- •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
A Spinors and Gamma Matrix Algebra |
409 |
of current research in Gravitational Theory at the beginning of the XXIst century. The author will also be satisfied of his own work if, while traveling along this path which was rich in conceptual developments, inclusion of complex mathematical structures and discovery of new physical phenomena, the reader has strengthened his belief in that the Universe is Gravitation, Gravitation is Geometry but Geometry is enormously variegated and full of yet undiscovered surprises.
Appendix A: Spinors and Gamma Matrix Algebra2
A.1 Introduction to the Spinor Representations of SO(1, D − 1)
The spinor representations of the orthogonal and pseudo-orthogonal groups have different structure in various dimensions. Starting from the representation of the Dirac gamma matrices one begins with a complex representation whose dimension is equal to the dimension of the gammas. A vector in this complex linear space is named a Dirac spinor. Typically Dirac spinors do not form irreducible representations. Depending on the dimensions, one can still impose SO(1, D − 1) invariant conditions on the Dirac spinor that separate it into irreducible parts. These constraints can be of two types:
(a)A reality condition which maintains the number of components of the spinor but relates them to their complex conjugates by means of linear relations. This reality condition is constructed with an invariant matrix C , named the charge conjugation matrix whose properties depend on the dimensions D.
(b)A chirality condition constructed with a chirality matrix ΓD+1 that halves the number of components of the spinor. The chirality matrix exists only in even dimensions.
Depending on which conditions can be imposed, besides Dirac spinors, in various dimensions D, one has Majorana spinors, Weyl spinors and, in certain dimensions, also Majorana-Weyl spinors. In this appendix we discuss the properties of gamma matrices and we present the various types of irreducible spinor representations in all relevant dimensions from D = 4 to D = 11. The upper bound D = 11 is dictated by supersymmetry since supergravity, i.e. the supersymmetric extension of Einstein gravity, can be constructed in all dimensions up to D = 11, which is maximal in this respect.
A.2 The Clifford Algebra
In order to describe spinors one needs the Dirac gamma matrices. These form the Clifford algebra:
{Γa , Γb} = 2ηab |
(A.2.1) |
2This appendix is present also in Volume 1. It is repeated in Volume 2 for reader’s convenience.
410 |
10 Conclusion of Volume 2 |
where ηab is the invariant metric of SO(1, D − 1), that we always choose according to the mostly minus conventions, namely:
ηab = diag(+, −, −, . . . , −) |
(A.2.2) |
To study the general properties of the Clifford algebra (A.2.1) we use a direct construction method.
We begin by fixing the following conventions. Γ 0 = Γ0 corresponding to the time direction is Hermitian:
Γ0† = Γ0 |
(A.2.3) |
while the matrices Γi = −Γ i corresponding to space directions are anti-Hermitian:
Γi† = −Γi |
(A.2.4) |
In the study of Clifford algebras it is necessary to distinguish the case of even and odd dimensions.
A.2.1 Even Dimensions
When D = 2ν is an even number the representation of the Clifford algebra (A.2.1) has dimension:
D |
= 2ν |
(A.2.5) |
dim Γa = 2 2 |
In other words the gamma matrices are 2ν × 2ν . The proof of such a statement is easily obtained by iteration. Suppose that we have the gamma matrices γa corresponding to the case ν = ν − 1, satisfying the Clifford algebra (A.2.1) in D − 2 dimensions and that they are 2ν -dimensional. We can write down the following
representation for the gamma matrices in D-dimension by means of the following 2ν × 2ν matrices:
Γa = |
γa |
|
0 |
|
; |
ΓD−2 |
= |
0 |
0 |
i |
||||
|
0 |
|
γa |
|
|
|
|
|
i |
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
− |
|
|
(A.2.6)
ΓD−1 = |
0 |
1 |
; |
a = 0, 1, . . . , D − 3 |
|
− |
1 0 |
||||
|
|
|
|
|
|
which satisfy the correct anticommutation relations and have the correct hermiticity properties specified above. This representation admits the following interpretation in terms of matrix tensor products:
Γa = γa σ1; |
ΓD−2 = 1 iσ3; |
ΓD−1 = 1 iσ2 |
(A.2.7) |
|||||||||||
where σ1,2,3 denote the Pauli matrices: |
0 |
|
|
|
0 |
|
1 |
|
||||||
|
= |
1 |
0 ; |
|
= |
i |
; |
|
= |
|
(A.2.8) |
|||
σ1 |
|
0 |
1 |
σ2 |
|
0 |
−i |
|
σ3 |
|
1 |
0 |
||
|
|
|
|
|
|
|
|
|
|
|
|
− |
|
|
A Spinors and Gamma Matrix Algebra |
411 |
To complete the proof of our statement we just have to show that for ν = 2, corresponding to D = 4 we have a 4-dimensional representation of the gamma matrices. This is well established. For instance we have the representation:
γ0 |
= |
1 |
0 |
; |
γ1,2,3 = |
|
σ1,2,3 |
10 |
|
||
|
|
0 |
1 |
|
|
|
|
0 |
σ ,2,3 |
|
|
|
|
|
|
|
|
|
|
− |
|
|
|
In D = 2ν one can construct the chirality matrix defined as follows:
ΓD+1 = αD Γ0Γ1Γ2 . . . ΓD−1; |
|αD |2 = 1 |
where αD is a phase-factor to be fixed in such a way that:
ΓD2+1 = 1
By direct evaluation one can verify that:
(A.2.9)
(A.2.10)
(A.2.11)
{Γa , ΓD+1} = 0 a = 0, 1, 2, . . . , D − 1
The normalization αD is easily derived. We have:
Γ0Γ1 . . . ΓD−1 = (−) 12 D(D−1)ΓD−1ΓD−2ΓD−1
so that imposing (A.2.11) results into the following equation for αD :
αD2 (−) 12 D(D−1)(−)(D−1) = 1
which has solution:
αD = 1 if ν = 2μ + 1 odd αD = i if ν = 2μ even
With the same token we can show that the chirality matrix is Hermitian:
(A.2.12)
(A.2.13)
(A.2.14)
(A.2.15)
Γ |
† |
= |
α ( |
) |
12 D(D−1)( |
)(D−1)Γ Γ Γ |
2 |
. . . Γ |
D−1 = |
Γ |
D+1 |
(A.2.16) |
|
D+1 |
− |
|
− |
0 1 |
|
|
|
A.2.2 Odd Dimensions
When D = 2ν + 1 is an odd number, the Clifford algebra (A.2.1) can be represented by 2ν × 2ν matrices. It suffices to take the matrices Γa corresponding to the even case D = D − 1 and add to them the matrix ΓD = iΓD +1, which is anti-Hermitian and anti-commutes with all the other ones.