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

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444 R. Gambini and J. Pullintum Mechanics, Castello di Piombino, Tuscany, Italy, 1-4 Sep 2004, H. T.Elze, editor [arXiv:gr-qc/0501027].20. T. Thiemann, “The Phoenix project: Master constraint programme for loopquantum gravity,” arXiv:gr-qc/0305080.21. A. Ashtekar, J. Lewandowski Class. Quant. Grav. 21, R53 (2004) [arXiv:grqc/0404018].22. R. Gambini and J. Pullin, Phys. Rev. Lett. 94, 101302 (2005) [arXiv:grqc/0409057].
CHAPTER 16CAUSAL SETS ANDTHE DEEP STRUCTURE OF SPACETIMEFAY DOWKERBlackett Laboratory, Imperial College,Prince Consort Road, London SW7 2AZ, UKThe causal set approach to quantum gravity embodies the concepts ofcausality and discreteness. This article explores some foundational andconceptual issues within causal set theory.1. IntroductionThe problem of quantum gravity in its narrow sense is the problem of findinga theory that incorporates both Quantum Mechanics and General Relativity.A broader vision is that a theory of quantum gravity would restore tophysics a unified framework, a framework in which there is no fundamentaldivision in principle between observer and observed nor between matter andspacetime. One reading of Einstein’s writings on Quantum Mechanics andUnification is that he viewed success in the broad quest as a prerequisiteto a solution of the narrow problem.Every approach to quantum gravity in this broad conception must embodyanswers to two fundamental questions: “What is quantum mechanics?”and “What is the deep structure of spacetime?” This article will touchon the former question and focus on the latter and the answer to it providedby the approach known as causal set theory which marries the two conceptsof discreteness (or atomicity) and causality.The view that causality is a more fundamental organising principle, eventhan space and time, is an ancient tradition of thought. Corresponding tothis view, momentary events have a better claim to be basic than objectsextended in time like particles, the latter being understood as persistentpatterns of events rather than enduring “substances”. Within Relativity,the recognition that almost all of the geometrical properties of Minkowski445
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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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xA. Ashtekarhappy, schizophrenic at
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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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24 J. Stachel(b) General relativity
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26 J. Stachelfour-dimensional analo
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28 J. Stachelply the lessons learne
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30 J. Stachelmanifold appropriate f
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32 J. Stachelwhich merely defines t
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34 J. Stacheldynamical process as a
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36 J. Stachel9. J. Stachel, Special
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Part IIEinstein’s UniverseRamific
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40 H. Nicolai• Higher dimensions
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42 H. Nicolaiadvance, and possibly
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44 H. NicolaiBianchi identities are
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46 H. NicolaiIt does not appear pos
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48 H. NicolaiThe other SL(2, R), of
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50 H. Nicolai3.1. BKL dynamics and
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52 H. Nicolaidegrees of freedom, th
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54 H. Nicolaiin the valley. The Ham
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56 H. NicolaiThe main feature of th
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58 H. Nicolai4. Basics of Kac Moody
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60 H. Nicolaifor indefinite A. It i
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62 H. NicolaiThe level l = 0 sector
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64 H. NicolaiConsequently, the repr
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66 H. Nicolaiwhere the abelian part
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68 H. Nicolaiconstant momenta Π yi
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70 H. Nicolaiσ-model side is suppo
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72 H. NicolaiReferences1. H.A. Buch
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74 H. Nicolai44. G. Neugebauer and
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CHAPTER 3THE NATURE OF SPACETIME SI
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78 A. D. Rendallspacetime to define
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80 A. D. RendallOne of the most imp
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82 A. D. Rendalltheorems was an elu
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84 A. D. Rendallsymmetric solution
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86 A. D. Rendallback into a curvatu
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88 A. D. Rendallspace and there are
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The Nature of Spacetime Singulariti
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CHAPTER 4BLACK HOLES - AN INTRODUCT
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Black Holes 95It follows that ˙v d
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Black Holes 973221100-1-1-2-200-5-3
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Black Holes 99calculation of the de
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Black Holes 101family of null geode
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Black Holes 105(3) Ω is positive o
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Black Holes 107In dimension 3 + 1 t
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Black Holes 109A fundamental theore
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Black Holes 111(4) The Kerr black h
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Black Holes 113˚g ab =˚g ab (x a
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Black Holes 115It is a folklore the
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Black Holes 117⃗x 4 = −⃗x 3
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Black Holes 119Work partially suppo
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Black Holes 121totically flat space
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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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Page 421 and 422: Loop Quantum Cosmology 403λ max or
Page 423 and 424: Loop Quantum Cosmology 405This happ
Page 425 and 426: Loop Quantum Cosmology 407Loop quan
Page 427 and 428: Loop Quantum Cosmology 409to obtain
Page 429 and 430: Loop Quantum Cosmology 411familiar
Page 431 and 432: Loop Quantum Cosmology 41334. M. Bo
Page 433 and 434: CHAPTER 15CONSISTENT DISCRETE SPACE
Page 435 and 436: Consistent Discrete Space-Time 417v
Page 437 and 438: Consistent Discrete Space-Time 419I
Page 439 and 440: Consistent Discrete Space-Time 421a
Page 441 and 442: Consistent Discrete Space-Time 423F
Page 443 and 444: Consistent Discrete Space-Time 4250
Page 445 and 446: Consistent Discrete Space-Time 4272
Page 447 and 448: Consistent Discrete Space-Time 429i
Page 449 and 450: Consistent Discrete Space-Time 431a
Page 451 and 452: Consistent Discrete Space-Time 433n
Page 453 and 454: Consistent Discrete Space-Time 435t
Page 455 and 456: Consistent Discrete Space-Time 437w
Page 457 and 458: Consistent Discrete Space-Time 439f
Page 459 and 460: Consistent Discrete Space-Time 441(
Page 461: Consistent Discrete Space-Time 443R
Page 465 and 466: Causal Sets and the Deep Structure
Page 483 and 484: CHAPTER 17THE TWISTOR APPROACH TOSP
Page 485 and 486: The Twistor Approach to Space-Time
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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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