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Gravity, a Geometrical Course: Volume 2: Black Holes, Cosmology and Introduction to Supergravity
Gravity, a Geometrical Course: Volume 2: Black Holes, Cosmology and Introduction to Supergravity
Pietro Giuseppe Frè
‘Gravity, a Geometrical Course’ presents general relativity (GR) in a systematic and exhaustive way, covering three aspects that are homogenized into a single texture: i) the mathematical, geometrical foundations, exposed in a self consistent contemporary formalism, ii) the main physical, astrophysical and cosmological applications, updated to the issues of contemporary research and observations, with glimpses on supergravity and superstring theory, iii) the historical development of scientific ideas underlying both the birth of general relativity and its subsequent evolution. The book is divided in two volumes. Volume Two is covers black holes, cosmology and an introduction to supergravity. The aim of this volume is twofold. It completes the presentation of GR and it introduces the reader to theory of gravitation beyond GR, which is supergravity. Starting with a short history of the black hole concept, the book covers the Kruskal extension of the Schwarzschild metric, the causal structures of Lorentzian manifolds, Penrose diagrams and a detailed analysis of the KerrNewman metric. An extensive historical account of the development of modern cosmology is followed by a detailed presentation of its mathematical structure, including nonisotropic cosmologies and billiards, de Sitter space and inflationary scenarios, perturbation theory and anisotropies of the Cosmic Microwave Background. The last three chapters deal with the mathematical and conceptual foundations of supergravity in the frame of free differential algebras. Branes are presented both as classical solutions of the bulk theory and as worldvolume gauge theories with particular emphasis on the geometrical interpretation of kappasupersymmetry. The rich bestiary of special geometries underlying supergravity lagrangians is presented, followed by a chapter providing glances on the equally rich collection of special solutions of supergravity. Pietro Frè is Professor of Theoretical Physics at the University of Torino, Italy and is currently serving as Scientific Counsellor of the Italian Embassy in Moscow. His scientific passion lies in supergravity and all allied topics, since the inception of the field, in 1976. He was professor at SISSA, worked in the USA and at CERN. He has taught General Relativity for 15 years. He has previously two scientific monographs, “Supergravity and Superstrings” and “The N=2 Wonderland”, He is also the author of a popular science book on cosmology and two novels, in Italian.
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2012
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2013
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Gravity, a Geometrical Course Pietro Giuseppe Frè Gravity, a Geometrical Course Volume 2: Black Holes, Cosmology and Introduction to Supergravity Pietro Giuseppe Frè Dipartimento di Fisica Teorica University of Torino Torino, Italy Additional material to this book can be downloaded from http://extras.springer.com. ISBN 9789400754423 ISBN 9789400754430 (eBook) DOI 10.1007/9789400754430 Springer Dordrecht Heidelberg New York London Library of Congress Control Number: 2012950601 © Springer Science+Business Media Dordrecht 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publicati; on, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acidfree paper Springer is part of Springer Science+Business Media (www.springer.com) This book is dedicated to my beloved daughter Laura and to my darling wife Olga. Preface This book grew out from the Lecture Notes of the course in General Relativity which I gave for more than 15 years at the University of Torino. That course has a long tradition since it was attached to the Chair of Relativity created at the beginning of the 1960s for prof. Tullio Regge. In the years 1990–1996, while prof. Regge was Member of the European Parliament the course was given by my long time excellent friend and collaborator prof. Riccardo D’Auria. In 1996 I had the honor to be appointed on Regge’s chair1 and I left SISSA of Trieste to take this momentous and challenging legacy. Feeling the burden of the task laid on my shoulders I humbly tried to do my best and create a new course which might keep alive the tradition established by my so much distinguished predecessors. In my efforts to cope with the expected standards, I obviously introduced my own choices, viewpoints and opinions that are widely reflected in the present book. The length of the original course was of about 120 hours (without exercises). In the new 3 + 2 system introduced by the Bologna agreements it was split in two courses but, apart from minor readjustments, I continued to consider them just as part one and part two of a unique entity. This was not a random choice but it sprang from the views that inspired my teaching and the present book. I always held the opinion that University courses should be long, complex and articulated in many aspects. They should not aim at a quick transmission of calculating abilities and of ready to use information, rather they should be as much formative as informative. They should offer a general overview of a subject as seen by the professor, in this way giving the students the opportunity of developing their own opinions through the critical absorption of those of the teacher. One aspect that I always considered essential is the historical one, concerning on one side the facts, the life and the personalities of the scientists who shaped our present understanding, on the other hand concerning the usually intricate development of fundamental ideas. The second aspect to which I paid a lot of attention is the use of an updated and as much as possible rigorous mathematical formalism. Moreover I always tried to 1 At that time Regge had shifted from the University to the Politecnico of Torino. vii viii Preface convey the view that Mathematics should not be regarded as a technical tool for the solution of Physical Problems or simply as a language for the formulation of Physical Laws, rather as an essential integral part of the whole game. The third aspect taken not only into account but also into prominence, is the emphasis on important physical applications of the theory: not just exercises, from which I completely refrained, but the fullfledged ab initio development of relevant applications in Astrophysics, Cosmology or Particle Theory. The aim was that of showing, from A to Z, as one goes from the first principles to the actual prediction of experimentally verifiable numbers. For the reader’s or student’s convenience I included the listing of some computer codes, written in MATHEMATICA, that solve some of the posed problems or parts thereof. The aim was, once again formative. In the course of their theoretical studies the students should develop the ability to implement formal calculations on a machine, freeing themselves from the slavery to accidental errors and focusing instead all their mental energies on conceptual points. Furthermore implementation of formulae in a computer code is the real test of their comprehension by the learners, more efficient in its task than any adhoc prepared exercise. As for the actual choice of the included and developed material, I was inspired by the following view on the role of the course I used to gave, which I extended as a mission to the present book. General Relativity, Quantum Mechanics, Gauge Theories and Statistical Mechanics are the four pillars of the Physical Thought developed in the XXth century. That century laid also the foundations for new theories, whose actual relations with the experimental truth and with observations will be clarified only in the present millennium, but whose profound influence on the current thought is so profound that noone approaching theoretical studies can ignore them: I refer to supersymmetry, supergravity, strings and branes. The role of the course in General Relativity which I assumed as given, was not only that of presenting Einstein Theory, in its formulation, historical development and applications, but also that of comparing the special structure of Gravity in relation with the structure of the GaugeTheories describing the other fundamental interactions. This was specially aimed at the development of critical thinking in the student and as a tool of formative education, preparatory to the study of unified theories. The present one is a Graduate Text Book but it is also meant to be a selfcontained account of Gravitational Theory attractive for the person with a basic scientific education and a curiosity for the topic who would like to learn it from scratch, being his/her own instructor. Just as the original course given in Torino after the implementation of the Bologna agreements, this book is divided in two volumes: 1. Volume 1: Development of the Theory and Basic Physical Applications. 2. Volume 2: Black Holes, Cosmology and Introduction to Supergravity. Volume 1, starting from a summary of Special Relativity and a sketchy historical introduction of its birth, given in Chap. 1, develops the general current description of the physical world in terms of gauge connections and sections of the bundles on Preface ix which such connections are constructed. The special role of Gravity as the gauge theory of the tangent bundle to the base manifold of all other bundles is emphasized. The mathematical foundations of the theory are contained in Chaps. 2 and 3. Chapter 2 introduces the basic notions of differential geometry, the definition of manifolds and fibrebundles, differential forms, vector fields, homology and cohomology. Chapter 3 introduces the theory of connections and metrics. It includes an extensive historical account of the development of mathematical and physical ideas which eventually lead to both general relativity and modern gauge theories of the nongravitational interactions. The notion of geodesics is introduced and exemplified with the detailed presentation of a pair of examples in two dimensions, one with Lorentzian signature, the other with Euclidian signature. Chapter 4 is devoted to the Schwarzschild metric. It is shown how geodesics of the Schwarzschild metric retrieve the whole building of Newtonian Physics plus corrections that can be very tiny in weak gravitational fields, like that of the Solar System, or gigantic in strong fields, where they lead to qualitatively new physics. The classical tests of General Relativity are hereby discussed, perihelion advance and the bending of light rays, in particular. Chapter 5 introduces the Cartan approach to differential geometry, the vielbein and the spin connection, discusses Bianchi identities and their relation with gauge invariances and eventually introduces Einstein field equations. The dynamical equations of gravity and their derivation from an action principle are developed in a parallel way to their analogues for electrodynamics and nonAbelian gauge theories whose structure and features are constantly compared to those of gravity. The linearization of Einstein field equations and the spin of the graviton are then discussed. After that the bottomup approach to gravity is discussed, namely, following Feynman’s ideas, it is shown how a special relativistic linear theory of the graviton field could be uniquely inferred from the conservation of the stressenergy tensor and its nonlinear upgrading follows, once the stressenergy tensor of the gravitational field itself is taken into account. The last section of Chap. 5 contains the derivation of the Schwarzschild metric from Einstein equations. Chapter 6 addresses the issue of stellar equilibrium in General Relativity, derives the Tolman Oppenheimer Volkhoff equation and the corresponding mass limits. Next, considering the role of quantum mechanics the Chandrasekhar mass limits for white dwarfs and neutron stars are derived. Chapter 7 is devoted to the emission of gravitational waves and to the tests of General Relativity based on the slowing down of the period of double star systems. Volume 2, after a short introductory chapter, the following two chapters are devoted to Black Holes. In Chap. 2 we begin with a historical account of the notion of black holes from Laplace to the present identification of stellar mass black holes in the galaxy and elsewhere. Next the Kruskal extension of the Schwarzschild solution is considered in full detail preceded by the pedagogical toy example of Rindler spacetime. Basic concepts about Future, Past and Causality are introduced next. Conformal Mappings, the Causal Structure of infinity and Penrose diagrams are discussed and exemplified. Chapter 3 deals with rotating blackholes and the KerrNewman metric. The usually skipped form of the spin connection and of the Riemann tensor of this metric is calculated and presented in full detail, together with the electric and magnetic x Preface field strengths associated with it in the case of a charged hole. This is followed by a careful discussion of the static limit, of locally nonrotating observers, of the horizon and of the ergosphere. In a subsequent section the geodesics of the Kerr metric are studied by using the Hamilton Jacobi method and the system is shown to be Liouville integrable with the derivation of the fourth Hamiltonian (the Carter constant) completing the needed shell of four, together with the energy, the angular momentum and the mass. The last section contains a discussion of the analogy between the Laws of Thermodynamics and those of Black Hole dynamics including the BekensteinHawking entropy interpretation of the horizon area. Chapters 4 and 5 are devoted to cosmology. Chapter 4 contains a historical outline of modern Cosmology starting from Kant’s proposal that nebulae might be different islanduniverses (galaxies in modern parlance) to the current space missions that have measured the Cosmic Microwave Background anisotropies. The crucial historical steps in building up the modern vision of a huge expanding Universe, which is even accelerating at the present moment, are traced back in some detail. From the Olbers paradox to the discovery of the stellar parallax by Bessel, to the Great Debate of 1920 between Curtis and Shapley, how the human estimation of the Universe’s size enlarged, is historically reported. The discovery of the Cepheides law by Henrietta Leavitt, the first determination of the distance to nearby galaxies by Hubble and finally the first measuring of the universal cosmic recession are the next episodes of this tale. The discovery of the CMB radiation, predicted by Gamow, the hunt for its anisotropies and the recent advent of the Inflationary Universe paradigm are the subsequent landmarks, which are reported together with biographical touches upon the life and personalities of the principal actors in this exciting adventure of the human thought. Chapter 5, entitled Cosmology and General Relativity: Mathematical Description of the Universe, provides a fullfledged introduction to Relativistic Cosmology. The chapter begins with a long mathematical interlude on the geometry of coset manifolds. These notions are necessary for the mathematical formulation of the Cosmological Principle, stating homogeneity and isotropy, but have a much wider spectrum of applications. In particular they will be very important in the subsequent chapters about Supergravity. Having prepared the stage with this mathematical preliminaries, the next sections deal with homogeneous but not isotropic cosmologies. Bianchi classification of three dimensional Lie groups is recalled, Bianchi metrics are defined and, within Bianchi type I, the Kasner metrics are discussed with some glimpses about the cosmic billiards, realized in Supergravity. Next, as a pedagogical example of a homogeneous but not isotropic cosmology, an exact solution, with and without matter, of Bianchi type II spacetime is treated in full detail. After this, we proceed to the Standard Cosmological Model, including both homogeneity and isotropy. Freedman equations, all their implications and known solutions are discussed in detail and a special attention is given to the embedding of the three type of standard cosmologies (open, flat and closed) into de Sitter space. The concept of particle and event horizons is next discussed together with the derivation of exact formulae for readshift distances. The conceptual problems (horizon and flatness) of the Standard Cosmological Model are next discussed as an introduction to the new Preface xi inflationary paradigm. The basic inflationary model based on one scalar field and the slow rolling regime are addressed in the following sections with fully detailed calculations. Perturbations, the spectrum of fluctuations up to the evaluation of the spectral index and the principles of comparison with the CMB data form the last part of this very long chapter. The last four chapters of the book provide a conceptual, mathematical and descriptive introduction to Supergravity, namely to the Beyond GR World. Chapter 6 starts with a historical outline that describes the birth of supersymmetry both in String Theory and in Field Theory, touching also on the biographies and personalities of the theorists who contributed to create this entire new field through a complicated and, as usual, far from straight, path. The chapter proceeds than with the conceptual foundations of Supergravity, in particular with the notion of Free Differential Algebras and with the principle of rheonomy. Sullivan’s structural theorems are discussed and it is emphasized how the existence of pforms, that close the supermultiplets of fundamental fields appearing in higher dimensional supergravities, is at the end of the day a consequence of the superPoincaré Lie algebras through their cohomologies. The structure of Mtheory, the constructive principles to build supergravity Lagrangians and the fundamental role of Bianchi identities is emphasized. The last two sections of the chapter contain a thorough account of type IIA and type IIB supergravities in D = 10, the structure of their FDAs, the rheonomic parameterization of their curvatures and the fullfledged form of their field equations. Chapter 7 deals with the brane/bulk dualism. The first section contains a conceptual outline where the three sided view of branes as 1) classical solitonic solutions of the bulk theory, 2) world volume gaugetheories described by suitable worldvolume actions endowed with κsupersymmetry and 3) boundary states in the superconformal field theory description of superstring vacua is spelled out. Next a New First Order Formalism, invented by the author of this book at the beginning of the XXIst century and allowing for an elegant and compact construction of κsupersymmetric BornInfeld type worldvolume actions on arbitrary supergravity backgrounds is described. It is subsequently applied to the case of the D3brane, both as an illustration and for the its intrinsic relevance in the gauge/gravity correspondence. The last sections of the chapter are devoted to the presentations of branes as classical solitonic solutions of the bulk theory. General features of the solutions in terms of harmonic functions are presented including also a short review of domain walls and some sketchy description of the RandallSundrun mechanism. Chapter 8 is a bestiary of Supergravity Special Geometries associated with its scalar sector. The chapter clarifies the codifying role of the scalar geometry in constructing the bosonic part of a supergravity Lagrangian. The dominant role among the scalar manifolds of homogeneous symmetric spaces is emphasized illustrating the principles that allow the determination of such U/H cosets for any supergravity theory. The mechanism of symplectic embedding that allows to extend the action of Uisometries from the scalar to the vector field sector are explained in detail within the general theory of electric/magnetic duality rotations. Next the chapter provides a selfcontained summary of the most important special geometries appearing in xii Preface D = 4 and D = 5 supergravity, namely Special Kähler Geometry, Very Special Real Geometry and Quaternionic Geometry. Chapter 9 presents a limited anthology of supergravity solutions aimed at emphasizing a few relevant new concepts. Relying on the special geometries described in Chap. 8 a first section contains an introduction to supergravity spherical Black Holes, to the attraction mechanism and to the interpretation of the horizon area in terms of a quartic symplectic invariant of the U duality group. The second and third sections deal instead with flux compactifications of both Mtheory and type IIA supergravity. The main issue is that of the relation between supersymmetry preservation and the geometry of manifolds of restricted holonomy. The problem of supergauge completion and the role of orthosymplectic superalgebras is also emphasized. Appendices contain the development of gamma matrix algebra necessary for the inclusion of spinors, details on superalgebras and the user guide to Mathematica codes for the computer aided calculation of Einstein equations. Moscow, Russia University of Torino presently Scientific Counselor of the Italian Embassy in Moscow Pietro Giuseppe Frè Acknowledgements With great pleasure I would like to thank my collaborators and colleagues Pietro Antonio Grassi, Igor Pesando and Mario Trigiante for the many suggestions and discussions we had during the writing of the present book and also for their critical reading of several chapters. Similarly I express my gratitude to the Editors of SpringerVerlag, in particular to Dr. Maria Bellantone, for their continuous assistance, constructive criticism and suggestions. My thoughts, while finishing the writing of these volumes, that occurred during solitary winter weekends in Moscow, were frequently directed to my late parents, whom I miss very much and I will never forget. To them I also express my gratitude for all what they taught me in their life, in particular to my father who, with his own example, introduced me, since my childhood, to the great satisfaction and deep suffering of writing books. Furthermore it is my pleasure to thank my very close friend and collaborator Aleksander Sorin for his continuous encouragement and for many precious consultations. xiii Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams . . 2.1 Introduction and a Short History of Black Holes . . . . . . . . . . 2.2 The Kruskal Extension of Schwarzschild SpaceTime . . . . . . . 2.2.1 Analysis of the Rindler SpaceTime . . . . . . . . . . . . . 2.2.2 Applying the Same Procedure to the Schwarzschild Metric 2.2.3 A First Analysis of Kruskal SpaceTime . . . . . . . . . . 2.3 Basic Concepts about Future, Past and Causality . . . . . . . . . . 2.3.1 The LightCone . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Future and Past of Events and Regions . . . . . . . . . . . 2.4 Conformal Mappings and the Causal Boundary of SpaceTime . . 2.4.1 Conformal Mapping of Minkowski Space into the Einstein Static Universe . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Asymptotic Flatness . . . . . . . . . . . . . . . . . . . . . 2.5 The Causal Boundary of Kruskal SpaceTime . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 10 10 14 17 19 20 22 28 Rotating Black Holes and Thermodynamics . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The KerrNewman Metric . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Riemann and Ricci Curvatures of the KerrNewman Metric 3.3 The Static Limit in KerrNewman SpaceTime . . . . . . . . . . . 3.4 The Horizon and the Ergosphere . . . . . . . . . . . . . . . . . . 3.5 Geodesics of the Kerr Metric . . . . . . . . . . . . . . . . . . . . 3.5.1 The Three Manifest Integrals, E , L and μ . . . . . . . . . 3.5.2 The HamiltonJacobi Equation and the Carter Constant . . 3.5.3 Reduction to First Order Equations . . . . . . . . . . . . . 3.5.4 The Exact Solution of the Schwarzschild Orbit Equation as an Application . . . . . . . . . . . . . . . . . . . . . . 3.5.5 About Explicit Kerr Geodesics . . . . . . . . . . . . . . . 43 43 43 45 49 53 55 56 58 60 3 29 36 37 42 62 65 xv xvi Contents 3.6 The Kerr Black Hole and the Laws of Thermodynamics . . . . . 3.6.1 The Penrose Mechanism . . . . . . . . . . . . . . . . . . 3.6.2 The Bekenstein Hawking Entropy and Hawking Radiation References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 67 69 70 4 Cosmology: A Historical Outline from Kant to WMAP and PLANCK 71 4.1 Historical Introduction to Modern Cosmology . . . . . . . . . . . 71 4.2 The Universe Is a Dynamical System . . . . . . . . . . . . . . . . 71 4.3 Expansion of the Universe . . . . . . . . . . . . . . . . . . . . . . 72 4.3.1 Why the Night is Dark and Olbers Paradox . . . . . . . . . 73 4.3.2 Hubble, the Galaxies and the Great Debate . . . . . . . . . 73 4.3.3 The Discovery of Hubble’s Law . . . . . . . . . . . . . . . 81 4.3.4 The Big Bang . . . . . . . . . . . . . . . . . . . . . . . . 84 4.4 The Cosmological Principle . . . . . . . . . . . . . . . . . . . . . 86 4.5 The Cosmic Background Radiation . . . . . . . . . . . . . . . . . 91 4.6 The New Scenario of the Inflationary Universe . . . . . . . . . . . 97 4.7 The End of the Second Millennium and the Dawn of the Third Bring Great News in Cosmology . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5 Cosmology and General Relativity: Mathematical Description of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Mathematical Interlude: Isometries and the Geometry of Coset Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Isometries and Killing Vector Fields . . . . . . . . . . . . 5.2.2 Coset Manifolds . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 The Geometry of Coset Manifolds . . . . . . . . . . . . . 5.3 Homogeneity Without Isotropy: What Might Happen . . . . . . . 5.3.1 Bianchi Spaces and Kasner Metrics . . . . . . . . . . . . . 5.3.2 A Toy Example of Cosmic Billiard with a Bianchi II SpaceTime . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Einstein Equation and Matter for This Billiard . . . . . . . 5.3.4 The Same Billiard with Some Matter Content . . . . . . . 5.3.5 ThreeSpace Geometry of This Toy Model . . . . . . . . . 5.4 The Standard Cosmological Model: Isotropic and Homogeneous Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Viewing the Coset Manifolds as Group Manifolds . . . . . 5.5 Friedman Equations for the Scale Factor and the Equation of State 5.5.1 Proof of the Cosmological RedShift . . . . . . . . . . . . 5.5.2 Solution of the Cosmological Differential Equations for Dust and Radiation Without a Cosmological Constant . 5.5.3 Embedding Cosmologies into de Sitter Space . . . . . . . . 5.6 General Consequences of Friedman Equations . . . . . . . . . . . 5.6.1 Particle Horizon . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Event Horizon . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 RedShift Distances . . . . . . . . . . . . . . . . . . . . . 107 107 108 108 109 114 125 125 130 132 137 141 146 149 150 152 154 159 162 166 168 171 Contents 5.7 Conceptual Problems of the Standard Cosmological Model . . 5.8 Cosmic Evolution with a Scalar Field: The Basis for Inflation . 5.8.1 de Sitter Solution . . . . . . . . . . . . . . . . . . . . 5.8.2 SlowRolling Approximate Solutions . . . . . . . . . . 5.9 Primordial Perturbations of the Cosmological Metric and of the Inflaton . . . . . . . . . . . . . . . . . . . . . . . . 5.9.1 The Conformal Frame . . . . . . . . . . . . . . . . . . 5.9.2 Deriving the Equations for the Perturbation . . . . . . . 5.9.3 Quantization of the Scalar Degree of Freedom . . . . . 5.9.4 Calculation of the Power Spectrum in the Two Regimes 5.10 The Anisotropies of the Cosmic Microwave Background . . . . 5.10.1 The SachsWolfe Effect . . . . . . . . . . . . . . . . . 5.10.2 The TwoPoint Temperature Correlation Function . . . 5.10.3 Conclusive Remarks on CMB Anisotropies . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 xvii . . . . . . . . 172 174 176 177 . . . . . . . . . . . . . . . . . . . . 187 187 188 195 198 203 203 206 208 209 Supergravity: The Principles . . . . . . . . . . . . . . . . . . . . . 6.1 Historical Outline and Introduction . . . . . . . . . . . . . . . . 6.1.1 Fermionic Strings and the Birth of Supersymmetry . . . . 6.1.2 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Supergravity . . . . . . . . . . . . . . . . . . . . . . . . 6.2 AlgebroGeometric Structure of Supergravity . . . . . . . . . . . 6.3 Free Differential Algebras . . . . . . . . . . . . . . . . . . . . . 6.3.1 Chevalley Cohomology . . . . . . . . . . . . . . . . . . 6.3.2 General Structure of FDAs and Sullivan’s Theorems . . . 6.4 The Super FDA of M Theory and Its Cohomological Structure . . 6.4.1 The Minimal FDA of MTheory and Cohomology . . . . 6.4.2 FDA Equivalence with Larger (Super) Lie Algebras . . . 6.5 The Principle of Rheonomy . . . . . . . . . . . . . . . . . . . . 6.5.1 The Flow Chart for the Construction of a Supergravity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Construction of D = 11 Supergravity, Alias MTheory . . 6.6 Summary of Supergravities . . . . . . . . . . . . . . . . . . . . 6.7 Type IIA Supergravity in D = 10 . . . . . . . . . . . . . . . . . 6.7.1 Rheonomic Parameterizations of the Type IIA Curvatures in the String Frame . . . . . . . . . . . . . . . . . . . . 6.7.2 Field Equations of Type IIA Supergravity in the String Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Type IIB Supergravity . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 The SU(1, 1)/U(1) ∼ SL(2, R)/O(2) Coset . . . . . . . 6.8.2 The Free Differential Algebra, the Supergravity Fields and the Curvatures . . . . . . . . . . . . . . . . . . . . . 6.8.3 The Bosonic Field Equations and the Standard Form of the Bosonic Action . . . . . . . . . . . . . . . . . . . 6.9 About Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 211 215 218 221 223 227 228 230 233 235 236 239 . . . . 242 243 246 248 . 251 . 253 . 254 . 254 . 256 . 259 . 261 . 261 xviii 7 8 Contents The Branes: Three Viewpoints . . . . . . . . . . . . . . . . . . . . 7.1 Introduction and Conceptual Outline . . . . . . . . . . . . . . . 7.2 pBranes as World Volume GaugeTheories . . . . . . . . . . . 7.3 From 2nd to 1st Order and the Rheonomy Setup for to κ Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 NambuGoto, BornInfeld and Polyakov Kinetic Actions for pBranes . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 κSupersymmetry and the Example of the M2Brane . . . 7.3.3 With DpBranes We Have a Problem: The WorldVolume Gauge Field A[1] . . . . . . . . . . . . . . . . . . . . . . 7.4 The New First Order Formalism . . . . . . . . . . . . . . . . . . 7.4.1 An Alternative to the Polyakov Action for pBranes . . . 7.4.2 Inclusion of a WorldVolume Gauge Field and the BornInfeld Action in First Order Formalism . . . . . . . . . . 7.4.3 Explicit Solution of the Equations for the Auxiliary Fields for F and h−1 . . . . . . . . . . . . . . . . . . . . . . . 7.5 The D3Brane Example and κSupersymmetry . . . . . . . . . . 7.5.1 κSupersymmetry . . . . . . . . . . . . . . . . . . . . . 7.6 The D3Brane: Summary . . . . . . . . . . . . . . . . . . . . . 7.7 Supergravity pBranes as Classical Solitons: General Aspects . . 7.8 The Near Brane Geometry, the Dual Frame and the AdS/CFT Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Domain Walls in Diverse SpaceTime Dimensions . . . . . . . . 7.9.1 The Randall Sundrum Mechanism . . . . . . . . . . . . 7.9.2 The Conformal Gauge for Domain Walls . . . . . . . . . 7.10 Conclusion on This Brane Bestiary . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 . 263 . 268 . 269 . 269 . 272 . 273 . 275 . 275 . 277 . . . . . 280 281 283 287 288 . . . . . . 291 292 295 296 299 299 Supergravity: A Bestiary in Diverse Dimensions . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Supergravity and Homogeneous Scalar Manifolds G/H . . . . . . 8.2.1 How to Determine the Scalar Cosets G/H of Supergravities from Supersymmetry . . . . . . . . . . . . . . . . . . . . 8.2.2 The Scalar Cosets of D = 4 Supergravities . . . . . . . . . 8.2.3 Scalar Manifolds of Maximal Supergravities in Diverse Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Duality Symmetries in Even Dimensions . . . . . . . . . . . . . . 8.3.1 The Kinetic Matrix N and Symplectic Embeddings . . . . 8.3.2 Symplectic Embeddings in General . . . . . . . . . . . . . 8.4 General Form of D = 4 (Ungauged) Supergravity . . . . . . . . . 8.5 Summary of Special Kähler Geometry . . . . . . . . . . . . . . . 8.5.1 HodgeKähler Manifolds . . . . . . . . . . . . . . . . . . 8.5.2 Connection on the Line Bundle . . . . . . . . . . . . . . . 8.5.3 Special Kähler Manifolds . . . . . . . . . . . . . . . . . . 8.5.4 The Vector Kinetic Matrix NΛΣ in Special Geometry . . . 303 303 304 305 307 309 310 317 319 322 323 324 325 326 328 Contents 8.6 Supergravities in Five Dimension and More Scalar Geometries 8.6.1 Very Special Geometry . . . . . . . . . . . . . . . . . 8.6.2 The Very Special Geometry of the SO(1, 1) × SO(1, n)/SO(n) Manifold . . . . . . 8.6.3 Quaternionic Geometry . . . . . . . . . . . . . . . . . 8.6.4 Quaternionic, Versus HyperKähler Manifolds . . . . . 8.7 N = 2, D = 5 Supergravity Before Gauging . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 xix . . 329 . . 334 . . . . . . . . . . 336 338 338 342 342 Supergravity: An Anthology of Solutions . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Black Holes Once Again . . . . . . . . . . . . . . . . . . . . . . 9.2.1 The σ Model Approach to Spherical Black Holes . . . . 9.2.2 The Oxidation Rules . . . . . . . . . . . . . . . . . . . . 9.2.3 General Properties of the d = 4 Metric . . . . . . . . . . 9.2.4 Attractor Mechanism, the Entropy and Other Special Geometry Invariants . . . . . . . . . . . . . . . . . . . . 9.2.5 Critical Points of the Geodesic Potential and Attractors . 9.2.6 The N = 2 Supergravity S 3 Model . . . . . . . . . . . . 9.2.7 Fixed Scalars at BPS Attractor Points: The S 3 Explicit Example . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.8 The Attraction Mechanism Illustrated with an Exact NonBPS Solution . . . . . . . . . . . . . . . . . . . . . 9.2.9 Resuming the Discussion of Critical Points . . . . . . . . 9.2.10 An Example of a Small Black Hole . . . . . . . . . . . . 9.2.11 Behavior of the Riemann Tensor in Regular Solutions . . 9.3 Flux Vacua of MTheory and Manifolds of Restricted Holonomy 9.3.1 The Holonomy Tensor from D = 11 Bianchi Identities . . 9.3.2 Flux Compactifications of MTheory on AdS4 × M7 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 MTheory Field Equations and 7Manifolds of Weak G2 Holonomy i.e. Englert 7Manifolds . . . . . . . . . . . . 9.3.4 The SO(8) Spinor Bundle and the Holonomy Tensor . . . 9.3.5 The Well Adapted Basis of Gamma Matrices . . . . . . . 9.3.6 The so(8)Connection and the Holonomy Tensor . . . . . 9.3.7 The Holonomy Tensor and Superspace . . . . . . . . . . 9.3.8 Gauged Maurer Cartan 1Forms of OSp(84) . . . . . . . 9.3.9 Killing Spinors of the AdS4 Manifold . . . . . . . . . . . 9.3.10 Supergauge Completion in Mini Superspace . . . . . . . 9.3.11 The 3Form . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Flux Compactifications of Type IIA Supergravity on AdS4 × P3 . 9.4.1 Maurer Cartan Forms of OSp(64) . . . . . . . . . . . . 9.4.2 Explicit Construction of the P3 Geometry . . . . . . . . . 9.4.3 The Compactification Ansatz . . . . . . . . . . . . . . . 9.4.4 Killing Spinors on P3 . . . . . . . . . . . . . . . . . . . . . . . . . 345 345 349 349 351 354 . 356 . 357 . 359 . 364 . . . . . . 367 368 369 371 372 373 . 375 . . . . . . . . . . . . . . 376 382 382 382 384 386 387 388 390 391 391 392 396 397 xx Contents 9.4.5 9.4.6 9.4.7 Gauge Completion in Mini Superspace . . . . . . . . Gauge Completion of the B[2] Form . . . . . . . . . . Rewriting the MiniSuperspace Gauge Completion as Maurer Cartan Forms on the Complete Supercoset 9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 . . . 401 . . . 401 . . . 403 . . . 404 10 Conclusion of Volume 2 . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 The Legacy of Volume 1 . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Story Told in Volume 2 . . . . . . . . . . . . . . . . . . . . . Appendix A Spinors and Gamma Matrix Algebra1 . . . . . . . . . . . A.1 Introduction to the Spinor Representations of SO(1, D − 1) A.2 The Clifford Algebra . . . . . . . . . . . . . . . . . . . . A.3 The Charge Conjugation Matrix . . . . . . . . . . . . . . . A.4 Majorana, Weyl and MajoranaWeyl Spinors . . . . . . . . A.5 A Particularly Useful Basis for D = 4 γ Matrices . . . . . Appendix B Auxiliary Tools for pBrane Actions . . . . . . . . . . . B.1 Notations and Conventions . . . . . . . . . . . . . . . . . B.2 The κSupersymmetry Projector for D3Branes . . . . . . Appendix C Auxiliary Information About Some Superalgebras . . . . C.1 The OSp(N  4) Supergroup, Its Superalgebra and Its Supercosets . . . . . . . . . . . . . . . . . . . . . . . . . C.2 The Relevant Supercosets and Their Relation . . . . . . . . C.3 D = 6 and D = 4 Gamma Matrix Bases . . . . . . . . . . C.4 An so(6) Inversion Formula . . . . . . . . . . . . . . . . . Appendix D MATHEMATICA Package NOVAMANIFOLDA . . . . . Appendix E Examples of the Use of the Package NOVAMANIFOLDA References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 407 407 409 409 409 412 413 414 415 415 416 419 419 422 426 429 430 436 444 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 Chapter 1 Introduction The Two Most Powerful Warriors Are Patience and Time Leo Tolstoy The goal of this second volume is twofold. On one hand we want to complete the presentation of General Relativity by analyzing two of its main fields of application: 1. Black Holes, 2. Cosmology. On the other hand we want to introduce the reader to Theory of Gravitation Beyond General Relativity which is Supergravity. The latter invokes, in a way which we hope to be able to explain, Superstrings and also other Branes. Sticking to the method followed in Volume 1 we will trace the conceptual development of fundamental ideas through history. At the same time we will recast all equations in a mathematical formalism adapted to the embedding of General Relativity into its modern extensions like Supergravity. This is done in order to retrieve the logical development of ideas, which differs from the historical one and constantly requires revisiting Old Theories from the standpoint of New Ones. This was the motivation for the particular and sometimes unconventional way of presenting General Relativity we adopted in the first volume. The reader will fully appreciate the relevance of this strategy when coming to Chap. 6 and to the constructive principles underlying supergravity. The prominence given to the Cartan formulation in terms of vielbein and spin connection and to the role of Bianchi identities will reveal its profound rationale in that chapter. There the reader will find the endpoint of a long argument that, starting from Lorentz symmetry leads first to the distinctive features of a gauge theory of the Poincaré connection and then, if one admits the supersymmetry charges, to a new algebraic category, that of Free Differential Algebras encompassing pforms and a totally new viewpoint on gauging. The pforms open the window on the world of branes and on their dualism with the gravitational theories living in the bulk. In the rich and complex new panorama provided by the Bestiary of Supergravities and of their solutions also Black Holes and Cosmology acquire new perspectives and possibilities. P.G. Frè, Gravity, a Geometrical Course, DOI 10.1007/9789400754430_1, © Springer Science+Business Media Dordrecht 2013 1 2 1 Introduction Introducing step by step the necessary mathematical structures and framing historically the development of ideas we promise our patient reader to conduct him smoothly and, hopefully without logical jumps, to the current frontier of Gravitational Theory. Chapter 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams O radiant Dark! O darkly fostered ray Thou hast a joy too deep for shallow Day! George Eliot (The Spanish Gypsy) 2.1 Introduction and a Short History of Black Holes It seems that the first to conceive the idea of what we call nowadays a blackhole was the English Natural Philosopher and Geologist John Michell (1724–1793). Member of the Royal Society, Michell already before 1783 invented a device to measure Newton’s gravitational constant, namely the torsion balance that he built independently from its coinventor Charles Augustin de Coulomb. He did not live long enough to put into use his apparatus which was inherited by Cavendish. In 1784 in a letter addressed precisely to Cavendish, John Michell advanced the hypothesis that there could exist heavenly bodies so massive that even light could not escape from their gravitational attraction. This letter surfaced back to the attention of contemporary scientists only in the later seventies of the XXth century [1]. Before that finding, credited to be the first inventor of blackholes was Pierre Simon Laplace (see Fig. 2.1). In the 1796 edition of his monumental book Exposition du Système du Monde [2] he presented exactly the same argument put forward in Michell’s letter, developing it with his usual mathematical rigor. All historical data support the evidence that Michell and Laplace came to the same hypothesis independently. Indeed the idea was quite mature for the physics of that time, once the concept of escape velocity ve had been fully understood. Consider a spherical celestial body of mass M and radius R and let us pose the question what is the minimum initial vertical velocity that a pointlike object located on its surface, for instance a rocket, should have in order to be able to escape to infinite distance from the center of gravitational attraction. Energy conservation provides the immediate answer to such a problem. At the initial moment t = t0 the energy of the missile is: 1 GMmm (2.1.1) E = mm ve2 − 2 R where G is Newton’s constant. At a very late time, when the missile has reached R = ∞ with a final vanishing velocity its energy is just 0 + 0 = 0. Hence E vanished P.G. Frè, Gravity, a Geometrical Course, DOI 10.1007/9789400754430_2, © Springer Science+Business Media Dordrecht 2013 3 4 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams Fig. 2.1 Pierre Simon Laplace (1749–1827) was born in Beaumont en Auge in Normandy in the family of a poor farmer. He could study thanks to the generous help of some neighbors. Later with a recommendation letter of d’Alembert he entered the military school of Paris where he became a teacher of mathematics. There he started his monumental and original research activity in Mathematics and Astronomy that made him one of the most prominent scientists of his time and qualified him to the rank of founder of modern differential calculus, his work being a pillar of XIXth century Mathematical Physics. A large part of his work on Astronomy was still done under the Ancien Regime and dates back to the period 1771–1787. He proved the stability of the Solar System and developed all the mathematical tools for the systematic calculus of orbits in Newtonian Physics. His results were summarized in the two fundamental books Mecanique Cèleste and Exposition du Système du Monde. Besides introducing the first idea of what we call nowadays a blackhole, Laplace was also the first to advance the hypothesis that the Solar System had formed through the cooling of a globularshaped, rotating, cluster of very hot gas (a nebula). In later years of his career Laplace gave fundamental and framing contributions to the mathematical theory of probability. His name is attached to numberless corners of differential analysis and function theory. He received many honors both in France and abroad. He was member of all most distinguished Academies of Europe. He also attempted the political career serving as Minister of Interiors in one of the first Napoleonic Cabinets, yet he was soon dismissed by the First Consul as a person not qualified for that administrative job notwithstanding Napoleon’s recognition that he was a great scientist. Politically Laplace was rather cynic and ready to change his opinions and allegiance in order to follow the blowing wind. Count of the First French Empire, after the fall of Napoleon he came on good terms with the Bourbon Restoration and was compensated by the King with the title of marquis also at the beginning, which yields: ve = 2 GM R (2.1.2) If we assume that light travels at a finite velocity c, then there could exist heavenly bodies so dense that: GM >c (2.1.3) 2 R 2.1 Introduction and a Short History of Black Holes 5 In that case not even light could escape from the gravitational field of that body and noone on the surface of the latter could send any luminous signal that distant observers could perceive. In other words by no means distant observers could see the surface of that supermassive object and even less what might be in its interior. Obviously neither Michell nor Laplace had a clear perception that the speed of light c is always the same in every reference frame, since Special Relativity had to wait its own discovery for another century. Yet Laplace’s argument was the following: let us assume that the velocity of light is some constant number a on the surface of the considered celestial body. Then he proceeded to an estimate of the speed of light on the surface of the Sun, which he could do using the annual light aberration in the EarthSun system. The implicit, although unjustified, assumption was that light velocity is unaffected, or weakly affected, by gravity. Analyzing such an assumption in fulldepth it becomes clear that it was an anticipation of Relativity in disguise. Actually condition (2.1.3) has an exact intrinsic meaning in General Relativity. Squaring this equation we can rewrite it as follows: R > rS ≡ 2 GM ≡ 2m c2 (2.1.4) where rS is the Schwarzschild radius of a body of mass M, namely the unique parameter which appears in the Schwarzschild solution of Einstein Equations. So massive bodies are visible and behave qualitatively according to human common sense as long as their dimensions are much larger then their Schwarzschild radius. Due to the smallness of Newton’s constant and to the hugeness of the speed of light, this latter is typically extremely small. Just of the order of a kilometer for a star, and about 10−23 cm for a human body. Nevertheless, as we extensively discussed in Chap. 6 of Volume 1, sooner or later all stars collapse and regions of spacetime with outrageously large energydensities do indeed form, whose typical linear size becomes comparable to rS . The question of what happens if it is smaller than rS is not empty, on the contrary it is a fundamental one, related with the appropriate interpretation of what lies behind the apparent singularity of the Schwarzschild metric at r = rS . As all physicists know, any singularity is just the signal of some kind of criticality. At the singular point a certain description of physical reality breaks down and it must be replaced by a different one: for instance there is a phasetransition and the degrees of freedom that capture most of the energy in an ordered phase become negligible with respect to other degrees of freedom that are dominating in a disordered phase. What is the criticality signaled by the singularity r = rS of the Schwarzschild metric? Is it a special feature of this particular solution of Einstein Equations or it is just an instance of a more general phenomenon intrinsic to the laws of gravity as stated by General Relativity? The answer to the first question is encoded in the wording event horizon. The answer to the second question is that event horizons are a generic feature of static solutions of Einstein equations. An eventhorizon H is a hypersurface in a pseudoRiemannian manifold (M , g) which separates two submanifolds, one E ⊂ M , named the exterior, can communicate with infinity by sending signals to distant observers, the other BH ⊂ M , named the blackhole, is causally disconnected from infinity, since no signal produced in 6 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams BH can reach the outside region E. The blackhole is the region deemed by Michell and Laplace where the escape velocity is larger than the speed of light. In order to give a precise mathematical sense to the above explanation of eventhorizons a lot of things have to be defined and interpreted. First of all what is infinity and is it unique? Secondly which kind of hypersurface is an eventhorizon? Thirdly can we eliminate the horizon singularity by means of a suitable analytic extension of the apparently singular manifold? Finally, how do we define causal relations in a curved Lorentzian spacetime? The present chapter addresses the above questions. The answers were found in the course of the XXth century and constitute the principal milestones in the history of blackholes. Although Schwarzschild metric was discovered in 1916, less than six months after the publication of General Relativity, its analytic extension, that opened the way to a robust mathematical theory of blackholes, was found only fortyfive years later, six after Einstein’s death. In 1960, the American theorist Martin Kruskal (see Fig. 2.2) found a onetomany coordinate transformation that allowed him to represent Schwarzschild spacetime as a portion of a larger spacetime where the locus r = rS is nonsingular, rather it is a welldefined lightlike hypersurface constituting precisely the eventhorizon [6]. A similar coordinate change was independently proposed the same year also by the AustralianHungarian mathematician Georges Szekeres [7]. These mathematical results provided a solid framework for the description of the final state in the gravitational collapse of those stars that are too massive to stop at the stage of whitedwarfs or neutronstars. In Chap. 6 of Volume 1 we already mentioned the intuition of Robert Openheimer and H. Snyder who, in their 1939 paper, wrote: When all thermonuclear sources of energy are exhausted, a sufficiently heavy star will collapse. Unless something can somehow reduce the star’s mass to the order of that of the sun, this contraction will continue indefinitely...past white dwarfs, past neutron stars, to an object cut off from communication with the rest of the universe. Such an object, could be identified with the interior of the event horizon in the newly found Kruskal spacetime. Yet, since the KruskalSchwarzschild metric is spherical symmetric such identification made sense only in the case the parent star had vanishing angular momentum, namely was not rotating at all. This is quite rare since most stars rotate. In 1963 the New Zealand physicist Roy Kerr, working at the University of Texas, found the long sought for generalization of the Schwarzschild metric that could describe the endpoint equilibrium state in the gravitational collapse of a rotating star. Kerr metric, that constitutes the main topic of Chap. 3, introduced the third missing parameter characterizing a blackhole, namely the angular momentum J . The first is the mass M, known since Schwarzschild’s pioneering work, the second, the charge Q (electric, magnetic or both) had been introduced already in the first two years of life of General Relativity. Indeed the ReissnerNordström metric,1 which 1 Hans Jacob Reissner (1874–1967) was a German aeronautical engineer with a passion for mathematical physics. He was the first to solve Einstein’s field equations with a charged electric source 2.1 Introduction and a Short History of Black Holes 7 Fig. 2.2 Martin David Kruskal (1925–2006) on the left and George Szekeres (1911–2005) on the right. Student of the University of Chicago, Kruskal obtained his Ph.D from New York University and was for many years professor at Princeton University. In 1989 he joined Rutgers University were he remained the rest of his life. Mathematician and Physicist, Martin Kruskal gave very relevant contributions in theoretical plasma physics and in several areas of nonlinear science. He discovered exact integrability of some nonlinear differential equations and is reported to be the inventor of the concept of solitons. Kruskal 1960 discovery of the maximal analytic extension of Schwarzschild spacetime came independently and in parallel with similar conclusions obtained by Georges Szekeres. Born in Budapest, Szekeres graduated from Budapest University in Chemistry. As a Jewish he had to escape from Nazi persecution and he fled with his family to China where he remained under Japanese occupation till the beginning of the Communist Revolution. In 1948 he was offered a position at the University of Adelaide in Australia. In this country he remained the rest of his life. Notwithstanding his degree in chemistry Szekeres was a Mathematician and he gave relevant contributions in various of its branches. He is among the founders of combinatorial geometry solves coupled EinsteinMaxwell equations for a charged spherical body, dates back to 1916–1918. The long time delay separating the early finding of the spherical symmetric solutions and the construction of the axial symmetric Kerr metric is explained by the high degree of algebraic complexity one immediately encounters when spherical and he did that already in 1916 [3]. Emigrated to the United States in 1938 he taught at the Illinois Institute of Technology and later at the Polytechnic Institute of Brooklyn. Reissner’s solution was retrieved and refined in 1918 by Gunnar Nordström (1881–1923) a Finnish theoretical physicist who was the first to propose an extension of spacetime to higher dimensions. Independently from Kaluza and Klein and as early as 1914 he introduced a fifth dimension in order to construct a unified theory of gravitation and electromagnetism. His theory was, at the time, a competitor of Einstein’s theory. Working at the University of Leiden in the Netherlands with Paul Ehrenfest, in 1918 he solved Einstein field equations for a spherically symmetric charged body [4] thus extending the Hans Reissner’s results for a point charge. 8 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams symmetry is abandoned. Kerr’s achievement would have been impossible without the previous monumental work of the young Russian theoretician A.Z. Petrov [5]. Educated in the same University of Kazan where, at the beginning of the XIXth century Lobachevskij had first invented nonEuclidian geometry, in his 1954 doctoral dissertation, Petrov conceived a classification of Lorentzian metrics based on the properties of the corresponding Weyl tensor. This leads to the concept of principal nulldirections. According to Petrov there are exactly six types of Lorentzian metrics and, in current nomenclature, Schwarzschild and Reissner Nordström metrics are of Petrov type D. This means that they have two double principal null directions. Kerr made the hypothesis that the metric of a rotating blackhole should also be of Petrov type D and searching in that class he found it. The decade from 1964 to 1974 witnessed a vigorous development of the mathematical theory of blackholes. Brandon Carter solved the geodesic equations for the Kerrmetric, discovering a fourth hidden first integral which reduces these differential equations to quadratures. In the same time through the work of Stephen Hawking, George Ellis, Roger Penrose and several others, general analytic methods were established to discuss, represent and classify the causal structure of spacetimes. Slowly a new picture emerged. Similarly to soliton solutions of other nonlinear differential equations, blackholes have the characteristic features of a new kind of particles, mass, charge and angular momentum being their unique and defining attributes. Indeed it was proved that, irrespectively from all the details of its initial structure, a gravitational collapsing body sets down to a final equilibrium state parameterized only by (M, J, Q) and described by the so called KerrNewman metric, the generalization of the Kerr solution which includes also the Reissner Nordström charges (see Chap. 3, Sect. 3.2). This introduced the theoretical puzzle of information loss. Through gravitational evolution, a supposedly coherent quantum state, containing a detailed fine structure, can evolve to a new state where all such information is unaccessible, being hidden behind the event horizon. The information loss paradox became even more severe when Hawking on one side demonstrated that blackholes can evaporate through a quantum generated thermic radiation and on the other side, in collaboration with Bekenstein, he established, that the horizon has the same properties of an entropy and obeys a theorem similar to the second principle of thermodynamics. Hence from the theoretical viewpoint blackholes appear to be much more profound structures than just a particular type of classical solutions of Einstein’s field equations. Indeed they provide a challenging clue into the mysterious realm of quantum gravity where causality is put to severe tests and needs to be profoundly revised. For this reason the study of blackholes and of their higher dimensional analogues within the framework of such candidates to a Unified Quantum Theory of all Interactions as Superstring Theory is currently a very active stream of research. Ironically such a Revolution in Human Thought about the Laws of Causality, whose settlement is not yet firmly acquired, was initiated two century ago by the observations of Laplace, whose unshakable faith in determinism is well described by the following quotation from the Essai philosophique sur les probabilités. In 2.1 Introduction and a Short History of Black Holes 9 Fig. 2.3 J1655 is a binary system that harbors a black hole with a mass seven times that of the sun, which is pulling matter from a normal star about twice as massive as the sun. The Chandra observation revealed a bright Xray source whose spectrum showed dips produced by absorption from a wide variety of atoms ranging from oxygen to nickel. A detailed study of these absorption features shows that the atoms are highly ionized and are moving away from the black hole in a highspeed wind. The system J1655 is a galactic object located at about 11,000 light years from the Sun that book he wrote: We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes. The vast intellect advocated by Pierre Simon and sometimes named the Laplace demon might find some problems in reconstructing the past structure of a star that had collapsed into a black hole even if that intellect had knowledge of all the conditions of the Universe at that very instant of time. From the astronomical viewpoint the existence of blackholes of stellar mass has been established through many overwhelming evidences, the best being provided by binary systems where a visible normal star orbits around an invisible companion which drags matter from its mate. An example very close to us is the system J1655 shown in Fig. 2.3. Giant blackholes of millions of stellar masses have also been indirectly revealed in the core of active galactic nuclei and also at the center of our Milky Way a black hole is accredited. In the present chapter, starting from the Kruskal extension of the Schwarzschild metric we establish the main framework for the analysis of the causal structure of spacetimes and we formulate the general definition of blackholes. In the next chapter we study the Kerr metric and the challenging connection between the laws of blackhole mechanics and those of thermodynamics. 10 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams 2.2 The Kruskal Extension of Schwarzschild SpaceTime According to the outlined programme in this section we come back to the Schwarzschild metric (2.2.1) that we rewrite here for convenience m m −1 2 dt 2 + 1 − 2 (2.2.1) dr + r 2 dθ 2 + sin2 θ dφ 2 ds 2 = − 1 − 2 r r and we study its causal properties. In particular we investigate the nature and the significance of the coordinate singularity at the Schwarzschild radius r = rS ≡ 2m which, as anticipated in the previous section, turns out to correspond to an event horizon. This explains the nomenclature Schwarzschild emiradius that in Chap. 4 of Volume 1 we used for the surface r = m. 2.2.1 Analysis of the Rindler SpaceTime Before analyzing the Kruskal extension of the Schwarzschild spacetime, as a preparatory exercise we begin by considering the properties of a twodimensional toymodel, the so called Rindler spacetime. This is R2 equipped with the following Lorentzian metric: 2 = −x 2 dt 2 + dx 2 dsRindler (2.2.2) ⊂ R2 singled out by the equation which, apparently, has a singularity on the line H x = 0. A careful analysis reveals that such a singularity is just a coordinate artefact since the metric (2.2.2) is actually flat and can be brought to the standard form of the Minkowski metric via a suitable coordinate transformation: ξ : R2 → R2 (2.2.3) The relevant point is that the diffeomorphism ξ is not surjective since it maps the whole of Rindler spacetime, namely the entire R2 manifold into an open subset I = ξ(R2 ) ⊂ R2 = Mink2 of Minkowski space. This means that Rindler spacetime is incomplete and can be extended to the entire 2dimensional Minkowski space Mink2 . The other key point is that the image ξ(H ) ⊂ Mink2 of the singularity in the extended spacetime is a perfectly regular nulllike hypersurface. These features are completely analogous to corresponding features of the Kruskal extension of Schwarzschild spacetime. Also there we can find a suitable coordinate transformation ξK : R4 → R4 which removes the singularity displayed by the Schwarzschild metric at the Schwarzschild radius r = 2m and such a map is not surjective, rather it maps the entire Schwarzschild spacetime into an open submanifold ξK (Schwarzschild) ⊂ Krusk of a larger manifold named the Kruskal spacetime. Also in full analogy with the case of the Rindler toymodel the image ξK (H ) of the coordinate singularity H defined by the equation r = 2m is a regular nulllike hypersurface of Kruskal spacetime. In this case it has the interpretation of eventhorizon delimiting a blackhole region. 2.2 The Kruskal Extension of Schwarzschild SpaceTime 11 The basic question therefore is: how do we find the appropriate diffeomorphism ξ or ξK ? The answer is provided by a systematic algorithm which consists of the following steps: 1. derivation of the equations for geodesics, 2. construction of a complete system of incoming and outgoing null geodesics, 3. transition to a coordinate system where the new coordinates are the affine parameters along the incoming and outgoing null geodesics, 4. analytic continuation of the new coordinate patch beyond its original domain of definition. We begin by showing how this procedure works in the case of the metric (2.2.2) and later we apply it to the physically significant case of the Schwarzschild metric. The metric (2.2.2) has a coordinate singularity at x = 0 where the determinant det gμν = −x 2 has a zero. In order to understand the real meaning of such a singularity we follow the programme outlined above and we write the equation for null geodesics: dx μ dx ν = 0; −x 2 t˙2 + ẋ 2 = 0 dλ dλ from which we immediately obtain: 2 1 dx dx = ± ln x + const = 2 ⇒ t =± dt x x gμν (x) (2.2.4) (2.2.5) Hence we can introduce the null coordinates by writing: t + ln x = v; v = const ⇔ (incoming null geodesics) t − ln x = u; u = const ⇔ (outgoing null geodesics) (2.2.6) The shape of the corresponding null geodesics is displayed in Fig. 2.4. The first change of coordinates is performed by replacing x, t by u, v. Using: x 2 = exp[v − u]; dx dv − du = ; x 2 dt = dv + du 2 (2.2.7) the metric (2.2.2) becomes: 2 dsRindler = − exp[v − u] du dv (2.2.8) Next step is the calculation of the affine parameter along the null geodesics. Here we use a general property encoded in the following lemma: Lemma 2.2.1 Let k be a Killing vector for a given metric gμν (x) and let t = be the tangent vector to a geodesic. Then the scalar product: E ≡ −(t, k) = −gμν is constant along the geodesic. dx μ ν k dλ dx μ dλ (2.2.9) 12 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams Fig. 2.4 Null geodesics of the Rindler metric. The thin curves are incoming (v = const), while the thick ones are outgoing (u = const) Proof The proof is immediate by direct calculation. If we take the d/dλ derivative of E we get: dE dx ρ dx μ ν dx μ dx ρ ν = − ∇ρ gμν k − gμν ∇ρ k dλ dλ dλ dλ dλ − = 0 since metric is cov. const. dx ρ dx μ gμν ∇ρ k ν , = 0 for the geodesic eq. (2.2.10) dλ dλ = 0 for the Killing vec. eq. So we obtain the sum of three terms that are separately zero for three different reasons. dt Relying on Lemma 2.2.1 in Rindler space time we can conclude that E = x 2 dλ d is constant along geodesics. Indeed the vector field k ≡ dt is immediately seen to be a Killing vector for the metric (2.2.2). Then by means of straightforward manipulations we obtain: dλ = du + dv 1 exp[v − u] E 2 λ= exp[−u] 2E exp[v] exp[v] − 2E exp[−u] ⇒ on u = const outgoing null geodesics on v = const incoming null geodesics (2.2.11) The third step in the algorithm that leads to the extension map corresponds to a coordinate transformation where the new coordinates are proportional to the affine parameters along incoming and outgoing null geodesics. Hence in view of (2.2.11) we introduce the coordinate change: U = −e−u ⇒ dU = e−u du; V = ev ⇒ dV = ev dv (2.2.12) 2.2 The Kruskal Extension of Schwarzschild SpaceTime 13 Fig. 2.5 The image of Rindler spacetime in twodimensional Minkowski spacetime is the shaded region I bounded by the two null surfaces X = T (X > 0) and X = −T (X > 0). These latter are the image of the coordinate singularity x = 0 of the original metric by means of which the Rindler metric (2.2.8) becomes: 2 dsRindler = −dU ⊗ dV (2.2.13) Finally, with a further obvious transformation: T= V +U ; 2 X= V −U 2 (2.2.14) the Rindler metric (2.2.13) is reduced to the standard twodimensional Minkowski metric in the plane {X, T }: 2 dsRindler = −dT 2 + dX 2 (2.2.15) Putting together all the steps, the coordinate transformation that reduces the Rindler metric to the standard form (2.2.15) is the following: T x = X2 − T 2 ; (2.2.16) t = arctanh X In this way we have succeeded in eliminating the apparent singularity x = 0 since the metric (2.2.15) is perfectly regular in the whole {X, T } plane. The subtle point of this procedure is that by means of the transformation (2.2.12) we have not only eliminated the singularity, but also extended the spacetime. Indeed the definition (2.2.12) of the U and V coordinates is such that V is always positive and U always negative. This means that in the {U, V } plane the image of Rindler spacetime is the quadrant U < 0; V > 0. In terms of the final X, T variables the image of the original Rindler spacetime is the angular sector I depicted in Fig. 2.5. Considering the coordinate transformation (2.2.16) we see that the image in the extended spacetime of the apparent singularity x = 0 is the locus X 2 = T 2 which is perfectly regular but has the distinctive feature of being a nulllike surface. This surface is also the boundary of the image I of Rindler spacetime in its maximal extension. Furthermore setting X = ±T we obtain t = ±∞. This means that in the original Rindler space any test particle takes an infinite amount of coordinate time to reach the boundary locus x = 0: this is also evident from the plot of null geodesics in Fig. 2.4. On the other hand the proper time taken by a test particle to reach such a locus from any other point is just finite. 14 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams All these features of our toy model apply also to the case of Schwarzschild spacetime once it is extended with the same procedure. The image of the coordinate singularity r = 2m will be a nulllike surface, interpreted as event horizon, which can be reached in a finite propertime but only after an infinite interval of coordinate time. What will be new and of utmost physical interest is precisely the interpretation of the locus r = 2m as an event horizon H which leads to the concept of BlackHole. Yet this interpretation can be discovered only through the Kruskal extension of Schwarzschild spacetime and this latter can be systematically derived via the same algorithm we have applied to the Rindler toy model. 2.2.2 Applying the Same Procedure to the Schwarzschild Metric We are now ready to analyze the Schwarzschild metric (2.2.1) by means of the tokens illustrated above. The first step consists of reducing it to twodimensions by fixing the angular coordinates to constant values θ = θ0 , φ = φ0 . In this way the metric (2.2.1) reduces to: 2 dsSchwarz. 2m 2m −1 2 2 dt + 1 − =− 1− dr r r (2.2.17) Next, in the reduced space spanned by the coordinates r and t we look for the nullgeodesics. From the equation: 2m −1 2 2m 2 ṙ = 0 − 1− t˙ + 1 − r r (2.2.18) we obtain: r dt =± dr r − 2m ⇒ t = ±r ∗ (r) (2.2.19) where we have introduced the so called ReggeWheeler tortoise coordinate defined by the following indefinite integral: r r ∗ r (r) ≡ dr = r + 2m log −1 (2.2.20) r − 2m 2m Hence, in full analogy with (2.2.6), we can introduce the null coordinates t + r ∗ (r) = v; v = const ⇔ (incoming null geodesics) − r ∗ (r) = u; u = const ⇔ (outgoing null geodesics) t (2.2.21) and the analogue of Fig. 2.4 is now given by Fig. 2.6. Inspection of this picture reveals the same properties we had already observed in the case of the Rindler toy model. What is important to stress in the present model is that each point of the 2.2 The Kruskal Extension of Schwarzschild SpaceTime 15 Fig. 2.6 Null geodesics of the Schwarzschild metric in the r, t plane. The thin curves are incoming (v = const), while the thick ones are outgoing (u = const). Each point in this picture represents a 2sphere, parameterized by the angles θ0 and φ0 . The thick vertical line is the surface r = rS = 2m corresponding to the coordinate singularity. As in the case of the Rindler toy model the null– geodesics incoming from infinity reach the coordinate singularity only at asymptotically late times t →> +∞. Similarly outgoing nullgeodesics were on this surface only at asymptotically early times t → −∞ diagram actually represents a 2sphere parameterized by the two angles θ and φ that we have freezed at the constant values θ0 and φ0 . Since we cannot make fourdimensional drawings some pictorial idea of what is going on can be obtained by replacing the 2sphere with a circle S1 parameterized by the azimuthal angle φ. In this way we obtain a threedimensional spacetime spanned by coordinates t, x = r cos φ, y = r sin φ. In this space the nullgeodesics of Fig. 2.6 become twodimensional surfaces. Indeed these nullsurfaces are nothing else but the projections θ = θ0 = π/2 of the true null surfaces of the Schwarzschild metric. In Fig. 2.7 we present two examples of such projected null surfaces, one incoming and one outgoing. Having found the system of incoming and outgoing nullgeodesics we go over to point (iii) of our programme and we make a coordinate change from t, r to u, v. By straightforward differentiation of (2.2.20), (2.2.21) we obtain: rS du − dv dr = − 1 − ; r 2 dt = du + dv 2 (2.2.22) so that the reduced Schwarzschild metric (2.2.17) becomes: 2 dsSchwarz. rS =− 1− r du ⊗ dv (2.2.23) 16 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams Fig. 2.7 An example of two null surfaces generated by null geodesics of the Schwarzschild metric in the r, t plane Using the definition (2.2.20) of the tortoise coordinate we can also write: r rS v−u exp − (2.2.24) 1− = − exp r 2rS rS which combined with (2.2.22) yields: v − u rS r 2 dsSchwarz. exp du ⊗ dv = exp − rS 2rS r (2.2.25) In complete analogy with (2.2.12) we can now introduce the new coordinates: u u U = − exp − ; V = exp − (2.2.26) 2rS 2rS that play the role of affine parameters along the incoming and outgoing null geodesics. Then by straightforward differentiation of (2.2.26) the reduced Schwarzschild metric (2.2.25) becomes: rS3 r 2 dsSchwarz. = −4 exp − dU ⊗ dV (2.2.27) r rS where the variable r = r(U, V ) is the function of the independent coordinates U , V implicitly determined by the transcendental equation: r r + rS log − 1 = rS log(−U V ) (2.2.28) rS In analogy with our treatment of the Rindler toy model we can make a final coordinate change to new variables X, T related to U , V as in (2.2.14). These, together 2.2 The Kruskal Extension of Schwarzschild SpaceTime 17 with the angular variables θ , φ make up the Kruskal coordinate patch which, putting together all the intermediate steps, is related to the original coordinate patch t, r, θ , φ by the following transition function: ⎧ θ =θ polar ⎪ ⎪ ⎪ ⎨φ = φ versus ( rrS − 1) exp[ rrS ] = T 2 − X 2 Kruskal ⎪ ⎪ ⎪ coord. ⎩ t = log( T +X ) ≡ 2 arctanh X rS T −X (2.2.29) T In Kruskal coordinates the Schwarzschild metric (2.2.1) takes the final form: 2 dsKrusk rS3 r −dT 2 + dX 2 + r 2 dθ 2 + sin2 θ dφ 2 = 4 exp r rS (2.2.30) where the r = r(X, T ) is implicitly determined in terms of X, T by the transcendental equations (2.2.29). 2.2.3 A First Analysis of Kruskal SpaceTime Let us now consider the general properties of the spacetime (MKrusk , gKrusk ) identified by the metric (2.2.30) and by the implicit definition of the variable r contained in (2.2.29). This analysis is best done by inspection of the twodimensional diagram displayed in Fig. 2.8. This diagram lies in the plane {X, T }, each of whose points represents a two sphere spanned by the anglecoordinates θ and φ. The first thing to remark concerns the physical range of the coordinates X, T . The Kruskal manifold MKrusk does not coincide with the entire plane, rather it is the infinite portion of the latter comprised between the two branches of the hyperbolic locus: T 2 − X 2 = −1 (2.2.31) This is the image in the X, T plane of the r = 0 locus which is a genuine singularity of both the original Schwarzschild metric and of its Kruskal extension. Indeed from (5.9.6)–(5.9.11) of Volume 1 we know that the intrinsic components of the curvature tensor depend only on r and are singular at r = 0, while they are perfectly regular at r = 2m. Therefore no geodesic can be extended in the X, T plane beyond (2.2.31) which constitutes a boundary of the manifold. Let us now consider the image of the constant r surfaces. Here we have to distinguish two cases: r > rS or r < rS . We obtain: r {X, T } = {h cosh p, h sinh p}; h = e rS rrS − 1 for r > rS (2.2.32) r {X, T } = {h sinh p, h cosh p}; h = e rS 1 − rrS for r < rS 18 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams Fig. 2.8 A twodimensional diagram of Kruskal spacetime These are the hyperbolae drawn in Fig. 2.8. Calculating the normal vector N μ = {∂p T , ∂p X, 0, 0} to these surfaces, we find that it is timelike N μ N ν gμν < 0 for r > rS and spacelike N μ N ν gμν > 0 for r < rS . Correspondingly, according to a discussion developed in the next section, the constant r surfaces are spacelike outside the sphere of radius rS and timelike inside it. The dividing locus is the pair of straight lines X = ±T which correspond to r = rS and constitute a nullsurface, namely one whose normal vector is lightlike. This nullsurface is the event horizon, a concept whose precise definition needs, in order to be formulated, a careful reconsideration of the notions of Future, Past and Causality in the context of General Relativity. The next two sections pursue such a goal and by their end we will be able to define BlackHoles and their Horizons. Here we note the following. If we solve the geodesic equation for timelike or nulllike geodesics with arbitrary initial data inside region II of Fig. 2.8 then the end point of that geodesic is always located on the singular locus T 2 − X 2 = −1 and the whole development of the curve occurs inside region II. The formal proof of this statement is involved and it will be overcome by the methods of Sects. 2.3 and 2.4. Yet there is an intuitive argument which provides the correct answer and suffices to clarify the situation. Disregarding the angular variables θ and φ the Kruskal metric (2.2.30) reduces to: 2 dsKrusk = F (X, T ) −dT 2 + dX 2 ; rS3 r F (X, T ) = 4 exp r rS (2.2.33) 2 so that it is proportional to twodimensional Minkowski metric dsMink = −dT 2 + dX 2 through the positive definite function F (X, T ). In the language of Sect. 2.4 this fact means that, reduced to twodimensions, Kruskal and Minkowski metrics are conformally equivalent. According to Lemma 2.4.1 proved later on, conformally equivalent metrics share the same lightlike geodesics, although the timelike and spacelike ones may be different. This means that in twodimensional Kruskal spacetime light travels along straight lines of the form X = ±T + k where k is some constant. This is the same statement as saying that at any point p of the {X, T } plane the tangent vector to any curve is timelike or lightlike and oriented to the future if 2.3 Basic Concepts about Future, Past and Causality 19 Fig. 2.9 The lightcone orientations in Kruskal spacetime and the difference between physical geodesics in regions I and II its inclination α with respect to the X axis is in the following range 3π/4 ≥ α ≥ π/4. This applies to the whole plane, yet it implies a fundamental difference in the destiny of physical particles that start their journey in region I (or IV) of the Kruskal plane, with respect to the destiny of those ones that happen to be in region II at some point of their life. As it is visually evident from Fig. 2.9, in region I we can have curves (and in particular geodesics) whose tangent vector is timelike and future oriented at any of their points which nonetheless avoid the singular locus and escape to infinity. In the same region there are also future oriented timelike curves which cross the horizon X = ±T and end up on the singular locus, yet these are not the only ones, as already remarked. On the contrary all curves that at some point happen to be inside region II can no longer escape to infinity since, in order to be able to do so, their tangent vector should be spacelike, at least at some of their points. Hence the horizon can be crossed from region I to region II, never in the opposite direction. This leads to the existence of a BlackHole, namely a spacetime region, (II in our case) where gravity is so strong that not even light can escape from it. No signal from region II can reach a distant observer located in region I who therefore perceives only the presence of the gravitational field of the black hole swapping infalling matter. To encode the ideas intuitively described in this section into a rigorous mathematical framework we proceed next to implement our already announced programme. This is the critical review of the concepts of Future, Past and Causality within General Relativity, namely when we assume that all physical events are points p in a pseudoRiemannian manifold (M , g) with a Lorentzian signature. 2.3 Basic Concepts about Future, Past and Causality Our discussion starts by reviewing the basic properties of the lightcone (see Fig. 2.10). In Special Relativity, where spacetime is Minkowskispace, namely a pseudoRiemannian manifold which is also affine, the light cone has a global meaning, while in General Relativity lightcones can be defined only locally, namely at each point p ∈ M . In any case the Lorentzian signature of the metric implies that ∀p ∈ M , the tangent space Tp M is isomorphic to Minkowski space and it admits the same decomposition in timelike, nulllike and spacelike submanifolds. Hence 20 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams Fig. 2.10 The structure of the lightcone the analysis of the lightcone properties has a general meaning also in General Relativity, although such analysis needs to be repeated at each point. All the complexities inherent with the notion of global causality arise from the need of gluing together the locally defined lightcones. We will develop appropriate conceptual tools to manage such a gluing after our review of the local lightcone properties. 2.3.1 The LightCone When a metric has a Lorentzian signature, vectors t can be of threetypes: 1. Timelike, if (t, t) < 0 in mostly plus convention for gμν . 2. Spacelike, if (t, t) > 0 in mostly plus convention for gμν . 3. Nulllike, if (t, t) = 0 both in mostly plus and mostly minus convention for gμν . At any point p ∈ M the lightcone Cp is composed by the set of vectors t ∈ Tp M which are either timelike or nulllike. In order to study the properties of the lightcones it is convenient to review a few elementary but basic properties of vectors in Minkowski space. Theorem 2.3.1 All vectors orthogonal to a timelike vector are spacelike. Proof Using a mostly plus signature, we can go to a diagonal basis such that: g(X, Y ) = g00 X 0 Y 0 + (X, Y) (2.3.1) where g00 < 0 and ( , ) denotes a nondegenerate, positivedefinite, Euclidian bilinear form on Rn−1 . In this basis, if X⊥T and T is timelike we have: −g00 T 0 T 0 > (T, T) −g00 T 0 X 0 = (T, X) ≤ √ (T, T)(X, X) (2.3.2) Then we get: −g00 T 0 X 0 (T, X) ≤ <√ (T, T) −g00 T 0 T 0 (X, X) (2.3.3) 2.3 Basic Concepts about Future, Past and Causality 21 Squaring all terms in (2.3.3) we obtain −g00 X 0 X 0 < (X, X) ⇒ g(X, X) > 0 namely the fourvector X is spacelike as asserted by the theorem. (2.3.4) Another useful property is given by the following Lemma 2.3.1 The sum of two futuredirected timelike vectors is a futuredirected timelike vector. Proof Let t and T be the two vectors under considerations. By hypothesis we have g(t, t) < 0; g(T , T ) < 0; Since: t0 > 0 T0 >0 √ −g00 t 0 > (t, t) √ −g00 T 0 > (T, T) √ √ −g00 t 0 T 0 > (t, t)(T, T) > (t, T) (2.3.5) (2.3.6) we have: g(t + T , t + T ) = g(t, t) + g(T , T ) + 2g(t, T ) ⇓ 0 2 0 2 −g00 t + T + 2t 0 T 0 > (t, t) + (T, T) + 2(t, T) (2.3.7) which proves that t + T is timelike. Moreover t 0 + T 0 > 0 and so the sum vector is also futuredirected as advocated by the lemma. On the other hand with obvious changes in the proof of Theorem 2.3.1 the following lemma is established Lemma 2.3.2 All vectors X, orthogonal to a lightlike vector L are either lightlike or spacelike. Let us now consider in the manifold (M , g) surfaces Σ defined by the vanishing of some smooth function of the local coordinates: p∈Σ ⇔ f (p) = 0 where f ∈ C∞ (M ) (2.3.8) By definition the normal vector to the surface Σ is the gradient of the function f : n(Σ) μ = ∇μ f = ∂μ f (2.3.9) 22 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams Indeed any tangent vector to the surface is by construction orthogonal to n(Σ) : g t (Σ) , n(Σ) = 0 (2.3.10) Definition 2.3.1 A surface Σ is said to be spacelike if its normal vector n(Σ) is everywhere timelike on the surface. Conversely Σ is timelike if n(Σ) is spacelike. We name null surfaces those Σ whose normal vector n(Σ) is nulllike. Null surfaces have very intriguing properties. First of all their normal vector is also tangent to the surface. This follows from the fact that the normal vector is orthogonal to itself. Furthermore we can prove that any nullsurface is generated by nullgeodesics. Indeed we can easily prove that the normal vector n(Σ) to a null surface is the tangent vector to a nullgeodesics. Indeed we have: 0 = ∇μ ∇ν f ∇ ν f = 2∇ ν f ∇ν ∇μ f = n ν ∇ν n μ (2.3.11) and the last equality is precisely the geodesic equation satisfied by the integral curve to the normal vector n(Σ) . A typical nullsurface is the eventhorizon of a blackhole. 2.3.2 Future and Past of Events and Regions Let us now consider the pseudoRiemannian spacetime manifold (M , g) and at each point p ∈ M introduce the local lightcone Cp ⊂ Tp M . In this section we find it convenient to change convention and use a mostly minus signature where g00 > 0. Definition 2.3.2 The local lightcone Cp (see Fig. 2.11) is the set of all tangent vectors t ∈ Tp M , such that: gμν t μ t ν ≥ 0 (2.3.12) and it is the union of the future lightcone with the past lightcone: Cp = Cp+ Cp− (2.3.13) where t ∈ Cp+ ⇔ g(t, t) ≥ 0 and t 0 > 0 t ∈ Cp− ⇔ g(t, t) ≥ 0 and t 0 < 0 (2.3.14) The vectors in Cp+ are named futuredirected, while those in Cp− are named pastdirected. 2.3 Basic Concepts about Future, Past and Causality 23 Fig. 2.11 At each point of the spacetime manifold, the tangent space Tp M contains the submanifold Cp of timelike and nullvectors which constitutes the local lightcone We can now transfer the notions of time orientation from vectors to curves by means of the following definitions: Definition 2.3.3 A differentiable curve λ(s) on the spacetime manifold M is named a futuredirected timelike curve if at each point p ∈ λ, the tangent vector to the curve t μ is futuredirected and timelike. Conversely λ(s) is pastdirected timelike if such is t μ . Similarly we have: Definition 2.3.4 A differentiable curve λ(s) on the spacetime manifold M is named a futuredirected causal curve if at each point p ∈ λ, the tangent vector to the curve t μ is either a futuredirected timelike or a futuredirected nulllike vector. Conversely λ(s) is a pastdirected causal curve when the tangent t μ , timelike or nulllike, is past directed. Relying on these concepts we can introduce the notions of Chronological Future and Past of a point p ∈ M . Definition 2.3.5 The Chronological Future (Past) of a point p, denoted I ± (p) is the subset of points of M , defined by the following condition: ⎧ ⎫ ∃ future (past)directed timelike ⎬ ⎨ curve λ(s) such that I ± (p) = q ∈ M (2.3.15) ⎩ ⎭ λ(0) = p; λ(1) = q In other words the Chronological Future or Past of an event are all those events that can be connected to it by a futuredirected or pastdirected timelike curve. Let now S ⊂ M be a region of spacetime, namely a continuous submanifold of the spacetime manifold. Definition 2.3.6 The Chronological Future (Past) of the region S, denoted I ± (S) is defined as follows: I ± (S) = I ± (p) (2.3.16) p∈S 24 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams Fig. 2.12 The union of two timelike futuredirected curves is still a timelike future directed curve An elementary property of the Chronological Future is the following: I ± I ± (S) = I ± (S) (2.3.17) The proof is illustrated in Fig. 2.12. If q ∈ I ± (I ± (S)) then, by definition, there exists at least one point q ∈ I ± (S) to which q is connected by a timelike future directed curve λ2 (s). On the other hand, once again by definition, q is connected by a futuredirected timelike curve λ1 (s) to at least one point p ∈ S. Joining λ1 with λ2 we obtain a futuredirected timelike curve that connects q to p, which implies that q ∈ I + (S). In a similar way, if S denotes the closure, in the topological sense, of the region S, we prove that: I + (S) = I + (S) (2.3.18) In perfect analogy with Definition 2.3.5 we have: Definition 2.3.7 The Causal Future (Past) of a point p, denoted J ± (p) is the subset of points of M , defined by the following condition: ⎧ ⎫ ∃ future (past)directed causal ⎬ ⎨ curve λ(s) such that J ± (p) = q ∈ M (2.3.19) ⎩ ⎭ λ(0) = p; λ(1) = q and the Causal Future(Past) of a region S, denoted J ± (S) is: J ± (S) = J ± (p) (2.3.20) p∈S An important point which we mention without proof is the following. In flat Minkowski space J ± (p) is always a closed set in the topological sense, namely it contains its own boundary. In general curved spacetimes J ± (p) can fail to be closed. 2.3 Basic Concepts about Future, Past and Causality 25 Fig. 2.13 In twodimensional Minkowski space we show an example of achronal set. In the picture on left the segment S parallel to the space axis is achronal because it does not intersect its chronological future. On the other hand, in the picture on the right, the line S, although one dimensional is not achronal because it intersects its own chronological future Achronal Sets Definition 2.3.8 Let S ⊂ M be a region of spacetime. S is said to be achronal if and only if S=∅ (2.3.21) I + (S) The relevance of achronal sets resides in the following. When considering classical or quantum fields φ(x), conditions on these latter specified on an achronal set S are consistent, since all the events in S do not bear causal relations to each other. On the other hand one cannot freely specify initial conditions for fields on regions that are not achronal because their points are causally related to each other. In Fig. 2.13 we illustrate an example and a counterexample of achronal sets in twodimensional Minkowski space. TimeOrientability We mentioned above the splitting of the local lightcones in the future Cp+ and past Cp− cones. Clearly, just as all the tangent spaces are glued together to make a fibrebundle, the same is true of the local lightcones. The subtle point concerns the nature of the transition functions. Those of the tangent bundle T M → M to an ndimensional manifold take values in GL(n, R). The lightcone, on the other hand, is leftinvariant only by the subgroup O(1, n − 1) ⊂ GL(n, R). Furthermore the past and future cones are left invariant only by the subgroup of the former connected with the identity, namely SO(1, n − 1) ⊂ O(1, n − 1). Hence the tipping of the lightcones from one point to the other of the spacetime manifold are described by those transition functions of the tangent bundle that take values in the cosets GL(n, R)/O(1, n − 1) or GL(n, R)/SO(1, n − 1). The difference is subtle. Let Hp ⊂ GL(n, R) be the subgroup isomorphic to SO(1, n − 1), which leaves invariant the future and past lightcones at p ∈ M and let Hq ⊂ GL(n, R) be the subgroup, also isomorphic to SO(1, n − 1), which leaves invariant the future and past light cones at the point q ∈ M . The question is the following. Are Hp and Hq conjugate to each other under the transition function g(p, q) ∈ GL(n, R) of the tangent bundle, that connects the tangent plane at p with the tangent plane at q, namely is it true that Hq = g(p, q)Hp g −1 (p, q)? If the answer is yes for all pair of points 26 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams Fig. 2.14 The edge of an achronal set in twodimensional Minkowski space. Notwithstanding how small can be the neighborhood O of the end point of the segment S, which we singled out with the dashed line, it contains a pair of points q and p, the former in the past of the endpoint, the latter in its future, which can be connected by a timelike curve getting around the segment S and not intersecting it. Clearly this property does not hold for any of the interior points of the segment p, q in M , then the manifold (M , g) is said to be timeorientable. In this case the definition of future and past orientations varies continuously from one point to the other of the manifold without singular jumps. Yet there exist cases where the answer is no. When this happens the corresponding manifold is not timeorientable and all global notions of causality loose their meaning. In all the sequel we assume timeorientability. For time orientable spacetimes we have the following theorem that we mention without proof Theorem 2.3.2 Let (M , g) be timeorientable and let S ⊂ M be a continuous connected region. The boundary of the chronological future of S, denoted ∂I + (S) is an achronal (n − 1)dimensional submanifold. Domains of Dependence The future domains of dependence are those submanifolds of spacetime which are completely causally determined by what happens on a certain achronal set S. Alternatively the past domains of dependence are those that completely causally determine what happens on S. To discuss them we begin by introducing one more concept, that of edge. Definition 2.3.9 Let S be an achronal and closed set. We define edge of S the set of points a ∈ S such that for all open neighborhoods Oa of a, there exists two points q ∈ I − (a) and p ∈ I + (a) both contained in Oa which are connected by at least one timelike curve that does not intersect S. The definition of edge is illustrated in Fig. 2.14. A very important theorem that once again we mention without proof is the following: 2.3 Basic Concepts about Future, Past and Causality 27 Fig. 2.15 Two examples of Future and Past domains of dependence for an achronal region S of twodimensional Minkowski space Theorem 2.3.3 Let S ⊂ M be an achronal closed region of a timeorientable ndimensional spacetime (M , g) with Lorentz signature. Let us assume that edge(S) = ∅. Then S is an (n − 1)dimensional submanifold of M . The relevance of this theorem resides in that it establishes the appropriate notion of places in spacetime, where one can formulate initial conditions for the time development. These are achronal sets without an edge and, as intuitively expected, they correspond to the notion of space ((n − 1)dimensional submanifolds) as opposed to time. These ideas are made more precise introducing the appropriate mathematical definitions of domains of dependence. Definition 2.3.10 Let S be a closed achronal set. We define the Future (Past) Domain of Dependence of S, denoted D ± (S) as follows: every past (future)directed timelike (2.3.22) D ± (S) = p ∈ M curve through p intersects S The above definition is illustrated in Fig. 2.15. The meaning of D ± (S) was already outlined above. What happens in the points p ∈ D + (S) is completely determined by the knowledge of what happened in S. Conversely what happened in S is completely determined by the knowledge of what happened in all points of p ∈ D − (S). The Complete Domain of Dependence of the achronal set S is defined below: D − (S) (2.3.23) D(S) ≡ D + (S) All the introduced definitions were preparatory for the appropriate formulation of the main concept, that of Cauchy surface. Cauchy surfaces Definition 2.3.11 A closed achronal set Σ ⊂ M of a Lorentzian spacetime manifold (M , g) is named a Cauchy surface if and only if its domain of dependence 28 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams coincides with the entire spacetime, as follows: D(Σ) = M (2.3.24) A Cauchy surface is without edge by definition. Hence it is an (n − 1)dimensional hypersurface. If a Cauchy surface Σ exists, data on Σ completely determine their future development in time. This is true for all fields lying on M but also for the metric. Knowing for instance the perturbations of the metric on a Cauchy surface we can calculate (analytically or numerically) their future evolution without ambiguity. Definition 2.3.12 A Lorentzian spacetime (M , g) is named Globally Hyperbolic if and only if it admits at least one Cauchy surface. Globally Hyperbolic spacetimes are the good, nonpatological solutions of Einstein equations which allow a consistent and global formulation of causality. A major problem of General Relativity is to pose appropriate conditions on matter fields such that Global Hyperbolicity of the metric is selected. Unified theories should possess such a property. 2.4 Conformal Mappings and the Causal Boundary of SpaceTime Given the appropriate definitions of Future and Past discussed in the previous section, in order to study the causal structure of a given spacetime (M , g), one has to cope with a classical problem met in the theory of analytic functions, namely that of bringing the point at infinity to a finite distance. Only in this way the behavior at infinity can be mastered and understood. Behavior of what? This is the obvious question. In complex function theory the behavior under investigation is that of functions, in our case is that of geodesics or, more generally, of causal curves. These latter are those that can be traveled by physical particles and the issue of causality is precisely the question of who can be reached by what. Infinity plays a distinguished role in this game because of an intuitively simple feature that characterizes those systems which the spacetimes (M , g) under consideration here are supposed to describe. The feature alluded above corresponds to the concept of an isolated dynamical system. A massive star, planetary system or galaxy is, in any case, a finite amount of energy concentrated in a finite region which is separated from other similar regions by extremely large spatial distances. The basic idea of General Relativity foresees that spacetime is curved by the presence of energy or matter so that, far away from concentrations of the latter, the metric should become the flat one of empty Minkowski space. This was the boundary condition utilized in the solution of Einstein equations which lead to the Schwarzschild metric and it is the generic one assumed whenever we use Einstein equations to describe any type of star or of other localized energy lumps. Mathematically, the property of (M , g) 2.4 Conformal Mappings and the Causal Boundary of SpaceTime 29 which encodes such a physical idea is named asymptotic flatness. The point at infinity corresponds to the regions of the considered spacetime (M , g) where the metric g becomes indistinguishable from the Minkowski metric gMink and, by hypothesis, these are at very large distances from the center of gravitation. We would like to study the structure of such an asymptotic boundary and its causal relations with the finite distance spacetime regions. Before proceeding in this direction it is mandatory to stress that asymptotic flatness is neither present nor required in other physical contexts, notably that of cosmology. When we apply General Relativity to the description of the Universe and of its Evolution, energy is not localized rather it is overall distributed. There is no asymptotically far empty region and most of what we discuss here has to be revised. This being clarified let us come back to the posed problem. Assuming that a flat boundary at infinity exists how can we bring it to a finite distance and study its structure? The answer is suggested by the analogy with the theory of analytic functions we already anticipated and it is provided by the notion of conformal transformations. In the complex plane, conformal transformations change distances but preserve angles. In the same way the conformal transformations we want to consider here are allowed to change the metric, that is the instrument to calculate distances, yet they should preserve the causal structure. In plain words this means that timelike, spacelike and nulllike vector fields should be mapped into vector fields with the same properties. Under these conditions causal curves are mapped into causal curves, although geodesics are not necessarily mapped into geodesics. Shortening the distances, infinity can come close enough to be inspected. We begin by presenting an explicit instance of such conformal transformations corresponding to a specifically relevant case, namely that of Minkowski space. From the analysis of this example we will extract the general rules of the game to be applied also to the other cases. 2.4.1 Conformal Mapping of Minkowski Space into the Einstein Static Universe Let us consider flat Minkowski metric in polar coordinates: 2 dsMink = −dt 2 + dr 2 + r 2 dθ 2 + sin2 θ dφ 2 and let us perform the following change of coordinates: T +R t + r = tan 2 T −R t − r = tan 2 θ =θ φ=φ (2.4.1) (2.4.2) (2.4.3) (2.4.4) (2.4.5) 30 2 Extended SpaceTimes, Causal Structure and Penrose Diagrams where T , R are the new coordinates replacing t, r. By means of straightforward calculations we find that in the new variables the flat metric becomes: 2 2 dsMink = Ω −2 (T , R) dsESU = −dT + dR + sin R dθ 2 + sin2 θ dφ 2 1 T +R T +R Ω(T , R) = cos cos 2 2 2 2 dsESU 2 2 2 (2.4.6) (2.4.7) (2.4.8) This apparently trivial calculation leads to many important conclusions. 2 , First of all let us observe that, considered in its own right, the metric dsESU 3 named the Einstein Static Universe, is the natural metric on a manifold R × S . To see this it suffices to note that because of its appearance as argument of a sine, the variable R is an angle, furthermore, parameterizing the points of a threesphere: 1 = X12 + X22 + X32 + X42 (2.4.9) as follows: X1 = cos R X2 = sin R cos θ X3 = sin R sin θ cos φ (2.4.10) X4 = sin R sin θ sin φ another straightforward calculation reveals that: 4 dXi2 = dR 2 + sin2 R dθ 2 + sin2 θ dφ 2 (2.4.11) i=1 2 2 = −dT 2 + dsS23 . The metric dsESU receives the name This demonstrates that dsESU of Einstein Static Universe for the following reason. It is just an instance of a family of metrics, which we will consider in later chapters while studying cosm