Pagina principale 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

‘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 two-fold. 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 Kerr-Newman metric. An extensive historical account of the development of modern cosmology is followed by a detailed presentation of its mathematical structure, including non-isotropic 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 world-volume gauge theories with particular emphasis on the geometrical interpretation of kappa-supersymmetry. 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.
Anno:
2012
Edizione:
2013
Editore:
Springer
Lingua:
english
Pagine:
476 / 465
ISBN 10:
9400754426
ISBN 13:
9789400754423
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PDF, 7.04 MB
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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.
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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, view-points
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.
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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 full-fledged 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 ad-hoc 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 no-one 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
Gauge-Theories 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 self-contained
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 fibre-bundles, 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
non-gravitational 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 non-Abelian 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 bottom-up 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 stress-energy tensor and its
non-linear upgrading follows, once the stress-energy 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
space-time. 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 black-holes and the Kerr-Newman 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 non-rotating 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 Bekenstein-Hawking 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 island-universes (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 full-fledged 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 space-time 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 read-shift distances. The conceptual problems (horizon and flatness) of
the Standard Cosmological Model are next discussed as an introduction to the new

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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 p-forms, 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 M-theory, 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 full-fledged 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 gauge-theories described by suitable world-volume
actions endowed with κ-supersymmetry 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 κ-supersymmetric
Born-Infeld type world-volume actions on arbitrary supergravity backgrounds is described. It is subsequently applied to the case of the D3-brane, 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 Randall-Sundrun 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
U-isometries 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 self-contained summary of the most important special geometries appearing in

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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 M-theory 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
Springer-Verlag, 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 week-ends 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.

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Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2

Extended Space-Times, Causal Structure and Penrose Diagrams . .
2.1 Introduction and a Short History of Black Holes . . . . . . . . . .
2.2 The Kruskal Extension of Schwarzschild Space-Time . . . . . . .
2.2.1 Analysis of the Rindler Space-Time . . . . . . . . . . . . .
2.2.2 Applying the Same Procedure to the Schwarzschild Metric
2.2.3 A First Analysis of Kruskal Space-Time . . . . . . . . . .
2.3 Basic Concepts about Future, Past and Causality . . . . . . . . . .
2.3.1 The Light-Cone . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Future and Past of Events and Regions . . . . . . . . . . .
2.4 Conformal Mappings and the Causal Boundary of Space-Time . .
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 Space-Time . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Rotating Black Holes and Thermodynamics . . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Kerr-Newman Metric . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Riemann and Ricci Curvatures of the Kerr-Newman Metric
3.3 The Static Limit in Kerr-Newman Space-Time . . . . . . . . . . .
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 Hamilton-Jacobi 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 . . . . . . . . . . . . . . .

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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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
Space-Time . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Einstein Equation and Matter for This Billiard . . . . . . .
5.3.4 The Same Billiard with Some Matter Content . . . . . . .
5.3.5 Three-Space 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 Red-Shift . . . . . . . . . . . .
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 Red-Shift Distances . . . . . . . . . . . . . . . . . . . . .

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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 Slow-Rolling 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 Sachs-Wolfe Effect . . . . . . . . . . . . . . . . .
5.10.2 The Two-Point Temperature Correlation Function . . .
5.10.3 Conclusive Remarks on CMB Anisotropies . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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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 Algebro-Geometric 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 M-Theory 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 M-Theory . .
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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xviii

7

8

Contents

The Branes: Three Viewpoints . . . . . . . . . . . . . . . . . . . .
7.1 Introduction and Conceptual Outline . . . . . . . . . . . . . . .
7.2 p-Branes as World Volume Gauge-Theories . . . . . . . . . . .
7.3 From 2nd to 1st Order and the Rheonomy Setup for to κ
Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Nambu-Goto, Born-Infeld and Polyakov Kinetic Actions
for p-Branes . . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 κ-Supersymmetry and the Example of the M2-Brane . . .
7.3.3 With Dp-Branes We Have a Problem: The World-Volume
Gauge Field A[1] . . . . . . . . . . . . . . . . . . . . . .
7.4 The New First Order Formalism . . . . . . . . . . . . . . . . . .
7.4.1 An Alternative to the Polyakov Action for p-Branes . . .
7.4.2 Inclusion of a World-Volume 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 D3-Brane Example and κ-Supersymmetry . . . . . . . . . .
7.5.1 κ-Supersymmetry . . . . . . . . . . . . . . . . . . . . .
7.6 The D3-Brane: Summary . . . . . . . . . . . . . . . . . . . . .
7.7 Supergravity p-Branes 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 Space-Time 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
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. 269
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. 272
. 273
. 275
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281
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288

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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 Hodge-Kähler 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
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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
Non-BPS 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 M-Theory and Manifolds of Restricted Holonomy
9.3.1 The Holonomy Tensor from D = 11 Bianchi Identities . .
9.3.2 Flux Compactifications of M-Theory on AdS4 × M7
Backgrounds . . . . . . . . . . . . . . . . . . . . . . . .
9.3.3 M-Theory Field Equations and 7-Manifolds of Weak G2
Holonomy i.e. Englert 7-Manifolds . . . . . . . . . . . .
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 1-Forms of OSp(8|4) . . . . . . .
9.3.9 Killing Spinors of the AdS4 Manifold . . . . . . . . . . .
9.3.10 Supergauge Completion in Mini Superspace . . . . . . .
9.3.11 The 3-Form . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Flux Compactifications of Type IIA Supergravity on AdS4 × P3 .
9.4.1 Maurer Cartan Forms of OSp(6|4) . . . . . . . . . . . .
9.4.2 Explicit Construction of the P3 Geometry . . . . . . . . .
9.4.3 The Compactification Ansatz . . . . . . . . . . . . . . .
9.4.4 Killing Spinors on P3 . . . . . . . . . . . . . . . . . . .

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Contents

9.4.5
9.4.6
9.4.7

Gauge Completion in Mini Superspace . . . . . . . .
Gauge Completion of the B[2] Form . . . . . . . . . .
Rewriting the Mini-Superspace Gauge Completion
as Maurer Cartan Forms on the Complete Supercoset
9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . 400
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. . . 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 Majorana-Weyl Spinors . . . . . . . .
A.5
A Particularly Useful Basis for D = 4 γ -Matrices . . . . .
Appendix B Auxiliary Tools for p-Brane Actions . . . . . . . . . . .
B.1
Notations and Conventions . . . . . . . . . . . . . . . . .
B.2
The κ-Supersymmetry Projector for D3-Branes . . . . . .
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 two-fold.
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 stand-point 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 end-point 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 p-forms and a totally new viewpoint on gauging. The p-forms 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 10.1007/978-94-007-5443-0_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 Space-Times, 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 black-hole 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 co-inventor 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 black-holes 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 point-like 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 10.1007/978-94-007-5443-0_2,
© Springer Science+Business Media Dordrecht 2013

3

4

2

Extended Space-Times, 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 black-hole,
Laplace was also the first to advance the hypothesis that the Solar System had formed through the
cooling of a globular-shaped, 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 no-one 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 super-massive 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 Earth-Sun system. The implicit, although unjustified, assumption was that light
velocity is unaffected, or weakly affected, by gravity. Analyzing such an assumption
in full-depth 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 energy-densities 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 phase-transition 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 event-horizon H is a hypersurface in a pseudo-Riemannian manifold (M , g)
which separates two sub-manifolds, one E ⊂ M , named the exterior, can communicate with infinity by sending signals to distant observers, the other BH ⊂ M , named
the black-hole, is causally disconnected from infinity, since no signal produced in

6

2

Extended Space-Times, Causal Structure and Penrose Diagrams

BH can reach the outside region E. The black-hole 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 event-horizon? 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 space-time?
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 black-holes.
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 black-holes, was found only forty-five years
later, six after Einstein’s death. In 1960, the American theorist Martin Kruskal (see
Fig. 2.2) found a one-to-many coordinate transformation that allowed him to represent Schwarzschild space-time as a portion of a larger space-time where the locus
r = rS is non-singular, rather it is a well-defined light-like hypersurface constituting precisely the event-horizon [6]. A similar coordinate change was independently
proposed the same year also by the Australian-Hungarian 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 white-dwarfs or neutron-stars. 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 space-time. Yet, since the Kruskal-Schwarzschild 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 end-point 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 black-hole, 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 Reissner-Nordström 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 non-linear science. He
discovered exact integrability of some non-linear differential equations and is reported to be the
inventor of the concept of solitons. Kruskal 1960 discovery of the maximal analytic extension of
Schwarzschild space-time 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 Einstein-Maxwell 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 space-time 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 Space-Times, 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 non-Euclidian 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
null-directions. 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 black-hole 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 black-holes. Brandon Carter solved the geodesic equations for the
Kerr-metric, 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 space-times.
Slowly a new picture emerged. Similarly to soliton solutions of other non-linear
differential equations, black-holes 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 Kerr-Newman 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 black-holes 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 view-point black-holes 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 black-holes 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 X-ray 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
high-speed 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 view-point the existence of black-holes 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 black-holes 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
space-times and we formulate the general definition of black-holes. In the next chapter we study the Kerr metric and the challenging connection between the laws of
black-hole mechanics and those of thermodynamics.

10

2

Extended Space-Times, Causal Structure and Penrose Diagrams

2.2 The Kruskal Extension of Schwarzschild Space-Time
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 Space-Time
Before analyzing the Kruskal extension of the Schwarzschild space-time, as a
preparatory exercise we begin by considering the properties of a two-dimensional
toy-model, the so called Rindler space-time. 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 space-time, 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 2-dimensional Minkowski
space Mink2 . The other key point is that the image ξ(H ) ⊂ Mink2 of the singularity in the extended space-time is a perfectly regular null-like hypersurface.
These features are completely analogous to corresponding features of the Kruskal
extension of Schwarzschild space-time. 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 space-time into an open
sub-manifold ξK (Schwarzschild) ⊂ Krusk of a larger manifold named the Kruskal
space-time. Also in full analogy with the case of the Rindler toy-model the image
ξK (H ) of the coordinate singularity H defined by the equation r = 2m is a regular
null-like hypersurface of Kruskal space-time. In this case it has the interpretation of
event-horizon delimiting a black-hole region.

2.2 The Kruskal Extension of Schwarzschild Space-Time

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 Space-Times, 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 Space-Time

13

Fig. 2.5 The image of
Rindler space-time in
two-dimensional Minkowski
space-time 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 two-dimensional 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 space-time. 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 space-time is the
quadrant U < 0; V > 0. In terms of the final X, T variables the image of the original Rindler space-time 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 space-time
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 null-like surface. This surface is also the boundary of the image I of Rindler space-time 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 Space-Times, 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 null-like surface, interpreted as event horizon, which can
be reached in a finite proper-time 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
Black-Hole. Yet this interpretation can be discovered only through the Kruskal extension of Schwarzschild space-time 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 two-dimensions 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 Regge-Wheeler 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 Space-Time

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
2-sphere, 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 null-geodesics were on this surface only at asymptotically early
times t → −∞

diagram actually represents a 2-sphere 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 2-sphere with a circle S1 parameterized by the azimuthal angle φ.
In this way we obtain a three-dimensional space-time spanned by coordinates t,
x = r cos φ, y = r sin φ. In this space the null-geodesics of Fig. 2.6 become twodimensional surfaces. Indeed these null-surfaces 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 null-geodesics 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 Space-Times, 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 Space-Time

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 Space-Time
Let us now consider the general properties of the space-time (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 two-dimensional 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 angle-coordinates 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 Space-Times, Causal Structure and Penrose Diagrams

Fig. 2.8 A two-dimensional
diagram of Kruskal
space-time

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 time-like N μ N ν gμν < 0 for
r > rS and space-like N μ N ν gμν > 0 for r < rS . Correspondingly, according to a
discussion developed in the next section, the constant r surfaces are space-like outside the sphere of radius rS and time-like inside it. The dividing locus is the pair
of straight lines X = ±T which correspond to r = rS and constitute a null-surface,
namely one whose normal vector is light-like. This null-surface 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 Black-Holes and their Horizons. Here we note the following. If we
solve the geodesic equation for time-like or null-like 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 two-dimensional 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 two-dimensions, Kruskal and Minkowski metrics
are conformally equivalent. According to Lemma 2.4.1 proved later on, conformally equivalent metrics share the same light-like geodesics, although the time-like
and space-like ones may be different. This means that in two-dimensional Kruskal
space-time 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 time-like or light-like and oriented to the future if

2.3 Basic Concepts about Future, Past and Causality

19

Fig. 2.9 The light-cone
orientations in Kruskal
space-time 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 time-like 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 time-like 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 space-like, 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 Black-Hole, namely a space-time 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
pseudo-Riemannian 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 light-cone (see
Fig. 2.10). In Special Relativity, where space-time is Minkowski-space, namely a
pseudo-Riemannian manifold which is also affine, the light cone has a global meaning, while in General Relativity light-cones 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 time-like, null-like and space-like sub-manifolds. Hence

20

2

Extended Space-Times, Causal Structure and Penrose Diagrams

Fig. 2.10 The structure of
the light-cone

the analysis of the light-cone 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 light-cones. We will develop appropriate conceptual tools to manage
such a gluing after our review of the local light-cone properties.

2.3.1 The Light-Cone
When a metric has a Lorentzian signature, vectors t can be of three-types:
1. Time-like, if (t, t) < 0 in mostly plus convention for gμν .
2. Space-like, if (t, t) > 0 in mostly plus convention for gμν .
3. Null-like, if (t, t) = 0 both in mostly plus and mostly minus convention for gμν .
At any point p ∈ M the light-cone Cp is composed by the set of vectors t ∈ Tp M
which are either time-like or null-like. 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 time-like vector are space-like.
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 non-degenerate, positive-definite, Euclidian bilinear form on Rn−1 . In this basis, if X⊥T and T is time-like 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 four-vector X is space-like as asserted by the theorem.

(2.3.4)


Another useful property is given by the following
Lemma 2.3.1 The sum of two future-directed time-like vectors is a future-directed
time-like 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 time-like. Moreover t 0 + T 0 > 0 and so the sum vector
is also future-directed 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 light-like vector L are either light-like
or space-like.
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 Space-Times, 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 space-like if its normal vector n(Σ) is
everywhere time-like on the surface. Conversely Σ is time-like if n(Σ) is space-like.
We name null surfaces those Σ whose normal vector n(Σ) is null-like.
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 null-surface is generated
by null-geodesics. Indeed we can easily prove that the normal vector n(Σ) to a null
surface is the tangent vector to a null-geodesics. 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 null-surface is the event-horizon of a black-hole.

2.3.2 Future and Past of Events and Regions
Let us now consider the pseudo-Riemannian space-time manifold (M , g) and at
each point p ∈ M introduce the local light-cone 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 light-cone 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 light-cone with the past light-cone:

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 future-directed, 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 space-time manifold, the
tangent space Tp M contains
the sub-manifold Cp of
time-like and null-vectors
which constitutes the local
light-cone

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 space-time manifold M is
named a future-directed time-like curve if at each point p ∈ λ, the tangent vector
to the curve t μ is future-directed and time-like. Conversely λ(s) is past-directed
time-like if such is t μ .
Similarly we have:
Definition 2.3.4 A differentiable curve λ(s) on the space-time manifold M is
named a future-directed causal curve if at each point p ∈ λ, the tangent vector to
the curve t μ is either a future-directed time-like or a future-directed null-like vector.
Conversely λ(s) is a past-directed causal curve when the tangent t μ , time-like or
null-like, 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 time-like ⎬
⎨
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 future-directed or past-directed time-like curve.
Let now S ⊂ M be a region of space-time, namely a continuous sub-manifold of
the space-time 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 Space-Times, Causal Structure and Penrose Diagrams

Fig. 2.12 The union of two
time-like future-directed
curves is still a time-like
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 time-like future directed curve λ2 (s). On the other hand,
once again by definition, q is connected by a future-directed time-like curve λ1 (s)
to at least one point p ∈ S. Joining λ1 with λ2 we obtain a future-directed time-like
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 space-times J ± (p) can fail to be
closed.

2.3 Basic Concepts about Future, Past and Causality

25

Fig. 2.13 In two-dimensional 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 space-time. 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 two-dimensional
Minkowski space.
Time-Orientability We mentioned above the splitting of the local light-cones in
the future Cp+ and past Cp− cones. Clearly, just as all the tangent spaces are glued
together to make a fibre-bundle, the same is true of the local light-cones. The subtle
point concerns the nature of the transition functions. Those of the tangent bundle
T M → M to an n-dimensional manifold take values in GL(n, R). The light-cone,
on the other hand, is left-invariant 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 light-cones from one point to the other of the space-time 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 light-cones 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 Space-Times, Causal Structure and Penrose Diagrams

Fig. 2.14 The edge of an achronal set in two-dimensional 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 end-point, the latter
in its future, which can be connected by a time-like 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 time-orientable. 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 time-orientable
and all global notions of causality loose their meaning. In all the sequel we assume
time-orientability.
For time orientable space-times we have the following theorem that we mention
without proof
Theorem 2.3.2 Let (M , g) be time-orientable 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 sub-manifold.
Domains of Dependence The future domains of dependence are those submanifolds of space-time 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
time-like 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
two-dimensional Minkowski space

Theorem 2.3.3 Let S ⊂ M be an achronal closed region of a time-orientable
n-dimensional space-time (M , g) with Lorentz signature. Let us assume that
edge(S) = ∅. Then S is an (n − 1)-dimensional sub-manifold of M .
The relevance of this theorem resides in that it establishes the appropriate notion of places in space-time, 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 sub-manifolds) 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 time-like
(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 space-time manifold (M , g) is named a Cauchy surface if and only if its domain of dependence

28

2

Extended Space-Times, Causal Structure and Penrose Diagrams

coincides with the entire space-time, 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 space-time (M , g) is named Globally Hyperbolic
if and only if it admits at least one Cauchy surface.
Globally Hyperbolic space-times are the good, non-patological 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 Space-Time
Given the appropriate definitions of Future and Past discussed in the previous section, in order to study the causal structure of a given space-time (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 space-times (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 space-time 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 Space-Time

29

which encodes such a physical idea is named asymptotic flatness. The point at infinity corresponds to the regions of the considered space-time (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 space-time 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, space-like and null-like 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)

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2

Extended Space-Times, 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 three-sphere:
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