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

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154 P. Lagunaprovides a basic overview and is intended for non-experts in numerical relativity.The following reviews are recommended for those interested in amore technical view:Numerical Relativity and Compact Binaries by Thomas W. Baumgarteand Stuart L. Shapiro 12 This is comprehensive review with specificfocus on traditional numerical relativity, namely those aspects of numericalrelativity that have been designed for or applicable to the two body problemin numerical relativity. In my opinion this review is an excellent startingpoint for students and researchers interested in numerical relativity.Initial Data for Numerical Relativity by Greg B. Cook. 25 This reviewgives an excellent introduction to the construction of Cauchy initial datafor space-times containing binary systems of black holes and/or neutronstars. The emphasis is on those methods to obtain initial data representingbinary systems in quasi-equilibrium orbits. The review also addresses theissue of the astrophysically content of initial data sets.Hyperbolic Methods for Einstein Equations by Oscar A. Reula. 49Reula’s review is a most for those interested in the mathematical propertiesof evolution equations arising in 3+1 formulations of the Einstein equations.The review also examines the techniques to obtaining symmetric hyperbolicsystems.Characteristic Evolution and Matching by Jeff Winicour. 69 Characteristicformulations, although not widely used in numerical evolutions,have attractive properties for wave extraction. This review discusses thedevelopments of characteristic codes and, in particular, the application ofcharacteristic evolutions to Cauchy-characteristic matching.Conformal Field Equations by Jörg Frauendiener. 28 This is a generalreview on the notion of conformal infinity. Of direct relevance to numericalrelativity is the discussion on Friedrich’s 29 conformal field equationsand how this formulation of the field equations could provide a naturalframework to study isolated systems and gravitational radiation.2. Geometrodynamics and Numerical EvolutionsThe most popular approach in numerical relativity is to view spacetime asthe time history of the geometry of a space-like hypersurface or Geometro-
Probing Space-Time Through Numerical Simulations 155dynamics, as first referred by J.A. Wheeler. 44 Under the geometrodynamicspoint of view, the Einstein equations, G µν = 8 π T µν for the spacetime metricg µν , can be reformulated or rewritten in a space+time or 3+1 structureby suitable projections on and orthogonal to the space-like hypersurfaces.The Arnowitt-Deser-Misner 6 (ADM) equations are perhaps the simplest3+1 formulation (see Appendix A). In the ADM formulation, the 10 Einsteinequations for the spacetime metric g µν become 12 first order in timeevolution equations for the metric h ij and metric velocity K ij or moreformally the extrinsic curvature of the space-like hypersurfaces. a The remaining4 equations are constraints (Hamiltonian and momentum) thatthe pair (h ij , K ij ) must satisfy.Obtaining long-term stable numerical evolutions of spacetimes containingblack holes and neutron stars with codes based on the ADM equationshas been extremely difficult. It is not completely clear the reasons behindthe numerical problems with the ADM system. Most likely these problemshave to do with the weakly hyperbolic properties of the system. 46 Theseimpediments have motivated researchers to develop other 3+1 formulationsof the Einstein equations.The interest in numerical relativity on investigating 3+1 formulationshas reinvigorated the field. Numerical relativity is no longer dominated bytour de force efforts. The field is receiving a balanced injection of formalmathematical ideas and creative engineering. This excitement has produceda zoo of formulations of the Einstein equations, formulations designed withthe sole purpose of finding the silver bullet against numerical instabilities.The emphasis has been on manifestly hyperbolic formulations (seeReula’s review 49 ). These formulations are supported by mathematical theoremsthat provide valuable information, for instance, on the dependence ofevolved data on initial conditions. Hyperbolicity alone is not enough. Thereare other crucial factors. For example, lower order non-liner terms could beresponsible for triggering or facilitating the development of instabilities. 37Only through numerical experimentation one is going to able to tell whichformulation or formulations are more advantageous.I will focus the discussion on three formulations. These are currentlythe most successful formulations used in 3D simulations involving nontrivialconfigurations of compact objects. The first of these formulationsa The following index notation has been adopted: 4D indices will be denoted with Greekletters and 3D indices with Latin letters. Latin indices are to be understood as pull-backsto the hypersurfaces in the foliation. Units are such that G = c = 1.
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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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Page 125 and 126: Black Holes 107In dimension 3 + 1 t
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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
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Page 135 and 136: Black Holes 117⃗x 4 = −⃗x 3
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Page 193 and 194: CHAPTER 7UNDERSTANDING OUR UNIVERSE
Page 195 and 196: 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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