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1 UNITEXT for Physics Series editors Michele Cini, Roma, Italy Attilio Ferrari, Torino, Italy Stefano Forte, Milano, Italy Guido Montagna, Pavia, Italy Oreste Nicrosini, Pavia, Italy Luca Peliti, Napoli, Italy Alberto Rotondi, Pavia, Italy Paolo Biscari, Milano, Italy Nicola Manini, Milano, Italy

2 UNITEXT for Physics series, formerly UNITEXT Collana di Fisica e Astronomia, publishes textbooks and monographs in Physics and Astronomy, mainly in English language, characterized of a didactic style and comprehensiveness. The books published in UNITEXT for Physics series are addressed to graduate and advanced graduate students, but also to scientists and researchers as important resources for their education, knowledge and teaching. More information about this series at

3 Maurizio Gasperini Theory of Gravitational Interactions Second Edition 123

4 Maurizio Gasperini Department of Physics University of Bari Bari Italy ISSN ISSN (electronic) UNITEXT for Physics ISBN ISBN (ebook) DOI / Library of Congress Control Number: Springer International Publishing AG 2013, 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

5 To my parents

6 Preface to the Second Edition The second edition of this book closely follows the spirit of the first edition, keeping all its main features. It is a book primarily intended for graduate/advanced undergraduate students with a theoretical high-energy physics or astrophysics curriculum, but it is also accessible to students with different interests. It provides the reader with all information about gravitational theory that every graduate in physics nowadays should posses: starting from the basic notions of general relativity, the classical theory of gravity is developed up to include advanced topics such as the supersymmetric and higher-dimensional extension of the Einstein equations. Since the appearance of the first edition, however, many important events in physics have occurred. The LHC experiment has produced more and more precision data confirming the discovery of the Higgs boson and may be pointing at some new, unexpected result such as the recently observed 750 GeV diphoton excess (December 2015, to be confirmed). In the field of cosmology and quantum gravity, there was the announcement (later confuted!) of the BICEP2 experiment (March 2014) about the polarization of the cosmic microwave radiation, possibly produced by primordial gravitons of inflationary origin. But, above all, we should recall the direct detection of gravitational waves from merging black holes reported by the LIGO Observatory on 11 February and 15 June All these exciting news, together with the need of revising and improving various points of the original book, have stimulated the preparation of this second edition which differs from the first one for the addition of topics of strong current interest. We have expanded, in particular, the Appendix B, in order to provide more details on the Dirichlet branes and their important gravitational application, the higher-dimensional brane-world scenario. We have added a new section to Chap. 10 for a deeper discussion of time dilatation effects in the presence of a static gravitational field, and their possible impact on the so-called twin paradox. We should mention, finally, the new sections of Chap. 9 introducing the study of the relic background of cosmic gravitational waves and discussing its possible detection, either directly (through interferometric antennas) or indirectly (through the vii

7 viii Preface to the Second Edition polarization of the cosmic electromagnetic radiation). We have also added throughout the book six new original figures. I believe that this new edition provides a more complete reference textbook, better suited to the needs of students following modern academic plans to graduate in physics. Let me thank again Marina Forlizzi and Barbara Amorese for their continuous and excellent support during the preparation of this book and all stages of the publishing process. Finally, I wish to thank in advance all those readers (students or not) who will report me about errors, inaccuracies and important omissions found in this book (as well as those readers wishing to present their personal comments/criticism). They can send an to gasperini@ba.infn.it, and they are welcomed for their useful collaboration. Cesena, Romagna July 2016 Maurizio Gasperini

8 Preface to the First Edition This book grew out of lectures given by the author at the University of Turin and at the University of Bari. It is primarily intended for undergraduate students taking classes in gravitational theory, as prescribed by modern academic plans to graduate in physics with a theoretical/high-energy physics or astrophysics curriculum. The challenge is to provide students with a textbook which, on one hand, can represent a self-contained reference for a semester cycle of lectures and, on the other hand, may be accessible and of profitable use also for students having different interests and following different academic tracks. To this aim, the book includes a first, conventional, part introducing general relativity as a geometric theory of the macroscopic gravitational field, and a second, more advanced, part connecting general relativity to the gauge theories of fundamental interactions. A discussion of the deep analogies (and of the physical differences) existing between gravity and the other standard model interactions fills a gap which is present within the traditional geometric approach to general relativity, and which usually puzzles students about the role of gravity in the context of a unified model of all interactions. In this spirit, the formalism of differential geometry has been reduced to the necessary minimum, leaving more room to current interesting aspects of gravitational physics of both applicative/observational type (such as the phenomenology of gravitational waves) and theoretical/fundamental type (such as the gravitational interactions of spinors, supergravity and higher-dimensional gravity). We have included, however, a final appendix introducing the so-called Cartan calculus of exterior (or differential) forms, in view of the important applications of this formalism not only to the gravitational theory but also to many other fields of theoretical physics. A second appendix introduces various possible approaches to the problem of embedding a four-dimensional theory of gravity in the context of a higher-dimensional space-time manifold. For most profitable use of this book, the reader is expected to have a basic knowledge of special relativity, electromagnetic theory and classical theory of fields. Except for the above input, however, the book aims at being self-contained as much as possible, following the informal style of class lectures where all the ix

9 x Preface to the First Edition required notions and techniques are explicitly recalled and/or introduced whenever necessary. Also, for a better pedagogic efficiency, all computations are explicitly carried out in the main text (leaving no voids to be filled by the readers) or presented as solved exercises at the end of each chapter. The present book is certainly not intended to represent a complete reference for a rigorous and comprehensive study of all theoretical aspects of the gravitational interaction. Its main purpose is to provide students with the basic starting notions, enabling them to do further independent work and subsequent deeper studies on more professional textbooks and papers. The readers interested in advanced discussions of some specific topic are strongly advised to refer to the list of specialistic books presented in the bibliography. Finally, it should be noted that this book deliberately avoids any gravitational application to cosmology and large-scale astrophysics, because according to modern academic plans of studies they are a matter of specific courses and lectures, well separated from a course on the theory gravity. The field of relativistic cosmology is today so extended, with so many branches and applications, as to deserve by itself a dedicated book. We refer, for this purpose, to the excellent books quoted in the bibliography, as well as to an introduction to theoretical cosmology which represents the natural continuation of this book and which currently exists as a Springer Italian edition. 1 Acknowledgements It is a pleasure, as well as a duty, to thank all my colleagues and students for their comments, suggestions and criticism that contributed over the years to correct and improve these lecture notes. Listing all of them would be an impossible task, so let me thank them jointly for their important help. A warm acknowledgement is also due to Venzo De Sabbata, who was one of my Professors when (many years ago!) I was a student of physics at the University of Bologna. Professor De Sabbata introduced me to the study of gravitation and cosmology, and the interest he was able to stimulate towards those branches of physics was so intense as to be still alive, and fully effective, even today in my present scientific activity. Finally, I wish to thank Marina Forlizzi, Executive Editor, Springer-Verlag, for her kind encouragement, advice and many useful suggestions. Cesena, Romagna Maurizio Gasperini 1 M. Gasperini, Lezioni di Cosmologia Teorica (Springer-Verlag, Milano, 2012).

10 Contents 1 Elementary Notions of Relativistic Field Theory Symmetries and Conservation Laws Global Translations and Canonical Energy-Momentum Tensor A Comment on the Non-uniqueness of the Definition Lorentz Transformations and Canonical Angular Momentum Symmetrization of the Energy-Momentum Tensor Examples of Energy-Momentum Tensor Scalar Field Electromagnetic Field Point-Like Particle Perfect Fluid Exercises Chap Solutions Towards a Relativistic Theory of Gravity The Postulates of the Riemannian Geometry The Equivalence Principle Exercises Chap Solutions Tensor Calculus in a Riemann Manifold Covariant and Contravariant Tensors Tensor Densities Infinitesimal Transformations, Isometries and Killing Vectors Second-Order Infinitesimal Transformations Covariant Derivative and Affine Connection Autoparallel Curves Torsion, Non-metricity and Christoffel Symbols xi

11 xii Contents 3.6 Useful Rules of Covariant Differentiation Trace of the Christoffel Connection Covariant Derivatives of Tensor Densities Covariant Divergence and d Alembert Operator Exercises Chap Solutions Maxwell Equations and Riemann Geometry The Minimal Coupling Principle Coupling Geometry to Electromagnetic Fields The Generalized Maxwell Equations Analogy with the Equations in an Optical Medium Exercises Chap Solutions Test Bodies and Signals in a Riemann Space Time Geodesic Motion of Free Particles The Newtonian Limit Time Dilatation and Frequency Shifts The Frequency Shift in a Newtonian Field Exercises Chap Solutions Geodesic Deviation and Curvature Tensor The Equation of Geodesic Deviation The Riemann Curvature Tensor A Simple Example: Constant-Curvature Manifolds Exercises Chap Solutions The Einstein Equations for the Gravitational Field Gravitational Action and Field Equations Boundary Contributions Contribution of the Matter Sources Einstein Equations The Dynamical Energy-Momentum Tensor Examples: Scalar and Vector Fields, Point-Like Sources The Einstein Equations with a Cosmological Constant Energy-Momentum Conservation and Motion of Test Bodies Exercises Chap Solutions The Weak-Field Approximation Linearized Einstein Equations The Harmonic Gauge

12 Contents xiii 8.2 Space Time Metric for a Weak and Static Field The Bending of Light Rays The Radar-Echo Delay Velocity Measurements in the Presence of Gravity Exercises Chap Solutions Gravitational Waves Propagation of Metric Fluctuations in Vacuum Polarization and Helicity States Radiation Emission in the Quadrupole Approximation Gravitational Field in the Radiation Zone Energy-Momentum Tensor of a Gravitational Wave Radiated Power Example: A Binary Star System The Interaction of Monochromatic Waves with Massive Bodies The Damped Oscillator as a Gravitational Detector The Presently Operating Detectors The Relic Background of Cosmic Gravitational Waves The Polarization of the CMB Radiation Exercises Chap Solutions The Schwarzschild Solution Spherically Symmetric Einstein Equations in Vacuum The Birkhoff Theorem and the Schwarzschild Solution The Weak-Field Limit Perihelion Precession Event Horizon and Kruskal Coordinates Causal Structure of the Black Hole Geometry Proper Time for Moving Observers in a Static Field Exercises Chap Solutions The Kasner Solution Einstein Equations for Homogeneous and Anisotropic Metrics Higher-Dimensional Solutions in Vacuum Exercises Chap Solutions Vierbeins and Lorentz Connection Projection on the Flat Tangent Space Local Symmetries and Gauge Fields

13 xiv Contents 12.2 Local Lorentz Invariance and Covariant Derivative The Metricity Condition for the Vierbeins The Levi-Civita Connection and the Ricci Rotation Coefficients The Curvature Tensor and the Gravitational Action Exercises Chap Solutions The Dirac Equation in a Gravitational Field A Concise Summary of the Spinor Formalism A Covariant and Locally Lorentz-Invariant Dirac Equation Geometric Couplings to the Vector and Axial-Vector Currents Symmetrized Form of the Covariant Dirac Action Exercises Chap Solutions Supersymmetry and Supergravity Global Supersymmetry in Flat Space Time Example: The Wess Zumino Model The Rarita Schwinger Field Global Supersymmetry in the Graviton-Gravitino System N ¼ 1 Supergravity In D ¼ 4 Dimensions Field Equations for the Metric and the Gravitino Exercises Chap Solutions Appendix A: The Language of Differential Forms Appendix B: Higher-Dimensional Gravity References Index

14 Notations, Units and Conventions Throughout this book we will use the index 0 for the time-like components of vector and tensor objects, while the indices 1, 2, 3 will refer to the space-like components. For the space-time metric g we will adopt the signature with a positive time-like eigenvalue, namely: g ¼ diagðþ ; ; ; Þ: Our conventions for the curvature and the covariant derivatives will be as follows. Riemann tensor: R fi fl ¼ o Γ fi fl þ Γ fl Γ fi f $ g; where the symbol f $ g means that we must insert all the preceding terms with the indices and interchanged between themselves. Ricci tensor: Covariant derivative: R fi ¼ R fi : r V fi ¼ o V fi þ Γ fl fi V fl ; r V fi ¼ o V fi Γ fi fl V fl ; Lorentz covariant derivative: D V a ¼ o V a þ ω a b V b ; D V a ¼ o V a ω b a V b : Also, the symbol h will denote the usual d Alembert operator of the flat Minkowski space time, that is, h ¼ o o ¼ 1 o 2 c 2 ot 2 r2 ; xv

15 xvi Notations, Units and Conventions where is the Minkowski metric and r 2 ¼ ij o i o j is the Laplacian operator of the three-dimensional Euclidean space. Unless otherwise stated, we will use small Latin letters i; j; k;... for the spatial indices 1; 2; 3; small Greek letters ; ; fi;...for the space-time indices 0; 1; 2; 3. In higher-dimensional space-times, with a number d [ 3 of spatial dimensions, the space-time indices will be denoted instead by capital Latin letters: A; B; C;...¼ 0; 1; 2; 3;...; d. Two (or more) indices, when enclosed in round or square brackets, will satisfy, respectively, the symmetry or antisymmetry property defined by: T ðfiflþ 1 2 T fifl þ T flfi ; T½fifl 1 2 T fifl T flfi : In the presence of more than two indices, two symmetric (or antisymmetric) indices which are not contiguous will be separated from the others by a vertical bar. We write, for instance: T ð jfij Þ 1 2 T fi þ T fi ; T ½ jfij fl 1 2 T fi fl T fi fl ; to mean that the first object is symmetric with respect to and (at fixed fi), while the second object is antisymmetric with respect to and (at fixed fi and fl) and so on for a higher number of indices. Such a symmetrization/antisymmetrization procedure can be easily extended to an arbitrary number of indices n 2, by including all their possible permutations and then dividing by the total number of permutations, n!. For the symmetrization procedure, all permutations are to be added with the same sign, while, for the antisymmetrization procedure, even permutations are to be added with the plus sign and odd permutations with the minus sign. For instance: T ð fiþ ¼ 1 T fi þ T fi þ T fi þ T fi þ T fi þ T fi ; 3! T ½ fi ¼ 1 T fi þ T fi þ T fi T fi T fi T fi : 3! And so on for a higher number of indices. Finally, the fully antisymmetric tensor (also called the Levi-Civita symbol) of the Minkowski space-time, fifl, will be defined according to the following conventions: 0123 ¼þ1; fifl ¼ fifl : Its generalization to the case of an arbitrary Riemann manifold will be introduced in Chap. 3, Sect. 3.2.

16 Notations, Units and Conventions xvii The units that will be adopted for our numerical estimates and for the electromagnetic variables are the so-called (unrationalized) c.g.s units, where the Maxwell equations take the form: o F ¼ 4 c J ; F ¼ o A o A ; A ¼ ð`; AÞ: When dealing with scalar and spinor fields, we will instead refer to the more convenient natural units, where both the light velocity c and the Planck constant h are set to one, c ¼ h ¼ 1. In such units, the Newton constant G acquires length-squared (or inverse mass-squared) dimensions and is related to the Planck mass M P and to the Planck length P by: In c.g.s units: ð8 GÞ 1 ¼ M 2 P ¼ 2 P : M P ¼ hc 1=2 0: g; 8 G P ¼ 8 G h 1=2 c cm: The energy associated with the Planck mass is E P ¼ M P c 2 2: GeV, where 1 Gev = 10 9 ev is the energy scale associated with the mass of the proton. As evident from the above definitions, the Planck energy scale characterizes the strength of the gravitational coupling and also controls the importance of the quantum corrections to the equations of the classical gravitational theory. Other special notations, different from the ones reported here, will be introduced directly where needed (for instance, in Chap. 12 for the so-called tangent space formalism, in Chap. 14 for models of supersymmetry and supergravity, in Appendix A for the calculus of differential forms and in Appendix B for models of higher-dimensional gravity).