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Higgs–Field and Gravity 103 is defined by and the non–trivial Higgs–field condition which we resolve as with the isospinorial unit vector N: The general Higgs–field is different form (11) by a local unitary transformation: with where represents the real valued excited Higgs-field normalized with respect to the ground-state value Now we use the possibility of a unitary gauge transformation which is inverse to (13), so that Goldstone– bosons are avoided and (13) goes over in In the following all calculations are performed in the gauge (15) (unitary gauge). Using (11) up to (15) the field equations (3) through (5) take the form wherein is the mass–matrix of the matter-fields
104 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY the symmetric matrix of the mass-square of the gauge-fields and the square of the mass of the excited Higgs–field corresponding to the Higgs–particle. The diagonalization procedure of (20) is not given here explicitly but results in the usual Weinberg–mixing with the masses and of the electro–weak gauge bosons gauge coupling constants belonging to the group U(1) × SU(2)). Obviously in the field equations (16) up to (18) the Higgs–field plays the role of an attractive scalar gravitational potential between the massive particles [1]: According to (18) the source of is the mass of the fermions and of the gauge–bosons. (The second term in the bracket on the right hand side of Eq.(18) is positive with respect to the signature of the metric.), whereby this equation linearized with respect to results in a potential equation of Yukawa–type. Accordingly the potential has finite range given by the mass of the Higgs-particle and has the meaning of a gravitational constant, so that is valid, where G is the Newtonian gravitational constant and a dimensionless factor, which compares the strength of the Newtonian gravity with that of the Higgs–field. On the other hand, the gravitational potential acts back on the mass of the fermions and of the gauge–bosons according to the field equations (16) and (17). Simultaneously the equivalence between inertial and passive as well as active gravitational mass is guaranteed. This feature results from the fact that by the symmetry breaking only one type of mass is introduced. We mention that this Higgs–gravity fulfils an old idea of Einstein known as ”Mach’s principle of relativity of inertia” [2], according to which the masses should be produced by the interaction with a gravitational field. Finally the factor in (23) can be estimated as follows: Taking into consideration the unified theory of electroweak interaction the value of is correlated according to (20) with the mass of the according to Combination with (23) results in
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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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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
Page 124 and 125: REFERENCES 99 (1979) 515; H. Dehnen
Page 126 and 127: HIGGS-FIELD AND GRAVITY Heinz Dehne
Page 130 and 131: Higgs-Field and Gravity 105 Consequ
Page 132 and 133: Higgs-Field and Gravity 107 From th
Page 134 and 135: REFERENCES 109 Evidently by inserti
Page 136 and 137: THE ROAD TO GRAVITATIONAL S-DUALITY
Page 138 and 139: The road to Gravitational S-duality
Page 146 and 147: REFERENCES 121 The partition functi
Page 148 and 149: EXACT SOLUTIONS IN MULTIDIMENSIONAL
Page 150 and 151: Exact solutions in multidimensional
Page 156 and 157: REFERENCES 131 For possible physica
Page 158 and 159: EFFECTIVE FOUR-DIMENSIONAL DILATON
Page 160 and 161: Dilaton gravity from Chern-Simons g
Page 166 and 167: A PLANE-FRONTED WAVE SOLUTION IN ME
Page 168 and 169: A plane-fronted wave solution in me
Page 176 and 177: 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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