastní objem. Metriku uvnitř takovéto konfi- gurace lze plně určit pomocí hustoty hmoty podél osy rotace. Posledními studovaími modely hmoty jsou stacionární, rotující řešení s ideální tekutinou, která nejsou axiálně symetrická. Používáme zde postnewtonovskou aproximaci (PN) Dedekindových elipsoidů do prvního řádu. Důkladně zkoumáme dvě limity této 1-PN posloupnosti, kdy se z 1-PN Dedekindových elipsoidů stávají ax- iálně symetrické 1-PN Maclaurinovy sféroidy nebo kdy degenerují na tyčovou singularitu. Keywords: Dipólové vrstvy na zakřiveném pozadí, prach rotující jako tuhé těleso, Dedekindovy elipsoidy, postnewtonovská aproximace CONTENTS List of Figures iv List of Tables v 1 Introduction 1 Results . 3 Organization of the thesis . 4 2 Spherical condensers 5 Paper I: Spherical gravitating condensers in general relativity ............... 6 I Introduction . 6 II The classical system . 7 III The Einstein-Maxwell system . 7 IV Condensers . 8 IV.A Classical condensers . 8 IV.B Einstein-Maxwell condensers . 9 References . 11 3 Monopole and dipole layers 12 Paper II: Monopole and dipole layers in curved spacetimes: formalism and examples .. 13 I Introduction . 13 II Monopole and dipole layers in general . 14 II.A The 4-currents for charges or dipoles distributed on a shell . 14 II.B Discontinuities in the potential and the fields . 16 II.C The equivalence of electric charges and magnetic dipoles . 17 III Schwarzschild disks with electric/magnetic charge and dipole density . 18 III.A Asymptotically homogeneous electric and magnetic field . 18 III.B Disks generated by point charges . 19 References . 21 Paper III: Electromagnetic sources distributed on shells in a Schwarzschild background . 23 1 Introduction . 23 2 Stationary fields in a Schwarzschild background . 25 3 A direct approach for spherical shells . 28 4 Discontinuities in the electric field and the potential . 31 References . 33 CONTENTS 4 Axially symmetric, stationary and rigidly rotating dust 35 Paper IV: The interior solution of axially symmetric, stationary and rigidly rotating dust configurations ....................................... 36 1 Introduction . 36 2 Dust configurations in Newton’s theory of gravity . 38 3 Rigidly rotating dust configurations in general relativity . 40 4 The solution of the field equations . 41 5 The non-existence of homogeneous dust configurations . 44 References . 45 4.1 Addendum to Paper IV: The metric in the interior . 46 4.2 The mass density close to the axis of rotation . 46 5 The Newtonian Dedekind ellipsoids 48 5.1 The field equations . 49 5.2 Ellipsoidal figures of equilibrium . 49 5.2.1 Maclaurin ellipsoids . 50 5.2.2 Jacobi ellipsoids . 51 5.2.3 Riemann ellipsoids . 51 5.3 Dedekind ellipsoids . 52 5.3.1 The rod-limit of the Dedekind ellipsoids . 52 5.3.2 The exterior solution . 55 6 The 1-PN corrections to the Newtonian Dedekind ellipsoids 58 6.1 What do we expect of 1-PN Dedekind ellipsoids? . 59 6.2 The 1-PN field equations . 59 6.3 The ansatz for the 1-PN velocity field and the surface . 61 6.4 The 1-PN metric . 63 6.5 The Bianchi identity . 64 6.6 Properties of the solution . 65 6.6.1 The 1-PN corrections to the mass and the angular momentum . 68 6.6.2 Removing the singularity . 70 6.6.3 The surface and the gravitomagnetic effect . 73 6.6.4 The motion of the fluid . 75 6.7 The axially symmetric limit of the 1-PN Dedekind ellipsoids . 76 Paper V: The axisymmetric case for the post-Newtonian Dedekind ellipsoids .... 77 1 Introduction . 77 2 The axisymmetric solution of a generalization to Chandrasekhar and Elbert’s paper ....................................... 77 3 The axisymmetric limit of a generalization to Chandrasekhar and Elbert’s paper ....................................... 79 4 Discussion . 79 A A detailed discussion of Chandrasekhar & Elbert’s work . 81 B Explicit expressions for S1, S3, and r1 ..................... 84 References . 85 6.8 The rod-limit of the 1-PN Dedekind ellipsoids . 86 6.9 An arbitrary coordinate volume . 91 6.10 Concluding remarks . 92 Summary 93 ii CONTENTS Appendix 96 A Index symbols 96 A.1 Definitions . 96 A.2 Properties . 97 A.3 Dimensionless index symbols . 97 A.4 Explicit formulas of the index symbols .
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