- •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
114 |
5 Cosmology and General Relativity |
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(5.2.21) |
This guarantees that L(y) are elements of SO(n, 1), secondly observe that the image x(y) of the standard vector x0 through L(y),
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(5.2.23) |
and has n linearly independent entries (the first n) parameterized by y. Hence the lateral classes can be labeled by y and this concludes our argument to show that (5.2.19) is a good coset parameterization. L(0) = 1(n+1)×(n+1) corresponds to the identity class which is usually named the origin of the coset.
5.2.3 The Geometry of Coset Manifolds
In order to study the geometry of a coset manifold G/H, the first important step is provided by the orthogonal decomposition of the corresponding Lie algebra, namely by
G = H K |
(5.2.24) |
where G is the Lie algebra of G and the subalgebra H G is the Lie algebra of the subgroup H and where K denotes a vector space orthogonal to H with respect to the Cartan Killing metric of G. By definition of subalgebra we always have:
[H, H] H |
(5.2.25) |
while in general one has: |
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[H, K] H K |
(5.2.26) |
Definition 5.2.3 Let G/H be a Lie coset manifold and let the orthogonal decomposition of the corresponding Lie algebra be as in (5.2.24). If the condition:
[H, K] K |
(5.2.27) |
applies, the coset G/H is named reductive.
5.2 Mathematical Interlude: Isometries and the Geometry of Coset Manifolds |
115 |
Equation (5.2.27) has an obvious and immediate interpretation. The complementary space K forms a linear representation of the subalgebra H under its adjoint action within the ambient algebra G.
Almost all of the “reasonable” coset manifolds which occur in various provinces of Mathematical Physics are reductive. Violation of reductivity is a sort of pathology whose study we can disregard in the scope of this book. We will consider only reductive coset manifolds.
Definition 5.2.4 Let G/H be a reductive coset manifold. If in addition to (5.2.27) also the following condition:
[K, K] H |
(5.2.28) |
applies, then the coset manifold G/H is named a symmetric space.
Let TA (A = 1, . . . , n) denote a complete basis of generators for the Lie algebra G:
[TA, TB ] = CCAB TC |
(5.2.29) |
and Ti (i = 1, . . . , m) denote a complete basis for the subalgebra H G. We also introduce the notation Ta (a = 1, . . . , n − m) for a set of generators that provide a basis of the complementary subspace K in the orthogonal decomposition (5.2.24). We nickname Ta the coset generators. Using such notations, (5.2.29) splits into the following three ones:
[Tj , Tk ] = Cij k Ti |
(5.2.30) |
[Ti , Tb] = Caib Ta |
(5.2.31) |
[Tb, Tc] = Cibc Ti + Cabc Ta |
(5.2.32) |
Equation (5.2.30) encodes the property of H of being a |
subalgebra. Equa- |
tion (5.2.31) encodes the property of the considered coset of being reductive. Finally if in (5.2.32) we have Cabc = 0, the coset is not only reductive but also symmetric.
We will be able to provide explicit formulae for the Riemann tensor of reductive coset manifolds equipped with G-invariant metrics in terms of such structure constants. Prior to that we consider the infinitesimal transformation and the very definition of the Killing vectors with respect to which the metric has to be invariant.
5.2.3.1 Infinitesimal Transformations and Killing Vectors
Let us consider the transformation law (5.2.17) of the coset representative. For a group element g infinitesimally close to the identity, we have:
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+ εATA |
(5.2.33) |
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− εAWAi (y)Ti |
(5.2.34) |
y α yα + εAkAα |
(5.2.35) |
116 |
5 Cosmology and General Relativity |
The induced h transformation in (5.2.17) depends in general on the infinitesimal G-parameters εA and on the point in the coset manifold y, as shown in (5.2.34). The y-dependent rectangular matrix WAi (y) is usually named the H-compensator. The shift in the coordinates yα is also proportional to εA and the vector fields:
kA = kAα |
∂ |
(5.2.36) |
(y) ∂yα |
are named the Killing vectors of the coset. The reason for such a name will be justified when we will show that on G/H we can construct a (pseudo-)Riemannian metric which admits the vector fields (5.2.36) as generators of infinitesimal isometries. For the time being those in (5.2.36) are just a set of vector fields that, as we prove few lines below, close the Lie algebra of the group G.
Inserting (5.2.33)–(5.2.35) into the transformation law (5.2.17) we obtain:
TAL(y) = kAL(y) − WAi (y)L(y)Ti |
(5.2.37) |
Consider now the commutator g2−1g1−1g2g1 acting on L(y). If both group elements g1,2 are infinitesimally close to the identity in the sense of (5.2.33), then we obtain:
g2−1g1−1g2g1L(y) 1 − ε1Aε2B [TA, TB ] L(y) |
(5.2.38) |
By explicit calculation we find:
[TA, TB ]L(y) = TATB L(y) − TB TAL(y)
= [kA, kB ]L(y) − kAWBi − kB WAi + 2Cij k WAj WBk L(y)Ti
(5.2.39)
On the other hand, using the Lie algebra commutation relations we obtain:
[TA, TB ]L(y) = CCAB TC L(y) = CCAB kC L(y) − WCi L(y)Ti |
(5.2.40) |
By equating the right hand sides of (5.2.39) and (5.2.40) we conclude that:
[kA, kB ] = CCAB kC |
(5.2.41) |
kAWBi − kB WAi + 2Cij k WAj WBk = CCAB WCi |
(5.2.42) |
where we separately compared the terms with and without W’s, since the decomposition of a group element into L(y)h is unique.
Equation (5.2.41) shows that the Killing vector fields defined above close the commutation relations of the G-algebra.
Equation (5.2.42) will be used to construct a consistent H-covariant Lie derivative.
In the case of the spaces H(n,− 1), which we choose as illustrative example, the Killing vectors can be easily calculated by following the above described procedure
5.2 Mathematical Interlude: Isometries and the Geometry of Coset Manifolds |
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step by step. For later purposes we find it convenient to present such a calculation in a slightly more general set up by introducing the following coset representative that depends on a discrete parameter κ = ±1:
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Namely L−1(y) is an SO(n, 1) matrix, while L1(y) is an SO(n + 1) group element. Furthermore defining, as in (5.2.22):
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xκ (y)T ηκ xκ (y) = κ |
(5.2.46) |
Hence by means of L1(y) we parameterize the points of the n-sphere Sn, while by means of L−1(y) we parameterize the points of H(n,− 1) named also the n-pseudo- sphere or the n-hyperboloid. In both cases the stability subalgebra is so(n) for which a basis of generators is provided by the following matrices:
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(5.2.47) |
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j th column |
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the (n + 1) × (n + 1) matrices whose only non-vanishing entry is the ij th one, equal to 1.