Book contents
- Frontmatter
- Dedication
- Contents
- Preface to the second edition
- Preface to the first edition
- Part I Introduction to gravity and supergravity
- Part II Gravitating Point-Particles
- 11 The Schwarzschild Black Hole
- 12 The Reissner–Nordström black hole
- 13 The Taub–NUT solution
- 14 Gravitational pp-waves
- 15 The Kaluza–Klein black hole
- 16 Dilaton and dilaton/axion black holes
- 17 Unbroken supersymmetry I: supersymmetric vacua
- 18 Unbroken supersymmetry II: partially supersymmetric solutions
- 19 Supersymmetric black holes from supergravity
- Part III Gravitating extended objects of string theory
- Appendix A Lie groups, symmetric spaces, and Yang–Mills fields
- Appendix B The irreducible, non-symmetric Riemannian spaces of special holonomy
- Appendix C Miscellanea on the symplectic group
- Appendix D Gamma matrices and spinors
- Appendix E Kähler geometry
- Appendix F Special Kähler geometry
- Appendix G Quaternionic-Kähler geometry
- Appendix H Real special geometry
- Appendix I The generic scalar manifolds of N ≥ 2, d = 4 SUEGRAs
- Appendix J Gauging isometries of non-linear σ-models
- Appendix K n-spheres
- Appendix L Palatini's identity
- Appendix M Conformal rescalings
- Appendix N Connections and curvature components
- Appendix O The harmonic operator on ℝ3 × S1
- References
- Index
15 - The Kaluza–Klein black hole
from Part II - Gravitating Point-Particles
Published online by Cambridge University Press: 05 April 2015
- Frontmatter
- Dedication
- Contents
- Preface to the second edition
- Preface to the first edition
- Part I Introduction to gravity and supergravity
- Part II Gravitating Point-Particles
- 11 The Schwarzschild Black Hole
- 12 The Reissner–Nordström black hole
- 13 The Taub–NUT solution
- 14 Gravitational pp-waves
- 15 The Kaluza–Klein black hole
- 16 Dilaton and dilaton/axion black holes
- 17 Unbroken supersymmetry I: supersymmetric vacua
- 18 Unbroken supersymmetry II: partially supersymmetric solutions
- 19 Supersymmetric black holes from supergravity
- Part III Gravitating extended objects of string theory
- Appendix A Lie groups, symmetric spaces, and Yang–Mills fields
- Appendix B The irreducible, non-symmetric Riemannian spaces of special holonomy
- Appendix C Miscellanea on the symplectic group
- Appendix D Gamma matrices and spinors
- Appendix E Kähler geometry
- Appendix F Special Kähler geometry
- Appendix G Quaternionic-Kähler geometry
- Appendix H Real special geometry
- Appendix I The generic scalar manifolds of N ≥ 2, d = 4 SUEGRAs
- Appendix J Gauging isometries of non-linear σ-models
- Appendix K n-spheres
- Appendix L Palatini's identity
- Appendix M Conformal rescalings
- Appendix N Connections and curvature components
- Appendix O The harmonic operator on ℝ3 × S1
- References
- Index
Summary
Kaluza's [848] and Nordström's [994] original idea/observation that electromagnetism could be seen as part of five-dimensional gravity, combined with Klein's curling up of the fifth dimension in a tiny circle [862], constitutes one of the most fascinating and recurring themes of modern physics. Kaluza–Klein (KK) theories are interesting both in their own right (in spite of their failure to produce realistic four-dimensional theories [1274], at least when the internal space is a manifold) and because of the usefulness of the techniques of dimensional reduction for treating problems in which the dynamics in one or several directions is irrelevant. We saw an example in Chapter 9, when we related four-dimensional instantons to monopoles.
On the other hand, the effective field theories of some superstring theories (which are supergravity theories) can be obtained by dimensional reduction of 11-dimensional supergravity, which is the low-energy effective field theory of (there is no real consensus on this point) M theory or one of its dual versions. In turn, string theory needs to be “compactified” to take a four-dimensional form and, to obtain the four-dimensional low-energy effective actions, one can apply the dimensional-reduction techniques.
Here we want to give a simple overview of the physics of compact dimensions and the techniques used to deal with them (dimensional reduction, etc.) in a non-stringy context. We will deal only with the compactification of pure gravity and vector fields, leaving aside compactification in the presence of more general matter fields (including fermions) until Part III. We will also leave aside many subjects such as spontaneous compactification and the issue of constructing realistic KK theories, which are covered elsewhere [476, 1271]. In addition to establishing the basic results, we want to study classical solutions of the original and dimensionally reduced theories and to see how KK techniques can be used to generate new solutions of both of them.
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- Gravity and Strings , pp. 402 - 463Publisher: Cambridge University PressPrint publication year: 2015