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Self–gravitating stationary axisymmetric perfect fluids 9 subspace, a 1-form which reduces in vacuum to ( being the Weyl canonical coordinate), and the definition (extended by linearity to any 1-form), the full set of equations to be considered is the following: This equation corresponds to (2.62) of reference [9]. Please note that there is a typographical error in the last sign of that equation. Equations (19)-(22) are the first structural equations of Cartan, (23)- (26) their integrability conditions (first Bianchi identities), (27)-(32) the field equations (in particular, (28) is the Raychaudhuri equation), and (33) the contracted second Bianchi identity (Euler equation for the fluid). While and are fixed, there is a remaining gauge freedom of spatial rotations in the subspace. Under such rotations, however, and remain invariant. The operation is not intrinsically defined, and gauge rotations change it, but the equations remain invariant in form as a set. Fluids with a barotropic equation of state are obviously characterized by the condition by using (23) and the exterior derivative of (33), it is easy to see that (disregarding the degenerate case ) is closed, and and are collinear:
10 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY The irrotational solution presented in [8] was found by imposing the Ansatz A word is in order concerning that choice. It was inspired by the fact that, for a Newtonian fluid rotating rigidly, one has (with the appropriate definitions of and in the underlying Euclidean metric); it seemed then natural to impose (34) as a simplifying assumption in the irrotational case, with the role of taken over by Although this Ansatz did work as an aid in finding the solution, we should bear in mind, however, that is satisfied due to the fact that in the Newtonian case the vorticity lines are straight lines, whereas in a relativistic fluid one would expect a dynamical (multipole-like) distribution for or In order to arrive at the solution in [8], a further Ansatz of a more mathematical nature was imposed; the end result was that all the quantities of the solution, including the metric, can be written in terms of a single function ( means here a coordinate!), which satisfies the second-order ordinary differential equation where is a constant, and subindices denote ordinary derivatives. Equation (35) can be transformed to a first-order one, which turns out to be an Abel equation of the first kind. The (non-constant) angular velocity distribution for the rotating fluid is given by where and are constants. The equation of state is that of stiff matter, and the solution is of Petrov type I. The surface area of the body is finite. The solution admits no further Killing vectors. A few other differentially rotating interior solutions have subsequently been found [10]. Needless to say, none of the differentially rotating interior solutions with no extra Killing fields has yet been shown to admit a matching asymptotically flat exterior vacuum field, a fact similar to that mentioned above for rigidly rotating solutions. 4. VACUUM AS A “PERFECT FLUID”. THE “RIGID ROTATION” AND THE “IRROTATIONAL” GAUGES Equations (19)-(32) still hold when (and (33) becomes a trivial identity.) One important difference, however, is the following.
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 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 82 and 83: The superposition of null dustbeams
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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
Page 352 and 353:
Prof. Dietrich Kramer: Brief Biogra
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