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REFERENCES 269 The boundary is constituted at its interior by the event horizon and at the infinity by the asymptotic regions. Since the potential U is constant over each one of the connected components of these boundaries, it can be pulled out from each one of the corresponding boundary integrals in the left hand side of (14). The asymptotic regions and the connected components of the horizon are all topological 2–spheres [8, 25], by this reason the left hand side of (14) vanishes; from Stokes theorem, the integral of an exact form over a manifold without boundary is zero. Thereby, it is satisfied that The integrand in (15) is non–negative, due to the fact that is a spacelike 1–form since it is orthogonal by definition to the timelike field (4). Hence, it follows that (15) is satisfied if and only if the integrand vanishes in and by stationarity it also vanishes in all the domain of outer communications consequently From the previous conclusion (16), it follows that in but by the continuity of in all of and in particular through the lower dimensional surfaces that constitute the regions vanishes also in and accordingly in all the domain of outer communications Hence, the staticity theorem is proved. 3. CONCLUSIONS Finally, it is concluded that for a non–rotating strictly stationary black hole with a self–interacting scalar field non–minimally coupled to gravity, the corresponding domain of outer communications is static if analytic field configurations are considered. As in the minimal case [15] this result remains valid when no horizon is present. Acknowledgments The author thanks Alberto García by its incentive in the study of this topics and by its support in the course of the investigation. This research was partially supported by the CONACyT Grant 32138E. The author also thanks all the encouragement and guide provided by his recently late father: Erasmo Ayón Alayo, Ibae Ibae Ibayen Torun.
270 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY M. Heusler, Black Hole Uniqueness Theorems (Cambridge Univ. Press, Cambridge 1996). J.D. Bekenstein, Proceedings of Second Sakharov Conference in Physics, Moscow, eds. I.M. Dremin, A.M. Semikhatov, (World Scientific, Singapore 1997) pp. 761. M. Heusler, Living Rev. Rel., 1, (1998) 6, http://www.livingreviews.org/Articles/Volume1/1998-6heusler. J.D. Bekenstein, Proceedings of the 9th Brazilian School of Cosmology and Gravitation, Rio de Janeiro, Brazil, gr-qc/9808028 (1998). B. Carter, Proceedings of 8th Marcel Grossmann Meeting, Jerusalem, ed. T. Piran (World Scientific, Singapore 1999). A. Lichnerowicz, Théories Relativistes de la Gravitation et de L’Électromagnétisme (Masson & Cie, Paris 1955). S.W. Hawking, Commun. Math. Phys., 25 (1972) 152. S.W. Hawking and G. F. Ellis, The Large Scale Structure of Space–Time (Cambridge Univ. Press 1973). B. Carter, in: Gravitation in Astrophysics (Cargèse Summer School 1986), eds. B. Carter and J.B. Hartle (Plenum, New York 1987). D. Sudarsky and R.M. Wald, Phys. Rev., D46 (1992) 1453. D. Sudarsky and R.M. Wald, Phys. Rev., D47 (1993) R3209. R.M. Wald, in: Proceedings: Directions In General Relativity, eds. B.L. Hu, M.P. Ryan, Jr., C.V. Vishveshwara, T.A. Jacobson (New York, Cambridge Univ. Press, 1993). and R.M. Wald, Commun. Math. Phys., 163 (1994) 561. M. Rogatko, Phys. Rev., D58 (1998) 044011. M. Heusler, Class. Quant. Grav., 10 (1993) 791. M. Heusler and N. Straumann, Class. Quant. Grav., 10 (1993) 1299. H. Müller zum Hagen, Proc. Camb. Phil. Soc., 67 (1970) 415. T. Zannias, J. Math. Phys., 39 (1998) 6651. H. Friedrich, I. Racz, and R.M. Wald, Commun. Math. Phys., 204 (1999) 691. T. Zannias, J. Math. Phys., 36 (1995) 6970. A. Saa, J. Math. Phys., 37 (1996) 2346. Mayo, A.E., Bekenstein, J.D., Phys. Rev., D54 (1996) 5059. A. Saa, Phys. Rev., D53 (1996) 7377. I. Racz and R.M. Wald, Class. Quant. Grav., 13 (1996) 539. and R.M. Wald, Class. Quant. Grav., 11 (1994) L147. H. Müller zum Hagen and D.C. Robinson, and H.J. Seifert, Gen. Rel. Grav., 4 (1973) 53.
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EXACT SOLUTIONS AND SCALAR FIELDS I
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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
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Contributing Authors xxiii L. Artur
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I EXACT SOLUTIONS
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SELF-GRAVITATING STATIONARY AXI- SY
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Self-gravitating stationary axisymm
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REFERENCES 13 be shown that the ana
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NEW DIRECTIONS FOR THE NEW MILLENNI
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New Directions for the New Millenni
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New Directions for the New Millenni
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REFERENCES 21 Acknowledgments The r
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ON MAXIMALLY SYMMETRIC AND TOTALLY
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On maximally symmetric and totally
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DISCUSSION OF THE THETA FORMULA FOR
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Theta formula for the Ernst potenti
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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
Page 244 and 245: EXACT SOLUTIONS AND SCALAR FIELDS I
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Page 248 and 249: ADAPTIVE CALCULATION OF A COLLAPSIN
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Page 260 and 261: REVISITING THE CALCULATION OF INFLA
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Page 272 and 273: CONFORMAL SYMMETRY AND DEFLATIONARY
Page 274 and 275: Conformal symmetry and deflationary
Page 284 and 285: REFERENCES 259 [16] R. Maartens, D.
Page 286 and 287: IV EXPERIMENTS AND OTHER TOPICS
Page 288 and 289: STATICITY THEOREM FOR NON-ROTATING
Page 290 and 291: Staticity Theorem for Non-Rotating
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Page 296 and 297: QUANTUM NONDEMOLITION MEASUREMENTS
Page 298 and 299: Quantum nondemolition measurements
Page 304 and 305: REFERENCES 279 [11] M. Kasevich and
Page 306 and 307: ON ELECTROMAGNETIC THIRRING PROBLEM
Page 308 and 309: On Electromagnetic Thirring Problem
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Page 320 and 321: ON THE EXPERIMENTAL FOUNDATION OF M
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Page 336 and 337: 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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