EXACT SOLUTIONS and SCALAR FIELDS in GRAVITY This Page Intentionally Left Blank EXACT SOLUTIONS and SCALAR FIELDS in GRAVITY RECENT DEVELOPMENTS

EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY This page intentionally left blank EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY RECENT DEVELOPMENTS Edited by ALFREDO MACIAS Universidad Autónoma Metropolitana–Itzapalapa Mexico City, Mexico JORGE L. CERVANTES-COTA Institute Nacional de Investigaciones Nucleares Mexico City, Mexico CLAUS LÄMMERZAHL University of Düsseldorf Düsseldorf, Germany KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW eBook ISBN: 0-306-47115-9 Print ISBN: 0-306-46618-X ©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2001 Kluwer Academic/Plenum Publishers New York All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: http://kluweronline.com and Kluwer's eBookstore at: http://ebooks.kluweronline.com To the 65th birthday of Heinz Dehnen and to the 60th birthday of Dietrich Kramer v This page intentionally left blank CONTENTS Preface xv Contributing Authors xvii Part I Exact Solutions Self–gravitating stationary axi-symmetric perfect fluids: Differential rotation and some geometric features 3 F.J. Chinea 1 Newtonian configurations 3 2 Dirichlet boundary problem vs. free-boundary problem 6 3 Interior solutions with differential rotation in general relativity 7 4 Vacuum as a “perfect fluid”. The “rigid rotation” and the “irrotational” gauges 10 5 Some geometric features of a rigidly rotating perfect fluid interior solution 12 New Directions for the New Millennium 15 Frederick J. Ernst 1 Introduction 15 2 The Quest for the Geroch Group and its Generalizations 16 3 The Perfect Fluid Joining Problem 18 4 Future Directions 19 4.1 General 19 4.2 Radiating Axisymmetric Systems 20 5 Time for a New World Order? 20 On maximally symmetric and totally geodesic spaces 23 Alberto Garcia 1 Introduction 23 2 Killing Equations and Integrability Conditions 24 3 Maximally Symmetric Spaces 26 4 Representation of Maximally Symmetric Spaces 29 5 Maximally Symmetric Sub–spaces 31 6 Totally Geodesic Hypersurfaces 32 vii viii EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY 6.1 Maximally Symmetric Totally Geodesic Hypersurface 34 7 Totally Geodesic Sub–spaces 35 7.1 Maximally Symmetric Totally Geodesic Sub–spaces 35 8 The 2+1 BTZ Solution as Hypersurface of the 3+1 PC Metric 36 8.1 The Static BTZ Solution as Anti– de–Sitter Metric 37 Discussion of the theta formula for the Ernst potential of the rigidly rotating disk of dust 39 Andreas Kleinwächter 1 The theta formula for the Ernst potential 39 2 Calculation of the arguments 42 3 Transformation of the theta formula 48 4 Applications 50 5 Appendix 50 The superposition of null dust beams in General Relativity 53 Dietrich Kramer 1 Introduction 53 2 The superposition of spherically symmetric beams of null dust 55 3 The gravitational field of two counter–moving beams of light 56 3.1 The problem to be solved. Basic assumptions 56 3.2 The static solution 57 3.3 The stationary solution 59 3.4 Other solutions 60 4 Discussion 60 Solving equilibrium problem for the double–Kerr spacetime 63 Vladimir S. Manko, Eduardo Ruiz 1 Introduction 63 2 A short history of the problem 63 3 A new approach to the double–Kerr equilibrium problem 64 4 The Komar masses and angular momenta 66 5 Towards the analysis of the multi–black hole equilibrium states 67 Rotating equilibrium configurations in Einstein’s theory of gravitation 69 Reinhard Meinel 1 Figures of equilibrium of rotating fluid masses 69 2 Rotating dust configurations – The Maclaurin disk 71 3 General–relativistic continuation 72 4 Black–hole limit 74 5 Discussion 75 Integrability of SDYM Equations for the Moyal Bracket Lie Algebra 77 M. Przanowski, J.F. Plebański, S. Formański 1 Introduction 77 2 Nonlocal conservation laws 79 3 Linear systems for ME 80 4 Twistor construction 82 Contents ix 5 The case of 84 Part II Alternative Theories and Scalar Fields The FRW Universes with Barotropic Fluids in Jordan–Brans–Dicke Theory 91 P. Chauvet–Alducin 1 Introduction 91 2 Barotropic Fluids 93 3 Radiation 95 4 Vacuum 97 5 Conclusions 98 Higgs–Field and Gravity 101 Heinz Dehnen 1 Higgs–field gravity 101 2 Higgs–field scalar tensor theory of gravity 105 The road to Gravitational S–duality 111 H. Garcia–Compeán, O. Obregón, C. Ramirez, M. Sabido 1 Introduction 112 2 S–Duality in Topological Gravity 113 3 S–Duality in MacDowell–Mansouri Gauge Theory of Gravity 116 4 (Anti)Self–duality of the Three–dimensional Chern–Simons Grav- ity 119 5 ChernSimons Gravity Dual Action in Three Dimensions 120 6 Concluding Remarks 121 Exact solutions in multidimensional gravity with p–branes and PPN Pa- rameters 123 V. D. Ivashchuk, V. N. Melnikov 1 Introduction 123 2 The model 124 2.1 Ansatz for composite p-branes 124 2.2 The sigma model 125 3 Toda–type solutions from null–geodesic method 126 4 Cosmological solutions 128 4.1 Solutions with Ricci-flat spaces 128 4.2 Solutions with one curved space 128 5 Black hole solutions 129 6 Post–Newtonian approximation 130 Effective four–dimensional dilaton gravity from five–dimensional Chern– Simons gravity 133 Alfredo Macias, Alberto Garcia 1 Introduction 133 2 Chern–Simons action in five dimensions 134 3 Five dimensional principal fiber boundle 136 4 Effective action 137 x EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY 5 Discussion 140 A plane–fronted wave solution in metric–affine gravity 141 Dirk Puetzfeld 1 Introduction 141 2 MAG in general 142 3 The triplet ansatz 144 4 Plane–fronted waves in GR 145 5 Plane-fronted waves in MAG 148 6 Summary 151 Part III Cosmology and Inflation New solutions of Bianchi models in scalar-tensor theories 155 Jorge L. Cervantes–Cota, M. A. Rodriguez–Meza, Marcos Nahmad 1 Introduction 155 2 Equations for FRW and Bianchi models 156 3 Solutions and asymptotic behavior in BD theory 158 3.1 FRW solutions 159 3.2 Bianchi type I model 160 3.3 Bianchi type V model 161 3.4 Bianchi type IX model 161 4 Conclusions 162 Scalar Field Dark Matter 165 Tonatiuh Matos, F. Siddhartha Guzmán, L. Arturo Ureña–López, Dario Núñez 1 Introduction 166 2 Cosmological Scalar Field Solutions 168 2.1 Radiation Dominated Era (RD) 169 2.2 Matter Dominated Era (MD) and Scalar Field Dom- inated Era 170 2.3 Scalar Power Spectrum for dark matter 171 3 Scalar dark matter and Planck scale physics 174 4 Spherical Scalar Field Fluctuations as Galactic Halos 176 5 The Galaxy Center 181 6 Conclusions 182 Inflation with a blue eigenvalue spectrum 185 Eckehard W. Mielke, Franz E. Schunck 1 Introduction 185 2 Spatially flat inflationary universe 187 3 Abel equation from second-order slow–roll approximation 188 3.1 Solutions with a blue spectrum 189 4 Complete eigenvalue spectrum 190 5 Discussion 192 Classical and Quantum cosmology with and without self–interacting scalar field in Bergmann–Wagoner theory 195 Contents xi Luis O. Pimentel, Cesar Mora 1 Introduction 195 2 Isotropic models 196 2.1 Gauge N=l 197 Case 197 Case 198 2.2 Gauge N=l/x 198 Classical solutions 199 Classical solutions for Brans–Dicke theory 199 3 Anisotropic models 200 3.1 Exact solutions for Bianchi I with 201 3.2 Solutions for Bianchi I with another potential 201 3.3 Exact solutions for Bianchi II with 202 4 Final remarks 202 The Big Bang in Gowdy Cosmological Models 205 Hernando Quevedo 1 Introduction 205 2 Gowdy Cosmological Models 206 3 Analogies and general results 208 4 The Big Bang 210 The influence of scalar fields in protogalactic interactions 213 M. A. Rodriguez–Meza, Jaime Klapp, Jorge L. Cervantes-Cota, Heinz Dehnen 1 Introduction 213 2 Scalar Fields and the Newtonian Approximation 215 3 Protogalactic Cloud Models and Results 216 4 Conclusions 220 Adaptive Calculation of a Collapsing Molecular Cloud Core: The Jeans Condition 223 Leonardo Di G. Sigalotti, Jaime Klapp 1 Introduction 223 2 Initial Model and Computational Methods 225 3 Results 228 4 Conclusions 232 Revisiting the calculation of inflationary perturbations 235 César A. Terrero–Escalante, Dominik J. Schwarz, Alberto Garcia 1 Introduction 235 2 The standard formulas 236 2.1 Amplitudes of inflationary perturbations 237 2.2 The spectral indices 239 3 Generalizing the Bessel approximation 241 3.1 Generalized power–law approximation 241 3.2 Generalized slow–roll approximation 242 3.3 The spectral indices 243 An alternative for more general models 243 xii EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY 4 Testing the expressions 244 5 Conclusions 245 Conformal symmetry and deflationary gas universe 247 Winfried Zimdahl, Alexander B. Balakin 1 Introduction 247 2 Fluid description 249 3 Gas dynamics 250 4 Cosmological dynamics and conformal symmetry 252 5 Phenomenological vacuum decay 255 6 Equivalent scalar field dynamics 256 7 Conclusions 257 Part IV Experiments and Other Topics Staticity Theorem for Non–Rotating Black Holes with Non–Minimally Cou- pled Self–Interacting Scalar Fields 263 Eloy Ayón–Beato 1 Introduction 263 2 The Staticity Theorem for Non–Minimally Coupled Scalar Fields 265 3 Conclusions 269 Quantum nondemolition measurements and non–Newtonian gravity 271 A. Camacho 1 Introduction 271 2 Yukawa term 273 3 Quantum Measurements 274 4 Quantum nondemolition measurements 275 5 QND and non–Newtonian gravity: Propagators and probabil- ities 276 6 Conclusions 277 On Electromagnetic Thirring Problems 281 Markus King, Herbert Pfister 1 Introduction 281 2 The Static Two–Shell Model 282 3 First Order Rotation of the Shells 285 4 Results 288 4.1 Results in first order of the charge 289 4.2 Results in second order of the charge 291 4.3 Results exact in mass M and charge 292 On the experimental foundation of Maxwell’s equations 295 C.

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