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On maximally symmetric and totally geodesic spaces 35 7. TOTALLY GEODESIC SUB–SPACES A immersed in a is called totally geodesic subspace of if all the geodesics of are geodesics of A necessary and sufficient condition for a immersed in a be totally geodesic is that where see (54), are given by where denote the components of a particular choice of mutually orthogonal one to another and normal to vectors. The functions and are invariants for transformations of coordinates in are the components of a symmetric tensor in the and are components of a vector. Hence it follows from (69), that for each value of the are the components of a symmetric tensor in One can write In E186 one encounters the following statement: the sub–spaces const., are totally geodesic in if the metric of can be decomposed as where the functions are independent of In general, are functions of and 7.1. MAXIMALLY SYMMETRIC TOTALLY GEODESIC SUB–SPACES If metric components of the above metric are independent of that is the sub–spaces are maximally symmetric totally geodesic in if the metric of can be decomposed as
36 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY 8. THE 2+1 BTZ SOLUTION AS HYPERSURFACE OF THE 3+1 PC METRIC Let us now consider the Weyl zero [A] metric [3, 4]. We shall show that this metric (being locally maximally symmetric by itself) admits locally as a (maximally symmetric) hypersurface the BTZ space-time [5], which occurs to be not totally geodesic. In order to establish this assertion, we start with the conformally flat [A] metric, written as with structural functions Accomplishing the coordinate transformations the metric (72) is reducible to where corresponds to the metric of the well known BTZ black hole of the (2+l)-gravity:
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EXACT SOLUTIONS AND SCALAR FIELDS I
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To the 65th birthday of Heinz Dehne
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CONTENTS Preface Contributing Autho
Page 10 and 11: Contents ix 5 The case of 84 Part I
Page 12 and 13: Contents xi Luis O. Pimentel, Cesar
Page 14 and 15: Contents xiii 2.2 String theory ind
Page 16 and 17: PREFACE This book is dedicated to h
Page 18 and 19: CONTRIBUTING AUTHORS Eloy Ayón-Bea
Page 20 and 21: Contributing Authors xix Jaime Klap
Page 22 and 23: Contributing Authors xxi A.P. 70-54
Page 24 and 25: Contributing Authors xxiii L. Artur
Page 26 and 27: I EXACT SOLUTIONS
Page 28 and 29: SELF-GRAVITATING STATIONARY AXI- SY
Page 30 and 31: Self-gravitating stationary axisymm
Page 38 and 39: REFERENCES 13 be shown that the ana
Page 40 and 41: NEW DIRECTIONS FOR THE NEW MILLENNI
Page 42 and 43: New Directions for the New Millenni
Page 44 and 45: New Directions for the New Millenni
Page 46 and 47: REFERENCES 21 Acknowledgments The r
Page 48 and 49: ON MAXIMALLY SYMMETRIC AND TOTALLY
Page 50 and 51: On maximally symmetric and totally
Page 64 and 65: DISCUSSION OF THE THETA FORMULA FOR
Page 66 and 67: Theta formula for the Ernst potenti
Page 76 and 77: REFERENCES 51 Rosenhain’s theta f
Page 78 and 79: THE SUPERPOSITION OF NULL DUST BEAM
Page 80 and 81: The superposition of null dustbeams
Page 86 and 87: REFERENCES 61 geodesic with respect
Page 88 and 89: SOLVING EQUILIBRIUM PROBLEM FOR THE
Page 90 and 91: Solving equilibrium problem for the
Page 92 and 93: REFERENCES 67 where the subindices
Page 94 and 95: ROTATING EQUILIBRIUM CONFIGURATIONS
Page 96 and 97: Rotating equilibrium configurations
Page 98 and 99: Rotating equilibrium configurations
Page 100 and 101: REFERENCES 75 5. DISCUSSION The ana
Page 102 and 103: INTEGRABILITY OF SDYM EQUATIONS FOR
Page 104 and 105: Integrability of SDYM Equations for
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Integrability of SDYM Equations for
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REFERENCES 87 [18] [19] [20] [21] [
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II ALTERNATIVE THEORIES AND SCALAR
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THE FRW UNIVERSES WITH BAROTROPIC F
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The FRW Universes with Barotropic F
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REFERENCES 99 (1979) 515; H. Dehnen
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HIGGS-FIELD AND GRAVITY Heinz Dehne
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Higgs-Field and Gravity 103 is defi
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Higgs-Field and Gravity 105 Consequ
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Higgs-Field and Gravity 107 From th
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REFERENCES 109 Evidently by inserti
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THE ROAD TO GRAVITATIONAL S-DUALITY
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The road to Gravitational S-duality
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REFERENCES 121 The partition functi
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EXACT SOLUTIONS IN MULTIDIMENSIONAL
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Exact solutions in multidimensional
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REFERENCES 131 For possible physica
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EFFECTIVE FOUR-DIMENSIONAL DILATON
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Dilaton gravity from Chern-Simons g
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A PLANE-FRONTED WAVE SOLUTION IN ME
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A plane-fronted wave solution in me
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REFERENCES 6. SUMMARY 151 We invest
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III COSMOLOGY AND INFLATION
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NEW SOLUTIONS OF BIANCHI MODELS IN
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New solutions of Bianchi models in
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REFERENCES 163 Further solutions of
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SCALAR FIELD DARK MATTER Tonatiuh M
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Scalar field dark matter 167 tional
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Scalar field dark matter 169 being
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Scalar field dark matter 171 of dar
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Scalar field dark matter 173 growin
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Scalar field dark matter 175 This s
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Scalar field dark matter 177 rotati
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Scalar field dark matter 179 where
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Scalar field dark matter 181 while
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REFERENCES 183 The hypothesis of th
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INFLATION WITH A BLUE EIGENVALUE SP
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Inflation with a blue eigenvalue sp
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REFERENCES 193 problem” for nonli
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CLASSICAL AND QUANTUM COSMOLOGY WIT
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Quantum cosmology with self-interac
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REFERENCES 203 [13] has considered
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THE BIG BANG IN GOWDY COSMOLOGICAL
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The Big Bang in Gowdy Cosmological
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THE INFLUENCE OF SCALAR FIELDS IN P
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The influence of scalar fields in p
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The influence of scalar fields in p
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REFERENCES 221 [3] by R.C. Kennicut
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ADAPTIVE CALCULATION OF A COLLAPSIN
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Adaptive Calculation of a Collapsin
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REFERENCES 233 [5] [6] [7] [8] [9]
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REVISITING THE CALCULATION OF INFLA
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Revisiting the calculation of infla
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CONFORMAL SYMMETRY AND DEFLATIONARY
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Conformal symmetry and deflationary
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REFERENCES 259 [16] R. Maartens, D.
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IV EXPERIMENTS AND OTHER TOPICS
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STATICITY THEOREM FOR NON-ROTATING
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Staticity Theorem for Non-Rotating
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Staticity Theorem for Non-Rotating
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REFERENCES 269 The boundary is cons
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QUANTUM NONDEMOLITION MEASUREMENTS
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Quantum nondemolition measurements
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REFERENCES 279 [11] M. Kasevich and
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ON ELECTROMAGNETIC THIRRING PROBLEM
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On Electromagnetic Thirring Problem
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REFERENCES 293 [8] [9] D. Brill and
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ON THE EXPERIMENTAL FOUNDATION OF M
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On the experimental foundation of M
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REFERENCES 309 [12] [13] [14] [15]
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LORENTZ FORCE FREE CHARGED FLUIDS I
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Lorentz force free charged fluids i
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REFERENCES 319 force can be foresee
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Index Axisymmetries and configurati
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INDEX 323 Theorem (cont.) staticity
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Prof. Heinz Dehnen: Brief Biography
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Prof. Dietrich Kramer: Brief Biogra
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