- •Preface
- •Acknowledgements
- •Contents
- •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
- •Achronal Sets
- •Time-Orientability
- •Domains of Dependence
- •Cauchy surfaces
- •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
- •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
- •Static Observers
- •3.4 The Horizon and the Ergosphere
- •The Horizon Area
- •3.5 Geodesics of the Kerr Metric
- •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
- •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
- •4.1 Historical Introduction to Modern Cosmology
- •4.2 The Universe Is a Dynamical System
- •4.3 Expansion of the Universe
- •4.3.1 Why the Night is Dark and Olbers Paradox
- •4.3.2 Hubble, the Galaxies and the Great Debate
- •4.3.4 The Big Bang
- •4.4 The Cosmological Principle
- •4.5 The Cosmic Background Radiation
- •4.6 The New Scenario of the Inflationary Universe
- •4.7 The End of the Second Millennium and the Dawn of the Third Bring Great News in Cosmology
- •References
- •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
- •5.2.3.1 Infinitesimal Transformations and Killing Vectors
- •5.2.3.2 Vielbeins, Connections and Metrics on G/H
- •5.2.3.3 Lie Derivatives
- •5.2.3.4 Invariant Metrics on Coset Manifolds
- •5.2.3.5 For Spheres and Pseudo-Spheres
- •5.3 Homogeneity Without Isotropy: What Might Happen
- •5.3.1 Bianchi Spaces and Kasner Metrics
- •5.3.1.1 Bianchi Type I and Kasner Metrics
- •5.3.2.1 A Ricci Flat Bianchi II Metric
- •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
- •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
- •5.8.2.1 Number of e-Folds
- •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
- •5.9.2.1 Meaning of the Propagation Equation
- •5.9.2.2 Evaluation of the Effective Mass Term in the Slow Roll Approximation
- •5.9.2.3 Derivation of the Propagation Equation
- •5.9.3 Quantization of the Scalar Degree of Freedom
- •5.9.4 Calculation of the Power Spectrum in the Two Regimes
- •5.9.4.1 Short Wave-Lengths
- •5.9.4.2 Long Wave-Lengths
- •5.9.4.3 Gluing the Long and Short Wave-Length Solutions Together
- •5.9.4.4 The Spectral Index
- •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
- •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
- •Contraction and Lie Derivative
- •Definition of FDA
- •Classification of FDA and the Analogue of Levi Theorem: Minimal Versus Contractible Algebras
- •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
- •6.6 Summary of Supergravities
- •Type IIA Super-Poicaré Algebra in the String Frame
- •The FDA Extension of the Type IIA Superalgebra in the String Frame
- •The Bianchi Identities
- •6.7.1 Rheonomic Parameterizations of the Type IIA Curvatures in the String Frame
- •Bosonic Curvatures
- •Fermionic Curvatures
- •6.7.2 Field Equations of Type IIA Supergravity in the String Frame
- •6.8 Type IIB Supergravity
- •SL(2, R) Lie Algebra
- •Coset Representative of SL(2, R)/O(2) in the Solvable Parameterization
- •The SU(1, 1)/U(1) Vielbein and Connection
- •6.8.2 The Free Differential Algebra, the Supergravity Fields and the Curvatures
- •The Curvatures of the Free Differential Algebra in the Complex Basis
- •The Curvatures of the Free Differential Algebra in the Real Basis
- •6.8.3 The Bosonic Field Equations and the Standard Form of the Bosonic Action
- •6.9 About Solutions
- •References
- •7.1 Introduction and Conceptual Outline
- •7.2 p-Branes as World Volume Gauge-Theories
- •7.4 The New First Order Formalism
- •7.4.1 An Alternative to the Polyakov Action for p-Branes
- •7.6 The D3-Brane: Summary
- •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
- •8.1 Introduction
- •8.2 Supergravity and Homogeneous Scalar Manifolds G/H
- •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
- •8.5 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
- •8.6 Supergravities in Five Dimension and More Scalar Geometries
- •8.6.1 Very Special Geometry
- •8.6.3 Quaternionic Geometry
- •8.6.4 Quaternionic, Versus HyperKähler Manifolds
- •References
- •9.1 Introduction
- •9.2 Black Holes Once Again
- •9.2.2 The Oxidation Rules
- •Orbit of Solutions
- •The Schwarzschild Case
- •The Extremal Reissner Nordström Case
- •Curvature of the Extremal Spaces
- •9.2.4 Attractor Mechanism, the Entropy and Other Special Geometry Invariants
- •9.2.5 Critical Points of the Geodesic Potential and Attractors
- •At BPS Attractor Points
- •At Non-BPS Attractor Points of Type I
- •At Non-BPS Attractor Points of Type II
- •9.2.6.2 The Quartic Invariant
- •9.2.7.1 An Explicit Example of Exact Regular BPS Solution
- •The Metric
- •The Scalar Field
- •The Electromagnetic Fields
- •The Fixed Scalars at Horizon and the Entropy
- •The Metric
- •The Scalar Field
- •The Electromagnetic Fields
- •The Fixed Scalars at Horizon and the Entropy
- •9.2.9 Resuming the Discussion of Critical Points
- •Non-BPS Case
- •BPS Case
- •9.2.10 An Example of a Small Black Hole
- •The Metric
- •The Complex Scalar Field
- •The Electromagnetic Fields
- •The Charges
- •Structure of the Charges and Attractor Mechanism
- •9.2.11 Behavior of the Riemann Tensor in Regular Solutions
- •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
- •9.4.1 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
- •9.4.5 Gauge Completion in Mini Superspace
- •9.4.6 Gauge Completion of the B[2] Form
- •9.5 Conclusions
- •References
- •10.1 The Legacy of Volume 1
- •10.2 The Story Told in Volume 2
- •Appendix A: Spinors and Gamma Matrix Algebra
- •A.2 The Clifford Algebra
- •A.2.1 Even Dimensions
- •A.2.2 Odd Dimensions
- •A.3 The Charge Conjugation Matrix
- •A.4 Majorana, Weyl and Majorana-Weyl Spinors
- •Appendix B: Auxiliary Tools for p-Brane Actions
- •B.1 Notations and Conventions
- •Appendix C: Auxiliary Information About Some Superalgebras
- •C.1.1 The Superalgebra
- •C.2 The Relevant Supercosets and Their Relation
- •C.2.1 Finite Supergroup Elements
- •C.4 An so(6) Inversion Formula
- •Appendix D: MATHEMATICA Package NOVAMANIFOLDA
- •Coset Manifolds (Euclidian Signature)
- •Instructions for the Use
- •Description of the Main Commands of RUNCOSET
- •Structure Constants for CP2
- •Spheres
- •N010 Coset
- •RUNCOSET Package (Euclidian Signature)
- •Main
- •Spin Connection and Curvature Routines
- •Routine Curvapack
- •Routine Curvapackgen
- •Contorsion Routine for Mixed Vielbeins
- •Calculation of the Contorsion for General Manifolds
- •Calculation for Cartan Maurer Equations and Vielbein Differentials (Euclidian Signature)
- •Routine Thoft
- •AdS Space in Four Dimensions (Minkowski Signature)
- •Lie Algebra of SO(2, 3) and Killing Metric
- •Solvable Subalgebra Generating the Coset and Construction of the Vielbein
- •Killing Vectors
- •Trigonometric Coordinates
- •Test of Killing Vectors
- •MANIFOLDPROVA
- •The 4-Dimensional Coset CP2
- •Calculation of the (Pseudo-)Riemannian Geometry of a Kasner Metric in Vielbein Formalism
- •References
- •Index
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.
xiii
Contents
1 |
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
1 |
|
2 |
Extended Space-Times, Causal Structure and Penrose Diagrams . . |
3 |
|
|
2.1 |
Introduction and a Short History of Black Holes . . . . . . . . . . |
3 |
|
2.2 |
The Kruskal Extension of Schwarzschild Space-Time . . . . . . . |
10 |
|
|
2.2.1 Analysis of the Rindler Space-Time . . . . . . . . . . . . . |
10 |
|
|
2.2.2 Applying the Same Procedure to the Schwarzschild Metric |
14 |
|
|
2.2.3 A First Analysis of Kruskal Space-Time . . . . . . . . . . |
17 |
|
2.3 |
Basic Concepts about Future, Past and Causality . . . . . . . . . . |
19 |
|
|
2.3.1 The Light-Cone . . . . . . . . . . . . . . . . . . . . . . . |
20 |
|
|
2.3.2 Future and Past of Events and Regions . . . . . . . . . . . |
22 |
|
2.4 |
Conformal Mappings and the Causal Boundary of Space-Time . . |
28 |
2.4.1Conformal Mapping of Minkowski Space into the Einstein
|
|
Static Universe . . . . . . . . . . . . . . . . . . . . . . . . |
29 |
|
2.4.2 |
Asymptotic Flatness . . . . . . . . . . . . . . . . . . . . . |
36 |
2.5 |
The Causal Boundary of Kruskal Space-Time . . . . . . . . . . . |
37 |
|
References . |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
42 |
|
3 Rotating Black Holes and Thermodynamics . . . . . . . . . . . . . . |
43 |
||
3.1 |
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
43 |
|
3.2 |
The Kerr-Newman Metric . . . . . . . . . . . . . . . . . . . . . . |
43 |
|
|
3.2.1 Riemann and Ricci Curvatures of the Kerr-Newman Metric |
45 |
|
3.3 |
The Static Limit in Kerr-Newman Space-Time . . . . . . . . . . . |
49 |
|
3.4 |
The Horizon and the Ergosphere . . . . . . . . . . . . . . . . . . |
53 |
|
3.5 |
Geodesics of the Kerr Metric . . . . . . . . . . . . . . . . . . . . |
55 |
|
|
3.5.1 The Three Manifest Integrals, E , L and μ . . . . . . . . . |
56 |
|
|
3.5.2 |
The Hamilton-Jacobi Equation and the Carter Constant . . |
58 |
|
3.5.3 Reduction to First Order Equations . . . . . . . . . . . . . |
60 |
|
3.5.4The Exact Solution of the Schwarzschild Orbit Equation
as an Application . . . . . . . . . . . . . . . . . . . . . . |
62 |
3.5.5 About Explicit Kerr Geodesics . . . . . . . . . . . . . . . |
65 |
xv
xvi |
Contents |
3.6 The Kerr Black Hole and the Laws of Thermodynamics . . . . |
. . 65 |
3.6.1 The Penrose Mechanism . . . . . . . . . . . . . . . . . . |
. 67 |
3.6.2The Bekenstein Hawking Entropy and Hawking Radiation . 69
References . |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
70 |
|
4 Cosmology: A Historical Outline from Kant to WMAP and PLANCK |
71 |
||
4.1 |
Historical Introduction to Modern Cosmology . . . . . . . . . . . |
71 |
|
4.2 |
The Universe Is a Dynamical System . . . . . . . . . . . . . . . . |
71 |
|
4.3 |
Expansion of the Universe . . . . . . . . . . . . . . . . . . . . . . |
72 |
|
|
4.3.1 Why the Night is Dark and Olbers Paradox . . . . . . . . . |
73 |
|
|
4.3.2 Hubble, the Galaxies and the Great Debate . . . . . . . . . |
73 |
|
|
4.3.3 The Discovery of Hubble’s Law . . . . . . . . . . . . . . . |
81 |
|
|
4.3.4 |
The Big Bang . . . . . . . . . . . . . . . . . . . . . . . . |
84 |
4.4 |
The Cosmological Principle . . . . . . . . . . . . . . . . . . . . . |
86 |
|
4.5 |
The Cosmic Background Radiation . . . . . . . . . . . . . . . . . |
91 |
|
4.6 |
The New Scenario of the Inflationary Universe . . . . . . . . . . . |
97 |
|
4.7 |
The End of the Second Millennium and the Dawn of the Third |
|
|
|
Bring Great News in Cosmology . . . . . . . . . . . . . . . . . . |
99 |
|
References . |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
105 |
|
5 Cosmology and General Relativity: Mathematical Description |
|
||
of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
107 |
||
5.1 |
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
107 |
|
5.2 |
Mathematical Interlude: Isometries and the Geometry of Coset |
|
|
|
Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
108 |
|
|
5.2.1 Isometries and Killing Vector Fields . . . . . . . . . . . . |
108 |
|
|
5.2.2 |
Coset Manifolds . . . . . . . . . . . . . . . . . . . . . . . |
109 |
|
5.2.3 The Geometry of Coset Manifolds . . . . . . . . . . . . . |
114 |
|
5.3 |
Homogeneity Without Isotropy: What Might Happen . . . . . . . |
125 |
|
|
5.3.1 Bianchi Spaces and Kasner Metrics . . . . . . . . . . . . . |
125 |
|
|
5.3.2 A Toy Example of Cosmic Billiard with a Bianchi II |
|
|
|
|
Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . |
130 |
|
5.3.3 Einstein Equation and Matter for This Billiard . . . . . . . |
132 |
|
|
5.3.4 The Same Billiard with Some Matter Content . . . . . . . |
137 |
|
|
5.3.5 Three-Space Geometry of This Toy Model . . . . . . . . . |
141 |
|
5.4 |
The Standard Cosmological Model: Isotropic and Homogeneous |
|
|
|
Metrics |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
146 |
|
5.4.1 |
Viewing the Coset Manifolds as Group Manifolds . . . . . |
149 |
5.5 |
Friedman Equations for the Scale Factor and the Equation of State |
150 |
|
|
5.5.1 Proof of the Cosmological Red-Shift . . . . . . . . . . . . |
152 |
|
|
5.5.2 Solution of the Cosmological Differential Equations |
|
|
|
|
for Dust and Radiation Without a Cosmological Constant . |
154 |
|
5.5.3 Embedding Cosmologies into de Sitter Space . . . . . . . . |
159 |
|
5.6 |
General Consequences of Friedman Equations . . . . . . . . . . . |
162 |
|
|
5.6.1 |
Particle Horizon . . . . . . . . . . . . . . . . . . . . . . . |
166 |
|
5.6.2 |
Event Horizon . . . . . . . . . . . . . . . . . . . . . . . . |
168 |
|
5.6.3 |
Red-Shift Distances . . . . . . . . . . . . . . . . . . . . . |
171 |
Contents |
|
|
xvii |
5.7 |
Conceptual Problems of the Standard Cosmological Model . |
. . . |
172 |
5.8 |
Cosmic Evolution with a Scalar Field: The Basis for Inflation |
. . . |
174 |
|
5.8.1 de Sitter Solution . . . . . . . . . . . . . . . . . . . |
. . . |
176 |
5.8.2 Slow-Rolling Approximate Solutions . . . . . . . . . . . . 177
5.9Primordial Perturbations of the Cosmological Metric
and of the Inflaton . . . . . . . . . . . . . . . . . . . . . . . . . . |
187 |
|
5.9.1 |
The Conformal Frame . . . . . . . . . . . . . . . . . . . . |
187 |
5.9.2 Deriving the Equations for the Perturbation . . . . . . . . . |
188 |
|
5.9.3 |
Quantization of the Scalar Degree of Freedom . . . . . . . |
195 |
5.9.4Calculation of the Power Spectrum in the Two Regimes . . 198
5.10 The Anisotropies of the Cosmic Microwave Background . . . . . . 203 5.10.1 The Sachs-Wolfe Effect . . . . . . . . . . . . . . . . . . . 203 5.10.2 The Two-Point Temperature Correlation Function . . . . . 206 5.10.3 Conclusive Remarks on CMB Anisotropies . . . . . . . . . 208 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6 Supergravity: The Principles . . . . . . . . . . . . . |
. . . . . . . . . |
211 |
||
6.1 |
Historical Outline and Introduction . . . . . . . . . |
. . . . . . . . |
211 |
|
|
6.1.1 Fermionic Strings and the Birth of Supersymmetry . . . . . |
215 |
||
|
6.1.2 |
Supersymmetry . . . . . . . . . . . . . . . |
. . . . . . . . |
218 |
|
6.1.3 |
Supergravity . . . . . . . . . . . . . . . . . |
. . . . . . . . |
221 |
6.2 |
Algebro-Geometric Structure of Supergravity . . . |
. . . . . . . . . |
223 |
|
6.3 |
Free Differential Algebras . . . . . . . . . . . . . . |
. . . . . . . . |
227 |
|
|
6.3.1 |
Chevalley Cohomology . . . . . . . . . . . |
. . . . . . . . |
228 |
6.3.2General Structure of FDAs and Sullivan’s Theorems . . . . 230
6.4 The Super FDA of M Theory and Its Cohomological Structure . . . 233
6.4.1The Minimal FDA of M-Theory and Cohomology . . . . . 235
|
6.4.2 |
FDA Equivalence with Larger (Super) Lie Algebras . . . . |
236 |
6.5 |
The Principle of Rheonomy . . . . . . . . . . . . . . . . . . . . . |
239 |
|
|
6.5.1 The Flow Chart for the Construction of a Supergravity |
|
|
|
|
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
242 |
|
6.5.2 Construction of D = 11 Supergravity, Alias M-Theory . . . |
243 |
|
6.6 |
Summary of Supergravities . . . . . . . . . . . . . . . . . . . . . |
246 |
|
6.7 |
Type IIA Supergravity in D = 10 . . . . . . . . . . . . . . . . . . |
248 |
|
|
6.7.1 Rheonomic Parameterizations of the Type IIA Curvatures |
|
|
|
|
in the String Frame . . . . . . . . . . . . . . . . . . . . . |
251 |
|
6.7.2 Field Equations of Type IIA Supergravity in the String |
|
|
|
|
Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
253 |
6.8 |
Type IIB Supergravity . . . . . . . . . . . . . . . . . . . . . . . . |
254 |
|
|
6.8.1 |
The SU(1, 1)/U(1) SL(2, R)/O(2) Coset . . . . . . . . |
254 |
|
6.8.2 The Free Differential Algebra, the Supergravity Fields |
|
|
|
|
and the Curvatures . . . . . . . . . . . . . . . . . . . . . . |
256 |
|
6.8.3 The Bosonic Field Equations and the Standard Form |
|
|
|
|
of the Bosonic Action . . . . . . . . . . . . . . . . . . . . |
259 |
6.9 |
About Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . |
261 |
|
References . |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
261 |
|
xviii |
|
Contents |
7 The Branes: Three Viewpoints . . . . . . . . . . . . . . . . . . . |
. . 263 |
|
7.1 |
Introduction and Conceptual Outline . . . . . . . . . . . . . . . |
. 263 |
7.2 |
p-Branes as World Volume Gauge-Theories . . . . . . . . . . . . 268 |
|
7.3 |
From 2nd to 1st Order and the Rheonomy Setup for to κ |
|
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Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . |
. 269 |
|
7.3.1 Nambu-Goto, Born-Infeld and Polyakov Kinetic Actions |
|
|
for p-Branes . . . . . . . . . . . . . . . . . . . . . . . . |
. 269 |
|
7.3.2 κ-Supersymmetry and the Example of the M2-Brane . . . |
. 272 |
7.3.3With Dp-Branes We Have a Problem: The World-Volume
Gauge Field A[1] . . . . . . . . . . . . . . . . . . . . . . . 273 7.4 The New First Order Formalism . . . . . . . . . . . . . . . . . . . 275
7.4.1An Alternative to the Polyakov Action for p-Branes . . . . 275
7.4.2Inclusion of a World-Volume Gauge Field and the Born-
Infeld Action in First Order Formalism . . . . . . . . . . . 277
7.4.3Explicit Solution of the Equations for the Auxiliary Fields
|
for F and h−1 . . . . . . . . . . . . . |
. . . . . . . . . . . |
280 |
7.5 |
The D3-Brane Example and κ-Supersymmetry |
. . . . . . . . . . . |
281 |
|
7.5.1 κ-Supersymmetry . . . . . . . . . . . . |
. . . . . . . . . . |
283 |
7.6 |
The D3-Brane: Summary . . . . . . . . . . . |
. . . . . . . . . . . |
287 |
7.7Supergravity p-Branes as Classical Solitons: General Aspects . . . 288
7.8The Near Brane Geometry, the Dual Frame and the AdS/CFT
Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . |
291 |
7.9 Domain Walls in Diverse Space-Time Dimensions . . . . . . . . . |
292 |
7.9.1 The Randall Sundrum Mechanism . . . . . . . . . . . . . |
295 |
7.9.2The Conformal Gauge for Domain Walls . . . . . . . . . . 296
7.10 |
Conclusion on This Brane Bestiary . . . . . . . . . . . . . . . . . |
299 |
|
References . |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
299 |
|
8 Supergravity: A Bestiary in Diverse Dimensions . . . . . . . . . . . . |
303 |
||
8.1 |
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
303 |
|
8.2 |
Supergravity and Homogeneous Scalar Manifolds G/H . . . . . . |
304 |
|
|
8.2.1 How to Determine the Scalar Cosets G/H of Supergravities |
|
|
|
|
from Supersymmetry . . . . . . . . . . . . . . . . . . . . |
305 |
|
8.2.2 The Scalar Cosets of D = 4 Supergravities . . . . . . . . . |
307 |
|
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8.2.3 Scalar Manifolds of Maximal Supergravities in Diverse |
|
|
|
|
Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . |
309 |
8.3 |
Duality Symmetries in Even Dimensions . . . . . . . . . . . . . . |
310 |
|
|
8.3.1 |
The Kinetic Matrix N and Symplectic Embeddings . . . . |
317 |
|
8.3.2 Symplectic Embeddings in General . . . . . . . . . . . . . |
319 |
|
8.4 |
General Form of D = 4 (Ungauged) Supergravity . . . . . . . . . |
322 |
|
8.5 |
Summary of Special Kähler Geometry . . . . . . . . . . . . . . . |
323 |
|
|
8.5.1 |
Hodge-Kähler Manifolds . . . . . . . . . . . . . . . . . . |
324 |
|
8.5.2 Connection on the Line Bundle . . . . . . . . . . . . . . . |
325 |
|
|
8.5.3 |
Special Kähler Manifolds . . . . . . . . . . . . . . . . . . |
326 |
|
8.5.4 |
The Vector Kinetic Matrix NΛΣ in Special Geometry . . . |
328 |
Contents |
|
|
|
|
xix |
8.6 |
Supergravities in Five Dimension and More Scalar Geometries |
. . |
329 |
||
|
8.6.1 |
Very Special Geometry . . . . . . . . . . |
. . . . . . . |
. . |
334 |
|
8.6.2 The Very Special Geometry |
|
|
|
|
|
|
of the SO(1, 1) × SO(1, n)/SO(n) Manifold . . . . . . |
. . |
336 |
|
|
8.6.3 |
Quaternionic Geometry . . . . . . . . . . . |
. . . . . . |
. . |
338 |
|
8.6.4 Quaternionic, Versus HyperKähler Manifolds . . . . . |
. . |
338 |
||
8.7 |
N = 2, D = 5 Supergravity Before Gauging . . . |
. . . . . . . |
. . |
342 |
|
References . |
. . . . . . . . . . . . . . . . . . . . . . . . |
. . . . . . |
. . |
342 |
|
9 Supergravity: An Anthology of Solutions . . . . . . . . |
. . . . . . |
. . |
345 |
||
9.1 |
Introduction . . . . . . . . . . . . . . . . . . . . . |
. . . . . . |
. . |
345 |
|
9.2 |
Black Holes Once Again . . . . . . . . . . . . . . . |
. . . . . . . |
. |
349 |
|
|
9.2.1 |
The σ -Model Approach to Spherical Black Holes . . . |
. . |
349 |
|
|
9.2.2 |
The Oxidation Rules . . . . . . . . . . . . . |
. . . . . . . |
. |
351 |
|
9.2.3 General Properties of the d = 4 Metric . . . |
. . . . . . . |
. |
354 |
|
|
9.2.4 Attractor Mechanism, the Entropy and Other Special |
|
|
||
Geometry Invariants . . . . . . . . . . . . . . . . . . . . . 356 9.2.5 Critical Points of the Geodesic Potential and Attractors . . 357 9.2.6 The N = 2 Supergravity S3-Model . . . . . . . . . . . . . 359
9.2.7Fixed Scalars at BPS Attractor Points: The S3 Explicit
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
9.2.8The Attraction Mechanism Illustrated with an Exact
Non-BPS Solution . . . . . . . . . . . . . . . . . . . . . . 367 9.2.9 Resuming the Discussion of Critical Points . . . . . . . . . 368 9.2.10 An Example of a Small Black Hole . . . . . . . . . . . . . 369 9.2.11 Behavior of the Riemann Tensor in Regular Solutions . . . 371 9.3 Flux Vacua of M-Theory and Manifolds of Restricted Holonomy . 372 9.3.1 The Holonomy Tensor from D = 11 Bianchi Identities . . . 373
9.3.2Flux Compactifications of M-Theory on AdS4 × M7
Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . 375
9.3.3M-Theory Field Equations and 7-Manifolds of Weak G2
|
Holonomy i.e. Englert 7-Manifolds . . . . . . . . . . . . . |
376 |
|
9.3.4 |
The SO(8) Spinor Bundle and the Holonomy Tensor . . . . |
382 |
|
9.3.5 |
The Well Adapted Basis of Gamma Matrices . . . . . . . . |
382 |
|
9.3.6 |
The so(8)-Connection and the Holonomy Tensor . . . . . . |
382 |
|
9.3.7 |
The Holonomy Tensor and Superspace . . . . . . . . . . . |
384 |
|
9.3.8 |
Gauged Maurer Cartan 1-Forms of OSp(8|4) . . . . . . . . |
386 |
|
9.3.9 |
Killing Spinors of the AdS4 Manifold . . . . . . . . . . . . |
387 |
|
9.3.10 |
Supergauge Completion in Mini Superspace . . . . . . . . |
388 |
|
9.3.11 |
The 3-Form . . . . . . . . . |
. . . . . . . . . . . . . . . . |
390 |
9.4 Flux Compactifications of Type IIA Supergravity on AdS4 × P3 . . |
391 |
||
|
6 4) . . . . . . . . . . . . . |
391 |
|
9.4.1 |
Maurer Cartan Forms of OSp(3 |
| |
392 |
9.4.2 |
Explicit Construction of the P |
Geometry . . . . . . . . . . |
|
9.4.3 |
The Compactification Ansatz |
. . . . . . . . . . . . . . . . |
396 |
9.4.4 |
Killing Spinors on P3 . . . . . |
. . . . . . . . . . . . . . . |
397 |
xx |
|
Contents |
|
|
9.4.5 |
Gauge Completion in Mini Superspace . . . . . . . . . . . |
400 |
|
9.4.6 |
Gauge Completion of the B[2] Form . . . . . . . . . . . . . |
401 |
|
9.4.7 |
Rewriting the Mini-Superspace Gauge Completion |
|
|
|
as Maurer Cartan Forms on the Complete Supercoset . . . |
401 |
9.5 |
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
403 |
|
References . |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
404 |
|
10 Conclusion of Volume 2 . . . . . . . . . . . . . . . . . . . . . . . . . |
407 |
||
10.1 |
The Legacy of Volume 1 . . . . . . . . . . . . . . . . . . . . . . . |
407 |
|
10.2 |
The Story Told in Volume 2 . . . . . . . . . . . . . . . . . . . . . |
407 |
|
Appendix A Spinors and Gamma Matrix Algebra1 . . . . . . . . . . . |
409 |
||
|
A.1 |
Introduction to the Spinor Representations of SO(1, D − 1) |
409 |
|
A.2 |
The Clifford Algebra . . . . . . . . . . . . . . . . . . . . |
409 |
|
A.3 |
The Charge Conjugation Matrix . . . . . . . . . . . . . . . |
412 |
|
A.4 |
Majorana, Weyl and Majorana-Weyl Spinors . . . . . . . . |
413 |
|
A.5 |
A Particularly Useful Basis for D = 4 γ -Matrices . . . . . |
414 |
Appendix B Auxiliary Tools for p-Brane Actions . . . . . . . . . . . |
415 |
||
|
B.1 |
Notations and Conventions . . . . . . . . . . . . . . . . . |
415 |
|
B.2 |
The κ-Supersymmetry Projector for D3-Branes . . . . . . |
416 |
Appendix C |
Auxiliary Information About Some Superalgebras . . . . |
419 |
|
|
C.1 |
The OSp(N | 4) Supergroup, Its Superalgebra and Its |
|
|
|
Supercosets . . . . . . . . . . . . . . . . . . . . . . . . . |
419 |
|
C.2 |
The Relevant Supercosets and Their Relation . . . . . . . . |
422 |
|
C.3 |
D = 6 and D = 4 Gamma Matrix Bases . . . . . . . . . . |
426 |
|
C.4 |
An so(6) Inversion Formula . . . . . . . . . . . . . . . . . |
429 |
Appendix D MATHEMATICA Package NOVAMANIFOLDA . . . . . |
430 |
||
Appendix E Examples of the Use of the Package NOVAMANIFOLDA |
436 |
||
References . |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
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, |
1 |
© Springer Science+Business Media Dordrecht 2013 |
|
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.