Antonio Romano • Mario Mango Furnari

The Physicaland MathematicalFoundations of the Theoryof RelativityA Critical Analysis

Antonio RomanoDipartimento di Matematica eApplicazioni “Renato Caccioppoli”Università degli Studi di NapoliFederico IINaples, Italy

Mario Mango FurnariIstituto di CiberneticaNaples, Italy

ISBN 978-3-030-27236-4 ISBN 978-3-030-27237-1 (eBook)https://doi.org/10.1007/978-3-030-27237-1

Mathematics Subject Classification (2010): 83AXX, 83BXX, 83CXX, 83FXX

© Springer Nature Switzerland AG 2019, corrected publication 2019This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof 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 transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe 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 thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, expressed or implied, with respect to the material containedherein or for any errors or omissions that may have been made. The publisher remains neutral with regardto jurisdictional claims in published maps and institutional affiliations.

This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registeredcompany Springer Nature Switzerland AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Writing a book on relativity today is equivalent to adding a grain of sand to a verylarge beach. In fact, the list of books about special and general relativity is veryextensive, and the reader can find many introductory and specialist books. Amongthe former, there are excellent texts that allow the reader to understand the prin-ciples of relativity, whereas in books of the second category, it is possible to deepenall the advanced topics of relativity (Cauchy problem, geometric properties ofspacetime in the large, cosmological models, star structure, etc.). Finally, there arebooks devoted to the most recent developments of the theory. Why, then, to addanother text to so extensive a list of books? Why add another grain of sand to sowide a beach?

We now make clear the factors that pushed us to write this introductory text-book. First, it is our rooted opinion that the reader should understand which clas-sical concepts he or she is leaving behind in following the path that will lead to theacceptance of the new ideas of relativity. In some cases, accepting the new ideascan be painful and confusing. With the aim of relieving this tiresome learningprocess, this book begins with two chapters in which the classical ideas that will beabandoned or saved are recalled in detail. Therefore, in these chapters the classicalprocedure to measure space and time, the basic ideas of classical dynamics, theGalilean relativity principle, and some fundamental results of Newtonian gravita-tion are recalled.

After introducing special relativity following the physical approach proposed byEinstein (Chap. 7), the Minkowski mathematical model is discussed (Chap. 8). Inthis regard, we recall that a model can be considered a mathematical transcription ofa physical reality if and only if the measurable physical quantities are uniquelyassociated with mathematical objects of the model. In Chap. 8, after defining thiscorrespondence for Minkowski’s model, all the characteristics and particularities ofthis correspondence are widely analyzed with the aim of putting in evidence thedeep differences existing between the correspondence that in special relativity allowone to attribute physical meaning to Minkowski’s model and the correspondenceproposed by Einstein in general relativity between the hyperbolic Riemannianspacetime and physical reality.

v

Chapters 9 and 10 are devoted to topics that are usually presented hastily; therelativistic thermodynamics of continua and the electromagnetic fields in matter. Inthese two classes of phenomena we face several ambiguities due to the fact thatmany quantities that are necessary to formulate these theories cannot be experi-mentally observed. This circumstance justifies the many proposals that can be foundin the literature about the momentum–energy tensor employed in thermodynamicsand in the electrodynamics of continua. In particular, we prove the macroscopicequivalence of the all transformation formulas of temperature and heat that havebeen proposed as well as the equivalence of all the models proposed to describe theinteraction between matter and electromagnetic fields. More precisely, we show thatall the models proposed in electrodynamics in matter are obtained by a suitablechoice of the variables we adopt to describe the electromagnetic field. In otherwords, they have no physical consistency but are useful mathematical models forevaluating observable macroscopic quantities.

In Chap. 11 we face the complex problem of analyzing the basic principles ofgeneral relativity with the awareness that there is no agreement on their formulation.One of the most controversial principles is that of general relativity. It should be thegeneralization of the special principle of relativity, since it states that all observershave the right to study nature. However, it is not explained what constitutes anarbitrary observer, how one measures physical quantities, and mostly how suchmeasurements are related to those of another observer. It is important to underlinethat only if this connection is made explicit can the observers realize a universalphysics, where “universal” has to be understood as the collection of the observers’descriptions together with the possibility of comparing them. A thoroughgoinganalysis of this problem can be found in Chap. 11.

Often, the general principle of relativity is identified with the mathematicalprinciple of general covariance. This principle states that physical laws must beformulated in a form independent of the coordinates adopted in the hyperbolicRiemannian spacetime that Einstein substitutes for Minkowski’s model. Thisconclusion is often justified by stating that there are coordinates in spacetime thatcan be considered a mathematical representation of physical frames of referenceand vice versa. Consequently, the covariance of physical laws represents themathematical version of the general principle of relativity. In Chap. 11, thisstatement is proved to be untrue.

Another basic assumption of general relativity is the equivalence principle,according to which the effects of any gravitational field on physical phenomena canbe eliminated in small spacetime regions. The local frames of reference in whichthat happens are called local inertial frames. This principle, together with theassumption that in these regions special relativity holds, allows the introduction of afirst partial correspondence between physical reality and geometric objects of thespacetime manifold V4. In fact, the geodesic coordinates in an arbitrary point of V4

are intended to be the mathematical representation of a local inertial frame.The absence of gravity in the local inertial frames stated by the equivalence

principle and the identification of these frames of reference with the geodeticcoordinates in which the metric assumes the Minkowski form pushed Einstein to

vi Preface

describe the gravitational field with the metric of a Riemannian manifold that in turnis related to the matter and energy occupying a region of spacetime. Then the mainproblem Einstein had to solve was to determine the equations relating the metriccoefficients to the distribution of matter and energy. Starting from reasonablehypotheses, which are discussed in Chap. 11, Einstein determined a system of 10nonlinear partial differential equations of the form Glm ¼ �vTlm, where the tensorGlm involves only the spacetime geometry, Tlm is the momentum–energy tensorsatisfying the conservation laws rmTlm ¼ 0, and v is an unknown constant.

In order to determine the constant v, Einstein resorted to a linear approximationof the field equations obtained on the assumption of weak gravitational fields andnonrelativistic velocities (see Chap. 12). The linear equations, in the staticapproximation, reduce to a single equation that is formally identical to Poisson’sequation, provided that v ¼ 8ph=c4. It should be noted that the identification ofEinstein’s equations and Poisson’s equation is purely formal, since the interpreta-tion of the gravitational potential in the two equations is completely different.

Furthermore, in the linear nonstatic case, every metric coefficient satisfiesd’Alemebert’s equation, so that the existence of gravitational waves is foreseen andthe gravitational potentials are obtained by retarded potentials (Chap. 12).

In Chap. 13, the hyperbolic character of Einstein’s equations is verified, andsome existence theorems under reasonable regularity spacetime conditions areproved for the exterior Cauchy problem.

It is very important to highlight a deep change of perspective in going fromspecial relativity to general relativity. In fact, in the former theory, the proceduresthat allow inertial observers to measure space and time are formulated before anyphysical law is determined. Furthermore, these procedures allow one to identify inMinkowski’s spacetime three-dimensional spaces and one-dimensional spaces thatare the geometric representations of space and time relative to an observer. Ingeneral relativity, the metric of spacetime is dynamic, i.e., it is determined by theevolution of matter and energy through the field equations. This means that wecannot speak about local or global space and time before solving the Einsteinequations and the conservation laws. In other words, before knowing the metric, wecannot speak about space, time, geodesics, etc. In particular, the definitions of localand global space and time will depend on the form of the metric.

To put in evidence another particularity of general relativity, we recall that theequivalence principle introduces a partial correspondence between local inertialframes and geodesic coordinates at points of the spacetime V4. This means thatcoordinates ðxaÞ must be defined on an open set U of V4 to which it is possible toassociate a frame of reference for an observer O. In particular, experimental pro-cedures must be defined that allow O to evaluate the coordinate ðxaÞ of an eventbelonging to a set E. After defining this one-to-one correspondence between pointsof U and physical events of E, we can also adopt in U arbitrary coordinates ðx0aÞ,provided that the coordinate transformation x0a ¼ x0aðxbÞ is known.

Preface vii

This is the approach we follow in Chaps. 14–16. Specifically, in Chap. 14, themodel of spacetime proposed by Schwarzschild is presented. This model describesthe relativistic gravitational field produced by a spherically symmetric mass dis-tribution S inside and outside S. In the coordinates introduced to solve separatelythe interior and exterior Einstein equations, the matching conditions on the gravi-tational potentials and their first derivatives cannot be satisfied. This result isachieved by adopting other coordinates to which it is possible to associate aphysical meaning.

After we have proved that the exterior Schwarzschild solution can be extendedto the whole spacetime V4, except for the origin r ¼ 0 of radial coordinates, inChap. 15 this metric is interpreted as representing the gravitational field of a massconcentrated at r ¼ 0. The event horizon r ¼ rs is defined, and the complex physicsinside the event horizon is described (black hole). Then it is explained why thissolution is supposed to describe the final state of a massive collapsing star.

In Chapter 16 we present the Friedmann equations and the different cosmo-logical models described by those equations. In particular, we show the existence ofglobal coordinates in the spacetime to which a physical meaning can be attributed.

In Chap. 17 we try to answer the following questions: how can we deduce thequantities relative to an observer from geometric objects? For instance, if theelectromagnetic tensor Fab is known, what is the relation between the componentsof Fab and the electric and magnetic fields as they are measured by an observer? Isit possible to formulate the tensor laws in V4 in terms of quantities and operatorsrelative to an observer? We show that the fundamental tools to answer the abovequestions are the spacetime projections and the Fermi–Walker derivative.

In Chaps. 1–4 the fundamental concepts of differential geometry are recalled:differential calculus, exterior algebra, differential manifolds, Riemannian manifolds,exterior derivation and integration, and transformation groups.

Naples, Italy Antonio RomanoMario Mango Furnari

viii Preface

Contents

Part I Elements of Differential Geometry

1 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Linear Forms and Dual Vector Spaces . . . . . . . . . . . . . . . . . . 31.2 Biduality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Covariant 2-Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 ðr; sÞ-Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.5 Contraction and Contracted Multiplication . . . . . . . . . . . . . . . 111.6 Skew-Symmetric (0, 2)-Tensors . . . . . . . . . . . . . . . . . . . . . . . 131.7 Skew-Symmetric ð0; rÞ-Tensors . . . . . . . . . . . . . . . . . . . . . . . 161.8 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.9 Oriented Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.10 Representation Theorems for Symmetric

and Skew-Symmetric ð0; 2Þ-Tensors . . . . . . . . . . . . . . . . . . . . 221.11 Degenerate and Nondegenerate ð0:2Þ-Tensors . . . . . . . . . . . . . 251.12 Pseudo-Euclidean Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 281.13 Euclidean Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.14 Eigenvectors of Euclidean 2-Tensors . . . . . . . . . . . . . . . . . . . 331.15 Orthogonal Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 351.16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2 Introduction to Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . 392.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.2 Elements of the Geometry of Curves . . . . . . . . . . . . . . . . . . . 412.3 Elements of Geometry of Surfaces . . . . . . . . . . . . . . . . . . . . . 442.4 The Second Fundamental Form . . . . . . . . . . . . . . . . . . . . . . . 462.5 Parallel Transport and Geodesics . . . . . . . . . . . . . . . . . . . . . . 502.6 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.7 Riemann’s Tensor and the Theorema Egregium . . . . . . . . . . . 572.8 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

ix

2.9 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.10 Differentiable Functions and Curves on Manifolds . . . . . . . . . 632.11 Tangent Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.12 Cotangent Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.13 Differential and Codifferential of a Map . . . . . . . . . . . . . . . . . 702.14 Tangent and Cotangent Fiber Bundles . . . . . . . . . . . . . . . . . . 732.15 Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.16 Geodesics over Riemannian Manifolds . . . . . . . . . . . . . . . . . . 772.17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3 Transformation Groups, Exterior Differentiationand Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.1 Global and Local One-Parameter Groups . . . . . . . . . . . . . . . . 833.2 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.3 Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.4 Closed and Exact Differential Forms . . . . . . . . . . . . . . . . . . . 923.5 Properties of the Exterior Derivative . . . . . . . . . . . . . . . . . . . 943.6 An Introduction to the Integration of r-Forms . . . . . . . . . . . . . 953.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4 Absolute Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.1 Preliminary Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.2 Affine Connection on Manifolds . . . . . . . . . . . . . . . . . . . . . . 1064.3 Parallel Transport and Autoparallel Curves . . . . . . . . . . . . . . . 1084.4 Covariant Differential of Tensor Fields . . . . . . . . . . . . . . . . . . 1104.5 Torsion Tensor and Curvature Tensor . . . . . . . . . . . . . . . . . . 1114.6 Properties of the Riemann Tensor . . . . . . . . . . . . . . . . . . . . . 1154.7 Geodesic Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.8 Levi-Civita Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.9 Ricci Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.10 Differential Operators on a Riemannian Manifold . . . . . . . . . . 1244.11 Riemann’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Part II Newtonian Dynamics, Gravitation, and Cosmology

5 Review of Classical Mechanics and Electrodynamics . . . . . . . . . . . 1315.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.2 Foundations of Classical Kinematics . . . . . . . . . . . . . . . . . . . 132

5.2.1 Change of the Frame of Reference . . . . . . . . . . . . . . 1335.2.2 Absolute and Relative Velocity

and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.3 Laws of Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . 137

5.3.1 Force Laws and the Action–Reaction Principle . . . . . . 1395.3.2 Newton’s Second Law . . . . . . . . . . . . . . . . . . . . . . . 140

x Contents

5.3.3 Dynamics in Noninertial Frames . . . . . . . . . . . . . . . . 1415.3.4 Restrictions on the Force Laws . . . . . . . . . . . . . . . . . 142

5.4 Collision Between Two Particles . . . . . . . . . . . . . . . . . . . . . . 1455.5 Galilean Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . 1465.6 Comments About the Galilean Principle of Relativity . . . . . . . 1475.7 Developments of Newtonian Mechanics . . . . . . . . . . . . . . . . . 1495.8 Classical Thermodynamics of Continua . . . . . . . . . . . . . . . . . 1505.9 Electromagnetic Fields and the Theory of Light . . . . . . . . . . . 1565.10 Incompatibility Between Newtonian Mechanics

and Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6 Newtonian Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1616.1 Newton’s Gravitational Law . . . . . . . . . . . . . . . . . . . . . . . . . 1616.2 Newton’s Theory of Gravitation of an Extended Body . . . . . . 1626.3 Asymptotic Behavior of the Gravitational Potential . . . . . . . . . 1666.4 Local Inertial Frames and Tidal Forces . . . . . . . . . . . . . . . . . . 1696.5 Equilibrium of Self-gravitating Bodies . . . . . . . . . . . . . . . . . . 1726.6 Evolution of a Spherically Symmetric Self-gravitating

Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.7 Polytropic Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 1796.8 Lane–Emden Equation for Polytropic Gases . . . . . . . . . . . . . . 1816.9 Evolution of a Spherically Symmetric Perfect Gas . . . . . . . . . 1836.10 Difficulties of Newtonian Gravitation . . . . . . . . . . . . . . . . . . . 1896.11 Newtonian Cosmology: Kinematics . . . . . . . . . . . . . . . . . . . . 1906.12 Mass Balance and Motion of a Substratum . . . . . . . . . . . . . . . 194

Part III Special Relativity

7 Physical Foundations of Special Relativity . . . . . . . . . . . . . . . . . . . 2017.1 The Optical Isotropy Principle . . . . . . . . . . . . . . . . . . . . . . . . 2017.2 The Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 202

7.2.1 The Special Lorentz Transformations . . . . . . . . . . . . . 2037.2.2 The General Lorentz Transformations . . . . . . . . . . . . 206

7.3 Relativistic Composition of Velocities and Accelerations . . . . . 2107.4 A Different Approach to Lorentz Transformations . . . . . . . . . . 2127.5 Some Consequences of the Lorentz Transformations . . . . . . . . 2147.6 The Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . 2187.7 Maxwell’s Equations in Vacuum . . . . . . . . . . . . . . . . . . . . . . 2207.8 Relativistic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2227.9 Transformation Formulas of Momentum and Energy . . . . . . . . 2257.10 Two Examples of Relativistic Dynamics . . . . . . . . . . . . . . . . . 2277.11 Proper Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Contents xi

7.12 Different Approaches to Relativistic Dynamics . . . . . . . . . . . . 2307.12.1 Approach to Relativistic Dynamics Based

on Particle Collision . . . . . . . . . . . . . . . . . . . . . . . . . 2307.12.2 Another Mechanical Formulation of Relativistic

Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2317.13 Collision of Two Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 2357.14 Experimental Verification of Relativistic Dynamics . . . . . . . . . 236

8 Special Relativity in Minkowski Space . . . . . . . . . . . . . . . . . . . . . . 2398.1 Minkowski Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2398.2 Physical Meaning of Minkowski Spacetime . . . . . . . . . . . . . . 2438.3 Classification of Lorentz Transformations . . . . . . . . . . . . . . . . 2458.4 Four-Dimensional Equation of Motion . . . . . . . . . . . . . . . . . . 2468.5 Tensor Formulation of Electromagnetism in a Vacuum . . . . . . 2488.6 Electromagnetic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 2508.7 The Electromagnetic Momentum–Energy Tensor

in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2528.8 Exterior Algebra and Maxwell’s Equations . . . . . . . . . . . . . . . 2538.9 Spacetime Decomposition of 4-Tensors . . . . . . . . . . . . . . . . . 2558.10 Infinitesimal Lorentz Transformations . . . . . . . . . . . . . . . . . . . 2588.11 Fermi’s Transport and Fermi’s Derivative of a 4-Vector . . . . . 2618.12 The Thomas Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

9 Continuous Systems in Special Relativity . . . . . . . . . . . . . . . . . . . . 2679.1 Relativistic Equations for Incoherent Matter . . . . . . . . . . . . . . 2679.2 Integral Laws of Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2699.3 The Momentum–Energy Tensor . . . . . . . . . . . . . . . . . . . . . . . 2749.4 Intrinsic Deformation Gradient . . . . . . . . . . . . . . . . . . . . . . . . 2779.5 Relativistic Dissipation Inequality . . . . . . . . . . . . . . . . . . . . . 2829.6 Thermoelastic Materials in Relativity . . . . . . . . . . . . . . . . . . . 2859.7 On the Physical Meaning of Relative Quantities . . . . . . . . . . . 289

10 Electrodynamics in Moving Media . . . . . . . . . . . . . . . . . . . . . . . . . 29310.1 Maxwell’s Equations in Matter . . . . . . . . . . . . . . . . . . . . . . . 29310.2 About the Equivalence of Formulations

of the Electrodynamics of Moving Bodies . . . . . . . . . . . . . . . 29510.3 Minkowski’s Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29610.4 Ampère’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30110.5 Boffi’s Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30610.6 Chu’s Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30710.7 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

xii Contents

Part IV General Relativity and Cosmology

11 Introduction to General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . 31311.1 Difficulties of Newtonian Gravitational Theory . . . . . . . . . . . . 31411.2 Attempts to Overcome the Difficulties of Newtonian

Gravitational Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31511.3 Principles of General Relativity and General Covariance . . . . . 31711.4 Principle of Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32211.5 The Spacetime of General Relativity . . . . . . . . . . . . . . . . . . . 32311.6 Einstein’s Gravitational Equations . . . . . . . . . . . . . . . . . . . . . 32511.7 Experimental Determination of gab . . . . . . . . . . . . . . . . . . . . . 32811.8 The Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33011.9 Variational Formulation of Gravitation . . . . . . . . . . . . . . . . . . 33511.10 Palatini’s Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . 33811.11 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . 341

12 Linearized Einstein’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 34312.1 Quasi-Minkowskian Spacetime . . . . . . . . . . . . . . . . . . . . . . . 34312.2 Einstein’s Linearized Equations . . . . . . . . . . . . . . . . . . . . . . . 34712.3 Momentum–Energy Tensor for Weak Fields . . . . . . . . . . . . . . 34912.4 Static Matter Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35112.5 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35312.6 Gravitational Wave Detection . . . . . . . . . . . . . . . . . . . . . . . . 35612.7 Gravitoelectromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

13 Cauchy’s Problem for Einstein’s Equations . . . . . . . . . . . . . . . . . . 36113.1 Cauchy’s Problem and First Considerations . . . . . . . . . . . . . . 36113.2 About the Uniqueness of the Solution of Cauchy’s

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36413.3 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 36513.4 Leray’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36713.5 Harmonic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36813.6 Einstein’s Equations in Harmonic Coordinates . . . . . . . . . . . . 370

14 Schwarzschild’s Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37314.1 Gaussian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37314.2 Matching Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37514.3 Static and Stationary Spacetime . . . . . . . . . . . . . . . . . . . . . . . 37614.4 Isometries and Killing’s Vector Fields . . . . . . . . . . . . . . . . . . 37814.5 Three-Dimensional Spherically Symmetric Manifolds . . . . . . . 37914.6 Schwarzschild’s Exterior Solution . . . . . . . . . . . . . . . . . . . . . 38314.7 Schwarzschild’s Interior Solution . . . . . . . . . . . . . . . . . . . . . . 38714.8 Matching Interior and Exterior Solutions . . . . . . . . . . . . . . . . 39014.9 Physical Remarks About Schwarzschild’s Solution . . . . . . . . . 39314.10 Planetary Orbits in a Schwarzschild Field . . . . . . . . . . . . . . . . 396

Contents xiii

14.11 Gravitational Deflection of Light . . . . . . . . . . . . . . . . . . . . . . 40214.12 Gravitational Shift of Spectral Lines . . . . . . . . . . . . . . . . . . . . 406

15 Schwarzschild’s Solution and Black Holes . . . . . . . . . . . . . . . . . . . 40915.1 On the Singularity of Schwarzschild’s Exterior Solution . . . . . 40915.2 Physical Interpretation of the Event Horizon . . . . . . . . . . . . . . 41415.3 Gravitational Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41715.4 Summary of Schwarzschild Spacetime and Black Holes . . . . . 42015.5 Heuristic Derivation of the Kerr Metric . . . . . . . . . . . . . . . . . 42215.6 Kerr Metric and Its Properties . . . . . . . . . . . . . . . . . . . . . . . . 42315.7 The Schwarzschild and Kerr Solutions . . . . . . . . . . . . . . . . . . 42415.8 Transformation of Ellipsoid Symmetric Orthogonal

Coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42515.9 A Solution of Einstein’s Equations in Vacuum . . . . . . . . . . . . 42715.10 Consequences of Kerr’s Solutions . . . . . . . . . . . . . . . . . . . . . 429

16 Elements of Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43116.1 Global Properties of Spacetime . . . . . . . . . . . . . . . . . . . . . . . 43116.2 On the Geometry of Space Sections . . . . . . . . . . . . . . . . . . . . 43316.3 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43716.4 Friedmann’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43816.5 Models of Universe for K ¼ p ¼ 0 . . . . . . . . . . . . . . . . . . . . 44016.6 Qualitative Analysis of Friedmann’s Equations

for K ¼ p ¼ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44216.7 Models of the Universe for p ¼ 0 and K 6¼ 0 . . . . . . . . . . . . . 444

17 Relative Formulation of Physical Laws . . . . . . . . . . . . . . . . . . . . . . 44717.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44717.2 Timelike Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45317.3 The Fermi–Walker Derivative . . . . . . . . . . . . . . . . . . . . . . . . 45617.4 The Fermi–Walker Covariant Derivative . . . . . . . . . . . . . . . . . 45817.5 F–W Derivation of 2-Tensors . . . . . . . . . . . . . . . . . . . . . . . . 46117.6 Frames of Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46217.7 Kinematic Characteristics of a Frame of Reference . . . . . . . . . 46417.8 Relative Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . 46717.9 Relative Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 47117.10 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47217.11 Divergence of a Skew-Symmetric Tensor . . . . . . . . . . . . . . . . 47417.12 Relative Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . 47817.13 Divergence of a Symmetric Tensor . . . . . . . . . . . . . . . . . . . . 47917.14 Momentum–Energy Tensor of Dust Matter . . . . . . . . . . . . . . . 480

Correction to: Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C1

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

xiv Contents

List of Figures

Fig. 2.1 Surface of <3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Fig. 2.2 Normal curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Fig. 2.3 The three possible points of a surface. . . . . . . . . . . . . . . . . . . . . 50Fig. 2.4 Coordinates ðh;/Þ on a hemisphere . . . . . . . . . . . . . . . . . . . . . . 54Fig. 2.5 Coordinates ðq;/Þ on a hemisphere . . . . . . . . . . . . . . . . . . . . . . 54Fig. 2.6 Coordinates ðr;/Þ on a hemisphere . . . . . . . . . . . . . . . . . . . . . . 55Fig. 2.7 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Fig. 2.8 Chart on a manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Fig. 2.9 Curve on a manifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Fig. 2.10 Map between manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Fig. 2.11 Differential of a map F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Fig. 2.12 Sphere in E3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Fig. 2.13 Ellipsoid in E3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Fig. 2.14 Stereographic projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Fig. 2.15 Central projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Fig. 3.1 Lie derivative of a vector field . . . . . . . . . . . . . . . . . . . . . . . . . . 87Fig. 3.2 Lie derivative of a 1-form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Fig. 3.3 An oriented 2-cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Fig. 3.4 An oriented 2-cube of E2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Fig. 3.5 An oriented 2-cube of E3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Fig. 5.1 Change of rigid frame of reference . . . . . . . . . . . . . . . . . . . . . . . 135Fig. 5.2 Collision in the lab frame of two identical particles . . . . . . . . . . 146Fig. 5.3 Reference and current configurations of a continuous body . . . . 151Fig. 6.1 Continuous mass distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Fig. 6.2 Arbitrary continuous mass distribution . . . . . . . . . . . . . . . . . . . . 167Fig. 6.3 Tidal forces on Earth due to the Moon. . . . . . . . . . . . . . . . . . . . 170Fig. 6.4 Rotating fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Fig. 7.1 Two arbitrary inertial frames. . . . . . . . . . . . . . . . . . . . . . . . . . . . 204Fig. 7.2 Star aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Fig. 7.3 Collision in the lab frame of two identical particles . . . . . . . . . . 236

xv

Fig. 8.1 Infinitesimal Lorentz transformation without rotation . . . . . . . . . 260Fig. 8.2 Fermi’s transport along a world line . . . . . . . . . . . . . . . . . . . . . . 262Fig. 8.3 F-transport and the Thomas precession . . . . . . . . . . . . . . . . . . . . 265Fig. 9.1 Space and time projection of the infinitesimal 4-vector dx . . . . . 279Fig. 9.2 Evolution of the infinitesimal space 4-vector dx . . . . . . . . . . . . . 280Fig. 12.1 Effects of the tidal forces produced by waves hþ on a circle

of test particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357Fig. 12.2 Effects of the tidal forces produced by waves h� on a circle

of test particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358Fig. 14.1 A surface with a symmetry of revolution about

the axis Ox3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382Fig. 14.2 grr � 1 versus r=r1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393Fig. 14.3 ðl� rÞ=r1 versus l=r1 inside S . . . . . . . . . . . . . . . . . . . . . . . . . . 394Fig. 14.4 Precession of the orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402Fig. 14.5 Gravitational deflection of light . . . . . . . . . . . . . . . . . . . . . . . . . 404Fig. 15.1 Trajectories of photons near the event horizon . . . . . . . . . . . . . . 414Fig. 16.1 Plot of �X0H2

0=2a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442Fig. 16.2 Solutions aðtÞ for different values of X0 . . . . . . . . . . . . . . . . . . . 443Fig. 17.1 Fermi–Walker transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

xvi List of Figures