- •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
412 |
10 Conclusion of Volume 2 |
A.3 The Charge Conjugation Matrix
Since Γa and their transposed ΓaT satisfy the same Clifford algebras it follows that there must be a similarity transformation connecting these two representations of the same algebra on the same carrier space. Such statement relies on Schur’s lemma and it is proved in the following way. We introduce the notation:
|
|
1 |
|
|
|
|
Γa1...an ≡ Γ[a1 Γa2 . . . Γan] = n |
! |
(A.3.1) |
||
|
|
(−)δP ΓaP (a1) . . . ΓaP (an) |
|||
|
D |
|
P |
|
|
of |
|
|
|
||
where |
P denotes the sum over the n! permutations of the indices and δP the parity |
||||
permutation P , i.e. the number of elementary transpositions of which it is composed. The set of all matrices 1, Γa , Γa1a2 , . . . , Γa1...aD constitutes a finite group of 2[D/2]-dimensional matrices. Furthermore the groups generated in this way by Γa , −Γa or ΓaT are isomorphic. Hence by Schur’s lemma two irreducible representations of the same group, with the same dimension and defined over the same vector space, must be equivalent, that is there must be a similarity transformation that connects the two. The matrix realizing such a similarity is called the charge conjugation matrix. Instructed by this discussion we define the charge conjugation matrix by means of the following equations:
C−Γa C−−1 = −ΓaT
(A.3.2)
C+Γa C+−1 = ΓaT
By definition C± connects the representation generated by Γa to that generated by ±ΓaT . In even dimensions both C− and C+ exist, while in odd dimensions only one of the two is possible. Indeed in odd dimensions ΓD−1 is proportional to Γ0Γ1 . . . ΓD−2 so that the C− and C+ of D − 1 dimensions yield the same result on ΓD−1. This decides which C exists in a given odd dimension.
Another important property of the charge conjugation matrix follows from iterating (A.3.2). Using Schur’s lemma one concludes that C± = αC±T so that iterating
Table A.1 Structure of charge conjugation matrices in various space-time dimensions
Charge conjugation matrices
DC+ = C+ (real)
4C+T = −C+; C+2 = −1
5C+T = −C+; C+2 = −1
6C+T = −C+; C+2 = −1
7
8C+T = C+; C+2 = 1
9C+T = C+; C+2 = 1
10C+T = C+; C+2 = 1
11
C− = C− (real)
C−T = −C−; C−2 = −1
C−T = C−; C−2 = 1 C−T = C−; C−2 = 1 C−T = C−; C−2 = 1
C−T = −C−; C−2 = −1 C−T = −C−; C−2 = −1
A Spinors and Gamma Matrix Algebra |
413 |
again we obtain α2 = 1. In other words C+ and C− are either symmetric or antisymmetric. We do not dwell on the derivation which can be obtained by explicit iterative construction of the gamma matrices in all dimensions and we simply collect below the results for the properties of C± in the various relevant dimensions (see Table A.1).
A.4 Majorana, Weyl and Majorana-Weyl Spinors
The Dirac conjugate of a spinor ψ is defined by the following operation:
ψ |
≡ ψ†Γ0 |
(A.4.1) |
and the charge conjugate of ψ is defined as:
ψc = C ψT |
(A.4.2) |
where C is the charge conjugation matrix. When we have such an option we can either choose C+ or C−. By definition a Majorana spinor λ satisfies the following condition:
λ = λc = CΓ0T λ |
(A.4.3) |
Equation (A.4.3) is not always self-consistent. By iterating it a second time we get the consistency condition:
C Γ0T C = Γ0 |
(A.4.4) |
There are two possible solutions to this constraint. Either C− is antisymmetric or C+ is symmetric. Hence, in view of the results displayed above, Majorana spinors exist only in
D = 4, 8, 9, 10, 11 |
(A.4.5) |
In D = 4, 10, 11 they are defined using the C− charge conjugation matrix while in D = 8, 9 they are defined using C+.
Weyl spinors, on the contrary, exist in every even dimension; by definition they are the eigenstates of the ΓD+1 matrix, corresponding to the +1 or −1 eigenvalue. Conventionally the former eigenstates are named left-handed, while the latter are named right-handed spinors:
ΓD+1ψ+ L |
, = ±ψ+ L , |
(A.4.6) |
R |
R |
|
In some special dimensions we can define Majorana-Weyl spinors which are both eigenstates of ΓD+1 and satisfy the Majorana condition (A.4.3). In order for this to be possible we must have:
C Γ0T ΓD+1ψ = ΓD+1ψ |
(A.4.7) |
414 |
10 Conclusion of Volume 2 |
which implies: |
|
C Γ0T ΓD+1Γ0T C −1 = ΓD+1 |
(A.4.8) |
With some manipulations the above condition becomes: |
|
C ΓD+1C −1 = −ΓDT+1 |
(A.4.9) |
which can be checked case by case, using the definition of ΓD+1 as product of all the other gamma matrices. In the range 4 ≤ D ≤ 11 the only dimension where (A.4.9) is satisfied is D = 10 which is the critical dimensions for superstrings. This is not a pure coincidence.
Summarizing we have:
Spinors in 4 ≤ D ≤ 11
D |
Dirac |
Majorana |
Weyl |
Majora-Weyl |
|
|
|
|
|
4 |
Yes |
Yes |
Yes |
No |
5 |
Yes |
No |
No |
No |
6 |
Yes |
No |
Yes |
No |
7 |
Yes |
No |
No |
No |
8 |
Yes |
Yes |
Yes |
No |
9 |
Yes |
Yes |
No |
No |
10 |
Yes |
Yes |
Yes |
Yes |
11 |
Yes |
Yes |
No |
No |
|
|
|
|
|
A.5 A Particularly Useful Basis for D = 4 γ -Matrices
In this section we construct a D = 4 gamma matrix basis which is convenient for various purposes. Let us first specify the basis and then discuss its convenient properties.
In terms of the standard matrices (A.2.8) we realize the so(1, 3) Clifford algebra:
{γa , γb} = 2ηab; |
ηab = diag(+, −, −, −) |
(A.5.1) |
by setting: |
|
|
γ0 = σ1 σ3; |
γ1 = iσ2 σ3 |
|
γ2 = i1 σ2; |
γ3 = iσ3 σ3 |
(A.5.2) |
γ5 = 1 σ1; |
C = iσ2 1 |
|
where γ5 is the chirality matrix and C the generators of the Lorentz algebra
is the charge conjugation matrix. In this basis so(1, 3), namely γab are particularly simple