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Theta formula for the Ernst potential of the rigidly rotating disk of dust 49 It is convenient to introduce the following combinations of the arguments. The results of the last section lead immediately to the equations (upper sign corresponds to and lower sign to From the definition of Rosenhain’s theta functions one can find the relations: Using these properties equation (66) leads to the result: For For
50 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY With one has 4. APPLICATIONS The arguments (67)–(70) of the equations (74), (73) are given by (47), (48), [(49)–(56) for or [(57)–(61) for and (63)–(65). These quantities can be calculated as shown above with the three types of integrals (43), (36)–(38), (39)–(40). These integrals themselves can be numerically evaluated very easily. If the reaches from the left or right the formulae (74) and (73) converge to the same result. The limiting procedure can be done in an analytical manner. In this region a little more attention must be paid to the numerical evaluation of the integrals. The formulae (74) and (73) are also very well suited to derive formulae for the special cases when one is in the disk or on the axis This can be found in [9]. Acknowledgments The author would like to thank Gernot Neugebauer, Reinhard Meinel and Marcus Ansorg for many discussions. The support of the DFG is gratefully acknowledged. 5. APPENDIX Jacobian theta functions
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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
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Contents ix 5 The case of 84 Part I
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Contents xi Luis O. Pimentel, Cesar
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Contents xiii 2.2 String theory ind
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PREFACE This book is dedicated to h
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CONTRIBUTING AUTHORS Eloy Ayón-Bea
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Contributing Authors xix Jaime Klap
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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
Page 112 and 113: REFERENCES 87 [18] [19] [20] [21] [
Page 114 and 115: II ALTERNATIVE THEORIES AND SCALAR
Page 116 and 117: THE FRW UNIVERSES WITH BAROTROPIC F
Page 118 and 119: 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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