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The Generic Black Hole Singularity Degree of Doctor of Philosophy University of Alberta The Generic Black Hole Singularity by Sharon Marijke Morsink 9 A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Theoret ical P hysics Depart ment of P hysics Edmonton, Alberta Spring 1997 Acquisitions and Acquisitions et BiMiqraphic Senrices , services bibfiogrsphiques The author has granted a non- L'auteur a accordé une licence non exclusive licence aIiowhg the exclusive permettant à la National Li"brary of Canada to Bibîiotbtqpe nationale du Canada de reptoduce, loan, dimi.or sell reproduire, prêter, distniuer ou copies of his/her thesis by any means vmQe des copies de sa thèse de and in any form or format. making quelque manière et sous quelque this thesis available to interesteci fome que ce soit pour mettre des persons. exemplauw de cette thése à la dispositon des personnes intéressées- The author retains ownership of t&e L'auteur conserve la propriété du copyright in Merthesis. Neither &oit d'auteur qui protège sa thèse. Ni the thesis nor substaritiai exnacts la thèse ni des extraits substantiels de fhm it may be printed or othewïse celle-ci ne doivent être imprimés ou reproduced with the author's autrement reproduits sans son permission. autorisation. Preface During my yeazs as a graduate student at the University of Alberta 1 have benefitted fkom colhborations with my supervisor, Werner bel, and my fellow graduate stu- dents, Warren Anderson, AEo Bonanno, Patrick Brady and Serge Droz. In order to reflect the collaborative nature of the research 1 have written the thesis in the first person plural. Chapter 2 is based on A. Bonanno, S. Droz, W. Israel and S.M. Morsink, Structure of the inner singu- lady of a sphericol black hole, Phys. Rev. D50, 7372 (1994) and A. Bonanno, S. Droz, W. Israel and S.M. Morsink, Stmcture of the chatged spher- ical black hole inte~ior,Proc. R. Soc. Lond. A 450, 553 (1995). Chapter 3 is based on P.R. Brady, S. Droz, W. Israel and S.M. Moainlr, Couanant double-nul1 dynamics: (2+2)-splitting of the Einstein equations, Class. Quant. Grav. 13, 2211 (1996). Chapter 4 is based on S.M. Morsink, Gravitational radiation, Cauchy horizons and the dynamics of the mass fùnction, to be submitted to Phys. Rev. D. Chapter 5 is based on P.R. Brady, S. Droz and S.M. Morsinlr, Mass inflation in non-spherical black holes. to be submitted to Phys. Rev. D. Chapter 6 is based on W.G. Anderson, P.R. Brady, W. hrael, S.M. Morsid, Quantum effects in black hole interiors, Phys. Rev. Lett. 70, 1041 (1993). Figures 1, 2, 7, 8 and 9 were drawn by Serge Droz, who has kindly aven me permission to use these figures with some alterations in this thesis. Abstract The gravitational collapse of a rotating star to a black hole genencally produces a weakly decaying gravitational wave tail whkh partially falls into the black hole. XI- though the influx of gravitational radiation is weak, its backreaction onto the black hole's interior geometry becomes significant near the black hole's inner horizon. The inner horizon of Kerr is a Cauchy horizon (CH), a lightlike hypeaurface of infinite blueshift, behind which lies a tunnel to other universes. At CH the influs of gravita- tional radiation is infinitely amplified by the geometry indicating that CH is unstable. Past work on a spherically symmetric mode1 of the interior has revealed a relatively simple scenario [Il. Wlen the influx is combined with any nominal outflux. the effect of backeaction is to cause the black hole's local mass function to diverge esponen- t ially at CH. Since the Weyl cunmture is completely determined by the mass funct ion in sphericai symmetry, this signals the appearance of a lightlike observer-independent curvature singularity at CH, which effectively se& off the CH tunnel. In this thesis we investigate whether the mass inflation picture can be extended to the non-spherical black hole interior. To do so, we solve the characteristic initial value problem for a general metric in the region near CH. Given initial conditions mhich correspond to the scat tering of the incoming gravitational wave t ail by the int erior Kerr geometry, we show that genericdly a lightlike singularity foms at CH. The general solution exhibits some features similar to the spherically symmetric solution. for the leading order divergences of Hawking's quasi-local mass function and the Kretschmann invariant are the same. However, in the general solution, al1 components of the Weyl tensor diverge, and the solution is not algebraically special. To leading order, the solution is very closely approximated by a colliding plane gravitational wave met ric. This leads to the following heuristic picture: the infalling gravit ational radiation is scattered by the bladr hole's interior curvature into two cross-flowing streams of gravitational radiation which are approximately plane symmetric. The interaction of the two streams in the region of high blueshift strongly focuses light rays and a cmture singularity results. Acknowledgement s The fact that my five years of rësearch at the University of Alberta have been a rewarding and enjoysble expenence is due to the warm and fnendly atmosphere in the Department of Physics and to my supervisor, Werner IsraeI. Werner, it has been a geat pleasure to have had the opportunity to work with, and lem nom you. kour enthusiasm for physics has been inspiring and contagious. 1 hope that you and Inge have many decades of hedth and happiness in your new home in Victoria. During my years in the depaztment 1 have had the opportunity to collaborate on many research projects with Werner and with my feUow graduate students Warren Anderson, Alfio Bonanno, Pat Brady and Serge Droz. Working with al1 of puhas been great learning experience and a lot of fun as well! 1 would dso like to thank al1 the graduate students, post-docs,professors and visitors who have discussed physics nit k me. An incomplete list includes Warren Anderson, Herbert Balasin, .Ufio Bonanno. Fat Brady, Steve Braham, Bruce Campbell, Bahman Darian, Sacha Davidson, Serge Droz, Valery Frolov, Shaun Hendy, Viqar Husain, Tomas Kopf, Pave1 &tous, Robb Mann, Erik Martinez, Dario Nunez, Don Page, Eric Poisson, Tom Steele, Robert Thacker, and hdrei Zelnikov. Discussions with ail of you have helped shape m. knowledge of physics. 1 wouid especially like to thank Robb Ev1an.n for encouraging me to work in the field of general relativity as an undergraduate and for suggesting the problems in lower dimensional gavity which gave me the confidencc to pursue a graduate degree. 1 am most gateful to June Bhatia for initiating the Avadh Bhatia fellowship and for agreeing to dow me to be the first person to hold the feIlowship. Thank you for your interest in supporting the research of women physicists. 1 would like to thank my family for their love and support. Dad, thank pufor al1 your questions about physics and the interest that you showed in my research. Mom, thanks for insisting that 1 take physics in hi& school. To my "lit tle" sister, Coreen. thanks for being a great fnend and correspondent. To Bubby Toby and (belatedly) to Zeidy Moe thanks for caring about me throughout my life. And, since 1 lilie to Save the best for the end, thanks to Erik Talvila for making the last three years of my life so happy. Contents 1 Introduction 1 1.1 Sphericdly symmetric gravit ational collapse .............. 3. 1.2 Gravitational collapse with perturbations ................ 3 . 1-3 Overview of the dissertation ....................... I The spherical black hole interior 11 2.1 The Reissner-Nordstrom bla& hole ................... 12 2.2 Asphencal collapse of a non-rotating star ................ 16 2.3 A simple mode1 of the black hole interior ................ iï 2.1 Field equations for spherical symrnetry ................. 20 2 -5 The Poisson-Israel mode1 ......................... 23 2.6 The Outflux ................................ 27 2.7 -4nalytic approximation for lightlike crossflow ............. 31 2.8 The Scalar Field Solution ........................ 34 2.9 Evolution of the singulanty ....................... 37 2.10 Conclusion ................................. 35 Double-nul1 dynamics 40 3.1 Geometrical ftamework .......................... 42 3.2 The Gauss- Weingarten relations ..................... 47 3.3 Commutation relations .......................... 49 3.4 The Gauss-Codazzi relations ....................... 51 3.5 Cornparison with the NP spin-coefficient formalism .......... 53 3.6 Bianchi identities ............................. 55 3 -7 Characteristic initial value problern ................... 56 3.8 Lagrangian ................................ 60 3.9 Conclusion ................................. 63 4 Dynamics of the mass funetion 64 4.1 The ADM and Bondi masses ....................... 66 4.2 Quasi-local definit ions of mas ...................... 69 4.3 Variation of Hawking's mas ....................... 73 .- 4.4 Variation of Hayward's mas ....................... li 4.5 Wave equation for the mas ....................... 79 4.6 Conclusion ................................. 81 5 The generic black hole singularity 83 1 CoUapse with angdar momentum .................... 85 5.2 The plane wave approximation ...................... 86 5.3 The general solution ........................... 95 5.4 Conclusion ................................ 10-2 6 Quantum effects in the bIa& hole interior 104 6.1 Particle creation in a blacb; hole spacetime ............... 106 6.2 Vacuum polarbation inside the bla& hole ..............
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