- •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
Index
A
Achronal set, 25–27 Active mode(s), 196
AdS/CFT correspondence, 291, 292, 329, 381 Akulov, 218
Andromeda, 77, 78, 81
Angular momentum, x, 6, 8, 43–45, 51, 52, 55–57, 64, 66, 68, 207, 240 Anisotropy(ies), x, 97, 100, 102–104, 108,
164, 172, 173, 187, 197, 202, 203, 207–209, 408
Anti-commutator, 217, 218, 225 Attraction mechanism, xii, 347, 349, 367 Automorphism group, algebra, subalgebra,
306–308, 333, 377
Auxiliary field(s), 268, 269, 271, 272, 274, 276, 278, 280, 286, 287, 416, 418
Azimuthal angle, 15, 44, 47, 52
B
Baryonic matter, 100, 164, 177
Base manifold, ix, 225, 239, 325, 335, 360, 362, 385
Bessel, x, 75 Beta-decay, β-decay, 94 Bianchi, 126, 127
Bianchi classification, x, 107, 147
Bianchi identity(ies), ix, xi, 1, 126, 149, 178, 190, 231, 233, 234, 241, 243, 250, 251, 253, 271, 292, 311, 313, 317, 351, 373, 391
Bianchi space(s), 125
Bianchi type, x, 127, 130, 131, 137, 147, 150 Big Bang, 83, 84, 86, 91, 94, 95, 97, 103, 140,
178, 203 Big Crunch, 91, 169 Binary system(s), 9
Black body, 94
Black body radiation, 94
Black hole(s), ix, x, xii, 1, 3, 9, 19, 43, 44, 49, 50, 52, 65, 67–70, 236, 347, 349, 356, 358, 368, 369, 407, 408
Blueshift, 90
Born-Infeld, xi, 268–271, 274, 275, 277, 280, 287
Bosonic 3-form, 233
Bosonic 6-form, 233
Boson(s), 215, 223, 224, 270, 329 Boundary operator, 232
BPS (state, black hole), 288, 290, 356–359, 364–367, 369, 371, 372
Brane solutions, 259, 267, 289, 290, 310, 317, 347, 403
Brane(s), viii, xi, 1, 107, 223, 236, 248, 263, 266, 267, 270, 290, 291, 293, 295, 299, 310, 347, 348, 358, 403, 408
Bulk (field) theory(ies), xi, 236
C
Calabi-Yau manifold(s), 403 Calabi-Yau three-folds, 323, 328 Cartan scalars, 310
Carter, x, 8, 56, 58, 60, 63
Castellani, 239, 254, 256–260, 304, 324, 431, 432
Cauchy surface, 27, 28
Causal boundary, 28, 32, 35, 37, 41 Causal future (past), 36, 39, 40 Causality, ix, 8, 18–20, 26, 28 Cavendish, 3
Cayley transformation, 257 Cepheides, x, 77–81, 100 CERN, 211, 213, 216, 219
Chandrasekhar mass (limit), ix, 101
P.G. Frè, Gravity, a Geometrical Course, DOI 10.1007/978-94-007-5443-0, |
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446
Charge conjugation matrix, 244, 247, 382, 409, 412–414, 420, 428
Charge(s), 1, 6–8, 43–45, 49, 56, 212, 215, 219, 249, 256, 264, 272, 282, 290, 310, 347, 348, 351, 352, 354, 356–358, 364–368, 370, 371, 420
Chern class(es), 324–326, 385 Chern Simons, 259, 330
Chevalley cohomology, 227–229, 231, 232 Chirality matrix, 247, 409, 411, 414, 426, 428 Christoffel, 33, 204
Chronological future (past), 23–26 Circular orbit(s), 50
Clebsch Gordan, 236
Clifford algebra(s), 244, 377, 382, 395, 409–412, 414, 426, 428
Closed universe, 90, 91, 154, 158, 167, 169, 172
Coboundary operator, 228
Cohomology, ix, xi, 228, 232, 233, 235–237, 242, 243, 249, 324, 346, 408
Cohomology group, 228, 229, 243 Compactification, 223, 305, 306, 309, 318,
322, 324, 345, 346, 348, 372–376, 378, 380–382, 385, 387–389, 391, 394, 396, 400, 403, 419, 426, 429
Conformal factor, 32, 36–38, 40 Conformal frame, 187, 188, 203 Conformal gauge, 296, 298
Conformal mapping (map), ix, 28, 29, 31, 32, 34–36
Congruence of geodesics, 144, 145 Connection one-form, 437
Connection(s), viii, ix, 1, 9, 118, 119, 122, 208, 224, 225, 227, 239, 241, 255, 299, 306, 324–326, 328, 332, 339–341, 345, 348, 363, 380, 382–384, 386, 387, 392, 395, 403, 407, 408, 423, 430
Contorsion, 433, 434, 441, 442 Contractible FDA, 230, 231 Contraction operator, 121, 229 Copernicus, 74
Coset generator(s), 115, 118, 119, 123, 393, 437, 438
Coset manifold(s), x, 88, 107–112, 114–116, 118–120, 122, 123, 125, 127, 147, 149, 159, 225, 254, 281, 292, 304, 306, 307, 310, 319, 323, 332, 345, 349, 359, 381, 386, 388, 391–393, 403, 408, 423, 430
Coset representative(s), 112, 113, 115, 117, 254, 255, 258, 321, 332–334, 359–361, 387, 392, 402, 422, 425, 426, 435
Coset space(s), 109, 110, 305, 323
Index
Cosmic background radiation, 91–93, 103, 164, 187, 197, 203, 206
Cosmic billiard(s), x, 129, 130, 347
Cosmic microwawe background (CMB), x, xi, 95, 96, 100, 102–104, 203, 205–209
Cosmological parameter(s), 163, 164, 167, 172, 174, 209
Cosmological principle, x, 86–88, 91, 95, 97, 103, 107, 109, 110, 125, 158, 408
Cosmological redshift, 94, 95 Coulomb, 3
Covariant derivative, 137, 258, 325, 326, 328, 335, 357, 382, 383
CPT symmetry, 308 Creatio ex nihilo, 72 Cremmer, 223, 324
Critical point(s), 357, 358, 364, 368, 369 Current(s), 218, 251, 256, 257, 264 Curtis, x, 74, 77, 81
Curvature(s), 17, 45–47, 88, 90, 91, 93, 108, 110, 124, 128, 133, 135, 140, 148, 152, 155, 157, 158, 162, 163, 167, 172, 174, 176, 177, 197, 227, 231, 233–235, 238, 239, 241, 243, 248–253, 256–259, 268, 271, 273, 294, 330, 338, 339, 341, 355, 372–374, 378, 387, 390, 396, 430, 432, 433, 440, 441, 443, 444
D
D3-brane, xi, 268, 269, 278, 280–282, 329, 416
d’Alembert, 4
Dark energy, 83, 100, 102 Dark matter, 100, 164
de Sitter solution, 176, 178, 188, 196
de Sitter space(s), x, 108, 157–159, 161, 162, 169, 176–178, 191, 192, 195, 196, 291, 293, 372, 376
Decoupling time, 94, 95 Deser, 222
Dicke, 92, 93, 95
Diffeomorphism(s), 10, 11, 108, 109, 120, 122, 189, 203, 225, 226, 239, 316, 317, 329
Differentiable manifold, 31, 121, 229, 376 Differential form(s), ix, 272, 283, 329 Differential geometry, ix, 107–109, 305, 408 Dilaton, 141, 142, 248, 249, 253, 255, 259,
281, 294, 310, 397
Dirac, 97, 103, 216, 222, 240, 409, 414 Dirac spinor(s), 409
Distance, x, 3, 28, 29, 34, 53, 71, 73–75, 77–79, 81, 82, 84, 85, 87–90, 101, 102, 166–168, 170–172, 176, 290, 291
Domain of dependence, 26, 27
Index
Domain wall(s), xi, 292–298, 303, 305, 329 Dp-brane(s), 264, 268–271, 273, 274, 278,
280, 290
Dual scattering amplitude(s), 212
Duality, xii, 130, 212, 228, 230, 264, 287, 317, 347, 349, 352, 391
Duality rotations, xi, 307, 311, 314–318 Duality symmetry(ies), 281, 306, 310, 311,
316, 317, 333, 345
Dust (filled) universe, 91, 151, 154–157, 162
E
E8 × E8, 263, 289 Eddington, 83 e-fold(s), 186, 187, 201
Einstein, 6, 30, 71, 72, 82, 83, 86, 97, 98, 149, 240, 404, 408
Einstein frame, 248, 253
Einstein manifold, 345, 346, 379, 381, 383 Einstein Static Universe, 29–31, 35, 37 Einstein tensor(s), 133, 134, 139, 148, 149,
189, 193 Electric current, 264 Electric field, 49, 314
Electromagnetism, 7, 270 Englert space(s), 379
Entropy, x, 8, 42, 43, 66, 69, 70, 347, 349, 356, 358, 364, 366, 368, 407, 408
Equation(s) of state, 108, 138, 139, 150, 162–164, 176
Euclidian geometry, 75 Euclidian space(s), 89, 188
Event horizon(s), x, 5, 6, 8, 10, 14, 18, 22, 39, 40, 42, 54, 55, 66, 68, 165, 168–170, 196, 200, 267, 347, 407
Event(s), 19, 22, 23, 25, 40, 101, 168–170, 211 Exterior derivative, 122, 148, 234
Exterior form(s), 230
F
Fermi, 241, 268
Fermion(s), 224, 252, 258, 283, 285, 287, 306, 307
Ferrara, 222, 304, 324, 347, 356 Fibre bundle, 119, 385
First integral(s), 8, 43, 52, 56, 57, 59, 61, 63, 154, 356
First order formalism, xi, 223, 268, 269, 271, 274, 275, 277, 280–282, 286, 330
Fixed scalar(s), 364–366, 368
Flat metric, 30, 34, 129, 133, 137, 160, 249, 291, 416
Flat universe, 90, 134, 154, 157, 158, 162, 167–169, 171, 176, 187, 203
447
FLRW metric, 89, 95, 97
Flux compactification(s), xii, 346, 348, 375, 376, 391
Flux vacuum solution(s), 348 Four-momentum, D-momentum, 153, 204 Fourier component(s), 200, 207, 208
Fred Hoyle, 83, 86
Free differential algebra(s), xi, 1, 223, 227, 230, 239, 241–243, 246, 248, 249, 254, 256, 257, 346, 373, 374, 389, 392, 408
Freedman, 222
Freund Rubin (parameter), 378, 379, 381 Friedman, 83, 86, 93, 94
Friedman equation(s), 89, 90, 96, 102, 108, 149, 150, 162, 163, 166, 171, 175–177
Frozen mode(s), 196 Fubini, 212, 213
Future- (past-)null infinity, 34, 40–42 Future- (past-)time infinity, 34
G
G-structure(s), 346, 348, 376 G2-structure, 377, 379
Gaillard Zumino, 284, 322, 323, 345, 349, 361, 363
Galaxy(ies), ix, x, 28, 71–73, 77, 78, 80–82, 84, 85, 87, 90, 100–102, 151, 164, 207
Gamma matrices, 243, 244, 270, 273, 377, 382, 384, 398, 409–411, 413, 414, 427
Gamow, x, 93–95 Gauge boson(s), 329
Gauge theory(ies), viii, ix, 1, 223, 240, 266, 267, 284, 312, 329, 407
Gauge/gravity correspondence, xi, 267, 348 Gauged supergravity(ies), 329
Gauging, 1, 227, 231, 243, 304, 306, 331, 342, 386
Gauss, 73, 76, 79, 404
Gaussian coordinates, curvilinear coordinates, 217
General relativity, vii–x, 1, 5, 6, 18–20, 28, 29, 71, 72, 82, 83, 97, 107, 151, 166, 208, 221, 223, 225, 234, 236, 240, 259, 267, 274, 299, 407, 408
Geodesic potential, 356, 357 Geodesic(s), ix, x, 11, 12, 17–19, 28, 29,
32–35, 41, 51, 55, 58, 59, 62, 65, 67, 68, 132, 141–146, 153, 165, 204, 208, 350, 356, 357
Gliozzi, 223 Golfand, 218, 219 Göttingen, 73, 94
Gravitational wave(s), ix
448
Gravitino(s), 221, 222, 225, 233, 236, 243, 245, 248, 252, 256, 273, 283, 306, 329, 331–333, 342, 373, 374, 384, 390, 399, 420, 421
Graviton(s), ix, 221, 222, 225, 243, 245, 246, 248, 256, 259, 295, 298, 308, 309, 325, 331, 332
Green function(s), 265, 266, 329 Guth, 97, 98
H
Hadron(s), 211–213
Hawking, 8, 69, 70 Herschel, 76, 77 Hertzsprung, 80
Hodge dual, 49, 260, 311, 312, 329 Holonomy tensor, 348, 373, 374, 382–384 Homogeneity, x, 84, 86, 88, 97, 99, 103, 107,
110, 125, 146, 149, 152, 158, 172 Homology, ix
Horizon, x, 6, 8, 18, 19, 42, 53–55, 67–70, 108, 146, 166–169, 173, 174, 184, 186, 196, 200, 215, 356, 365, 366, 368, 371, 372
Horizon area, x, xii, 43, 55, 65, 69, 365, 367, 369
Hubble, x, 73, 74, 76–78, 81–86, 94, 95, 99–102, 125, 170, 173
Hubble constant, 82, 86, 159, 162, 167, 168, 170
Hubble function, 86, 160, 162, 166, 168, 171, 175, 178, 179, 185, 188, 199, 201
Hubble radius, 170, 200, 201 Hyperbolic space(s), 112 Hypergeometric function(s), 134 HyperKähler manifolds, 338
Hypermultiplet(s), 304, 305, 318, 323, 330, 331, 338, 342
I
Immanuel Kant, Kant, x, 71, 72, 76, 77, 81 Inertial frames, 52, 53
Inflation, 99, 168, 174, 177–179, 181–187, 200, 201, 206, 408
Inflationary universe, x, 97–99, 103, 170, 174, 176, 408
Inflaton, 178, 187 Information loss, 8, 407
Inhomogeneity(ies), 97, 173, 203, 206 Irreducible representation(s) (irreps(s)), 123,
215, 236, 242, 243, 245, 246, 257, 306–308, 373, 376, 409, 438
Island-universe(s), x, 76–78, 81 Isometry(ies), xi, 87, 88, 107–110, 122, 142,
146, 147, 153, 158, 159, 291, 294, 304,
Index
306, 310, 311, 317, 319, 321, 323, 332–334, 336, 352, 353, 358, 381, 391, 420
Isotropy, x, 84, 86, 88, 97, 99, 103, 107, 108, 111, 122, 124, 125, 130, 146, 147, 149, 150, 152, 154, 158, 172, 257, 307, 308, 321
Isotropy subgroup(s), 111, 257, 321
J
Julia, 223
K
Kahler manifold, 304, 324, 339, 356 Kaluza Klein, 295, 375, 380–382, 397
κ-supersymmetry (kappa-supersymmetry), xi, 267–273, 280–283, 285–287, 416
Kasner epoch, 128–130, 136
Kasner metric(s), x, 125, 127, 128, 441 Kasner solution(s), 107
Kepler, 73 Kerr, 6, 8, 45
Kerr-Newman metric, ix, 8, 43–47, 49, 51 Killing spinor(s), 346, 348, 356, 373, 376, 379,
381, 383–385, 387–389, 392, 395–400, 430
Killing vector, 11, 12, 47, 50, 53–56, 67–69, 108, 109, 115, 116, 118, 122, 126, 132, 137, 142, 150, 152, 153, 158, 435, 436
Kinetic energy, 177, 181, 183
Kinetic (period) matrix N , 317, 320, 329, 333, 363
Klein-Gordon equation, 189
Kronecker, Kronecker delta, 124, 125, 193, 277, 384
Kruskal, 6, 7
Kruskal space-time, 6, 10, 17–19, 37, 38, 40–42
L
Lagrangian(s), xi, 51, 56, 58, 59, 61, 219, 220, 222, 223, 254, 263, 268, 275–277, 280, 284, 285, 297, 304–306, 311–317, 322, 325, 331, 333, 342, 345, 346, 349, 351, 352, 356, 357, 359, 362
Landau, 94
Laplace, ix, 3–6, 8, 9
Lateral class(es), 109–111, 114 Leavitt, x, 78–81
Left-handed, 247, 413
Left-invariant vector field, one-form, 126, 150, 228, 332, 425
Lemaitre, 83, 86, 93
Levi Civita connection, 341
Index
Lie derivative, 116, 120–122, 131, 132, 229, 239
Lie group, x, 88, 110, 112, 119, 126, 265, 304, 307, 309, 316, 319, 320, 381
Light-cone, 19, 20, 22, 23, 25 Likhtman, 218, 219
Linde, 97, 98
Line bundle, 324–326, 328 Little group, 242, 243 Lobachevskij, 8
Local trivialization, 325, 327, 335, 377 Lorentz algebra, 158, 225, 414 Lorentz bundle, 225
Lorentz group, 218, 236, 237, 268, 274 Lorentzian manifold(s), 159, 376
M
M-theory, xi, xii, 228, 233, 235, 239, 243, 246, 261, 264, 267, 290, 348, 372, 373, 375, 376, 378, 381, 388–391, 400, 419
M2-brane, 280
M5-brane, 236, 290 Magellanic cloud, 78, 80 Magnetic field, 49, 266 Magnitude(s) (of stars), 80, 81
Majorana spinor(s), 219, 272, 305, 409, 413, 421, 429
Majorana-Weyl spinor(s), 247, 249, 256, 409, 413
Manifold, ix, xii, 5, 6, 10, 17, 19, 21–23, 25–27, 30–32, 34, 39–41, 88, 89, 107–112, 119, 125, 127, 131, 147, 149, 152, 153, 158, 159, 177, 191, 225, 239, 248, 264, 293, 294, 304, 305, 312, 321–324, 326, 327, 329, 331, 334–340, 342, 345, 346, 348–350, 352, 356, 359, 361, 362, 372–389, 391–393, 395, 397, 402, 422, 430–434, 436, 443
Mass, x, 3, 5, 6, 8, 9, 43–45, 55, 56, 66, 68, 70, 97, 101, 151, 177, 191, 211, 220, 242, 348, 354
Mass term, 192, 195, 220
Matter dominated universe, 96, 166, 167, 169–172, 184, 200
Maurer, Maurer Cartan forms, 119, 126, 129, 348, 374, 386, 388, 389, 391, 401–403, 422, 423, 426, 434, 438
Maurer Cartan equations, 119, 124, 126, 225, 228, 230–232, 235, 237, 249, 258, 386, 387, 393, 421, 422, 425
Maximal supergravity(ies), 233, 309, 310, 319, 331
Maxwell equations, 7, 49, 264, 380
449
Metric(s), ix, x, 8, 10–14, 17–20, 28–34, 38, 40, 43–45, 48, 50, 51, 54–56, 58, 59, 70, 83, 86–88, 93, 103, 107–110, 114–116, 118, 122–132, 134, 138, 141, 146–148, 150, 152, 158, 159, 161, 162, 165, 175–177, 187–189, 191, 203–205, 239, 244, 264, 269, 271, 272, 274, 276, 277, 281, 291–294, 298, 303, 304, 311, 312, 317, 319, 323–325, 327, 329, 330, 332–334, 336, 338–342, 346, 349–355, 359, 360, 363, 365–367, 370, 371, 376, 382, 410, 421, 426, 435, 436, 441, 444
Michell, 3, 5, 6
Milky Way, 9, 71, 75–78, 80, 82, 87, 101 Mini superspace, 388, 400
Minimal FDA, 230–233, 235, 238, 242, 243 Minkowski metric, 10, 13, 18, 29, 31, 44 Minkowski space, 10, 19, 20, 24–29, 31, 32,
34–38, 158, 191, 195, 196, 225, 376 Moduli, 123, 196, 346
Moduli space(s), 305, 306, 323, 324, 331, 346 Momentum, 67–69, 153, 195, 203, 204, 244,
298
Monopole solutions, 347 Mukhanov, 208 Multipole (analysis), 103
Multipole (expansion), 207, 208
N
Naked singularity, 298, 350 Nambu, 212, 213, 215, 217, 269 Nambu-Goto, 268–272, 277 Near horizon geometry, 267 Ne’eman, 239
Neutron star(s), ix, 6 Neveu, 215, 216
New first order formalism, xi, 268, 269, 275, 277, 286
Newtonian potential, 203 Newton’s law, 295 Noether’s theorem, 56
Non-BPS, 356–359, 367–369, 371, 372 Null geodesic(s), 11–16, 22, 204
Null-like, 10, 13, 14, 18–20, 22, 23, 29, 65, 68, 69, 146, 205, 244, 350
O
Observer(s), x, 5, 19, 49–53, 84, 87, 153, 168–170, 204
Olbers, Olbers paradox, x, 73, 82, 166 Olive, 223
Open chart, 113
Open universe, 90, 158, 167, 171 Operatorial formalism, 212
450
Osp(N , 4), 381, 382, 386, 389, 391, 421, 422 Oxidation rule(s), 349–351
P
p-chain(s), 229 p-coboundary(ies), 228, 229 p-cochain, 228, 229, 232 p-cocycle(s), 228
Parallax, x, 74, 75 Particle horizon(s), x, 165 Penrose, 8, 39, 67
Penrose diagram(s), ix, 3, 35, 38, 40, 41, 407 Penrose mechanism, 39, 67
Penzias, 91, 92, 94, 95 Perfect fluid, 87, 89 Perlmutter, 99, 100 Pesando, xiii, 254, 256–260 Petrov, 8
Poincaré bundle, 225
Poincaré group, Poincaré algebra, 225, 242, 243, 248
Polar coordinate(s), 29, 43, 143, 147, 188, 291 Polyakov action, 271, 275
Power spectrum, 108, 197, 198, 200–202, 207–209
Pressure, 101, 137, 138, 163, 175, 176 Primeval atom, 83, 86
Primordial perturbation(s), 187 Principal bundle(s), 224, 225 Principal connection, 119, 225, 239 Proper time, 13, 58, 264
Pseudo-sphere(s), 89, 110, 124, 125, 127 Ptolemaic, 74
Pull-back, 32, 108, 147, 159, 161, 269, 270, 272, 276, 277, 329, 334
Push-forward, 108, 317
Q
Quantum chromodynamics, 211, 216, 312 Quartic (symplectic) invariant, xii, 347, 368 Quaternionic manifold(s), 304, 305, 318, 323,
331, 339, 341, 342
R
Radiation (filled) universe, 151, 154, 155 Ramond, 215–217, 267, 348, 349
Ramond Ramond, 249, 255, 256, 258, 260, 261, 278, 282
Randall-Sundrum, 298, 329 Red-shift distance(s), 171
Redshift(s), 82, 84, 85, 90, 94–96, 100–102, 151
Reductive, 114, 115, 118, 120 Reference frame, 5, 52, 67, 204
Index
Regge, vii, 239, 240 Regge calculus, 240 Regge poles, 240 Reiss, 99, 100
Reissner Nordström (solution, black hole, metric), 6, 8, 354
Representations of Lorentz group, algebra, 237 Restricted holonomy, xii, 322, 346, 348, 372 Rheonomic, 234, 238, 241, 243, 248, 250, 251,
253, 254, 258, 259, 268, 269, 271–273, 283, 287, 329, 348, 373, 390–392, 398–401
Rheonomy (principle), xi, 223, 226, 234, 239, 241
Riemann, 404
Riemann tensor, ix, 47, 115, 124, 125, 252, 259, 326, 340, 355, 371, 372, 374, 378, 384, 394, 430–433, 438, 439, 441, 443
Right-handed, 247, 413 Rindler space-time, 12, 13 Robertson, 83, 86, 93
Root(s), 51, 53, 54, 61–63, 65, 74, 154, 200, 238, 310, 347, 364, 365, 395
S
s channel, 212
Sachs Wolfe, 108, 207, 208 Salam (Abdus), 215
Scalar field(s), xi, 99, 108, 138, 139, 141, 164, 174–179, 182–185, 187–191, 202, 217, 219, 247, 268, 292, 298, 303, 304, 306, 310–312, 315, 317, 323, 330, 334, 338, 347, 349, 351, 352, 356–360, 362, 364–371, 375, 376, 408
Scalar manifold(s), xi, 254, 304–306, 308–311, 315–320, 323, 328, 331–334, 338, 345–347, 349, 352, 386
Scalar product, 11, 50, 67, 113, 137, 142, 153, 312
Scale factor(s), 30, 85, 86, 89–91, 95, 96, 101, 123, 124, 130, 134–136, 139–142, 150, 152, 154–158, 162, 166, 167, 169–171, 173, 174, 176, 177, 182–186, 191, 192, 195, 200, 206, 393, 394, 444
Scattering amplitude, 211, 212, 214, 240, 263 Scherck, 223
Schmidt, 99, 100 Schrödinger equation, 298 Schwarz, 215, 216, 263
Schwarzschild emiradius, 10, 64 Schwarzschild (metric), ix, 5, 6, 9–11, 14–17,
28, 37, 45, 50, 51, 57, 62, 65, 354 Schwarzschild radius, 5, 10, 365, 407 Semi-simple Lie algebra(s), 109
Index
Shapley, x, 74, 77, 78, 80, 81 σ model reduction, 346, 347
Signature, ix, 19, 20, 22, 27, 39, 110, 124, 158, 294, 350, 430, 432–435, 442
SL(2, R), 247, 248, 254–258, 260, 281, 282, 287, 310, 319, 352, 353, 359–364
Slow rolling, xi, 108, 178, 179, 181–185, 192 Small black hole(s), 369
SO(1, 10), 236, 243, 245, 373
SO(1, 3), 127, 147, 159, 389, 391, 392, 402, 421, 422
SO(9), 243, 245 SO(32), 263
Soldering, 225, 239, 243 Soliton(s), 7, 8, 264, 267, 288, 372
Solvable Lie algebra(s), 127, 255, 310 Space-like, 18–22, 29, 34, 36, 39, 67, 125,
126, 131, 137, 146, 152, 153, 350 Space-time bubble, 177
Spatial infinity, 32, 34, 36, 38, 40, 347 Spatially curved, 170
Spatially flat, 99, 103, 157, 167–169, 171, 174, 176, 177, 203, 209
Special Kähler, xii, 304, 305, 309, 322–327, 331, 335, 336, 349, 356, 359, 360, 362–364
Special relativity, viii, 5, 19 Spectral index, xi, 201, 202, 209
Speed of light, velocity of light, 5, 6, 102, 245 Sphere(s), 15, 17, 18, 30, 31, 34, 38, 44, 74,
85, 88, 89, 103, 110, 112, 113, 117, 124, 125, 127, 128, 147, 165, 206, 291, 292, 385, 431, 432, 434, 435
Spin connection(s), ix, 1, 46, 124, 128, 133, 137, 148, 223, 225, 233, 249, 274, 330, 355, 377, 380, 388, 392, 393, 395, 397, 420, 430–433, 437, 438, 442, 443
Spinor bundle(s), xix, 346, 377, 382–384, 386, 389
Spinor representations, xx, 226, 236, 237, 245, 377, 384, 409
Spinor(s), xii, 218, 220, 224, 249, 270, 273, 274, 373, 375, 377, 379, 380, 383, 387, 396, 399, 409, 413, 414, 429
Spinoza, 72
Standard candle(s), 78, 79, 101
Standard cosmological model, x, 83, 86, 97, 108, 130, 146, 168, 172, 174, 408
Standard model, 97, 348 Static limit, x, 49, 53, 54 Steinhardt, 97, 98
Stellar equilibrium, ix, 137 Stellar mass, ix, 9
Stereographic projection, 112, 113
451
Stress energy tensor(s), 48, 137, 138, 148, 149, 175, 277
String frame, 248–251, 253
String revolution(s), 216, 233, 263, 266 Structural group, structure group, 119, 225,
227, 326, 335, 376, 377
SU(1, 1), 254–258, 282, 284, 309, 336, 359, 361
Sullivan, xi, 227, 230
Sullivan’s theorem(s), xi, 227, 228, 230–233, 235, 242, 243, 249
Super-gauge completion, xii Super-Poincaré, xi, 223, 226, 236, 242, 243,
247–249, 400, 408
Supercharge(s), 218, 220, 221, 224, 225, 239, 244, 246–248, 263, 305, 309, 310
Supergravity, viii, x–xii, 1, 70, 98, 99, 107, 108, 130, 178, 187, 211, 221–223, 226–228, 230, 233, 235, 236, 238, 239, 241–243, 246–248, 250, 253–256, 259, 263, 264, 266–268, 270–272, 281, 282, 287–292, 297, 299, 303–311, 317–319, 322–325, 328–334, 336, 338, 342, 345–353, 356, 358, 359, 364, 366, 372, 380–382, 385, 391, 394, 397, 408, 409, 429
Supermutiplet(s), xi, 221, 242, 245–247, 306, 307
Supernova IA, 100–102 Supernova(e), 100
Superstring(s), superstring theory, 1, 8, 98, 105, 108, 211, 216, 233, 235, 236, 243, 246, 247, 263–265, 267, 270, 288, 289, 305, 306, 311, 347, 348, 358, 391, 408, 414, 419
Supersymmetry, viii, xi, xii, 215, 218–226, 239, 242–244, 246, 248, 254, 258, 260, 267–273, 280–288, 292, 303–306, 308, 309, 319, 323, 331–334, 338, 339, 342, 345–348, 356, 357, 374, 376, 384, 385, 387, 389, 391, 398, 409, 416, 419, 420, 423
Symmetric spaces(s), xi, 88, 107, 108, 125, 323
Symplectic embedding(s), xi, 317, 319–321, 333, 360, 361
Symplectic matrix(ces), 307, 321, 361, 421 Szekeres, 6, 7
T
t channel, 212
Tangent bundle(s), ix, 25, 225, 376 Tangent space(s), 19, 23, 25 Target space(s), 271, 273, 274
452
Taub-NUT charge, 351, 356, 367, 370 Time-like, 18–24, 26, 27, 29, 32, 34, 40, 51–53, 58, 65, 67, 146, 350
Tolman Oppenheimer Volkoff, ix
Toroidal compactification(s), 290, 309, 310, 322, 403
Torsion, 3, 250, 330, 431, 438 Torsion equation, 124, 133, 394 Tortoise coordinate, 14, 16 Triangle Galaxy, 77, 78
Type I theory(ies), x, 127, 129, 147, 150, 246, 263, 358, 359
Type II A, B theory(ies), x, 131, 137, 228, 246–248, 263, 281, 290, 358, 359, 369
U
U(1) group, factor, bundle, 308, 325
UIR (unitary irreducible representations), 242, 245
Ungauged supergravity(ies), 306, 317, 330, 332
Unitary irreducible representation, 242 Universal recession, x
V
Vacuum energy, 99, 100, 102, 103, 164, 167, 182
van Nieuwenhuizen, 222
Vector bundle(s), 119, 121, 122, 324–326, 335, 382
Vector field(s), ix, xi, 12, 29, 47, 50, 54, 67, 108, 109, 116, 118, 120–122, 126, 131, 277, 306–308, 318, 323, 325, 328, 331, 332, 334, 349, 353, 359, 386, 436
Vector multiplet(s), 246, 304, 305, 308, 319, 323–325, 331, 342, 358, 359
Veneziano, 211–213, 215, 240
Index
Veneziano duality, 212
Very special geometry, 305, 331, 334–336 Vielbein(s), ix, 1, 45, 46, 118, 124, 128, 132,
152, 225, 233, 239, 243, 249, 252, 255, 268, 270–272, 274, 276, 277, 330, 332, 340, 355, 375, 376, 380, 382, 388, 392, 393, 396, 397, 402, 403, 415, 416, 420, 423, 433–435, 441, 442
Virasoro, 215–218 Volkov, 218
W
Walker, 83, 86, 93, 165 Wave length(s), 200 Weight(s), 245, 246, 310, 326 Wess, 219
Wess-Zumino model, 220, 221 Wess-Zumino term(s) (WZT), 277, 278 Weyl rescaling, 250
Weyl spinor(s), 409, 413 Weyl transformation, 248, 253 Wheeler, 240
White dwarf, 101 White hole, 42 Wilson, 91, 92, 94, 95
WMAP, 71, 102–104, 207, 209 World sheet(s), 269
World volume (field) theory, 265 World-line, 49, 52, 226, 236, 264 Wronskian, 198
Y
Young tableau(x), 243
Z
Zumino, 219, 222, 322