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SELF–GRAVITATING STATIONARY AXI- SYMMETRIC PERFECT FLUIDS: DIFFERENTIAL ROTATION AND SOME GEOMETRIC FEATURES F.J. Chinea* Dept. de Física Teórica II, Ciencias Físicas, Universidad Complutense de Madrid E-28040 Madrid, Spain Keywords: Perfect fluids, differential rotations, Newtonian configurations. 1. NEWTONIAN CONFIGURATIONS The equations for the interior of a self-gravitating barotropic perfect fluid in the classical (Newtonian) theory in an inertial frame are the following: where is the mass density, the pressure, the velocity field of the fluid, and U the gravitational potential; all these functions may depend in principle on the space coordinates, and on time G is the gravitational constant (in the following, G = 1). Equation (1) is the continuity equation, (2) is the Euler equation for the fluid, (3) the Poisson equation for the gravitational potential U, and (4) the barotropic equation of state. Some closed-form solutions of fluid configurations * E–mail: chinea@eucmos.sim.ucm.es Exact Solutions and Scalar Fields in Gravity: Recent Developments Edited by Macias et al., Kluwer Academic/Plenum Publishers, New York, 2001 3
4 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY with a compact boundary are the Maclaurin ellipsoids in the stationary axisymmetric case, and the Jacobi, Dedekind, and Riemann ellipsoids in the (in general) non-axisymmetric case [1, 2]. It is remarkable that the Jacobi ellipsoids, triaxial figures rotating around a principal axis as a whole (rigidly) with constant angular velocity, can be considered formally as stationary, due to the fact that the equations (1)-(3) can be transformed to the rotating frame anchored to the principal axes so that does not appear explicitly; obviously, the analog of such nonaxisymmetric configurations in general relativity (whatever they might be) would not be stationary, as they would emit gravitational radiation. Another interesting stationary axisymmetric configuration is a torus-like figure rotating around its symmetry axis. It was shown to exist (among other highly interesting results) in a very important paper of Poincaré [3], and was further analyzed in [4] by means of power series expansions. In what follows, we shall concentrate on the stationary, axisymmetric case with azimuthal flow of equations (1)-(4); the is taken to be the symmetry axis. By imposing and (where denotes the Lie derivative in the direction of the vectorfield and being the azimuthal angle), one finds where is the angular velocity, depending in general on the position within the fluid (possible differential rotation). Taking this into account, one has and From these and the integrability conditions for the set of three scalar equations arising from (2), one gets (taking into account that and are functionally dependent, according to (4)), so that the angular velocity depends only on the distance to the rotation axis: where we have defined It is easy to see that the continuity equation (1) is automatically satisfied under the assumed symmetries (which also require and, in particular, the fluid flow has vanishing expansion. Equations (l)-(3) now reduce to the following ones:
Page 2 and 3: EXACT SOLUTIONS AND SCALAR FIELDS I
Page 4 and 5: EXACT SOLUTIONS AND SCALAR FIELDS I
Page 6 and 7: To the 65th birthday of Heinz Dehne
Page 8 and 9: 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 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
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THE SUPERPOSITION OF NULL DUST BEAM
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The superposition of null dustbeams
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REFERENCES 61 geodesic with respect
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SOLVING EQUILIBRIUM PROBLEM FOR THE
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Solving equilibrium problem for the
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REFERENCES 67 where the subindices
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ROTATING EQUILIBRIUM CONFIGURATIONS
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Rotating equilibrium configurations
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Rotating equilibrium configurations
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REFERENCES 75 5. DISCUSSION The ana
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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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EXACT SOLUTIONS AND SCALAR FIELDS I
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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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