100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

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74 H. Nicolai44. G. Neugebauer and D. Kramer, Annalen der Physik 24 (1969) 6245. H. Nicolai and A. Nagar, “Infinite dimensional symmetries in gravity”, inGravitational Waves, eds. I. Ciufolini, V. Gorini, U. Moschella and F. Fré,IoP, Bristol, 200146. H. Nicolai, “A Hyperbolic Lie Algebra From Supergravity,” Phys. Lett.B276, 333 (1992).47. V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, “Oscillatory approach toa singular point in the relativistic cosmology,” Adv. Phys. 19, 525 (1970).48. V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, “Construction of a generalcosmological solution of the Einstein equation with a time singularity”,Sov. Phys. JETP 35, 838-841 (1972).49. V.A. Belinskii, I.M. Khalatnikov and E.M. Lifshitz, “A general solution ofthe Einstein equations with a time singularity,” Adv. Phys. 31, 639 (1982).50. C.W. Misner, “Mixmaster universe,” Phys. Rev. Lett. 22, 1071-1074 (1969).51. C. W. Misner, “Quantum Cosmology. 1,” Phys. Rev. 186, 1319 (1969); “Minisuperspace,”In *J R Klauder, Magic Without Magic*, San Francisco 1972,441-473.52. D.M. Chitre, Ph. D. thesis, University of Maryland, 1972.53. M. P. Ryan, “Hamiltonian cosmology,” Springer-Verlag, Heidelberg (1972).54. M. P. Ryan and L. C. Shepley, “Homogeneous Relativistic Cosmologies,”Princeton, USA: Univ. Pr. ( 1975) 320 P. ( Princeton Series In Physics).55. R. T. Jantzen, “Spatially homogeneous dynamics: A unified picture,”arXiv:gr-qc/0102035.56. B. K. Berger, “Numerical Approaches to Spacetime Singularities,” arXiv:grqc/0201056.57. B. K. Berger and V. Moncrief, “Signature for local mixmaster dynamicsin U(1) symmetric cosmologies”, Phys. Rev. D62: 123501 (2000) arXiv:grqc/000607158. N. J. Cornish and J. J. Levin, “The mixmaster universe is chaotic,” Phys.Rev. Lett. 78, 998 (1997) [arXiv:gr-qc/9605029]; “The mixmaster universe:A chaotic Farey tale,” Phys. Rev. D 55, 7489 (1997) [arXiv:gr-qc/9612066].59. B. K. Berger, D. Garfinkle, J. Isenberg, V. Moncrief and M. Weaver, “The singularityin generic gravitational collapse is spacelike, local, and oscillatory,”Mod. Phys. Lett. A 13, 1565 (1998) [arXiv:gr-qc/9805063].60. L. Andersson and A.D. Rendall, “Quiescent cosmological singularities,” Commun.Math. Phys. 218, 479-511 (2001) [arXiv:gr-qc/0001047].61. E. Anderson, “Strong-coupled relativity without relativity,” arXiv:grqc/0205118.62. T. Damour and M. Henneaux, “Chaos in superstring cosmology,” Phys. Rev.Lett. 85, 920 (2000) [arXiv:hep-th/0003139]; [See also short version in Gen.Rel. Grav. 32, 2339 (2000).]63. T. Damour and M. Henneaux, “Oscillatory behaviour in homogeneous stringcosmology models,” Phys. Lett. B 488, 108 (2000) [arXiv:hep-th/0006171].64. T. Damour, M. Henneaux, B. Julia and H. Nicolai, “Hyperbolic Kac-Moodyalgebras and chaos in Kaluza-Klein models,” Phys. Lett. B 509, 323 (2001)[arXiv:hep-th/0103094].
Gravitational Billiards, Dualities and Hidden Symmetries 7565. T. Damour, M. Henneaux and H. Nicolai, Class. Quant. Grav. 20 (2003)R145-R200, hep-th/021225666. V.D. Ivashchuk and V.N. Melnikov, “Billiard Representation For MultidimensionalCosmology With Multicomponent Perfect Fluid Near The Singularity,”Class. Quantum Grav. 12, 809 (1995).67. V. D. Ivashchuk and V. N. Melnikov, “Billiard representation for multidimensionalcosmology with intersecting p-branes near the singularity,” J. Math.Phys. 41, 6341 (2000) [arXiv:hep-th/9904077].68. T. Damour, M. Henneaux, A. D. Rendall and M. Weaver, “Kasner-like behaviourfor subcritical Einstein-matter systems,” arXiv:gr-qc/0202069, to appearin Ann. Inst. H. Poincaré.69. T. Damour, S. de Buyl, M. Henneaux and C. Schomblond, “Einstein billiardsand overextensions of finite-dimensional simple Lie algebras,” arXiv:hepth/0206125.70. B. Julia, in Lectures in Applied Mathematics, AMS-SIAM, vol 21 (1985),p.355.71. H. Nicolai and T. Fischbacher, in: Kac Moody Lie Algebras and Related Topics,eds. N. Sthanumoorthy and K.C. Misra, Contemporary Mathematics 343,American Mathematical Society, 2004, hep-th/030101772. J. E. Humphreys, “Introduction to Lie Algebras and Representation Theory”,Graduate Texts in Mathematics 9, Springer Verlag, 198073. J.H. Conway and N.J.A. Sloane, “Sphere Packings, Lattices and Groups”,Grundlehren der mathematischen Wissenschaften 290, 2nd edition, SpringerVerlag, 199174. I. Schnakenburg and P. C. West, “Kac-Moody symmetries of IIB supergravity,”Phys. Lett. B 517, 421 (2001) [arXiv:hep-th/0107181].75. I. Schnakenburg and P. C. West, “Massive IIA supergravity as a non-linearrealisation,” Phys. Lett. B 540, 137 (2002) [arXiv:hep-th/0204207].76. A. Kleinschmidt, I. Schnakenburg and P. West, Very-extended Kac-Moodyalgebras and their interpretation at low levels, Class. Quant. Grav. 21 (2004)2493 [arXiv:hep-th/0309198].
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Published byWorld Scientific Publis
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viA. Ashtekarof space-time that eme
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viiiA. AshtekarGravitational waves
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xviContents9. Receiving Gravitation
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4 J. Stachel(2) the change over tim
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6 J. StachelFig. 2respectively. Fur
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8 J. Stachelof string theory, M-the
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10 J. StachelLogically, it would se
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12 J. Stachelupon by no (net) exter
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14 J. Stachelof the incompatibility
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16 J. Stachelis an important differ
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18 J. Stachelbetween physical conce
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20 J. Stachel10. Four-Dimensional F
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22 J. Stachelderivative of the metr
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Page 52 and 53: 34 J. Stacheldynamical process as a
Page 54 and 55: 36 J. Stachel9. J. Stachel, Special
Page 56 and 57: Part IIEinstein’s UniverseRamific
Page 58 and 59: 40 H. Nicolai• Higher dimensions
Page 60 and 61: 42 H. Nicolaiadvance, and possibly
Page 62 and 63: 44 H. NicolaiBianchi identities are
Page 64 and 65: 46 H. NicolaiIt does not appear pos
Page 66 and 67: 48 H. NicolaiThe other SL(2, R), of
Page 68 and 69: 50 H. Nicolai3.1. BKL dynamics and
Page 70 and 71: 52 H. Nicolaidegrees of freedom, th
Page 72 and 73: 54 H. Nicolaiin the valley. The Ham
Page 74 and 75: 56 H. NicolaiThe main feature of th
Page 76 and 77: 58 H. Nicolai4. Basics of Kac Moody
Page 78 and 79: 60 H. Nicolaifor indefinite A. It i
Page 80 and 81: 62 H. NicolaiThe level l = 0 sector
Page 82 and 83: 64 H. NicolaiConsequently, the repr
Page 84 and 85: 66 H. Nicolaiwhere the abelian part
Page 86 and 87: 68 H. Nicolaiconstant momenta Π yi
Page 88 and 89: 70 H. Nicolaiσ-model side is suppo
Page 90 and 91: 72 H. NicolaiReferences1. H.A. Buch
Page 94 and 95: CHAPTER 3THE NATURE OF SPACETIME SI
Page 96 and 97: 78 A. D. Rendallspacetime to define
Page 98 and 99: 80 A. D. RendallOne of the most imp
Page 100 and 101: 82 A. D. Rendalltheorems was an elu
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Page 104 and 105: 86 A. D. Rendallback into a curvatu
Page 106 and 107: 88 A. D. Rendallspace and there are
Page 109 and 110: The Nature of Spacetime Singulariti
Page 111 and 112: CHAPTER 4BLACK HOLES - AN INTRODUCT
Page 113 and 114: Black Holes 95It follows that ˙v d
Page 115 and 116: Black Holes 973221100-1-1-2-200-5-3
Page 117 and 118: Black Holes 99calculation of the de
Page 119: Black Holes 101family of null geode
Page 123 and 124: Black Holes 105(3) Ω is positive o
Page 125 and 126: Black Holes 107In dimension 3 + 1 t
Page 127 and 128: Black Holes 109A fundamental theore
Page 129 and 130: Black Holes 111(4) The Kerr black h
Page 131 and 132: Black Holes 113˚g ab =˚g ab (x a
Page 133 and 134: Black Holes 115It is a folklore the
Page 135 and 136: Black Holes 117⃗x 4 = −⃗x 3
Page 137 and 138: Black Holes 119Work partially suppo
Page 139 and 140: Black Holes 121totically flat space
Page 141 and 142: Black Holes 12376. R.C. Myers and M
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The Physical Basis of Black Hole As
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Probing Space-Time Through Numerica
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CHAPTER 7UNDERSTANDING OUR UNIVERSE
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Understanding Our Universe: Current
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CHAPTER 8WAS EINSTEIN RIGHT? TESTIN
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Was Einstein Right? 207We begin in
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Was Einstein Right? 20910 -810 -9E
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Was Einstein Right? 211On this 100t
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Was Einstein Right? 213The SME and
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Was Einstein Right? 215where d is t
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Was Einstein Right? 217of VLBI data
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Was Einstein Right? 219milliarcseco
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Was Einstein Right? 2215. Gravitati
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Was Einstein Right? 223be measured
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Was Einstein Right? 22510. J. G. Wi
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Was Einstein Right? 22761. C. M. Wi
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Receiving Gravitational Waves 229th
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Receiving Gravitational Waves 231gr
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Receiving Gravitational Waves 233Th
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Receiving Gravitational Waves 235We
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Receiving Gravitational Waves 23719
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Receiving Gravitational Waves 239th
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Receiving Gravitational Waves 241to
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Receiving Gravitational Waves 243In
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Receiving Gravitational Waves 245
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Receiving Gravitational Waves 247a
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Receiving Gravitational Waves 249ab
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Receiving Gravitational Waves 251Fi
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Receiving Gravitational Waves 253Eq
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Receiving Gravitational Waves 255In
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CHAPTER 10RELATIVITY IN THE GLOBAL
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Relativity in the Global Positionin
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Part IIIBeyond EinsteinUnifying Gen
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CHAPTER 11SPACETIME IN SEMICLASSICA
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Spacetime in Semiclassical Gravity
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CHAPTER 12SPACE TIME IN STRING THEO
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Space Time in String Theory 313limi
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Space Time in String Theory 315luti
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Space Time in String Theory 317In s
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Space Time in String Theory 319of t
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Space Time in String Theory 321an o
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Space Time in String Theory 323arbi
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Space Time in String Theory 325an e
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Space Time in String Theory 327in a
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Space Time in String Theory 329doma
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Space Time in String Theory 331a tu
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Space Time in String Theory 333bran
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Space Time in String Theory 335is a
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Space Time in String Theory 337thes
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Space Time in String Theory 339The
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Space Time in String Theory 341gene
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Space Time in String Theory 343are
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Space Time in String Theory 345tion
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Space Time in String Theory 34723.
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Space Time in String Theory 34955.
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Gravity, Geometry and the Quantum 3
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£¢¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡
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Loop Quantum Cosmology 383inflation
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Loop Quantum Cosmology 385gravitati
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Loop Quantum Cosmology 387Secondly,
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Loop Quantum Cosmology 389sically d
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Loop Quantum Cosmology 391states an
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Loop Quantum Cosmology 393represent
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Loop Quantum Cosmology 3954.2.1. Di
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Loop Quantum Cosmology 397question
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Loop Quantum Cosmology 399exist. Th
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Loop Quantum Cosmology 401in the ex
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Loop Quantum Cosmology 403λ max or
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Loop Quantum Cosmology 405This happ
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Loop Quantum Cosmology 407Loop quan
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Loop Quantum Cosmology 409to obtain
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Loop Quantum Cosmology 411familiar
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Loop Quantum Cosmology 41334. M. Bo
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CHAPTER 15CONSISTENT DISCRETE SPACE
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Consistent Discrete Space-Time 417v
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Consistent Discrete Space-Time 419I
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Consistent Discrete Space-Time 421a
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Consistent Discrete Space-Time 423F
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Consistent Discrete Space-Time 4250
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Consistent Discrete Space-Time 4272
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Consistent Discrete Space-Time 429i
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Consistent Discrete Space-Time 431a
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Consistent Discrete Space-Time 433n
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Consistent Discrete Space-Time 435t
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Consistent Discrete Space-Time 437w
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Consistent Discrete Space-Time 439f
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Consistent Discrete Space-Time 441(
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Consistent Discrete Space-Time 443R
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CHAPTER 16CAUSAL SETS ANDTHE DEEP S
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Causal Sets and the Deep Structure
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CHAPTER 17THE TWISTOR APPROACH TOSP
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The Twistor Approach to Space-Time
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INDEXσ models, 57, 65, 693+1 formu
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Index 509inflation, 383, 385-387, 4
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