The physical and mathematical foundations of the theory of relativity 9783030272364, 9783030272371

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Antonio Romano Mario Mango Furnari

The Physical and Mathematical Foundations of the Theory of Relativity A Critical Analysis

Antonio Romano Mario Mango Furnari •

The Physical and Mathematical Foundations of the Theory of Relativity A Critical Analysis

Antonio Romano Dipartimento di Matematica e Applicazioni “Renato Caccioppoli” Università degli Studi di Napoli Federico II Naples, Italy

Mario Mango Furnari Istituto di Cibernetica Naples, Italy

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

(eBook)

Mathematics Subject Classification (2010): 83AXX, 83BXX, 83CXX, 83FXX © Springer Nature Switzerland AG 2019 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, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The 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 very large beach. In fact, the list of books about special and general relativity is very extensive, and the reader can find many introductory and specialist books. Among the former, there are excellent texts that allow the reader to understand the principles of relativity, whereas in books of the second category, it is possible to deepen all the advanced topics of relativity (Cauchy problem, geometric properties of spacetime in the large, cosmological models, star structure, etc.). Finally, there are books devoted to the most recent developments of the theory. Why, then, to add another text to so extensive a list of books? Why add another grain of sand to so wide a beach? We now make clear the factors that pushed us to write this introductory textbook. First, it is our rooted opinion that the reader should understand which classical concepts he or she is leaving behind in following the path that will lead to the acceptance of the new ideas of relativity. In some cases, accepting the new ideas can be painful and confusing. With the aim of relieving this tiresome learning process, this book begins with two chapters in which the classical ideas that will be abandoned or saved are recalled in detail. Therefore, in these chapters the classical procedure to measure space and time, the basic ideas of classical dynamics, the Galilean relativity principle, and some fundamental results of Newtonian gravitation are recalled. After introducing special relativity following the physical approach proposed by Einstein (Chap. 7), the Minkowski mathematical model is discussed (Chap. 8). In this regard, we recall that a model can be considered a mathematical transcription of a physical reality if and only if the measurable physical quantities are uniquely associated with mathematical objects of the model. In Chap. 8, after defining this correspondence for Minkowski’s model, all the characteristics and particularities of this correspondence are widely analyzed with the aim of putting in evidence the deep differences existing between the correspondence that in special relativity allow one to attribute physical meaning to Minkowski’s model and the correspondence proposed by Einstein in general relativity between the hyperbolic Riemannian spacetime and physical reality. v

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Chapters 9 and 10 are devoted to topics that are usually presented hastily; the relativistic thermodynamics of continua and the electromagnetic fields in matter. In these two classes of phenomena we face several ambiguities due to the fact that many quantities that are necessary to formulate these theories cannot be experimentally observed. This circumstance justifies the many proposals that can be found in the literature about the momentum–energy tensor employed in thermodynamics and in the electrodynamics of continua. In particular, we prove the macroscopic equivalence of the all transformation formulas of temperature and heat that have been proposed as well as the equivalence of all the models proposed to describe the interaction between matter and electromagnetic fields. More precisely, we show that all the models proposed in electrodynamics in matter are obtained by a suitable choice of the variables we adopt to describe the electromagnetic field. In other words, they have no physical consistency but are useful mathematical models for evaluating observable macroscopic quantities. In Chap. 11 we face the complex problem of analyzing the basic principles of general 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 the generalization of the special principle of relativity, since it states that all observers have the right to study nature. However, it is not explained what constitutes an arbitrary observer, how one measures physical quantities, and mostly how such measurements are related to those of another observer. It is important to underline that only if this connection is made explicit can the observers realize a universal physics, where “universal” has to be understood as the collection of the observers’ descriptions together with the possibility of comparing them. A thoroughgoing analysis of this problem can be found in Chap. 11. Often, the general principle of relativity is identified with the mathematical principle of general covariance. This principle states that physical laws must be formulated in a form independent of the coordinates adopted in the hyperbolic Riemannian spacetime that Einstein substitutes for Minkowski’s model. This conclusion is often justified by stating that there are coordinates in spacetime that can be considered a mathematical representation of physical frames of reference and vice versa. Consequently, the covariance of physical laws represents the mathematical version of the general principle of relativity. In Chap. 11, this statement 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 can be eliminated in small spacetime regions. The local frames of reference in which that happens are called local inertial frames. This principle, together with the assumption that in these regions special relativity holds, allows the introduction of a first partial correspondence between physical reality and geometric objects of the spacetime 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 geodetic coordinates in which the metric assumes the Minkowski form pushed Einstein to

Preface

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describe the gravitational field with the metric of a Riemannian manifold that in turn is related to the matter and energy occupying a region of spacetime. Then the main problem Einstein had to solve was to determine the equations relating the metric coefficients to the distribution of matter and energy. Starting from reasonable hypotheses, which are discussed in Chap. 11, Einstein determined a system of 10 nonlinear partial differential equations of the form Glm ¼ vTlm , where the tensor Glm involves only the spacetime geometry, Tlm is the momentum–energy tensor satisfying the conservation laws rm T lm ¼ 0, and v is an unknown constant. In order to determine the constant v, Einstein resorted to a linear approximation of the field equations obtained on the assumption of weak gravitational fields and nonrelativistic velocities (see Chap. 12). The linear equations, in the static approximation, reduce to a single equation that is formally identical to Poisson’s equation, provided that v ¼ 8ph=c4 . It should be noted that the identification of Einstein’s equations and Poisson’s equation is purely formal, since the interpretation of the gravitational potential in the two equations is completely different. Furthermore, in the linear nonstatic case, every metric coefficient satisfies d’Alemebert’s equation, so that the existence of gravitational waves is foreseen and the gravitational potentials are obtained by retarded potentials (Chap. 12). In Chap. 13, the hyperbolic character of Einstein’s equations is verified, and some existence theorems under reasonable regularity spacetime conditions are proved for the exterior Cauchy problem. It is very important to highlight a deep change of perspective in going from special relativity to general relativity. In fact, in the former theory, the procedures that allow inertial observers to measure space and time are formulated before any physical law is determined. Furthermore, these procedures allow one to identify in Minkowski’s spacetime three-dimensional spaces and one-dimensional spaces that are the geometric representations of space and time relative to an observer. In general relativity, the metric of spacetime is dynamic, i.e., it is determined by the evolution of matter and energy through the field equations. This means that we cannot speak about local or global space and time before solving the Einstein equations and the conservation laws. In other words, before knowing the metric, we cannot speak about space, time, geodesics, etc. In particular, the definitions of local and global space and time will depend on the form of the metric. To put in evidence another particularity of general relativity, we recall that the equivalence principle introduces a partial correspondence between local inertial frames and geodesic coordinates at points of the spacetime V4 . This means that coordinates ðxa Þ must be defined on an open set U of V4 to which it is possible to associate a frame of reference for an observer O. In particular, experimental procedures must be defined that allow O to evaluate the coordinate ðxa Þ of an event belonging to a set E. After defining this one-to-one correspondence between points of U and physical events of E, we can also adopt in U arbitrary coordinates ðx0a Þ, provided that the coordinate transformation x0a ¼ x0a ðxb Þ is known.

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This is the approach we follow in Chaps. 14–16. Specifically, in Chap. 14, the model of spacetime proposed by Schwarzschild is presented. This model describes the relativistic gravitational field produced by a spherically symmetric mass distribution S inside and outside S. In the coordinates introduced to solve separately the interior and exterior Einstein equations, the matching conditions on the gravitational potentials and their first derivatives cannot be satisfied. This result is achieved by adopting other coordinates to which it is possible to associate a physical meaning. After we have proved that the exterior Schwarzschild solution can be extended to the whole spacetime V4 , except for the origin r ¼ 0 of radial coordinates, in Chap. 15 this metric is interpreted as representing the gravitational field of a mass concentrated at r ¼ 0. The event horizon r ¼ rs is defined, and the complex physics inside the event horizon is described (black hole). Then it is explained why this solution is supposed to describe the final state of a massive collapsing star. In Chapter 16 we present the Friedmann equations and the different cosmological models described by those equations. In particular, we show the existence of global 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 the quantities relative to an observer from geometric objects? For instance, if the electromagnetic tensor F ab is known, what is the relation between the components of F ab and the electric and magnetic fields as they are measured by an observer? Is it possible to formulate the tensor laws in V4 in terms of quantities and operators relative to an observer? We show that the fundamental tools to answer the above questions 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 Romano Mario Mango Furnari

Contents

Part I 1

2

Elements of Differential Geometry . . . . . . . . . .

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22 25 28 30 33 35 35

Introduction to Differentiable Manifolds . . . . . . . . . . 2.1 Historical Introduction . . . . . . . . . . . . . . . . . . . 2.2 Elements of the Geometry of Curves . . . . . . . . 2.3 Elements of Geometry of Surfaces . . . . . . . . . . 2.4 The Second Fundamental Form . . . . . . . . . . . . 2.5 Parallel Transport and Geodesics . . . . . . . . . . . 2.6 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Riemann’s Tensor and the Theorema Egregium 2.8 Curvilinear Coordinates . . . . . . . . . . . . . . . . . .

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39 39 41 44 46 50 53 57 59

Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Linear Forms and Dual Vector Spaces . . . . . 1.2 Biduality . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Covariant 2-Tensors . . . . . . . . . . . . . . . . . . 1.4 ðr; sÞ-Tensors . . . . . . . . . . . . . . . . . . . . . . . 1.5 Contraction and Contracted Multiplication . . 1.6 Skew-Symmetric (0, 2)-Tensors . . . . . . . . . . 1.7 Skew-Symmetric ð0; rÞ-Tensors . . . . . . . . . . 1.8 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . 1.9 Oriented Vector Spaces . . . . . . . . . . . . . . . . 1.10 Representation Theorems for Symmetric and Skew-Symmetric ð0; 2Þ-Tensors . . . . . . . 1.11 Degenerate and Nondegenerate ð0:2Þ-Tensors 1.12 Pseudo-Euclidean Vector Spaces . . . . . . . . . 1.13 Euclidean Vector Spaces . . . . . . . . . . . . . . . 1.14 Eigenvectors of Euclidean 2-Tensors . . . . . . 1.15 Orthogonal Transformations . . . . . . . . . . . . . 1.16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 3

4

Transformation Groups, Exterior Differentiation and Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Global and Local One-Parameter Groups . . . . 3.2 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Exterior Derivative . . . . . . . . . . . . . . . . . . . . 3.4 Closed and Exact Differential Forms . . . . . . . 3.5 Properties of the Exterior Derivative . . . . . . . 3.6 An Introduction to the Integration of r-Forms . 3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Absolute Differential Calculus . . . . . . . . . . . . . . . . . . . 4.1 Preliminary Considerations . . . . . . . . . . . . . . . . 4.2 Affine Connection on Manifolds . . . . . . . . . . . . 4.3 Parallel Transport and Autoparallel Curves . . . . . 4.4 Covariant Differential of Tensor Fields . . . . . . . . 4.5 Torsion Tensor and Curvature Tensor . . . . . . . . 4.6 Properties of the Riemann Tensor . . . . . . . . . . . 4.7 Geodesic Deviation . . . . . . . . . . . . . . . . . . . . . . 4.8 Levi-Civita Connection . . . . . . . . . . . . . . . . . . . 4.9 Ricci Decomposition . . . . . . . . . . . . . . . . . . . . . 4.10 Differential Operators on a Riemannian Manifold 4.11 Riemann’s Theorem . . . . . . . . . . . . . . . . . . . . .

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Review of Classical Mechanics and Electrodynamics . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Foundations of Classical Kinematics . . . . . . . . . . . . . . 5.2.1 Change of the Frame of Reference . . . . . . . . . 5.2.2 Absolute and Relative Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . 5.3 Laws of Newtonian Mechanics . . . . . . . . . . . . . . . . . . 5.3.1 Force Laws and the Action–Reaction Principle . 5.3.2 Newton’s Second Law . . . . . . . . . . . . . . . . . .

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Part II 5

Differentiable Manifolds . . . . . . . . . . . . . . . . . . . Differentiable Functions and Curves on Manifolds Tangent Vector Space . . . . . . . . . . . . . . . . . . . . . Cotangent Vector Space . . . . . . . . . . . . . . . . . . . Differential and Codifferential of a Map . . . . . . . . Tangent and Cotangent Fiber Bundles . . . . . . . . . Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . Geodesics over Riemannian Manifolds . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Newtonian Dynamics, Gravitation, and Cosmology

Contents

5.4 5.5 5.6 5.7 5.8 5.9 5.10 6

5.3.3 Dynamics in Noninertial Frames . . . . . . . . . 5.3.4 Restrictions on the Force Laws . . . . . . . . . . Collision Between Two Particles . . . . . . . . . . . . . . . Galilean Principle of Relativity . . . . . . . . . . . . . . . . Comments About the Galilean Principle of Relativity Developments of Newtonian Mechanics . . . . . . . . . . Classical Thermodynamics of Continua . . . . . . . . . . Electromagnetic Fields and the Theory of Light . . . . Incompatibility Between Newtonian Mechanics and Electromagnetism . . . . . . . . . . . . . . . . . . . . . . .

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177 179 181 183 189 190 194

Physical Foundations of Special Relativity . . . . . . . . . . . . . . . 7.1 The Optical Isotropy Principle . . . . . . . . . . . . . . . . . . . . 7.2 The Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . 7.2.1 The Special Lorentz Transformations . . . . . . . . . 7.2.2 The General Lorentz Transformations . . . . . . . . 7.3 Relativistic Composition of Velocities and Accelerations . 7.4 A Different Approach to Lorentz Transformations . . . . . . 7.5 Some Consequences of the Lorentz Transformations . . . . 7.6 The Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . 7.7 Maxwell’s Equations in Vacuum . . . . . . . . . . . . . . . . . . 7.8 Relativistic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9 Transformation Formulas of Momentum and Energy . . . . 7.10 Two Examples of Relativistic Dynamics . . . . . . . . . . . . . 7.11 Proper Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Newtonian Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Newton’s Gravitational Law . . . . . . . . . . . . . . . . . . . 6.2 Newton’s Theory of Gravitation of an Extended Body 6.3 Asymptotic Behavior of the Gravitational Potential . . . 6.4 Local Inertial Frames and Tidal Forces . . . . . . . . . . . . 6.5 Equilibrium of Self-gravitating Bodies . . . . . . . . . . . . 6.6 Evolution of a Spherically Symmetric Self-gravitating Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Polytropic Transformations . . . . . . . . . . . . . . . . . . . . 6.8 Lane–Emden Equation for Polytropic Gases . . . . . . . . 6.9 Evolution of a Spherically Symmetric Perfect Gas . . . 6.10 Difficulties of Newtonian Gravitation . . . . . . . . . . . . . 6.11 Newtonian Cosmology: Kinematics . . . . . . . . . . . . . . 6.12 Mass Balance and Motion of a Substratum . . . . . . . . .

Part III 7

xi

Special Relativity

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Contents

7.12

7.13 7.14 8

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Different Approaches to Relativistic Dynamics . . . . . . . . 7.12.1 Approach to Relativistic Dynamics Based on Particle Collision . . . . . . . . . . . . . . . . . . . . . 7.12.2 Another Mechanical Formulation of Relativistic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collision of Two Particles . . . . . . . . . . . . . . . . . . . . . . . Experimental Verification of Relativistic Dynamics . . . . .

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Special Relativity in Minkowski Space . . . . . . . . . . . . . . . . . 8.1 Minkowski Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Physical Meaning of Minkowski Spacetime . . . . . . . . . 8.3 Classification of Lorentz Transformations . . . . . . . . . . . 8.4 Four-Dimensional Equation of Motion . . . . . . . . . . . . . 8.5 Tensor Formulation of Electromagnetism in a Vacuum . 8.6 Electromagnetic Potentials . . . . . . . . . . . . . . . . . . . . . . 8.7 The Electromagnetic Momentum–Energy Tensor in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Exterior Algebra and Maxwell’s Equations . . . . . . . . . . 8.9 Spacetime Decomposition of 4-Tensors . . . . . . . . . . . . 8.10 Infinitesimal Lorentz Transformations . . . . . . . . . . . . . . 8.11 Fermi’s Transport and Fermi’s Derivative of a 4-Vector 8.12 The Thomas Precession . . . . . . . . . . . . . . . . . . . . . . . .

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Continuous Systems in Special Relativity . . . . . . . . . 9.1 Relativistic Equations for Incoherent Matter . . . 9.2 Integral Laws of Balance . . . . . . . . . . . . . . . . . 9.3 The Momentum–Energy Tensor . . . . . . . . . . . . 9.4 Intrinsic Deformation Gradient . . . . . . . . . . . . . 9.5 Relativistic Dissipation Inequality . . . . . . . . . . 9.6 Thermoelastic Materials in Relativity . . . . . . . . 9.7 On the Physical Meaning of Relative Quantities

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10 Electrodynamics in Moving Media . . . . . . . . . . 10.1 Maxwell’s Equations in Matter . . . . . . . . 10.2 About the Equivalence of Formulations of the Electrodynamics of Moving Bodies 10.3 Minkowski’s Description . . . . . . . . . . . . . 10.4 Ampère’s Model . . . . . . . . . . . . . . . . . . . 10.5 Boffi’s Formulation . . . . . . . . . . . . . . . . . 10.6 Chu’s Formulation . . . . . . . . . . . . . . . . . 10.7 Final Remarks . . . . . . . . . . . . . . . . . . . . .

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Contents

Part IV

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General Relativity and Cosmology . . . . . 313 . . . . . 314

11 Introduction to General Relativity . . . . . . . . . . . . . . . . . . . . 11.1 Difficulties of Newtonian Gravitational Theory . . . . . . . 11.2 Attempts to Overcome the Difficulties of Newtonian Gravitational Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Principles of General Relativity and General Covariance 11.4 Principle of Equivalence . . . . . . . . . . . . . . . . . . . . . . . 11.5 The Spacetime of General Relativity . . . . . . . . . . . . . . 11.6 Einstein’s Gravitational Equations . . . . . . . . . . . . . . . . 11.7 Experimental Determination of gab . . . . . . . . . . . . . . . . 11.8 The Rotating Frame . . . . . . . . . . . . . . . . . . . . . . . . . . 11.9 Variational Formulation of Gravitation . . . . . . . . . . . . . 11.10 Palatini’s Variational Principle . . . . . . . . . . . . . . . . . . . 11.11 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . .

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315 317 322 323 325 328 330 335 338 341

12 Linearized Einstein’s Equations . . . . . . . . . . . . . . 12.1 Quasi-Minkowskian Spacetime . . . . . . . . . . 12.2 Einstein’s Linearized Equations . . . . . . . . . . 12.3 Momentum–Energy Tensor for Weak Fields . 12.4 Static Matter Distribution . . . . . . . . . . . . . . . 12.5 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Gravitational Wave Detection . . . . . . . . . . . 12.7 Gravitoelectromagnetism . . . . . . . . . . . . . . .

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13 Cauchy’s Problem for Einstein’s Equations . . . . . . . . . 13.1 Cauchy’s Problem and First Considerations . . . . . 13.2 About the Uniqueness of the Solution of Cauchy’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Mathematical Preliminaries . . . . . . . . . . . . . . . . . 13.4 Leray’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Harmonic Coordinates . . . . . . . . . . . . . . . . . . . . . 13.6 Einstein’s Equations in Harmonic Coordinates . . .

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364 365 367 368 370

14 Schwarzschild’s Universe . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Gaussian Coordinates . . . . . . . . . . . . . . . . . . . . . . . 14.2 Matching Conditions . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Static and Stationary Spacetime . . . . . . . . . . . . . . . . 14.4 Isometries and Killing’s Vector Fields . . . . . . . . . . . 14.5 Three-Dimensional Spherically Symmetric Manifolds 14.6 Schwarzschild’s Exterior Solution . . . . . . . . . . . . . . 14.7 Schwarzschild’s Interior Solution . . . . . . . . . . . . . . . 14.8 Matching Interior and Exterior Solutions . . . . . . . . . 14.9 Physical Remarks About Schwarzschild’s Solution . . 14.10 Planetary Orbits in a Schwarzschild Field . . . . . . . . .

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373 373 375 376 378 379 383 387 390 393 396

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xiv

Contents

14.11 Gravitational Deflection of Light . . . . . . . . . . . . . . . . . . . . . . 402 14.12 Gravitational Shift of Spectral Lines . . . . . . . . . . . . . . . . . . . . 406 15 Schwarzschild’s Solution and Black Holes . . . . . . . . . . . . . . 15.1 On the Singularity of Schwarzschild’s Exterior Solution 15.2 Physical Interpretation of the Event Horizon . . . . . . . . . 15.3 Gravitational Collapse . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Summary of Schwarzschild Spacetime and Black Holes 15.5 Heuristic Derivation of the Kerr Metric . . . . . . . . . . . . 15.6 Kerr Metric and Its Properties . . . . . . . . . . . . . . . . . . . 15.7 The Schwarzschild and Kerr Solutions . . . . . . . . . . . . . 15.8 Transformation of Ellipsoid Symmetric Orthogonal Coordinate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9 A Solution of Einstein’s Equations in Vacuum . . . . . . . 15.10 Consequences of Kerr’s Solutions . . . . . . . . . . . . . . . . 16 Elements of Cosmology . . . . . . . . . . . . . . . . . . . . . . 16.1 Global Properties of Spacetime . . . . . . . . . . . 16.2 On the Geometry of Space Sections . . . . . . . . 16.3 Conservation Laws . . . . . . . . . . . . . . . . . . . . 16.4 Friedmann’s Equations . . . . . . . . . . . . . . . . . 16.5 Models of Universe for K ¼ p ¼ 0 . . . . . . . . 16.6 Qualitative Analysis of Friedmann’s Equations for K ¼ p ¼ 0 . . . . . . . . . . . . . . . . . . . . . . . . 16.7 Models of the Universe for p ¼ 0 and K 6¼ 0 .

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409 409 414 417 420 422 423 424

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431 431 433 437 438 440

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17 Relative Formulation of Physical Laws . . . . . . . . . . . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Timelike Congruences . . . . . . . . . . . . . . . . . . . . . 17.3 The Fermi–Walker Derivative . . . . . . . . . . . . . . . 17.4 The Fermi–Walker Covariant Derivative . . . . . . . . 17.5 F–W Derivation of 2-Tensors . . . . . . . . . . . . . . . 17.6 Frames of Reference . . . . . . . . . . . . . . . . . . . . . . 17.7 Kinematic Characteristics of a Frame of Reference 17.8 Relative Momentum Equation . . . . . . . . . . . . . . . 17.9 Relative Energy Equation . . . . . . . . . . . . . . . . . . 17.10 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . 17.11 Divergence of a Skew-Symmetric Tensor . . . . . . . 17.12 Relative Maxwell’s Equations . . . . . . . . . . . . . . . 17.13 Divergence of a Symmetric Tensor . . . . . . . . . . . 17.14 Momentum–Energy Tensor of Dust Matter . . . . . .

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447 447 453 456 458 461 462 464 467 471 472 474 478 479 480

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

List of Figures

Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig.

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 3.1 3.2 3.3 3.4 3.5 5.1 5.2 5.3 6.1 6.2 6.3 6.4 7.1 7.2 7.3

Surface of j

Ti j θ ⊗ θ + i

j

i< j



 

T ji θ j ⊗ θi

j>i

Ti j θ ⊗ θ − i

j

i< j



Ti j θ j ⊗ θi .

i< j

Therefore, we can write T = T(i j) θi ∧ θ j ,

(1.6.6)

T(i j) = Ti j , i < j.

(1.6.7)

where Relation (1.6.6) shows that the set of the (0, 2)-tensors (θ i ∧ θ j ) ∈ 2 (E n ) generates the whole subspace 2 (E n ). Consequently, to verify that they form a basis of 2 (E n ), it is sufficient to verify their linear independence. Now, from any linear combination a(i j) θi ∧ θ j = 0, i < j, we have the condition a(i j) θi ∧ θ j (eh , ek ) = 0, which, in view of (1.6.5) and (1.1.4), implies a(i j) = 0, and the proof is complete.  Definition 1.13 The quantities T(i j) , i < j, are called strict components of T relative to the basis (θi ∧ θ j ). To determine the transformation formulas of the strict components, we recall that they are components of a (0, 2)-tensor. Consequently, in view of (1.3.8), we can write

1.6 Skew-Symmetric (0, 2)-Tensors

15

T(i j) = A(ih Akj) Thk   = A(ih Akj) Thk + A(ih Akj) Thk hk

 = (A(ih Akj) − Ak(i Ahj) )T(hk) , h π( j).

1.7 Skew-Symmetric (0, r )-Tensors

17

A permutation π is said to be even or odd according to whether the total number of inversions contained in Sπ is even or odd. In the sequel, we denote by m(π) the total number of inversions of π. Definition 1.15 A tensor T ∈ Tr (E n ), r ≥ 2, is skew-symmetric or alternating if T(x1 , . . . , xr ) = (−1)m(π) T(xπ(1) , . . . , xπ(r ) ),

(1.7.1)

for all x1 , . . . , xr ∈ E n and for all π ∈ r . The preceding definition implies that the value of T vanishes every time T is evaluated on a set of vectors containing two equal vectors. In fact, it is sufficient to consider the permutation that exchanges the position of these two vectors without modifying the position of the others and then to apply (1.7.1). From this remark, it easily follows that the value of T vanishes if the vectors {x1 , . . . , xr } are linearly dependent. If the r vectors {x1 , . . . , xr } belong to a basis (ei ), then in view of (1.4.6), we express the skew-symmetry of T in terms of its components: Ti1 ···ir = (−1)m(π) Tπ(i1 )···π(ir ) .

(1.7.2)

In particular, this condition implies that all the components of T in which two indices have the same value vanish. It is quite obvious that the set of all the skew-symmetric tensors forms a subspace r (E n ) of Tr (E n ). Remark 1.6 Since m vectors, when m > n, cannot be linearly independent, we can state that r (E n ) = {0}, n < m. Definition 1.16 The exterior product of r covectors ω 1 , . . . , ωr is the (0, r )-tensor ω 1 ∧ · · · ∧ ωr =



(−1)m(π) ω π(1) ⊗ · · · ⊗ ω π(r ) .

(1.7.3)

π∈r

The tensor (1.7.3) is skew-symmetric, since in view of (1.4.3), we have ω 1 ∧ · · · ∧ ωr (x1 , . . . , xr ) =



(−1)m(π) ω π(1) (x1 ) · · · ⊗ ω π(r ) (xr )

π∈r

 1  ω (x1 ) · · · ω 1 (xr )    =  · · · · · · · · ·  . ωr (x1 ) · · · ωr (xr )

(1.7.4)

A permutation π of the vectors x1 , . . . , xr corresponds to a permutation of the columns of the determinant in (1.7.4). This operation changes or does not change the sign of the determinant according to whether the permutation is odd or even. But this is just the property expressed by (1.7.1). Again recalling the properties of a

18

1 Tensor Algebra

determinant, we can easily prove the following properties of the exterior product of r covectors: • it is a (0, r )-skew-symmetric tensor; • it vanishes if one vector depends linearly on the others. Now we extend Theorem 1.9 to skew-symmetric (0, r )-tensors. i Theorem 1.10 Let (ei ) be a basis of the vector n  space E n and denote by (θ ) the dual ∗ basis in E n . Then if r ≤ n, it follows that r is the dimension of the vector space r (E n ) of the skew-symmetric (0, r )-tensors. Further, the skew-symmetric tensors

θ i 1 ∧ · · · ∧ θ ir ,

(1.7.5)

where i 1 < · · · < ir is an arbitrary r -tuple of integers chosen in the set of indices {1, . . . , n}, form a basis of r (E n ). Proof First, when r ≤ n, every T ∈ r (E n ) can be written as follows: T = T j1 ··· jr θ j1 ⊗ · · · ⊗ θ jr . The summation on the right-hand side contains only terms in which all the indices are distinct, since T is skew-symmetric. This circumstance makes it possible to group all the terms as follows: first, we consider the terms obtained by extracting arbitrarily  r distinct indices i 1 < · · · < ir from the set {1, . . . , n}. We recall that there are nr distinct choices of these indices. Then for each choice, which is characterized by the indices i 1 < · · · < ir , we consider all the terms obtained by permuting in all possible ways the indices i 1 < · · · < ir . In view of (1.2.2) and (1.2.3), we can write T=

 

Tπ(i1 )···π(ir ) θ π(i1 ) ⊗ · · · ⊗ θ π(ir )

i 1 0, T(y, y) = −

 r +s 

(y i )2 < 0.

i=r  +1

i=1

We can easily prove that the intersection E + ∩ (E − ⊕ E 0 ) is equal to {0}. In fact, if x ∈ E + and x = 0, then (1.11.3)1 is satisfied; further, every y ∈ E + , y = 0, satisfies (1.11.3)2 . Consequently, the subspace E + ⊕ E − ⊕ E 0 has dimension r + n − r  ≤ n, so that r ≤ r  . Applying this reasoning to E + ⊕ E − ⊕ E 0 , we obtain  r = r . Definition 1.21 The integer r is called the index of T, whereas the difference r − s is the signature of T. Definition 1.22 Let T belong to T2 (E n ). The map q : E n →  such that q(x) = T(x, x) is called the quadratic form associated with T. Since q(x) vanishes identically when T is skew-symmetric, throughout the next sections we refer only to symmetric tensors. Definition 1.23 A symmetric tensor T ∈ T2 (E n ) is said to be positive semidefinite if for all x ∈ E n , one has q(x) ≡ T(x, x) ≥ 0. (1.11.4) If (1.11.4) assumes the value zero if and only if x = 0, then T is positive definite. By adopting a canonical basis in E n and resorting to Theorem 1.15, we conclude that s = 0 when T is positive semidefinite. Further, T is positive definite if and only if r = n. The positive definitiveness of T does not depend on the basis (ei ) of E n . In fact, in any basis, (1.11.4) can be written as Ti j x i x j ≥ 0, ∀ (x i ) ∈ n .

(1.11.5)

It is well known that a quadratic form is positive semidefinite (respectively positive definite) if and only if all the principal minors T i , i = 1, . . . , n, of the representative matrix T = (Ti j ) of T with respect to the basis (ei ) satisfy the following conditions: T i ≥ 0, (repspectively T i > 0), i = 1, . . . , n.

(1.11.6)

28

1 Tensor Algebra

Theorem 1.16 Let T ∈ T2 (E n ) be a symmetric positive semidefinite (0, 2)-tensor. Then for all x, y ∈ E n , the Cauchy–Schwarz inequality   (1.11.7) |T(x, y)| ≤ T(x, x) T(y, y) and Minkowski’s inequality 

T(x + y, x + y) ≤



T(x, x) +



T(y, y)

(1.11.8)

hold. Proof Note that for all a ∈  and for all x, y ∈ E n , the following results are obtained: T(ax + y, ax + y) = a 2 T(x, x) + 2aT(x, y) + T(y, y) ≥ 0.

(1.11.9)

Since the left-hand side of the above inequality is a second-degree polynomial in the variable a, we can state that its discriminant is not positive, that is, |T(x, y)|2 − T(x, x)T(y, y) ≤ 0,

(1.11.10)

and (1.11.7) is proved. In order to prove (1.11.8), we start from the inequality T(x + y, x + y) = T(x, x) + 2T(x, y) + T(y, y) ≤ T(x, x) + 2|T(x, y)| + T(y, y), which, in view of (1.11.7), implies  T(x + y, x + y) ≤ T(x, x) + 2 T(x, x)T(y, y) + T(y, y)   = [ T(x, x) + T(x, x)]2 , 

and (1.11.8) is proved.

1.12 Pseudo-Euclidean Vector Spaces Definition 1.24 A pair (E n , g) of a vector space E n and a symmetric nondegenerate (0, 2)-tensor is called a pseudo-Euclidean vector space. In such a space, the tensor g introduces the scalar product x · y of two arbitrary vectors x, y ∈ E n , which is a mapping (x, y) ∈ E n × E n → x · y ∈ ,

1.12 Pseudo-Euclidean Vector Spaces

29

where x · y = g(x, y).

(1.12.1)

The scalar product has the following properties: x · y = y · x, x · (y + z) = x · y + x · z, a(x · y) = (ax) · y,

(1.12.2)

x · y = 0, ∀ y ∈ E n ⇒ x = 0, for all x, y, z ∈ E n . In fact, the first property follows from the symmetry of g; since g is bilinear, the second and third properties hold; finally, the fourth property follows from the positive definiteness of g. A vector x is said to be a unit vector if x · x = ±1.

(1.12.3)

x · y = 0.

(1.12.4)

Two vectors are orthogonal if In a basis (ei ) of E n , the scalar product determines a symmetric nonsingular matrix G = (gi j ), whose coefficients are gi j = ei · e j .

(1.12.5)

In contrast, if a symmetric nonsingular matrix G = (gi j ) is given, then a scalar product can be defined by the relation x · y = gi j x i y j

(1.12.6)

in any basis (ei ) of E n . Theorem 1.14 guarantees the existence of canonical bases (ei ) in which the representative matrix G of the scalar product has the diagonal form  Ir O , O −Is

 G=

(1.12.7)

where r + s = n. In these bases, called generalized orthonormal bases, the scalar product assumes the following form: x·y=

r  i=1

x i yi −

r +s  i=r +1

x i yi .

(1.12.8)

30

1 Tensor Algebra

1.13 Euclidean Vector Spaces Definition 1.25 A Euclidean vector space is a pair (E n , g), where E n is a vector space and g is a symmetric positive definite (0, 2)-tensor. In a Euclidean vector space, the scalar product satisfies the further condition x · x ≥ 0, ∀ x ∈ E n ,

(1.13.1)

where equality holds if and only if x = 0. The property (1.13.1) makes it possible to define the length or modulus of a vector as follows: |x| =

√ x · x.

(1.13.2)

Also, in a Euclidean space, the Cauchy–Schwarz and Minkowski inequalities (1.11.7) and (1.11.8) hold. With the new notation, they assume the following form: |x · y| ≤ |x||y|, |x + y| ≤ |x| + |y|.

(1.13.3) (1.13.4)

Noting that (1.3.8) is equivalent to the condition −1≤

x·y ≤ 1, |x||y|

(1.13.5)

we can define the angle ϕ between two vectors x and y by the equality cos ϕ =

x·y . |x||y|

(1.13.6)

It is easy to verify that the vectors belonging to an orthonormal system are independent; moreover, they form a basis if their number is equal to the dimension of E n . In any basis (ei ) of E n , the length of x ∈ E n and the angle between two vectors x and y assume the form |x| =



gi j x i x j ,

cos ϕ = 

i

(1.13.7) j

gi j x y  . gi j x i x j gi j y i y j

(1.13.8)

Definition 1.26 Let (ei ) be a basis of the Euclidean vector space E n . We define the covariant components of the vector x relative to the basis (ei ) as the quantities xi = x · ei = gi j x j .

(1.13.9)

1.13 Euclidean Vector Spaces

31

When a basis (ei ) is given, there is a one-to-one map between vectors and their contravariant components. The same property holds for the covariant components. In fact, it is sufficient to note that det(gi j ) = 0 and to refer to the linear relations (1.13.9). All the above formulas assume their simplest form relative to an orthonormal basis (ei ). In fact, since in such a basis one has ei · e j = δi j ,

(1.13.10)

we obtain also x·y=

n 

xi y j ,

(1.13.11)

i=1

  n  |x| =  (x i )2 ,

(1.13.12)

i=1

xi = xi .

(1.13.13)

After checking the advantage of the orthonormal bases, we understand the importance of the Gram–Schmidt orthonormalization procedure, which allows one to obtain an orthonormal basis (ui ) starting from any other basis (ei ). First, we put u1 = e1 .

(1.13.14)

Then we search for a vector u2 such that u2 = a21 u1 + e2 , u1 · u2 = 0.

(1.13.15) (1.13.16)

Introducing (1.13.16) into (1.13.15), we obtain the condition a21 u1 · u1 + u1 · e2 = 0. Since u1 = 0, the above condition allows us to determine a21 , and the vector u2 is not zero, owing to the linear independence of e1 and e2 . Then we search for a vector u3 such that u3 = a31 u1 + a22 u2 + u3 u1 · u3 = 0, u2 · u3 = 0.

32

1 Tensor Algebra

These relations imply the linear system a31 u1 · u1 + u1 · e3 = 0, a32 u2 · u2 + u2 · 33 = 0, which determines the unknowns a31 and a32 , since the vectors u1 and u2 do not vanish. Finally, the system (u1 , u2 , u3 ) is orthogonal. After n steps, an orthogonal system (u1 , . . . , un ) is determined. On dividing each vector of this system by its length, we obtain an orthonormal system. Definition 1.27 Let V be a vector subspace of E n . Then the set V⊥ = {x ∈ E n , x · y = 0, ∀ y ∈ V } ,

(1.13.17)

containing all the vectors that are orthogonal to every vector of V , is called the orthogonal complement of V . Theorem 1.17 If E n is a Euclidean vector space and V any vector subspace of E n , then V⊥ is a vector subspace of E n ; further, one has E n = V ⊕ V⊥ .

(1.13.18)

Proof If x1 , x2 ∈ V⊥ , a1 , a2 ∈ , and y ∈ V , we have that (a1 x1 + a2 x2 ) · y = a1 x1 · y + a2 x2 · y = 0, and a1 x1 + a2 x2 ∈ V⊥ . Further, we note that if (e1 , . . . , em ) is a basis of V , then every vector that is orthogonal to all the vectors of this basis is an element of V⊥ . In fact, when x · ei = 0, i = 1, . . . , n, we obtain for all y ∈ V that x·y=x·

m 

y i ei =

i=1

m 

y i x · ei = 0,

i=1

and then x ∈ V⊥ . For all x ∈ E n , we now put x = (x · e1 )e1 + · · · (x · em )em , x = x − x . Since the vector x is in V⊥ , the decomposition x = x + x is such that x ∈ V and x ∈ V⊥ . To prove that this decomposition is unique, we suppose that there is another decomposition y + y . Then it must be the case that (y − x ) + (y − x ) = 0, where the vector inside the first set of parentheses belongs to V , while the vector inside the other parentheses belongs to V⊥ . Finally, in a Euclidean space, the sum of two orthogonal vectors vanishes if and only if each of them vanishes, and the theorem is proved. 

1.14 Eigenvectors of Euclidean 2-Tensors

33

1.14 Eigenvectors of Euclidean 2-Tensors Definition 1.28 Let T be a (1, 1)-tensor of a Euclidean vector space E n . We say that the number λ ∈  and the vector x = 0 are, respectively, an eigenvalue of T and an eigenvector belonging to λ if λ and x satisfy the eigenvalue equation T(x) = λx.

(1.14.1)

The property that T is linear implies that the set Vλ of all the eigenvectors belonging to the same eigenvalue λ form a vector subspace of E n . In fact, if a, b ∈  and x, y ∈ Vλ , then we have that T(ax + by) = aT(x) + bT(y) = λ(ax + by) so that ax + by ∈ Vλ . Definition 1.29 The dimension of the vector subspace associated with the eigenvalue λ is called the geometric multiplicity of the eigenvalue λ; in particular, an eigenvalue with multiplicity 1 is also said to be simple. The set of all the eigenvalues of T is called the spectrum of T. Finally, the eigenvalue problem relative to T consists in determining the whole spectrum of T. In order to find the eigenvalues of T, we begin by noting that in a basis (ei ) of E n , the Eq. (1.14.1) can be written as (T ji − λδ ij )x j = 0,

i = 1, . . . , n.

(1.14.2)

This is a homogeneous linear system of n equations in the n unknowns x 1 , . . . , x n , which admits a solution different from zero if and only if Pn (λ) ≡ det(T ji − λδ ij ) = 0.

(1.14.3)

We now prove a fundamental property of the preceding equation: although the components T ji of the tensor T depend on the choice of the basis (ei ), the coefficients of the Eq. (1.14.3) do not depend on it. In fact, in the basis change ei = Ai e j , j

with the usual meaning of the symbols, we have that Pn (λ) = det(T − λI ) = [det A−1 (T − λI)A] = det A−1 det APn (λ), and then we have

Pn (λ) = Pn (λ).

(1.14.4)

34

1 Tensor Algebra

Since the polynomial Pn (λ) does not depend on the basis (ei ), it is called the characteristic polynomial of T. Denoting by Ii the coefficient of the power λn−i and noting that I0 = (−1)n , we can write Pn (λ) as follows: Pn (λ) = (−1)n λn + I1 λn−1 + · · · + In .

(1.14.5)

Remark 1.7 It is possible to verify that Ii = (−1)i Ji , i = 1, . . . , n,

(1.14.6)

where Ji is the sum of all the determinants of the principal minors of order i of the matrix T. In particular, I1 = T11 + · · · + Tnn and In = det T. In conclusion, we have proved the following result. Theorem 1.18 The eigenvalues of a (1, 1)-tensor T are the real roots of the characteristic polynomial Pn (λ) = (−1)n λn + I1 λn−1 + · · · + In = 0.

(1.14.7)

Definition 1.30 The Eq. (1.14.7) is the characteristic equation of the tensor T. Further, the multiplicity of a root λ of (1.14.7) is called the algebraic multiplicity of the eigenvalue λ. Let λ be any real roots of (1.14.7), i.e., an eigenvalue of the spectrum of T. Introducing λ into (1.14.3), we obtain a linear homogeneous system whose solutions form a subspace Vλ of eigenvectors. The dimension of Vλ is equal to k = n − p, where p is the rank of the matrix T − λI. In other words, we can find k independent eigenvectors u1 , . . . , uk belonging to Vλ that form a basis of Vλ . In particular, if there exists a basis of E n formed by eigenvectors of T belonging to the eigenvalues λ1 , . . . , λn , then the corresponding matrix T, i.e., the representative matrix of T, assumes the following diagonal form: ⎛

⎞ λ1 · · · 0 T = ⎝· · · · · · · · ·⎠ . 0 · · · λn

(1.14.8)

The following theorem, whose proof we omit, is very useful in applications. Theorem 1.19 Let T be a symmetric tensor of a Euclidean vector space E n . Then all the eigenvalues of T are real, and the dimension of the subspace Vλ associated with the eigenvalue λ is equal to the multiplicity of λ. Further, eigenvectors belonging to different eigenvalues are mutually orthogonal, and there exists at least one basis of eigenvectors of T relative to which the representative matrix T of T is diagonal.

1.15 Orthogonal Transformations

35

1.15 Orthogonal Transformations Definition 1.31 Let E n be a Euclidean n-dimensional vector space. An endomorphism Q : E n → E n is an orthogonal transformation if Q(x) · Q(y) = x · y, ∀ x, y ∈ E n .

(1.15.1)

We remark that if the basis (ei ) is orthonormal, then the n vectors Q(ei ) are independent and consequently form a basis of E n . Therefore, Q is an isomorphism, and the representative matrix Q of Q in any basis (ei ) is nonsingular. The condition (1.15.1) can be written in one of the following forms: ghk Q ik Q kj = gi j ,

(1.15.2)

Q GQ = G.

(1.15.3)

T

In particular, relative to the orthonormal basis (ei ), in which G = I, the above relations can be written as follows: QT Q = I ⇔ QT = Q−1 .

(1.15.4)

A matrix satisfying one of the conditions (1.15.4) is said to be orthogonal. Since the composite of two orthogonal transformations is still orthogonal, and two such transformations are the identity transformation and the inverse transformation, the set of all orthogonal transformations forms a group O(n), which is called the orthogonal group. In view of (1.15.4), it follows that det Q = ±1.

(1.15.5)

The orthogonal transformations of a three-dimensional Euclidean space E 3 are also called rotations; in particular, the rotations for which det Q = 1 are called proper rotations. The group of rotations is denoted by O(3), while the subgroup of proper rotations is denoted by SO(3). Finally, the orthogonal transformation −I is called the central inversion.

1.16 Exercises 1. The components of a (1, 1)-tensor T of the vector space E 3 relative to a basis (ei ⊗ θ j ) are given by the matrix ⎛

⎞ 121 ⎝2 1 2 ⎠ . 121

36

1 Tensor Algebra

Determine the vector corresponding to x = (1, 0, 1) by the linear endomorphism determined by T. 2. In the basis (ei ) of the vector space E 3 , two vectors x and y have components (1, 0, 1) and (2, 1, 0), respectively. Determine the components of x ⊗ y relative to the basis (ei ⊗ ej ), where e1 = e1 + e3 , e2 = 2e1 − e2 , e3 = e1 + e2 + e3 . 3. Given the (0, 2)-tensor Ti j θi ⊗ θ j of T2 (E 2 ), where   12 (Ti j ) = , 10 determine whether there exists a new basis in which its components become (Tij )



 11 = . 01

4. Given the (1, 1)-tensor T ji ei ⊗ θ j of T2 (E 2 ), verify that T11 + T22 and det(T ji ) are invariant with respect to a change of basis. 5. Prove that if the components of a (0, 2)-tensor T satisfy either of the conditions Ti j = T ji , Ti j = −T ji , in a given basis, then they satisfy the same conditions in every other basis. 6. Given the (0, 2)-tensors that in the basis (ei ), i = 1, 2, 3, have the components ⎛

⎞ 1 0 −1 T1 = ⎝ 0 −1 2 ⎠ , −1 2 1 ⎛

⎞ 0 1 −1 T2 = ⎝−1 0 2 ⎠ , 1 −2 0 determine the covector ω u , depending on u, such that ω u (v) = T1 (u, v), ω u (v) = T2 (u, v) for all v. Further, find the vectors u such that

1.16 Exercises

37

T1 (u, v) = 0, T2 (u, v) = 0, for all v. 7. Verify that in a vector space E 3 , a skew-symmetric (0, 2)-tensor T has the following form with respect to the basis (ei ): T = T12 θ 1 ∧ θ 2 + T13 θ 1 ∧ θ 3 + T23 θ 2 ∧ θ 3 , where (θi ) is the dual basis in E 3∗ of (ei ). Prove that the skew-symmetric (0, 2)tensor T = θ1 ∧ θ2 + θ1 ∧ θ3 + θ2 ∧ θ3 associates to every pair of vectors x, y of E 3 the sum of the areas of the projections of the parallelogram formed by the vectors x, y onto the subspaces generated by (e1 , e2 ), (e1 , e3 ), (e2 , e3 ), respectively. 8. Prove that the exterior product of T ∈ 2 (E 5 ) and L ∈ 2 (E 5 ) has the component T12 L 45 − T14 L 25 + T15 L 24 + T24 L 15 − T25 L 14 + T45 L 12 along the basis vector θ 1 ∧ θ 2 ∧ θ 4 ∧ θ 5 . 9. Given the 1−forms α = θ1 − θ2 , β = θ1 − θ2 + θ3 , σ = θ3 , the 2−form η = θ1 ∧ θ3 + θ2 ∧ θ3 , and the 3−form ω = θ1 ∧ θ2 ∧ θ3 , calculate the exterior products α ∧ β,

α ∧ β ∧ σ,

α ∧ η,

α ∧ ω.

10. Evaluate the components of the forms of the above exercises under the basis change θ 1 = θ 1 − 2θ 2 , θ 2 = θ 1 + θ 3 , θ 3 = θ 3 .

38

1 Tensor Algebra

11. Let (ei ) be a basis of the vector space E 3 and denote by (θi ) the dual basis. Given the skew-symmetric tensors T = T12 θ 1 ∧ θ 2 + T13 θ 1 ∧ θ 3 + T23 θ2 ∧ θ 3 , L = T123 θ 1 ∧ θ 2 ∧ θ 3 , and the basis change e1 = e1 − e3 , e2 = e1 + 2e2 , e3 = e2 − e3 ,

12. 13. 14. 15.

16.

determine the components of the above tensors in the corresponding new basis of 2 (E 3 ) and 3 (E 3 ). Determine the ratio between the volumes of the parallelepipeds formed by the above two bases. Write arbitrary skew-symmetric tensors of 2 (E 4 ) and 2 (E 5 ). Multiply a skew-symmetric tensor of 2 (E 4 ) by a skew-symmetric tensor of 3 (E 4 ). Given the volume form θ 1 ∧ θ 2 ∧ θ 3 in a three-dimensional space E 3 , determine the volume of the parallelepiped whose edges are the vectors (1, 0, 2), (−1, 2, 1), and (1, 1, 0). Evaluate the volume of the parallelepiped of the previous exercise, adopting the volume form θ 1 ∧ θ 2 ∧ θ 3 , where θ 1 = θ 1 + 2θ 2 , θ 2 = θ 2 + θ 3 , θ 3 = θ 1 − 2θ 3 .

Chapter 2

Introduction to Differentiable Manifolds

In this chapter, after a brief survey of the historical development of geometry, differentiable manifolds are defined together with many geometric structures equipping them as differentiable curves and functions, tangent and cotangent spaces, differential and codifferential of a map, tangent and cotangent fiber bundles, Riemannian manifolds, and geodesics.

2.1 Historical Introduction Around the year 300 B.C.E., in Alexandria, the great Greek mathematician Euclid wrote a monumental treatise consisting of 13 volumes: Elements of Geometry. This work contained not only all of the geometric knowledge of that time, but also many new results in geometry and number theory. However, the new approach to mathematical problems proposed in this treatise is much more important than its contents. This approach was based on the axiomatic method, whereby certain statements, called postulates, are assumed to be true, and then their consequences, the theorems, are deduced from them using the rules of logic. The postulates on which the whole Euclidian geometric edifice is constructed are five in number: the last of them is known as the parallel postulate. This postulate states that if r is any straight line and P is a point in the plane that does not belong to r , then there exists one and only one straight line in the plane containing P and parallel to r . Euclid himself considered this postulate less evident than the others. Consequently, he resorted to it only in the last part of his work. Over the following centuries, many attempts were made to deduce the parallel axiom from the other four axioms, but none of them were successful. In the middle of nineteenth century, the Hungarian mathematician János Bolyai and the Russian mathematician Nikolay Ivanovich Lobachevsky tried to prove that © Springer Nature Switzerland AG 2019 A. Romano and M. Mango Furnari, The Physical and Mathematical Foundations of the Theory of Relativity, https://doi.org/10.1007/978-3-030-27237-1_2

39

40

2 Introduction to Differentiable Manifolds

the parallel axiom is necessary by showing that if we deny it, then we obtain a contradictory geometry. More precisely, they supposed that there are infinitely many straight lines containing a point P and parallel to a given straight line. However, they did not obtain a logical contradiction. Instead, they produced a logically consistent geometry containing results that were completely different from the statements of Euclidean geometry. Although the existence of another coherent geometry had been proved, the mathematicians of that time considered it a purely mental construction without any relationship to physical reality. Later, Felix Klein, Henri Poincaré, and Eugenio Beltrami supplied concrete examples of this geometry, which was called hyperbolic geometry (see Sect. 2.4). In addition to the long established Cartesian analytic geometry, a new kind of geometry arose: differential geometry. The most important contributions were given by the eminent mathematicians Gaspard Monge and Carl Friedrich Gauss. Monge studied surfaces as boundaries of solid bodies and consequently analyzed the properties of a surface with respect to the surrounding space, often finding possible relationships using the theory of partial differential equations. Gauss approached the theory of surfaces primarily as a result of his work on triangulation, where the emphasis is on measurement between points on the Earth’s surface. Consequently, he considered a surface not as the boundary of a solid but as a two-dimensional entity not necessarily attached to a three-dimensional body. Gauss published the work Disquisitiones generales circa superficies curvas (1827), in which a revolutionary approach to analyzing surfaces was presented. In this work, Gauss made two fundamental contributions: • He highlighted the characteristics of a surface that can be obtained by an intrinsic analysis, i.e., by geometric measurements carried out on the surface itself without any relationship to the exterior space. In particular, hypothetical beings that live on the surface of a sphere can understand the nature of the space on which they live without leaving the surface. • The second contribution concerns the existence of a perfect chart, i.e., a representation of a part of a terrestrial surface in a plane conserving angles and length ratios. Gauss’s theorema egregium established the impossibility of realizing such a chart, which had been sought for many centuries. Gauss’s results and the new hyperbolic geometry refer to two-dimensional spaces. Riemann, a student of Gauss, in his Ph.D. thesis extended Gauss’s results to spaces of any finite dimension, called differentiable manifolds. Further, Riemann supposed that these infinite geometric constructions could represent real geometries of physical space, at least when their dimension was three. In other words, Riemann stated that geometry is an experimental science, so that we can establish which geometry holds in our space only by measures carried out in the space itself. Finally, Riemann supplied a geometric model of elliptic geometry, that is, a model in which the Euclidean parallel postulate is substituted by the following: given a straight line r and a point P that does not belong to r , there is no straight line parallel to r and containing P.

2.1 Historical Introduction

41

During the nineteenth century, mathematicians and physicists encountered numerous questions that had natural interpretations in terms of spaces of dimension greater than three. In the twentieth century, the non-Euclidean geometries were no longer considered merely abstract mathematical constructions but useful tools for describing the world around us. Minkowski spacetime is a first example of the use of these new geometries. Subsequently, Einstein arrived at a revolutionary vision of gravitation, adopting as spacetime a Riemannian manifold. Furthermore, Lagrangian and Hamiltonian mechanics were also formulated in terms of manifolds: the configuration space and phase space. However, the physical use of these geometric models required the extension to general manifolds of differential calculus. This extension revealed very difficult problems, which were solved by the fundamental researches of Gauss, Elwin Bruno Christoffel, Gregorio Ricci-Curbastro, Luigi Bianchi, Beltrami, Tullio LeviCivita, and Hermann Weyl. In this chapter and Chaps. 3–4 we will present the ideas underlying differential geometry, because they are essential in formulating the general theory of relativity. The rest of the chapter is organized as follows. In Sect. 2.2, we collect the main results about regular curves, while in Sect. 2.3, we discuss the main results on regular surfaces. Sections 2.4 and 2.5 are devoted to an analysis of some exterior and intrinsic properties of a surface, respectively. In particular, we discuss the parallel transport of a vector along a curve on a surface and the definition of geodesic curves. In Sect. 2.6, the Riemann tensor and Gauss’s theorema egregium are presented. In Sect. 2.7, we introduce differentiable manifolds. In order to develop the calculus on a differentiable manifold, differentiable functions and curves are introduced in Sect. 2.8. We introduce tangent and cotangent vector spaces in Sect. 2.8, the differential and codifferential of a map in Sects. 2.9–2.11, tangent and cotangent fiber bundles in Sect. 2.12. Finally, Riemannian manifolds and geodesics are introduced in Sect. 2.13.

2.2 Elements of the Geometry of Curves Curves are usually represented in parametric form, and they differ one from another by the way they bend and twist. The geometric properties associated with a curve such as curvature and arc length, are quantitatively measured by geometric invariants called the curvature and torsion of the curve. A fundamental theorem states that the values of these invariants completely determine the curve (see Theorem 2.1). Definition 2.1 Let n, r be finite natural numbers and let I be a nonempty interval of real numbers. A vector-valued function γ : t ∈ I → n

42

2 Introduction to Differentiable Manifolds

of class C r is called a parametric curve of class C r . Here t is the parameter of the curve γ, and γ(I ) is the image of γ. If I is a closed interval [a, b], then γ(a) is the starting point and γ(b) the ending point of the curve γ. If γ(a) = γ(b), then γ is said to be a closed curve or a loop. If γ : (a, b) → n is injective, then γ is a simple curve. We denote by −γ the curve γ traversed in the opposite direction. A C k -curve γ : I → n is called regular of order m if for every t ∈ I , {γ  (t), γ  (t), . . . , γ (m) (t)}, are linearly independent in n . Notice that Definition 2.1 implies that the curve γ changes if we change the parameter. Are there properties that do not depend on parametrization? The length l of the curve γ : [a, b] → n of class C 1 is defined as  l=

b

|γ  (t)|dt.

(2.2.1)

a

It is quite evident that the length of a curve does not depend on the parametrization. For each C r regular curve (r ≥ 1) γ : [a, b] → n , the function  s(t) =

t

   γ (x) d x

(2.2.2)

t0

represents the arc length between the points of γ corresponding to the values t0 , t ∈ [a, b], and its derivative is |γ  (t)|. Let t  = t  (t) be a C 1 function with the property dt  /dt = 0. Then we have γ¯  (t) = u  (t) · γ  (u(t)). If we use as parameter the arc length (2.2.2), then we obtain |γ  (s)| = 1 for all s ∈ [a, b], and s is called the natural parameter for γ. From now on, we take s as a parameter along γ. If γ is a C 1 curve, then the vector γ  (s0 ) = (dγ/ds)|s=s0 is called the unit tangent vector at the point P = γ(s0 ), and it is denoted by t(s). This vector determines the orientation of the curve, where the forward direction corresponds to the increasing values of the parameter. Differentiating |γ  | = γ  · γ  = 1 with respect to s, we have γ  · γ  = t · t = 0. Therefore, the vector t = dt/ds = k is orthogonal to the tangent vector t at every point P of γ. The vector k is called the curvature vector of γ. The plane spanned by the vectors t and k at every point of γ is called the osculating plane. The curvature vector reflects the rate at which a curve moves off its tangent line in the osculating plane and indicates also the deviation of the curve from being a plane curve.

2.2 Elements of the Geometry of Curves

43

At every point P of γ, the curvature vector can be written as follows: k(s) =

dt = κ(s)n(s), ds

(2.2.3)

where the unit vector n is the principal normal, and the scalar κ is the curvature at P ∈ γ. Finally, the scalar 1/κ(s) is called the radius of curvature. For each point P of γ, the binormal vector is defined as follows: b(s) = t(s) × n(s).

(2.2.4)

It is evident that this vector is orthogonal to the osculating plane. Differentiating (2.2.4), recalling (2.2.3), and noting that n · n = 1 implies n · dn/ds = 0, we obtain db = −τ (s)n(s), ds

(2.2.5)

where the scalar τ is called the torsion of the curve; it measures the rate of change of the osculating plane along the curve. Finally, differentiating n = t × b, we have dt db dn = ×b+t× , ds ds ds and taking into account (2.2.3) and (2.2.5), we get dn = −κ(s)t + τ (s)b. ds

(2.2.6)

The set {t(s), n(s), b(s)} defines, at every point P ∈ γ, an orthonormal basis of 3 , called the Frenet trihedron. Equations (2.2.3), (2.2.5), (2.2.6) can be written in the following matrix form: ⎤ ⎡ ⎤⎡ ⎤ t dt/ds 0 κ 0 ⎣dn/ds ⎦ = ⎣−κ 0 τ ⎦ ⎣n⎦ , db/ds 0 −τ 0 b ⎡

(2.2.7)

which together with dx/ds = t describes the motion of the trihedron along the curve. The equations κ = κ(s), τ = τ (s) are the natural or intrinsic equations of the curve, since the following theorem holds. Theorem 2.1 (Frenet–Serret theorem) Let κ(s) and τ (s) be arbitrary continuous functions on s ∈ [a, b]. Then there exists, except for its position in space, one and only one curve γ for which κ(s) is the curvature, τ (s) is the torsion, and s is the natural parameter along γ. See, for example, [160] for a proof of this theorem.

44

2 Introduction to Differentiable Manifolds

2.3 Elements of Geometry of Surfaces Let 3 be a three-dimensional Euclidean space and (O, ei ), i = 1, 2, 3, an orthonormal frame of reference in 3 (see Fig. 2.1). If r = x i ei denotes the position vector of any point of 3 , then a regular surface S is defined by a vector equation r = r(u 1 , u 2 )

(2.3.1)

x i = x i (u 1 , u 2 )

(2.3.2)

such that its components are functions of class C 2 whose Jacobian matrix J = ∂(x 1 , x 2 )/∂(u 1 , u 2 ) has rank 2. The relations ∂r ∂x i = ei , (2.3.3) aα ≡ r,α = ∂u α ∂u α α = 1, 2, define two vectors that are tangent to the coordinate curves on S. These vectors are linearly independent at every point of S, since the above hypotheses imply that (2.3.4) |a1 × a2 | = 0. Consequently, at every point r ∈ S, they form a basis for the tangent space Tr to S at r. This basis is called a coordinate basis or holonomic basis at r associated with the curvilinear coordinates u α . From (2.3.1) and (2.3.3) we derive the square of the line element ds that connects the nearby points r and r + dr:

where aαβ

ds 2 ≡ (dr)2 = aαβ du α du β ,

(2.3.5)

3  ∂x i ∂x i = aα · aβ = = aαβ ∂u α ∂u β i=1

(2.3.6)

Fig. 2.1 Surface of 3

n

x

a

e e x

r

S

a u

e coor

u

x

2.3 Elements of Geometry of Surfaces

45

are called metric coefficients. Since |a1 × a2 | =



(a1 × a2 ) · (a1 × a2 )

(a1 · a1 )(a2 · a2 ) − (a1 · a2 )2

2 = a11 a22 − a12 , =

we have

√ a,

(2.3.7)

a = det aαβ > 0.

(2.3.8)

|a1 × a2 | = where

If we introduce the reciprocal metric coefficients via the relations a αβ = then

Aαβ , (Aαβ = cofactor of aαβ ), a a αλ aλβ = δβα .

(2.3.9)

(2.3.10)

We note that the vectors aα are neither unit vectors nor mutually orthogonal. When they form an orthonormal basis at a point, the coordinates u α are said to be orthogonal. It is often useful to consider, besides the basis (aα ), the reciprocal basis formed by the independent vectors (2.3.11) aα = a αβ aβ , which satisfy the conditions

aα · aβ = δβα .

(2.3.12)

Due to (2.3.12), if v is a tangent vector to S, then we can write

where

v = v α aα = vα aα ,

(2.3.13)

vα = aαβ v β , v α = a αβ vβ .

(2.3.14)

Therefore, the contravariant components of v with respect to the reciprocal basis aα coincide with the covariant components of v with respect to the basis aα . Starting from the quadratic form (2.3.5), which represents the metric on S, or the first fundamental form of S, we can deduce the metric properties of the surface S. Thus, if a curve γ is given on S by the parametric equations

46

2 Introduction to Differentiable Manifolds

u α = u α (t), t ∈ [a, b],

(2.3.15)

its length l(γ) is given by 



b

l(γ) =

aαβ a

du α du β dt. dt dt

(2.3.16)

Similarly, consider a region σ ⊂ S obtained by varying the curvilinear coordinates over the set  = [a1 , b1 ] × [a2 , b2 ] ⊂ 2 . The area of the surface element dσ is defined by the relation dσ = |du 1 a1 × du 2 a2 | = 

so that we have σ=



√ a du 1 du 2 ,

√ a du 1 du 2 .

(2.3.17)

The preceding considerations show that all the metric evaluations over S are obtained from the first quadratic form. In other words, they are intrinsic characteristics of the surface and can be evaluated by anyone living on S without referring to the exterior space in which the surface is embedded.

2.4 The Second Fundamental Form In this section we analyze some properties of a surface from the outside, i.e., instead of discovering its intrinsic properties, we study the properties that can be evaluated from an external point of view. However, in doing that, we discover other fundamental intrinsic properties. On a regular surface S, the position n=

a1 × a2 |a1 × a2 |

(2.4.1)

defines, at least locally, a unit vector field that is orthogonal to S. We say that S is locally oriented when it is equipped with this vector field. Since the functions (2.3.2) are of class C 2 and aα · n = 0, we can define the following quantities (see (2.3.3)): bαβ = −aα · n,β = aα ,β · n = bβα ,

(2.4.2)

αβγ = aα · aβ ,γ = aα · r,βγ = αγβ ,

(2.4.3)

α βγ

αλ

= a λβγ .

(2.4.4)

2.4 The Second Fundamental Form

47

The quadratic form bαβ du α du β is called the second fundamental form of S, while α represent the Christoffel symbols of the first and second kinds, respecαβγ and βγ tively. It is now possible to prove the Gauss–Weingarten equations γ

aβ ,α = βα aγ + bαβ n,

(2.4.5)

−bαγ aγ ,

(2.4.6)

n,α = where γ

βα = In fact, we can write

1 γλ a (aλβ ,α + aαλ ,β −aαβ ,λ ). 2 n,α = cαβ aβ + dα n,

(2.4.7)

(2.4.8)

since (aα , n) is a basis of the three-dimensional space 3 . But the condition n · n = 1 implies that n · n,α = 0. Consequently, the scalar product of (2.4.8) and n gives dα = 0, since aβ · n = 0. Then (2.4.6) follows from (2.4.2). Furthermore, due to (2.3.9), (2.4.3), we have only to prove (2.4.7). To this end, we differentiate the relations aαβ = aα · aβ with respect to u λ and use (2.4.3) to obtain βαλ + αβλ = aαβ ,λ .

(2.4.9)

Cyclic permutations of the indices lead to the equations λβα + βλα = aβλ ,α , αλβ + λαβ = aλα ,β .

(2.4.10) (2.4.11)

By adding (2.4.10) to (2.4.9) and subtracting (2.4.11) to the result, we derive (2.4.7) when the symmetry properties of the Christoffel symbols are taken into account. Let γ be a curve on the surface S and let r = r(u α (s)) be its equation (s is the arc length on γ). If ξ is the curvature of γ and μ its principal normal unit vector, then the curvature vector k is expressed by Frenet’s formula: k = ξμ =

dλ , ds

(2.4.12)

where λ = λα aα is the unit vector tangent to γ. By (2.4.5), the above relation becomes k=

dλα α ν β + νβ λ λ aα + bαβ λα λβ n. ds

(2.4.13)

Let us define the normal curvature of γ at (u α ) as the quantity ξn (λ) = k · n = bαβ λα λβ .

(2.4.14)

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2 Introduction to Differentiable Manifolds

Fig. 2.2 Normal curvature

n S

cn

1/ c 1/ n

This formula shows that all of the surface curves that pass through a point r(u α ) have the same normal curvature. If γn is a curve whose osculating plane is determined by λ and n, then we have (2.4.15) ξn ≡ k · n = χμ · n = χ, and therefore the normal curvature of γn coincides with the normal curvature. Moreover, from the obvious inequality |ξn | = |χμ · n| ≤ |χ|,

(2.4.16)

we conclude that among all of the curves on S with the same unit tangent vector λ, the curve γn has minimal curvature; moreover, the center of curvature cn of γn coincides with the projection onto its osculating plane of the center of curvature c of all the curves tangent to λ (see Fig. 2.2). Let S be a regular surface of class C 2 and T2 (r) its tangent plane at the point r. The first fundamental form, which is positive definite, allows us to regard T2 (r) as a Euclidean two-dimensional vector space. A basis for it is given by (a1 , a2 ), and the scalar product is defined by the metric coefficients aαβ at r. The tensor bαβ , which represents the second normal form, is symmetric, and so there are two real eigenvalues, ϕ1 and ϕ2 , which could also be equal, together with an orthonormal basis comprising two eigenvectors v1 and v2 corresponding to the above eigenvalues. These eigenvectors v1 and v2 are solutions of the homogeneous linear system (bαβ − ϕi aαβ )v β = 0,

i = 1, 2,

(2.4.17)

whereas the eigenvalues ϕ1 and ϕ2 satisfy the characteristic equation det(bαβ − ϕi aαβ ) = 0,

i = 1, 2,

(2.4.18)

which can also be put into the form ϕ2 − 2H ϕ + K = 0,

(2.4.19)

2.4 The Second Fundamental Form

49

where H and K are the two principal invariants1 of (bαβ ): H=

1 α b , 2 α

K =

1 b det(bαβ ) = . a a

(2.4.20)

In the basis (v1 , v2 ), the tensor bαβ is represented by the diagonal matrix

ϕ1 0 , 0 ϕ2

(2.4.21)

and the quadratic form (2.4.14) can be written as χn (λ) = ϕ1 (λ1 )2 + ϕ2 (λ2 )2 ,

(2.4.22)

where (λ1 , λ2 ) are the components of the tangent vector λ with respect to the basis (v1 , v2 ). If χn (λ) does not vanish identically, then three cases can be distinguished: 1. ϕ1 , ϕ2 = 0 have the same sign. The quadratic form χn (λ) then has a definite sign at r, and so the normal curvature always has the same sign on varying λ. All of the points of S lie on the same side of the plane T2 (r). In this case, r is said to be an elliptic point (see Fig. 2.3a). 2. ϕ1 , ϕ2 = 0 have opposite signs. This means that the curvature along the two orthogonal lines defined by v1 and v2 have opposite signs. Therefore, the tangent plane T2 (r) intersects S in a neighborhood of r known as a hyperbolic point. Moreover, two directions λ1 and λ2 exist along which the curvature vanishes. These lines, called asymptotic lines, divide T2 (r) into four regions where the curvature is positive or negative in turn (see Fig. 2.3b). 3. ϕ1 = 0, ϕ2 = 0. In this case, χn (λ) = ϕ1 (λ1 )2 . Therefore, the normal curvature has the same sign along any direction that differs from v2 , and it vanishes when λ is parallel to v2 . The point r is said to be a parabolic point (see Fig. 2.3c). The eigenvalues of bαβ , which we henceforth denote by χ(1) and χ(2) , are called the principal curvatures of S at r, while the eigenvectors of bαβ are the principal directions of S at r. By Descartes’s rule, we have the relations below: χ(1) + χ(2) = 2H, χ(1) χ(2) = K .

(2.4.23)

By definition, the scalar quantity H is the mean curvature of S, and K is the Gaussian curvature. Formulas (2.4.20) show that the mean curvature at a point of S depends only on the second fundamental form bαβ du α du β . In contrast, the Gaussian curvature appears to depend on both the fundamental forms.

1 Invariants

of tensors are coefficients of the characteristic polynomial of the tensor A.

50

2 Introduction to Differentiable Manifolds

(a) Elliptic point

(b) Hyperbolic point

(c) Parabolic point Fig. 2.3 The three possible points of a surface

2.5 Parallel Transport and Geodesics In this section we again assume an exterior point of view to deduce other fundamental intrinsic properties of surfaces. Denote by u α = u α (t) a regular curve on the surface S and let v = v α aα be a vector field tangent to the surface S. In view of (2.4.5) and (2.4.6), we derive  γ  du α du α dv = v,α = v;α aγ + bαβ v β n , dt dt dt

(2.5.1)

where we have introduced the notation γ

γ

v;α = v,γα +αβ v β .

(2.5.2)

The left-hand side of (2.5.1) is the derivative of the vector field v along the curve u α (t) evaluated by an observer outside the surface S. The Gauss–Weingarten formulas allow us to decompose this derivative into the sum of two vectors. The first vector, which is given by (2.5.2), depends on the first fundamental form, so that it is intrinsic. The second vector depends on both the second fundamental form and the unit normal n to S. Consequently, it can be evaluated by an observer outside the surface. In other words, the variation of v along the curve u α (t) that can be appreciated by the observer on the surface is given by the first term on the right-hand side of (2.5.1). This vector, which we denote by α Dv γ du ≡ v;α aγ , dt dt

(2.5.3)

is called the covariant derivative of the tangent field v = v α aα along the curve u α (t).

2.5 Parallel Transport and Geodesics

51

Consider the vector field v = v α (t)aα , which is defined along the curve u α (t). We say that v is parallel transported along the curve u α (t) if Dv = dt



α dv λ λ β du v + αβ dt dt

aλ = 0.

(2.5.4)

Condition (2.5.4) is equivalent to the first-order differential system dv λ du α λ + αβ = 0, vβ dt dt

λ = 1, 2,

(2.5.5)

which admits a unique solution when the initial conditions v λ (0) = v0λ are given. Denote by u α (0) and u α (t1 ) the initial and final points of a curve u α (t) and by v0 the initial vector at u α (0). Then there is one vector v1 at the point u α (t1 ) that is parallel transported along u α (t). It is easy to verify that the transport does not depend on the parametrization of the curve. However, the vector v1 could depend on the curve along which the transport is carried out. It can be proved that v1 does not depend on the curve if and only if the Gaussian curvature vanishes. Let τ = (du α /dt) be the vector tangent to the curve γ with parametric equations α u (t). This curve is said to be a geodesic if its tangent vector is parallel transported along γ: α β α dτ λ d 2uλ λ β du λ du du τ +  + αβ = = 0. (2.5.6) αβ dt dt dt 2 dt dt This is a second-order system of two differential equations that admits a unique solution when the initial data are given: u λ (0) = u λ0 ,

du λ (0) = u λ 0 , λ = 1, 2. dt

(2.5.7)

In other words, given a point p of S and a vector v0 tangent to S at p, there is one and only one geodesic passing through p and tangent to v0 at p. We now analyze what happens to system (2.5.7) if we change the parametric u λ (s), where the function representation of the curve γ. If we put u λ (t) = u λ (t (s)) =  t (s) is invertible, then we have d u λ ds du λ = , dt ds dt

d 2uλ u λ ds d uλ d 2s d d + = dt 2 dt ds dt ds dt 2

2 2 λ ds u d  d uλ d 2s = + . ds 2 dt ds dt 2 Consequently, system (2.5.6) assumes the form

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2 Introduction to Differentiable Manifolds

2 λ 2 uβ dt d 2 uλ u α d d u d s λ d =− + αβ . 2 ds ds ds ds ds dt 2

(2.5.8)

Since ds/dt = 0, Eq. (2.5.8) again becomes the geodesic equation if and only if the parameters t and s are linearly related. This analysis shows that a geodesic is not only a locus of points of the surface S but a pair of parametric equations u α (t) in which the parameter t is defined up to a linear transformation t = as + b, where a and b are constants. Note that when we change the parameter, we determine a change of the tangent vector to the curve γ. Therefore, the tangent vector corresponding to a choice of the parameter could be parallel transported along γ, while the tangent vector corresponding to a different choice of the parameter might not satisfy this condition. All the preceding considerations can be extended to any tensor tangent to the surface S. For instance, given the 2-tensor T = T αβ aα ⊗ aβ

(2.5.9)

and a curve with parametric equations u α (t), we have du α dT = T,α . dt dt

(2.5.10)

Applying the Gauss–Weingarten formulas (2.4.5), (2.4.6), we can easily verify that  du α dT  βλ = T;α aβ ⊗ aλ + T βλ bλα aβ ⊗ n + T βλ bλα n ⊗ aβ , dt dt where

The tangent tensor

βλ

β

(2.5.11)

δλ λ T;α = T,βλ + αδ T βδ . α + αδ T

(2.5.12)

α DT βλ du = T;α aβ ⊗ aλ dt dt

(2.5.13)

is the covariant derivative of the tangent tensor T. It is evident how to extend the definition of parallel transport of T. It is also easy to verify that (2.4.5) and (2.5.12) imply (2.5.14) aαβ;λ = 0. Exercise 2.1 Prove that along a geodesic, the tangent vector τ satisfies the condition d (aαβ τ α τ β ) = 0, dt

(2.5.15)

which in particular implies that the curvilinear abscissa can be used as a parameter along a geodesic.

2.5 Parallel Transport and Geodesics

53

We conclude this section with the following theorem, which is stated without proof. Theorem 2.2 The geodesics of a surface are curves whose principal unit normal at any point coincides with the normal at the surface at that point.

2.6 An Example In this section we apply the preceding considerations to a hemisphere S. Let r be the position vector of a point p ∈ S with respect to a Cartesian frame of reference O x 1 x 2 x 3 . Then the components (x 1 , x 2 , x 3 ) of r relative to O x 1 x 2 x 3 are given by the following parametric equations: x 1 = r sin θ cos φ,

(2.6.1)

x = r sin θ sin φ,

(2.6.2)

x = r cos θ,

(2.6.3)

2

3

where θ and φ are respectively the colatitude and the longitude of p, and r is the radius of the hemisphere (see Fig. 2.4). In this example, θ and φ represent the curvilinear coordinates on S, and the coordinate curves are, respectively, the meridians and parallels. The vectors a1 and a2 tangent to the coordinate curves at p are a1 = r,θ = r (cos θ cos φe1 + cos θ sin φe2 − sin θe3 ),

(2.6.4)

a2 = r,φ = r (− sin θ sin φe1 + sin θ cos φe2 ).

(2.6.5)

It is important to note that all that we have introduced up to now is defined in the Euclidean three-dimensional space in which S is embedded. In view of (2.3.6), (2.6.4), and (2.6.5), we obtain the components of the metric tensor a11 = 1, a12 = 0, a22 = sin2 θ,

(2.6.6)

and the square of the elementary distance is ds 2 = r 2 dθ2 + r 2 sin2 θdφ2 .

(2.6.7)

Besides the spherical coordinates θ and φ, we can choose the polar coordinates ρ and φ in the plane O x 1 x 2 (see Fig. 2.5). This choice leads to the following equations of S: x 1 = ρ cos φ,

(2.6.8)

x = ρ sin φ,

(2.6.9)

2

54

2 Introduction to Differentiable Manifolds

Fig. 2.4 Coordinates (θ, φ) on a hemisphere

x p

S e

a

r

e

a x

e

x Fig. 2.5 Coordinates (ρ, φ) on a hemisphere

x p

S e e

e

r

a

a x

x

x3 =

r 2 − ρ2 ,

(2.6.10)

and the metric becomes ds 2 =

1−

1 2 2 2  ρ 2 dρ + ρ dφ .

(2.6.11)

r

All the characteristics of the intrinsic geometry of S can equivalently be derived from (2.6.7), (2.6.11) or by adopting another arbitrary pair (u 1 , u 2 ) of curvilinear coordinates on S that are related to (θ, φ) or (ρ, φ) by differentiable and invertible functions α = 1, 2. (2.6.12) u α = u α (θ, φ), u α = u α (ρ, φ), In view of Theorem 2.2, the family  of all the meridians crossing the north pole N are geodetics. Each geodesic of  is determined by the starting point N and the tangent vector at N . Denoting by σ the arc length from N to p along a geodesic, the pair (σ, φ) is a new system of curvilinear coordinates on the hemisphere S (see Fig. 2.6). The relation between (θ, φ) and (σ, φ) is given by the functions θ= and (2.6.7) becomes

σ , φ = φ, r

2.6 An Example

55

Fig. 2.6 Coordinates (σ, φ) on a hemisphere

N x

S

p a

e e

e

r

a x

x

ds 2 = dσ 2 + r 2 sin2

σ  r

dφ2 .

(2.6.13)

We now suppose that the hemisphere S is the two-dimensional world in which live two-dimensional beings. They set themselves the goal of discovering the (intrinsic) geometric properties of their world by geometric measures. The inhabitants are supposed to be equipped with inextensible2 one-dimensional rulers. To realize their task, they must succeed in determining the metric at every point p of S in an arbitrary system of curvilinear coordinates. Suppose that they carry out three measures of elementary lengths in three independent directions starting from an arbitrary point p. In such a way, they obtain three values of ds 2 that, inserted into the first fundamental form (2.3.5), supply a system of three equations in the unknowns a11 , a12 , and a22 . It is evident that this procedure requires knowledge of the coordinates of p. The meaning of the coordinates (θ, φ) or (ρ, φ) cannot be recognized by the beings living on S, since these coordinates are defined by resorting to the three-dimensional space. Put another way, (σ, φ) can be evaluated by the observers on S. These considerations lead us to classify the curvilinear coordinates on S in two classes: the physical coordinates that can be attributed to the points of S by the observers living on S and the mathematical coordinates that are defined by their mathematical relations with the physical coordinates. We now propose another hypothetical situation. Consider a two-dimensional world C given by the circle whose boundary ∂C is the intersection of S with the plane O x 1 x 2 in Fig. 2.6 and suppose that the two-dimensional inhabitants of C are equipped with rulers, which they again assume to be inextensible to make consistent any geometric measure. We now imagine that C is embedded into a three-dimensional space 3 whose inhabitants can observe the operations carried out by the inhabitants of C. Suppose that the observers of 3 discover a physical agent modifying the length of the rulers of C in a manner that depends on their position in C but not on their physical constitution. It is evident that this phenomenon cannot be revealed by the inhabitants of S. Denote by  : S → C the orthogonal projection of the points of S onto C and suppose that the rulers deform in such a way that the elementary distance ds between two arbitrarily close points p, q ∈ S and the distance between the image points ( p) and (q) satisfy the condition 2 This

is a fundamental assumption of any measure theory (see [19, 20, 132]).

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2 Introduction to Differentiable Manifolds

ds(( p), (q)) = ds( p, q).

(2.6.14)

Then the inhabitants of C will state that their world is a hemisphere, the half-lines from the center of C are geodesics, and the coordinates (ρ, φ) are physical coordinates in the circle C. The preceding example is analogous to Einstein’s example of a disk C under which a thermic source heats the area around the center O of C more than the boundary, which is held at a constant temperature less than the temperature in the central area. Let us dispose a certain number of rulers along a radius of a circle having its center at O. The rulers closer to O undergo an elongation greater than the elongation of the rulers close to the circumference. Therefore, the measure of the radius will be less than the Euclidean one. Consequently, the ratio between the circumference and the radius will be greater than 2π. We summarize the preceding considerations with the following remarks: Remark 2.1 The admissible curvilinear coordinates on a surface can be classified as mathematical coordinates and physical coordinates. The latter are the only coordinates to be used by an observer on the surface. It is possible to resort to the others when their relation with physical coordinates is known. Remark 2.2 In both the preceding examples, the circle and the disk are supposed to be uninfluenced by the physical agent deforming the rulers; in other words, they are intended to be immaterial. In the opposite case, they would, like the rulers, be influenced by this agent, and no effect of that agent on the geometry could be noted. We conclude this section with an exercise. Exercise 2.2 Evaluate the parallel transport of the vector v = (1, 0) tangent to the sphere S at the point A along the two curves ABC and ADC of Fig. 2.7. Show that the transport depends on the curve. The Christoffel symbols, which do not vanish, of the metric (2.6.7) are 1 2 = − sin θ cos θ, 11 = 22

Fig. 2.7 Parallel transport

cos θ . sin θ

x S

C

D r

x

A

v x

B

2.6 An Example

57

Consequently, the equations of parallel transport of a vector v become dφ dv 1 − sin θ cos θv 2 =0 dt dt

cos θ dθ dφ dv 2 + v2 + v1 = 0. dt sin θ dt dt The parametric equations of the curves ABC and ADC are (AB) θ = π/2, 0 ≤ φ ≤ α,

(BC) π/2 ≥ θ ≥ π/4, φ = α,

(AD) π/2 ≥ θ ≥ π/4 φ = 0,

(DC) θ = π/4, 0 ≤ φ ≤ α.

We limit ourselves to remarking that along the curve BC, the system of parallel transport reduces to dv 2 cos θ 2 dv 1 = 0, + v = 0, dθ dθ sin θ whose solution is v 2 = C/ sin θ, where C is an integration constant. Along the curve DC, we have v2 dv 2 dv 1 − = 0, + v 1 = 0, dφ 2 dφ whose solution is φ C2 φ v 1 = C1 cos √ + √ sin √ , 2 2 2

√ φ φ v 2 = C2 cos √ − 2C1 sin √ . 2 2

2.7 Riemann’s Tensor and the Theorema Egregium The great interest of the Gauss–Weingarten equations (2.4.5), (2.4.6) resides in the following considerations. They represent an overdetermined system of five vectorial partial differential equations in two vector unknowns r(u α ) and n(u α ). One can suppose that under suitable integrability conditions, the aforesaid system determines the surface S provided that the coefficients aαβ and bαβ of the two fundamental forms are known. The integrability conditions are relative to the right-hand sides, and therefore they restrict the functions aαβ and bαβ . To determine these conditions, we observe that owing to the supposed integrability, two C 3 functions r(u α ) and n(u α ) exist that satisfy (2.4.5), (2.4.6). From the evident necessary conditions aα,βλ = r,αβλ = aα,λβ ,

(2.7.1)

we deduce the necessary integrability conditions that have to be satisfied by the righthand sides of (2.4.5), (2.4.6). In fact, evaluating the partial derivatives of (2.4.5) and taking into account (2.4.6), we have

58

2 Introduction to Differentiable Manifolds γ

γ

ν ν aα ,βλ = (αβ γλ + αβ ,λ −bαβ bλν )aν + (αβ bγλ + bαβ ,λ )n

aα ,λβ =

γ ν (αλ γβ

+

ν αλ ,β

−bαλ bβν )aν

+

γ (αλ bγβ

+ bαλ ,β )n

(2.7.2) (2.7.3)

so that (2.7.1) takes the form  where

  ν − bαβ bλν − bαλ bβν aν + (bαβ;λ − bαλ;β )n = 0, Rαβλ γ

γ

ν ν ν ν ν = αβ,λ − αλ,β + αβ γλ − αλ γβ Rαβλ

(2.7.4)

(2.7.5)

are the components of the Riemann tensor. Conditions (2.7.4) are equivalent to the Codazzi–Mainardi equations (2.7.6) bαβ;λ = bαλ;β and the Gauss equations

ν = bαβ bλν − bαλ bβν . Rαβλ

(2.7.7)

It is possible to prove that (2.7.6) and (2.7.7) are also sufficient conditions for the integrability of the system (2.4.5), (2.4.6). In conclusion, the following theorem holds. Theorem 2.3 If the functions aαβ (u λ ) and bαβ (u λ ) of class C 3 satisfy the conditions (2.7.6), (2.7.7), then there exists (locally) one and only one surface that admits aαβ du α du β and bαβ du α du β as quadratic fundamental forms to within a rigid displacement. We are now in a position to prove the following fundamental theorem. Theorem 2.4 (Theorema Egregium (Gauss)) The Gaussian curvature K of a surface is invariant by local isometry. Proof If we introduce the covariant components of Riemann tensor δ Rναβγ = aνδ Rαβγ ,

(2.7.8)

2 − b11 b22 = −b. R1212 = b12

(2.7.9)

from (2.7.7) we obtain

In view of this result, we can write (2.4.20)2 as follows K =−

R1212 . a

(2.7.10)

In other words, the Gaussian curvature depends only on the first fundamental form, and the theorem is proved. 

2.7 Riemann’s Tensor and the Theorema Egregium

59

This theorem implies that if two surfaces are isometric, then their Gaussian curvatures are necessarily equal. A sphere with radius R has Gaussian curvature 1/R 2 , and a plane has Gaussian curvature equal to zero. Thus the perfect geographic chart does not exists.

2.8 Curvilinear Coordinates Let E 3 be a 3-dimensional Euclidean space. The position vector r of a point P ∈ E 3 , when Cartesian coordinates (x1 , x2 , x3 ) are adopted, is given by r = x1 e1 + x2 e2 + x3 e2 , where e1 , e2 , e3 are unit vectors along the Cartesian axes. Dropping the restriction that coordinate axes must be straight lines, we obtain curvilinear coordinates. These coordinates may be defined by applying an invertible transformation to the Cartesian coordinates ⎧ 1 ⎧ 1 1 2 3 1 ⎪ ⎪ ⎪q = g (x1 , x2 , x3 ), ⎪x1 = f (q , q , q ), ⎨ ⎨ ⇔ x2 = f 2 (q 1 , q 2 , q 3 ), q 2 = g 2 (x1 , x2 , x3 ), ⎪ ⎪ ⎪ ⎪ ⎩ 3 ⎩ x3 = f 3 (q 1 , q 2 , q 3 ), q = g 3 (x1 , x2 , x3 ). The surfaces q1 = constant, q2 = constant, q3 = constant are called coordinate surfaces; and the curves formed by the intersection of two coordinate surfaces are called the coordinate curves. In Cartesian coordinates, the standard basis vectors ei are obtained by differentiating the position vector with respect to xi . Similarly, in curvilinear coordinates the derivatives at P with respect to q i of the position vector define the natural basis vectors hi = ∂r/∂q1 , for i = 1, 2, 3. The basis vectors hi , which change their direction and/or magnitude from point to point, constitute what is called a local basis. Bases whose vectors are constant at every point are called global bases, and they exist only in rectilinear coordinates. In curvilinear coordinates, the vectors hi of a natural basis are neither perpendicular to each other nor of unit length. If they are orthogonal at all the points, the vectors bi = hi /|hi | define an orthonormal basis at every point. Since at every point we have both three intersecting coordinate surfaces and three intersecting coordinate curves, two different bases can be defined at any point P. A basis that is formed by vectors hi = ∂r/∂qi that are locally tangent to the coordinate curves at P, or a basis whose vectors hi = ∇qi are locally orthogonal to the surface defined by q i equal to a constant. Here ∇ is the gradient operator. Consequently, to any system of curvilinear coordinates, two sets of bases can be introduced at any point: the covariant basis (h1 , h2 , h3 } and the contravariant basis or reciprocal basis (h1 , h2 , h3 }. The covariant and contravariant bases have the identical direction for orthogonal curvilinear coordinate systems. It is evident that hi · h j = δ ij , where δ ij is the Kronecker symbol.

60

2 Introduction to Differentiable Manifolds

A vector v can be written as a linear combination of the vectors of both bases: v = v 1 h1 + v 2 h2 + v 3 h3 , v = v1 h1 + v2 h2 + v3 h3 . The quantities v i are called contravariant components (raised indices), while the quantities vi are called covariant components (lowered indices). It is easy to extend the above considerations to a Euclidean space E n , where n is a finite arbitrary positive integer. In the next section we generalize the preceding results to n-dimensional manifolds.

2.9 Differentiable Manifolds Let U be an open set of n . The real-valued function f : U →  is said to be of class C k (U ) or a C k function in U , where k ≥ 0, if it is continuous with its partial derivatives up to order k. In particular, a C 0 function in U is a continuous one. A map f : (x 1 , . . . , x n ) ∈ U → (y 1 , . . . , y m ) ∈ m is of class C k if every ith projection pr i ◦ f y i = pr i ◦ f (x 1 , . . . , x n ) ≡ y i (x 1 , . . . , x n ) is a C k function. A homeomorphism f : U → V , where V is an open set of n , is a continuous map with a continuous inverse. Finally, the map f is a diffeomorphism of class C k if both f and f −1 are C k maps. A diffeomorphism is represented by an invertible system of functions y i (x 1 , . . . , x n ), (x 1 , . . . , x n ) ∈ U , i = 1, . . . , n, of class C k together with the inverse functions. It is well known that the condition i ∂y = 0 det ∂x j 0 at the point (x01 , . . . , x0n ) ∈ U is a sufficient condition for the invertibility of these functions in a neighborhood of the point (x01 , . . . , x0n ). A differentiable manifold can be roughly defined as an n-dimensional surface embedded in m , n < m. This approach to the analysis of differentiable manifolds is more intuitive but not convenient for the reasons that we have already listed in Sect. 2.1. First, determining the lowest dimension n of the space n in which we can embed the manifold is not an easy task. For instance, plane curves can be embedded in 2 , whereas skew curves can be embedded in 3 . Further, in this approach the geometric objects on the manifold are defined starting from the space n in which they are embedded. It is much more interesting to build the geometry of a manifold in an intrinsic way. This approach makes it possible to answer questions like the

2.9 Differentiable Manifolds

61

Fig. 2.8 Chart on a manifold

X

U

xn

O U) x

following: Is it possible to recognize whether a manifold is a sphere staying on it? Is it possible recognize the geometric structure of the three-dimensional space in which we live by measures that are necessarily internal to our space? Definition 2.2 Let n be a positive integer and denote by X a Hausdorff3 paracompact topological space.4 Such a space X is said to be an n-dimensional manifold if for all x ∈ X , there exist an open neighborhood U of x and a homeomorphism ϕ : U → ϕ(U ) ⊆ n . The pair (U, ϕ) is called a chart of the domain U and coordinate map ϕ. Finally, the n numbers (x 1 , . . . , x n ) = ϕ(x) ∈ ϕ(U ) are the coordinates in the chart (U, ϕ) (see Fig. 2.8). Definition 2.3 An atlas of class C k on an n-dimensional manifold X is a collection α of charts on X satisfying the following conditions: • The collection of the domains of the charts of α is an open covering of X . • For all (U, ϕ), (V, ψ) ∈ α, the map ψ ◦ φ−1 : ϕ(U ∩ V ) → ψ(U ∩ V )

(2.9.1)

is a C k diffeomorphism, called coordinate transformation. A chart (V, ψ) is compatible with the atlas α if for all (U, ϕ) ∈ α, the map (2.9.1) is of class C k . We call the collection of the all charts compatible with α the maximal atlas α of X . Definition 2.4 The pair Vn = (X, α) is called an n-dimensional differentiable manifold of class C k . X is a Hausdorff  space if for all x, y ∈ X , x = y, there are neighborhoods U , V of x, y, respectively, such that U V = ∅. 4 A Hausdorff space is paracompact if every open covering contains a locally finite subcovering. 3 A topological space

62

2 Introduction to Differentiable Manifolds

A difficult theorem of Whitney proves that every C k manifold Vn , k ≤ 1, becomes an analytic manifold (i.e., the coordinate transformations between charts of an atlas are analytic diffeomorphisms) by discarding a suitable collection of C k charts belonging to the original maximal atlas. It is even more difficult to show that a C 0 manifold may fail to become a C 1 manifold. Now we show how we can obtain differentiable manifolds. • Let U ⊂ n be an open set and denote by (u 1 , . . . , u n ) a point of U . Let S be the locus of the points (x 1 , . . . , x l ) ∈ l , n < l, given by the set of C 1 functions x 1 = x 1 (u 1 , . . . , u n ), ................................ x l = x l (u 1 , . . . , u n ),

whose Jacobian matrix J=

∂x i ∂u α

(2.9.2)

,

i = 1, . . . , l, α = 1, . . . , n, has rank n at every point of U . In other words, S is a regular n-dimensional surface of l defined by the parametric equations (2.9.2). In particular, for l = 3 and n = 1, 2, we obtain regular curves and surfaces of 3 , respectively. We sketch the proof that all these regular surfaces are n-dimensional differentiable manifolds. First, S becomes a topological space when it is equipped with the topology induced by l . It is well known that the open sets of this topology are obtained by intersecting the open sets of l with S. Further, suppose that at the point x0 ∈ S, which is the image of (u 10 , . . . , u n0 ), the determinant of the minor

∂x i ∂u α

,

i = 1, . . . , n, α = 1, . . . , n, does not vanish. Then the first n Eq. (2.7.2) define a homeomorphism ϕ between a neighborhood U of x0 and a neighborhood ϕ(U ) of (u 10 , . . . , u n0 ). We don’t prove that all the coordinate transformations among these charts are of class C k . • Differentiable manifolds can also be obtained by the implicit representation of C k n-dimensional surfaces of l . Let S be such a surface implicitly defined by the following system: f 1 (x 1 , . . . , x l ) = 0, ............................, (2.9.3) f m (x 1 , . . . , x l ) = 0, where m < l, the functions f α , α = 1, . . . , m, are of class C k , and the Jacobian matrix

∂ fα , (2.9.4) J= ∂x i

2.9 Differentiable Manifolds

63

i = 1, . . . , l, has rank m. Again, S becomes a topological space with the topology induced by l . Further, let x0 = (x 1 , . . . , x l ) be a point of S and suppose that the determinant of the minor formed with the first m rows and m columns of (2.9.4) does not vanish at x0 . Then the m Eq. (2.9.3) can be written x 1 = x 1 (x m+1 , . . . , x l ), ................................... x m = x m (x m+1 , . . . , x l ), in a neighborhood V ⊂ n , n = l − m, of (x0m+1 , . . . , x l ). The above equations define a homeomorphism between V and the neighborhood  U = (x 1 (x m+1 , . . . , x l ), . . . , x m (x m+1 , . . . , x l ), x m+1 , . . . , x l ),  |(x m+1 , . . . , x l ) ∈ V

(2.9.5)

on S. • A manifold can be obtained by the topological product of two manifolds. Let Vn be a C k n-dimensional manifold and let Wm be a C k m-dimensional manifold. First, we equip Vn × Wm with the product topology. If α = {(Ui , ϕi )}i∈I is a C k atlas of Vn and β = {(V j , ψ j )} j∈J is a C k atlas of Wm , then it is easy to verify that {Ui × V j , (ϕi , ψ j )}(i, j)∈I ×J is a C k atlas of Vn × Wm , which becomes a C k (n + m)-dimensional manifold. • A manifold can be defined by a collection (Ui )i∈I of an open set of n and a set of diffeomorphisms between parts of them.

2.10 Differentiable Functions and Curves on Manifolds Definition 2.5 Let Vn be an n-dimensional differentiable manifold, and denote by α an atlas on Vn . We say that the real-valued function f : Vn →  is a C k function on Vn if the function (2.10.1) f ◦ ϕ−1 : ϕ(U ) →  is a C k function for all (U, ϕ) ∈ α. Definition 2.6 A C h curve γ on the C k manifold Vn , h ≤ k, is a map γ : [a, b] ⊆  → Vn such that for all (U, ϕ) ∈ α, where α is a C k atlas of Vn , the map ϕ ◦ γ : [a, b] → n is a C h map (see Fig. 2.9). The real-valued functions x i (t) = pr ◦ ϕ ◦ γ(t), t ∈ [a, b],

(2.10.2)

64

2 Introduction to Differentiable Manifolds

Vn

xn

U

U) O

a

O

t

b

x

Fig. 2.9 Curve on a manifold

Vn

F

U

V

xn

O U)

O

F

Wm

ym V)

y

x Fig. 2.10 Map between manifolds

are the parametric equations of γ in the chart (U, ϕ). The curve γ is closed if γ(a) = γ(b). Let (x i ) be the coordinates defined by the chart (U, ϕ). If x0 = (x0i ) ∈ U is a point of U , the ith coordinate curve at x0 is the curve with the following parametric equations: x 1 = x01 , ............., x i = t, (2.10.3) .............., x n = x0n . Definition 2.7 Let Vn be an n-dimensional manifold with a C k atlas α, and let Wm be an m-dimensional manifold with a C k atlas β. The map F : Vn → Wm is of class C k if

ψ ◦ F ◦ ϕ−1 : ϕ(U ) ⊆ n → ψ(V ) ⊆ m

is C k for all (U, ϕ) ∈ α and for all (V, ψ) ∈ β (see Fig. 2.10).

(2.10.4)

2.10 Differentiable Functions and Curves on Manifolds

65

If (x i ), i = 1, . . . , n, are the coordinates relative to the chart (U, ϕ) and (y α ), α = 1, . . . , m, the coordinates relative to (V, ψ), then the map (2.10.4) is equivalent to a system of m C k functions of n real variables: y α = y α (x 1 , . . . , x n ), α = 1, . . . , m.

(2.10.5)

2.11 Tangent Vector Space We denote by γ : [a, b] → Vn a C k curve on a differentiable manifold Vn and by F(x) the -vector space of the C k functions in a neighborhood of a point x = γ(t). Definition 2.8 The tangent vector to the curve γ at x is the map Xx : F(x) → 

such that Xx f =

d ( f ◦ γ(t)) dt

(2.11.1)

.

(2.11.2)

t

In other words, a tangent vector is defined as an operator that associates to every C k function about the point x of the curve γ the directional derivative along γ(t) at the point x. In the coordinates (x i ) relative to the chart (U, ϕ) on Vn , we have that f ◦ γ(t) = f ◦ ϕ−1 ◦ ϕ ◦ γ(t) = f ◦ ϕ−1 (x 1 (t), . . . , x n (t)), where (x 1 (t), . . . , x n (t)) are the parametric equations of γ in the chart (U, ϕ). Then (2.11.2) gives

dxi ∂ −1 . (2.11.3) Xx f = ( f ◦ ϕ ) ∂x i ϕ(x) dt To better understand the preceding definition, we consider a curve γ(t) = (x 1 (t), x 2 (t), x 3 (t)) in the Euclidean three-dimensional space E3 . The directional derivative of a C 1 function f (x 1 , x 2 , x 3 ) along γ(t) is given by d ( f (x 1 (t), x 2 (t), x 3 (t)) = (T · ∇)x f ≡ Xx f, dt

(2.11.4)

where T = (d x i (t)/dt) is the tangent vector to γ at the point x = (x i (t)). In other words, by (2.11.4), a derivation operator corresponds to each vector T. It is evident that if the directional derivatives of three independent functions are given at x, then (2.11.4) leads to a unique vector T. We note that our definition of tangent vector as derivative operator does not require an environment containing the manifold Vn .

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2 Introduction to Differentiable Manifolds

Consider the set Tx Vn of all the maps (2.11.1) obtained by varying the curve γ(t) at the point x ∈ Vn . This set, equipped with the operations (Xx + Yx ) f = Xx f + Yx f, (aXx ) f = aXx f, becomes an -vector space, which is called the tangent vector space to the manifold Vn at the point x. Theorem 2.5 Let (x i ) be a coordinate system relative to the chart (U, ϕ) of the manifold Vn . Then the relations

∂ ∂x i



f = x0

∂ ( f ◦ ϕ−1 ) ∂x i

(2.11.5) ϕ(x0 )

define n independent vectors tangent to the coordinate curves. Proof Denoting by γ i (t) the ith coordinate curve crossing x0 , that is, the curve with the parametric equations x 1 = x01 , . . . , x i = t, . . . , x n = x0n , the directional derivative along γ i (t) is j



dx d ∂ −1 ( f ◦ γ i (t)) = ( f ◦ ϕ ) j dt ∂x dt t0 t0 ϕ(x )



0 ∂ ∂ j ( f ◦ ϕ−1 ) δi = ( f ◦ ϕ−1 ) , i ∂x j ∂x ϕ(x0 ) ϕ(x0 )

and the relations (2.11.5) define n vectors tangent to the coordinate curves. Their linear independence is proved by applying the linear combination λ

j

∂ ∂x j

=0 x0

to the coordinate function x i = pr i ◦ ϕ and recalling (2.10.2). In fact, we obtain λj

∂ ∂x j

x i = λ j δ ij = λi = 0. x0

 From this result and (2.11.3) we obtain the following theorem. Theorem 2.6 The tangent space Tx Vn is an n-dimensional -vector space, and the vectors (∂/∂x i )x form a basis of Tx Vn , which is called the holonomic basis or

2.11 Tangent Vector Space

67

the natural basis relative to the coordinates (x i ). Therefore, for all Xx ∈ Tx Vn , the following result is obtained:

∂ i , (2.11.6) Xx = X ∂x i x where the real numbers X i are the components of Xx relative to the natural basis. It is fundamental to determine the transformation formulas of the natural bases and the components of a tangent vector for a change (x i ) → (x  j ) of local coordinates. From (2.10.2) we obtain that

∂ ∂x i

x

∂x j = ∂x i

and consequently, X i =



∂ ∂x j

≡ x

j Ai

∂ ∂x j

,

(2.11.7)

x

∂x i j X ≡ (A−1 )ij X j . ∂x j

(2.11.8)

2.12 Cotangent Vector Space Definition 2.9 The dual vector space of Tx Vn (see Sect. 2.1) is called the cotangent vector space Tx∗ Vn . If Tx Vn is referred to as the natural basis (∂/∂x i )x relative to the coordinates (x i ), the dual basis (θix ) is characterized by the conditions θix (Xx ) = X xi ,

(2.12.1)

where X xi are the components of Xx relative to the basis (∂/∂x i )x . Definition 2.10 If f ∈ F(x), the differential (d f )x of f at the point x ∈ Vn is the linear map (2.12.2) (d f )x : Tx Vn →  such that (d f )x Xx = Xx f, Xx ∈ Tx Vn .

(2.12.3)

In a natural basis relative to the coordinates (x i ) of the chart (U, ϕ) of Vn , (2.12.3) gives (2.12.4) (d f )x Xx = ai X xi ,

where ai =

∂ ∂x i

f. x

(2.12.5)

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2 Introduction to Differentiable Manifolds

In particular, for the differentials of the coordinate functions we obtain (d x )x Xx = Xx x = i

i

X xh

∂ ∂x i

x i = X xh δhi = X xi .

(2.12.6)

x

Comparing (2.12.6) and (2.12.2), we can state the following result. Theorem 2.7 The differentials (dx i )x of the coordinate functions form the dual basis of the cotangent space Tx∗ Vn . Consequently, every covector ω x ∈ Tx∗ Vn can be written as (2.12.7) ω x = ωi (dx i )x ,

where ωi = ω x

∂ ∂x i

.

(2.12.8)

x

Owing the results of Sect. 2.2, we can state that under the coordinate change (x i ) → (x i ), the following transformation formulas of the dual bases and the components of a covector hold (see (2.11.7)): (dx i )x =

∂x i (dx j )x = (A−1 )ij (dx j )x , ∂x j

ωi =

∂x j j ω j = Ai ω j . ∂x i

(2.12.9)

(2.12.10)

Starting from Tx Vn and Tx∗ Vn , it is  possible to build the whole tensor algebra as well as the exterior algebra ( s )x Vn at every point x ∈ Vn . In particular, the transformation formulas under a change of local coordinates (x i ) → (x i ) of the ···ir of any (r, s)-tensor belonging to (Tsr )x Vn , components T ji11···i s

(Tsr )x Vn

···ir T = T ji11···i s

∂ ∂ ⊗ · · · ⊗ i ⊗ dx j1 ⊗ · · · ⊗ dx js , i 1 ∂x ∂x 1

are T ji1 1···i···is r =

∂x i1 ∂x ir ∂x k1 ∂x ks h 1 ···hr · · · · · · T . ∂x h 1 ∂x hr ∂x  j1 ∂x is k1 ···ks

(2.12.11)

(2.12.12)

The preceding definitions can be extended to the whole manifold. A C k vector field is a map (2.12.13) X : x ∈ Vn → Xx ∈ Tx Vn . In local coordinates (x i ), the map (2.12.13) assumes the form X = X i (x 1 . . . x n )

∂ , ∂x i

(2.12.14)

2.12 Cotangent Vector Space

69

which differs from (2.11.6), since the components X i are C k functions of the coordinates. Similarly, a C k tensor field is a map T : x ∈ Vn → Tx ∈ (Tx )rs Vn .

(2.12.15)

In local coordinates (x i ), the map (2.12.13) assumes the form T = T ji11······ijsr (x 1 , . . . , x n )

∂ ∂ ⊗ · · · i ⊗ dx j1 ⊗ · · · ⊗ dx js , ∂x i1 ∂x r

(2.12.16)

where the components T ji11······ijsr (x 1 , . . . , x n ) are C k functions of the coordinates. In particular, a p-form is a map ω : x ∈ Vn → ω x ∈

  s x

Vn ,

(2.12.17)

which locally has the coordinate form ω =  j1 ··· js (x 1 , . . . , x n )dx i1 ∧ · · · ∧ dx is .

(2.12.18)

We conclude this section by introducing the Lie algebra of a vector field on a manifold Vn . Let Vn be a C ∞ n-dimensional manifold and denote by F ∞ Vn the -vector space of the C ∞ functions on Vn and by χ∞ Vn the -vector space of the C ∞ vector fields on Vn . Then to every C ∞ vector field X : x ∈ Vn → Xx ∈ Tx Vn we can associate the linear map X : f ∈ F ∞ Vn → X f ∈ F ∞ Vn

(2.12.19)

(X f )(x) = Xx f.

(2.12.20)

such that

It can be easily proved that the map (2.12.19) satisfies the following derivation property: (2.12.21) X( f g) = gX f + f Xg, ∀ f, g ∈ F ∞ Vn .

Definition 2.11 Let X, Y ∈ χ∞ Vn be two C ∞ vector fields. The bracket of X and Y is the C ∞ vector field [X, Y] such that [X, Y] f = (XY − YX) f, ∀ f ∈ F Vn .

(2.12.22)

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2 Introduction to Differentiable Manifolds

From (2.12.20) and (2.12.14), we obtain the following coordinate form of (2.12.22): [X, Y] =

i ∂Y i j ∂X X − Y ∂x j ∂x j j

∂ . ∂x i

(2.12.23)

It is not difficult to prove the following theorem. Theorem 2.8 The bracket operation satisfies the following properties: [X, Y] = −[Y, X], a[X, Y] = [aX, Y] = [X, aY], [X, Y + Z] = [X, Y] + [X, Z] , [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0,

(2.12.24)

for all a ∈  and for all X, Y, Z ∈ χ∞ Vn . This theorem proves that χ∞ Vn , equipped with the addition of vector fields, the product of a real number and a vector field, and the bracket operation is a Lie algebra.

2.13 Differential and Codifferential of a Map Definition 2.12 Let Vn and Wm be two C k manifolds with dimensions n and m, respectively, and let γ(t) be an arbitrary curve on Vn containing the point x ∈ Vn (see Fig. 2.11). The differential at the point x ∈ Vn of the C k map F : Vn → Wm is the linear map F∗x : Xx ∈ Tx Vn → Y F(x) ∈ TF(x) Wm

(2.13.1)

such that the image of the tangent vector Xx at x to the curve γ(t) is the tangent vector to the curve F(γ(t)) at the point F(x). Formally, we have Y F(x) g =

d g ◦ F ◦ γ(t), ∀g ∈ F(F(x)). dt

(2.13.2)

In order to find the coordinate representation of (2.13.2), we introduce a chart (U, ϕ) with coordinates (x i ), i = 1, . . . , n, in a neighborhood of x ∈ Vn and a chart (V, ψ), with coordinates (y α ), α = 1, . . . , m, in a neighborhood V of F(x). We denote by y α = y α (x 1 , . . . , x n ) the coordinate form of the map F and by x i (t) the parametric equations of the curve γ(t) in the coordinates (x i ). Then (2.13.2) can be written as follows: Y α (y β (x i ))

α ∂ ∂ d x i ∂ yα ∂ i ∂y g = g ≡ X g, ∀ f ∈ F(F(x)). ∂x α dt ∂x i ∂ y α ∂x i ∂ y α

2.13 Differential and Codifferential of a Map

Vn

Xx

U

71

Fx F

x

V

YF(x) Wm

t xn

O

ym

O

F

U)

V) y

x Fig. 2.11 Differential of a map F

In conclusion, the coordinate form of (2.13.1) is F∗x : Xx = X

i

∂ ∂x i

x

∂ yα i ∈ Tx Vn → X ∂x i



∂ ∂ yα

∈ TF(x) Wm .

(2.13.3)

F(x)

Starting from the linear map (2.13.1), we can give the following definition: Definition 2.13 The linear map ∗ ∗ : σ F(x) ∈ TF(x) Wm → ω x ∈ Tx∗ Vn , FF(x)

(2.13.4)

∗ Wm and the dual space Tx∗ Vn , defined by the condition between the dual space TF(x)

ω x (Xx ) = σ F(x) (F∗x Xx ), ∀ Xx ∈ Tx Vn ,

(2.13.5)

is called the codifferential of F : Vn → Wm at F(x). Adopting coordinates (x i ) on Vn and (y α ) of Wm , we can write (2.13.5) as ωi X i = σα

∂ yα i X , ∂x i

and taking into account the arbitrariness of Xx , we obtain ωi = σα

∂ yα . ∂x i

(2.13.6)

In conclusion, the coordinate form of (2.13.4) is ∗ ∗ FF(x) : σα dy α ∈ TF(x) W m → σα

∂ yα i dx ∈ Tx∗ Vn . ∂x i

(2.13.7)

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2 Introduction to Differentiable Manifolds

We now consider the extension of the differential F∗ of F that is the new linear map, denoted by the same symbol, T F(x) ∈ (T0r ) F(x) Wm , F∗x : Tx ∈ (T0r )x Vn → 

(2.13.8)

F∗x (X1 ⊗ . . . ⊗ Xr ) = F∗x X1 ⊗ . . . ⊗ F∗x Xr ,

(2.13.9)

such that for all X1 , . . . , Xr ∈ Tx Vn . It can be easily verified that the map (2.13.8) transforms the (r, 0)-tensor ∂ ∂ T = T i1 ···ir i ⊗ . . . ⊗ i ∂x 1 ∂x r into the (r, 0)-tensor ∂ y α1 ∂ y αr i1 ···ir ∂ ∂  T= · · · T ⊗ ... ⊗ α . i i α 1 r 1 ∂x ∂x ∂y ∂y r

(2.13.10)

Similarly, we can extend (2.13.4) by the linear map ∗ FF(x) : T F(x) ∈ (Ts0 ) F(x) Wm →  Tx ∈ (Ts0 )x Vn

(2.13.11)

such that ∗ ∗ ∗ (σ 1 ⊗ · · · ⊗ σ s ) = FF(x) σ 1 ⊗ · · · ⊗ FF(x) σs , FF(x)

(2.13.12)

∗ ∗ . It is evident that FF(x) maps the (0, s)-tensor for all σ 1 , . . . σ s ∈ TF(x)

T = Tα1 ···αs dy α1 ⊗ · · · ⊗ dy αs ∈ (Ts0 ) F(x) Wm into the (0, s)-tensor ∂ y α1 ∂ y αs  T= · · · Tα ···α dx i1 ⊗ · · · ⊗ dx is ∈ (Ts0 )x Vn . ∂x i1 ∂x is 1 s

(2.13.13)

Finally, we can define the linear map Fx∗ : ω F(x) ∈ (s )x Wm → ω  ∈ (s )x Vn

(2.13.14)

∗ ∗ ∗ (σ 1 ∧ · · · ∧ σ s ) = FF(x) σ 1 ∧ · · · ∧ FF(x) σs , FF(x)

(2.13.15)

such that

∗ . for all σ1 , . . . , σs ∈ TF(x)

2.13 Differential and Codifferential of a Map

73

Again it is simple to verify that (2.13.14) maps ω = (α1 ···αs ) dy α1 ∧ · · · ∧ dy αs ∈ (s ) F(x) Wm into α1 αs  = ∂(y , . . . , y ) (α1 ···αs ) dx i1 ∧ · · · ∧ dx is ∈ (s )x Vn .  ∂(x i1 , . . . , x is )

(2.13.16)

Remark 2.3 It is important to note that in general, none of the preceding linear maps can be extended over the whole manifolds Vn and Wm , because F : Vn → Wm can be neither one-to-one nor onto. For instance, the vector field X : x ∈ Vn → F∗x Xx is defined on F(Vn ), and it can assume more values at the same point if F is not one-to-one. If F : Vn → Wm is a diffeomorphism, then n = m, and we can define an isomorphism (2.13.17) F∗x : (Tsr )x Vn → (Tsr ) F(x) Wn such that −1 ωs . F∗x (X1 ⊗ · · · ⊗ Xr ⊗ ω 1 ⊗ · · · ⊗ ω s ) = F∗x X1 ⊗ · · · ⊗ F∗x

(2.13.18)

Once again, it is simple to verify that the linearity of (2.13.17) implies that (2.13.17) maps the tensor T = T ji11······ijsr

∂ ∂ ⊗ · · · ⊗ i ⊗ dx j1 ⊗ · · · ⊗ dx js ∈ (Tsr )x Vn i 1 ∂x ∂x r

into the tensor ∂ y α1 ∂ y αr ∂x j1 ∂x js i1 ···ir ∂ · · · · · · T ⊗ ∂x ii ∂x ir ∂ y β1 ∂ y βs j1 ··· js ∂ y α1 ∂ · · · ⊗ α ⊗ dy β1 ⊗ · · · ⊗ dy βs ∈ (Tsr ) F(x) Wn . ∂y r

F∗x T =

(2.13.19)

In conclusion, if F : Vn → Wm is a diffeomorphism, then n = m and the tensor and exterior algebras on Vn and Wn at the points x ∈ Vn and F(x) ∈ Wn are isomorphic. Further, on varying x ∈ Vn , F∗x maps (r, s)-tensor fields of Vn onto (r, s)-tensor fields of Wn .

2.14 Tangent and Cotangent Fiber Bundles Given the C k manifold Vn , consider the set T Vn = {(x, Xx ), x ∈ Vn , Xx ∈ Tx Vn } .

(2.14.1)

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2 Introduction to Differentiable Manifolds

The map π : (x, Xx ) ∈ T Vn → x ∈ Vn

(2.14.2)

is called the projection map, and the preimage π −1 (x) = {x} × Tx Vn is called the fiber on x. If (U, ϕ) is a chart on Vn , then every x ∈ U is determined by its coordinates (x i ). Further, every vector Xx ∈ Tx Vn , where x ∈ U , is determined by its components X i relative to the natural basis (∂/∂x i )x . In this way, we have defined a one-to-one correspondence φ : U → ϕ(U ) × n , where U = {(x, Xx ), x ∈ U, Xx ∈ Tx Vn } ⊆ T Vn . The set T Vn becomes a Hausdorff topological space when we equip it with the topology whose open sets have the form U × I n , with U an open set of Vn and I n an open set of n . It is not difficult to verify that the map φ is a homeomorphism, so that T Vn is a topological manifold and (U × n , φ) is a chart of T Vn . Collecting the domains U of the charts of an atlas of Vn , we define an atlas of T Vn that becomes a 2n-dimensional topological manifold. We do not prove that if Vn is a C k manifold, then T Vn is a C k manifold. Definition 2.14 The 2n-dimensional manifold T Vn is called the tangent fiber bundle of Vn , and the above coordinates (x i , X i ) are called the natural coordinates of T Vn . For instance, the tangent fiber bundle T S 1 of a circumference S 1 is the collection of the all pairs (x, Xx ), where x ∈ S 1 and Xx is a tangent vector to S 1 at the point x. The fiber at x is the straight line tangent to S 1 at x. It is evident that T S 1 is diffeomorphic to a cylinder S 1 × . All that we have said about T Vn can be repeated starting from the set   T ∗ Vn = (x, ω x ), x ∈ Vn , ω x ∈ Tx∗ Vn .

(2.14.3)

The corresponding C k 2n-manifold is called the cotangent fiber bundle of Vn . Natural coordinates in the open set U × n , where U is an open set of a chart of an atlas of Vn , are given by (x i , ωi ), where (x i ) are the coordinates of a point x ∈ U , and (ωi ) are the components of a covector ω x ∈ Tx∗ Vn in the dual basis (dx i ).

2.15 Riemannian Manifolds Before giving the definition of a Riemannian manifold, we consider two examples. First, let S 2 be the unit sphere embedded in the three-dimensional Euclidean

2.15 Riemannian Manifolds

75

space E3 with Cartesian coordinates (x i ) (see Fig. 2.12). The parametric equations of S 2 are x 1 = sin θ cos ϕ, x 2 = sin θ sin ϕ, x 3 = cos θ. We have already said that every regular surface in Euclidean space is a differentiable manifold. Now we wish to equip the sphere with the metrics induced by the metrics of E3 . This means that the squared distance ds 2 between two points (x i ) and (x i + d x i ) of S 2 is assumed to be equal to the Pythagorean distance in E3 , ds = 2

3 

(d x i )2 ,

(2.15.1)

i=1

but expressed in terms of the coordinates (ϕ, θ) on S 2 . To this end, we differentiate the parametric equations of the surface and obtain ds 2 = sin2 θ dϕ2 + dθ2 ,

(2.15.2)

where ϕ ∈ [0, 2π] and θ ∈ [0, π]. As a second example we consider the ellipsoid  2 in E3 with parametric equations x 1 = sin θ cos ϕ, x 2 = sin θ sin ϕ, x 3 = a cos θ,

Fig. 2.12 Sphere in E3

x

O

x

x

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2 Introduction to Differentiable Manifolds

Fig. 2.13 Ellipsoid in E3

x

O

x

x

with axes 2, 2, and 2a. Again adopting the above viewpoint, we see that the squared distance between two points (x i ) and (x i + d x i ) of  2 becomes ds 2 = sin2 dϕ2 + (cos2 θ + a sin2 θ)dθ2 .

(2.15.3)

The sphere S 2 and the ellipsoid  2 are diffeomorphic, so that they are essentially the same manifold. In particular, the variables ϕ and θ are local coordinates for both surfaces. These two manifolds become different from each other when we introduce a metric structure determined by the squared distance ds 2 (Fig. 2.13). Definition 2.15 A C k differentiable n-dimensional manifold Vn is a Riemannian manifold if on Vn , a metric tensor, that is, a C k (0, 2)-symmetric nondegenerate tensor field g, is given. Owing to the properties of g, we can introduce in any tangent space Tx Vn a scalar product of two arbitrary tangent vectors Xx and Yx , Xx · Yx = gx (Xx , Yx ),

(2.15.4)

and Tx Vn becomes a pseudo-Euclidean vector space. In other words, the whole tensor algebra at each point of a Riemannian manifold becomes a pseudo-Euclidean tensor algebra. In local coordinates (x i ) relative to a chart (U, ϕ) of Vn , the tensor g assumes the following representation: g = gi j dx i ⊗ dx j , gi j = g ji , where det(gi j ) = 0, and the scalar product (2.15.4) becomes

(2.15.5)

2.15 Riemannian Manifolds

77

Xx · Yx = gi j X i Y j .

(2.15.6)

In view of the results of Chap. 1 about symmetric (0, 2)-tensors, it is always possible to find, about any point x ∈ Vn , a coordinate system such that at x, one has gi j =

∂ ∂ · = ±δi j . ∂x i ∂x j

(2.15.7)

The set {1, . . . , 1, −1, . . . , −1} is called the signature of the metric tensor, and it is independent of the coordinates. Finally, we define the squared distance ds 2 between two nearby points (x i ) and i (x + d x i ) as the quantity (2.15.8) ds 2 = gi j d x i d x j .

2.16 Geodesics over Riemannian Manifolds In this section we extend to Riemannian manifolds the notion of geodesic curve over surfaces embedded in Euclidean space 3 ; see Sect. 2.6. Definition 2.16 Let Vn be a Riemannian manifold and suppose that the tensor gx is positive definite at every point x ∈ Vn . The length of a C 1 curve γ(t) : [a, b] ⊆  → Vn is the real number  l(γ) =

ds.

(2.16.1)

γ

Denote by x i (t), t ∈ [a, b], the parametric equations of γ(t) in an arbitrary system of coordinates (x i ), whose domain contains the curve γ(t). Then the length l(γ) of γ(t) assumes the form  b

l(γ) = gi j (x h )x˙ i x˙ j dt. (2.16.2) a

It is evident that l(γ) depends neither on the choice of coordinates nor on the parametrization of the curve. Let (xai ) = (x i (a)) and (xbi ) = (x i (b)) be the initial and final points of γ(t) and consider the one-parameter family  of curves f i (s, t) : (−, ) × [a, b] → Vn

(2.16.3)

such that • it includes γ(t) for s = 0,

f i (0, t) = x i (t);

• every curve starts from (xai ) and ends at (xbi ), i.e.,

(2.16.4)

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2 Introduction to Differentiable Manifolds

f i (s, a) = xai ,

f i (s, b) = xbi , ∀ s ∈ (−, ).

(2.16.5)

It is evident that the length of every curve of  is given by the integral 

b

I (s) =



gi j ( f h (s, t)) f˙i f˙ j dt.

(2.16.6)

a

Definition 2.17 (Geodesic) The curve γ(t) between the points (xai ) and (xbi ) is a geodesic of the metrics g if the function I (s) is stationary for s = 0, for every family  of curves satisfying (2.16.4) and (2.16.5). In order to determine the parametric equations of the geodesic between the points (xai ) and (xbi ), we begin by analyzing the stationarity condition dI (0) = 0. ds Introducing the position L( f h , f˙h ) =

(2.16.7)

gi j f˙i f˙ j

(2.16.8)

and taking into account (2.16.6), we obtain that dI (s) = ds Since



b



a

∂ ∂ L ∂ f˙h = h ˙ ∂t ∂ f ∂s



∂L ∂ f h ∂ L ∂ f˙h + dt. ∂ f h ∂s ∂ f˙h ∂s

∂L ∂ f h ∂ f˙h ∂s

−

∂ ∂t



∂L ∂ f˙h

∂fh , ∂s

we have that dI (s) = ds

 a

b



∂L ∂ ∂L − h ∂f ∂t ∂ f˙h

 ∂fh ∂L ∂ f h (0, t)dt + ∂s ∂ f˙h ∂s

b

.

(2.16.9)

a

From this formula, when we recall (2.16.4) and (2.16.5), we deduce that the condition (d I /ds)(0) = 0 is equivalent to requiring the equation  a

b



∂L j j d ∂L j j ∂ f h (0, t)dt = 0, (x , x ˙ ) − (x , x ˙ ) ∂x h dt ∂ x˙ h ∂s

(2.16.10)

to be satisfied for every choice of the functions (∂ f h /∂s)(0, t), h = 1, . . . , n. It is possible to prove that this happens if and only if the parametric equations x i (t) of the curve γ(t) satisfy the Euler–Lagrange equations

2.16 Geodesics over Riemannian Manifolds

79

∂L j j d ∂L j j (x , x˙ ) − (x , x˙ ) = 0, h = 1, . . . , n, h ∂x dt ∂ x˙ h

(2.16.11)

together with the following boundary conditions: x h (a) = xah , x h (b) = xbh , h = 1, . . . , n.

(2.16.12)

Remark 2.4 In the Cauchy problem relative to the Eq. (2.16.11), we must assign the initial conditions x h (0) = xah and x˙ h (0) = X h , h = 1, . . . , n. It is well known that under general hypotheses on the function L(x i , x˙ i ), there is only one solution satisfying the Euler–Lagrange equations and the initial data, that is, there is only one geodesic starting from the point (xai ) and having at that point the tangent vector (X i ). In contrast, there is no general theorem about the boundary problem (2.16.11), (2.16.12), so that it could happen that there is one and only one geodesic between the points (xah ) and (xbh ), no geodesic, or infinitely many geodesics. In this last case, we say that (xah ) and (xbh ) are focal points. Remark 2.5 A curve γ(t) has been defined as a map γ : t ∈ [a, b] → Vn . Consequently, a change of the parameter in the parametric equations x h (t) leads to a different curve, although the locus of points is the same. On the other hand, the presence of the square root under the integral (2.16.2) implies that the value of the length l(γ) is not modified by a change of the parameter t; in other words, l(γ) has the same value for all curves determining the same set of the points of Vn . This remark implies that the Eq. (2.16.11) cannot be independent. To show that only n − 1 of these equations are independent, it is sufficient to notice that from the identity n 

∂L , ∂ x˙ i

(2.16.13)

2  ∂L i ∂ L = x ˙ , ∂x h ∂ x˙ i ∂x h i=1

(2.16.14)

2  ∂L ∂L i ∂ L = + x ˙ , ∂ x˙ h ∂ x˙ h ∂ x˙ i ∂ x˙ h i=1

(2.16.15)

L=

x˙ i

i=1

it follows that n

n

and then one has

n  i=1

x˙ i

∂2 L = 0. ∂ x˙ i ∂ x˙ h

(2.16.16)

80

2 Introduction to Differentiable Manifolds

From the above conditions follows the identity n  ∂L i=1

=

∂x i

n  i=1

−

d ∂L dt ∂ x˙ i

2 2   ∂L i ∂ L h i ∂ L − x ˙ x ˙ − x ˙ x¨ h , i ∂x h i ∂ x˙ h ∂x i ∂ x ˙ ∂ x ˙ i=1 i=1 n

x˙ i

n

which in view of (2.16.14) and (2.16.15), becomes n  ∂L i=1

d ∂L − i ∂x dt ∂ x˙ i

= 0.

(2.16.17)

This relation shows that if the parametric equations x h (t) satisfy the first n − 1 Euler–Lagrange equations, then they satisfy the last one. We use the arbitrariness of the parameter t to obtain a new form of the equations of a geodesic. Putting √ L = ϕ, (2.16.18) we can give (2.16.11) the following form: d 1 ∂ϕ − √ h 2 ϕ ∂x dt



1 ∂ϕ √ 2 ϕ ∂ x˙ h

= 0, h = 1, . . . , n.

(2.16.19)

If we choose a parameter s for which the tangent vector (x˙ h ) to the geodesic has unit length, i.e., a parameter s such that ϕ = gi j x˙ i x˙ j = 1, x˙ h =

dxh , ds

(2.16.20)

then the Euler–Lagrange equations assume the form 1 ∂ d (gi j x˙ i x˙ j ) − (gh j x˙ j ) = 0, h = 1, . . . , n. 2 ∂x h ds

(2.16.21)

These equations define a geodesic in terms of the Riemannian metric. Let x i (s) be a curve in Vn parametrized by its arc length s. Then the components of the principal normal vector μ are given by μi =

j k d2xi i dx dx = ∇ ti . +  jk ds 2 ds ds

(2.16.22)

The tangent vector t and the principal normal vector μ relative to a curve are mutually perpendicular. In fact, from t · t, we have

2.16 Geodesics over Riemannian Manifolds

81

d dt = t · μ = 0. (t · t) = 0 ⇒ t · ds ds Furthermore, it is natural to define the curvature of a curve to be the quantity 1 = ρ

  gi j μi μ j .

In view of (2.16.22), the curvature vanishes along a geodesic. The following theorem is evident. Theorem 2.9 If the metric of Vn is definite, then only geodesics have zero curvature. If the metric is indefinite, then curves with zero curvature are either geodesics or curves with a null tangent vector.

2.17 Exercises • This exercise shows how we can obtain a geographic chart using a map between two manifolds. Let S 2 − {N } be the unit sphere minus the north pole. The diffeomorphism F, called stereographic projection, between S 2 − {N } and the tangent plane π at the south pole S is shown in Fig. 2.14. It can be seen that F maps the point x ∈ S 2 − {N } into the point x  ∈ π corresponding to the intersection of the straight line N x with π. If we adopt spherical coordinates (ϕ, θ) on S 2 and polar coordinates (ϕ, r ) with center at the south pole on π, the map F has the following coordinate form (see Fig. 2.14):

θ π + . ϕ = ϕ, r = 2 tan 4 2

Determine how F transforms coordinate curves, tangent vectors, and covectors on S 2 . Analyze how it transforms distances and angles between tangent vectors. • As another example of a geographic chart, we consider a diffeomorphism F : S 2 − {S, N } → C, where C is the cylinder in Fig. 2.15 and F maps x ∈ S 2 to the point of C belonging to the straight line O x. Adopting spherical coordinates (ϕ, θ) on S 2 and cylindrical coordinates (ϕ, z) on C, the map F assumes the following coordinate form: ϕ = ϕ, z = tan θ. Determine how F transforms coordinate curves, tangent vectors, covectors, distances, and angles between tangent vectors.

82

2 Introduction to Differentiable Manifolds

Fig. 2.14 Stereographic projection

N S

x S

r x’

N

Fig. 2.15 Central projection

S O

x

x’ z

S • We conclude our examples of geographic charts with Mercator’s projection (1569). This representation is obtained by considering a one-to-one map between S 2 − {S, N } and the cylinder C that in the coordinates of Fig. 2.14, has the form

π θ ϕ = ϕ, z = ln tan + . 4 2 Determine how F transforms coordinate curves, tangent vectors, covectors, distances, and angles between tangent vectors.

Chapter 3

Transformation Groups, Exterior Differentiation and Integration

In this chapter, the one-parameter transformation groups and Lie derivative are defined. Furthermore, two fundamental topics of differential geometry are presented in introductory form: exterior derivative and integration of r -forms. The exterior derivative extends to r -forms the elementary definitions of gradient of a function, curl, and divergence of a vector field as well as the meaning of exact and closed 1-forms. The integration of r -forms allows one to extend the definitions of surface and volume integrals as well as the theorems of Gauss and Stokes.

3.1 Global and Local One-Parameter Groups Definition 3.1 A one-parameter global group of diffeomorphisms G on a manifold Vn of class C k , k > 0, is a C k map φ : (t, x) ∈  × Vn → φt (x) ∈ Vn

(3.1.1)

such that 1. for all t ∈ , the map φt : x ∈ Vn → φt (x) ∈ Vn is a C k -diffeomorphism of Vn ; 2. for all t, s ∈ , ∀x ∈ Vn , φt+s (x) = φt ◦ φs (x). In particular, from the second property we have φt (x) = φ0+t (x) = φ0 ◦ φt (x), so that (3.1.2) φ0 (x) = x, ∀x ∈ Vn . Similarly, from x = φ0 (x) = φt (x) ◦ φ−t (x), we obtain that φ−t (x) = (φt )−1 (x), ∀x ∈ Vn . © Springer Nature Switzerland AG 2019 A. Romano and M. Mango Furnari, The Physical and Mathematical Foundations of the Theory of Relativity, https://doi.org/10.1007/978-3-030-27237-1_3

(3.1.3) 83

84

3 Transformation Groups, Exterior Differentiation and Integration

Definition 3.2 For all x0 ∈ Vn , the C k curve φt (x0 ) :  → Vn on Vn is called the orbit of G determined by x0 . In view of (3.1.2), the orbit contains x0 . Theorem 3.1 Every point x0 ∈ Vn belongs to one and only one orbit, i.e., the orbits determine a partition of Vn . Proof. First, we show that if x2 ∈ φt (x1 ), then the orbit determined by x2 coincides with φt (x1 ) up to a change of parameter. In other words, an orbit is determined by any of its points. In fact, if x2 ∈ φt (x1 ), then there exists a value t2 of t such that x2 = φt2 (x1 ). In view of property (3.1.3), x1 = φ−t2 (x2 ). Consequently, the orbit φt (x1 ) can also be written in the form φt−t2 (x2 ), which, up to the change t → t − t2 of the parameter, gives the orbit determined by x2 . Now it remains to prove that if x2 does not belong to φt (x1 ), then the orbits φt (x1 ) and φt (x2 ) do not intersect each other. In fact, if there is a point x0 belonging to both orbits, then there exist two values t1 , t2 ∈  such that x0 = φt1 (x1 ), x0 = φt2 (x2 ). In view of (3.1.3), it is also the case that x2 = φ−t2 (x0 ), and then x2 = φ−t2 (x0 ) = φ−t2 (φt1 (x1 )) = φt1 −t2 (x1 ). In conclusion, x2 belongs to the orbit determined by x1 , contrary to the hypothesis.  Theorem 3.2 Let Xx be the tangent vector to the orbit φt (x) at the point x. The map x ∈ Vn → Xx ∈ Tx (Vn ) defines a C k−1 vector field X over Vn . Proof. Let (U, x i ) be a chart of Vn and denote by φ i (t, x 1 , . . . , n n ) the representation of the group in these coordinates. Then ∀x ∈ U , one has  Xx =

∂φ i ∂t

 t=0

∂ , ∂xi 

and the theorem is proved.

Definition 3.3 The vector field X is called the infinitesimal generator of the group of diffeomorphisms. Example 3.1 Let E2 be the Euclidean plane. It is easy to verify that the differential map φ :  × E2 → E2 such that

x  ≡ φt (x) = x + tu,

where u is a constant vector in E2 , is a one-parameter global group of diffeomorphisms of E2 , called the group of translations. Further, the orbit determined by the point x0

3.1 Global and Local One-Parameter Groups

85

is x  = x0 + tu. Finally, the constant vector field u = ∂φt (x)/∂t is the infinitesimal generator of the one-parameter group. Example 3.2 Let E3 be the Euclidean three-dimensional space with cylindrical coordinates r , α, and z. Then the family of diffeomorphisms r  = r, α  = α + t, z  = z, is a one-parameter group of diffeomorphisms, called the group of rotations about the axis Oz of the cylindrical coordinates. The orbits of the group are circumferences having their centers on the axis Oz, and the infinitesimal generator is the vector field X=

∂ . ∂α

Definition 3.4 Let U be an open region of Vn . A one-parameter local group of diffeomorphisms is a differential map φ : (−, ) × U → Vn ,

(3.1.4)

where  > 0, such that 1. for all t ∈ (−, ), φt : x ∈ U → φt (x) ∈ φt (U ) is a diffeomorphism; 2. if t, s, t + s ∈ (−, ), then φt+s (x) = φt (φs (x)). Theorem 3.3 Let X be a differentiable vector field on Vn . For every x ∈ Vn , there exist an open region U ⊂ Vn , a real number  > 0, and a one-parameter local group of diffeomorphisms φ : (−, ) × U → Vn whose infinitesimal generator is X. Proof. Consider the system of ordinary differential equations dxi = X i (x 1 , . . . , x n ), i = 1, . . . , n. dt If the vector field is differentiable, then from well-known theorems of analysis, for all x ∈ Vn there exist a neighborhood U of x and an interval (−, ) such that one and only one solution x i = φ(t, x0 ), t ∈ (−, ), of the preceding system exists satisfying the initial data φ i (0, x0 ) = x0 , for all x0 ∈ U . For the uniqueness theorem, x i = φ(t, x0 ) is a diffeomorphism for all t ∈ (−, ) and φ i (0, x0 ) = x0 . We omit  the proof of the property φ(t + s, x0 ) = φ(t, φ(s, x0 )). Definition 3.5 The vector field X is said to be complete if it is an infinitesimal generator of a global one-parameter group of diffeomorphisms. Theorem 3.4 On a compact manifold Vn , every differential vector field is complete.

86

3 Transformation Groups, Exterior Differentiation and Integration

Proof. The previous theorem allows one to state that for all x ∈ Vn , a map φ : (−(x), (x)) × Ux → Vn exists satisfying conditions 1 and 2 of Definition 3.4. Since Vn is compact, the cover (Ux )x∈Vn admits a finite subcover (Uxi )i=1,...,s . Set ting  = min {(x1 ), . . . , (xs )}, the map φ is defined on (−, ) × Vn .

3.2 Lie Derivative In Chap. 2, we proved that if the differentiable map F : Vn → Vn is a diffeomorphism, then its differential (F∗ )x : Tx Vn → TF(x) Vn is an isomorphism that can be extended to the tensor algebra and the exterior algebra at the point x ∈ Vn . In this section we show that this result, which holds for every diffeomorphism φt of a one-parameter transformation group, makes it possible to introduce a meaningful derivation operator on the manifold Vn . We denote by X a differentiable vector field on Vn , by F(Vn ) the vector space of the differentiable functions f : Vn → , and by φt the one-parameter (local or global) transformation group generated by X. Definition 3.6 The Lie derivative of the function f with respect to the vector field X is the map L X : F(Vn ) → F(Vn ) such that (L X f )x = lim

t−>0

f (φt (x)) − f (x) = t



d f (φt (x) dt

 .

(3.2.1)

s=0

Since X is tangent to the orbits of φt , the Lie derivative of f at x is the directional derivative of f along the vector Xx , that is, L X f = X f.

(3.2.2)

Henceforth, we denote by χ , χ ∗ , and χsr the F(Vn ) modules of C ∞ -vector fields, 1-forms, and (r, s)-tensor fields of Vn , respectively. Definition 3.7 (Lie derivative of a vector) The Lie derivative on χ with respect to the vector field X is the map (see Fig. 3.1) L X : Y ∈ χ → L XY ∈ χ such that ∀x ∈ Vn , one has 1 (L X Y)x = lim [(φ−t )∗ (φt (x))Yφt (x) − Yx ]. t→0 t

(3.2.3)

3.2 Lie Derivative

87

Fig. 3.1 Lie derivative of a vector field

Yx

t Y t x

x

t

Tx Vn Fig. 3.2 Lie derivative of a 1-form

t

T

t

x

t x

Y t x

orbits

x

t x

x Tx* Vn

T*

t

t x

t

x

t x

orbits

Definition 3.8 (Lie derivative of a form) The Lie derivative on χ ∗ with respect to the vector field X is the map (see Fig. 3.2) L X : ω ∈ χ ∗ → L Xω ∈ χ ∗ such that ∀x ∈ Vn , one has 1 (L X ω)x = lim [(φt )∗ (x)ωφt (x) − ωx ]. t→0 t

(3.2.4)

It is evident how the Lie derivative can be extended to the tensor fields of χsr . To find the coordinate expression of the Lie derivative, we introduce a chart (U, x i ) on Vn and denote by y i = φti (x 1 , . . . , x n ) the coordinate expression of the one-parameter group of diffeomorphisms φt (x) and by X i the components of its infinitesimal generator X. Then the maps φt (x) and φ−t (x) in a neighborhood of x are given by the expressions y i = x i + X i (x)t + O(t),

(3.2.5)

x = y − X (y)t + O(t),

(3.2.6)

i

i

i

respectively. The coordinate expressions of the codifferential of (3.2.5) and of the differential of (3.2.6) are given by    i ∂ yi ∂X = δ ij + t + O(t), ((φt )∗ )ij = d j ∂x x ∂x j x  i  i ∂x ∂X i i ((φ−t )∗ ) j = = δj − t + O(t). ∂y j y ∂y j y

(3.2.7) (3.2.8)

88

3 Transformation Groups, Exterior Differentiation and Integration

From the preceding relations, we have that 

   j  ∂Y h ((φ−t )∗ Yφt (x) ) = − t Y j (x) + X (x)t + O(t). ∂xh x y (3.2.9) In view of (3.2.9), the Lie derivative (3.2.4) assumes the following coordinate form: i



δ ij

∂ Xi ∂y j



i ∂Y i j ∂X L XY = X − Y ∂x j ∂x j j



∂ . ∂xi

(3.2.10)

By comparing (3.2.10) and (2.12.23), we derive the important result L X Y = [X, Y] .

(3.2.11)

Starting from (3.2.5), we derive the coordinate form of the codifferential (φt (x))∗ ,        ∂ω j ∂X j j h ((φt ) ωφt (x) )i = δi − t ω j (x) + X (x)t + O(t), ∂xi x ∂xh x (3.2.12) so that (3.2.4) becomes ∗

 ∂ωi ∂X j dxi . L Xω = X − ωj ∂x j ∂xi 

j

(3.2.13)

It is not a difficult task to prove that for an arbitrary tensor T ∈ χsr , the Lie derivative is expressed by the following formula: 

∂ i1 ···ir  i1 ···ik−1 hik+1 ···ir ∂ X ik T − T ∂ x h j1 ··· js k=1 j1 ··· js ∂xh  s  ∂ ∂ Xh ∂ r + T ji11······ijk−1 ⊗ ··· ⊗ i h jk+1 ··· js jh i1 ∂ x ∂ x ∂ xr k=1 r

L XT = X h

(3.2.14)

⊗ d x j1 ⊗ · · · ⊗ d x js . In particular, for an s-form , the above formula gives  

∂ ∂ Xh + ( j ··· j ) h( j 1 s 2··· j ) s ∂xh ∂ x j1 i 1 x 1 , so that the length of A B  relative to I is 1/γ times smaller than the length relative to I . Since the dimensions of the lengths in the direction perpendicular to the velocity do not change, a volume V is connected with the proper volume V  by the relation V  = γ V.

(7.5.2)

Consider now a rest or proper time interval t  = t2 − t1 evaluated by a clock C at  rest at the point x 1 A of the inertial frame I . To obtain the corresponding instants t1 and t2 evaluated in the inertial frame I , it is sufficient to apply the inverse of (7.2.25) and recall that I moves relative to I  with uniform velocity −u. In this way, we obtain   u2 , t1 − 2 x 1 c A   u 2 1  t 2 = γ t2 − 2 x A . c t1 = γ

Consequently, we have also

7.5 Some Consequences of the Lorentz Transformations

t  =

1 t < t, γ

215

(7.5.3)

where t = t2 − t1 . In other words, the moving clock goes more slowly than the clock at rest. We now show some simple but interesting applications of (7.5.1) and (7.5.3). (a) Let P be an unstable particle produced in Earth’s atmosphere at a distance l from the terrestrial surface. Denote by I  and I the rest inertial frame of P and a terrestrial frame, respectively. Suppose that the mean lifetime t  (evaluated in I  ) and the speed v of P (evaluated in I ) are such that l > vt  . If we assume classical kinematics, then it is impossible for P to reach the surface of the Earth. However, resorting to (7.5.1) and (7.5.3), we can justify the arrival of P at the terrestrial surface. In fact, for the observer at rest relative to P, the particle has to cover a shorter distance l  = l/γ < l in the time t  . In contrast, for the observer at rest on the Earth’s surface, the particle P has to cover the distance l but lives a longer time, i.e., t/γ . (b) Suppose that at the points x 1A < x B1 of the axis O x 1 of the inertial frame I , two events occur at the instants t A and t B , respectively. For the observer in the inertial frame I  , moving relative to I with velocity u, the same events occur at the instants t A and t B given by   u  u  t A = γ t A − 2 x A , t B = γ t B − 2 x B , c c so that t B − t A = γ

  xB − xA . tB − t A − u c2

(7.5.4)

This condition shows that two events that are simultaneous for I are not simultaneous for I  if x B = x A . The following question arises: supposing that t B > t A , is it possible to find an observer I  for which the order of the events in inverted? This circumstance will obtain if and only if the following condition is satisfied: tB − t A
0, the 4-vector U⊥ is spacelike. Apply the same decomposition to a 1 4-vector V orthogonal to U. Then the following relations hold: U · U = U⊥ · U⊥ − (U 4 )2 < 0, U · V = U⊥ · V⊥ − U 4 V 4 = 0, V · V = V⊥ · V⊥ − (V 4 )2 . Consequently, we can write V · V = V⊥ · V⊥ −

(U⊥ · V⊥ )2 , (U 4 )2

so that by applying Schwartz’s inequality (U⊥ · V⊥ )2 < (U⊥ · U⊥ )(V⊥ · V⊥ ), we obtain (U⊥ · U⊥ )(V⊥ · V⊥ ) . V · V ≥ V⊥ · V⊥ − (U 4 )2 Finally, this inequality implies that V·V≥− and the proposition is proved.

V⊥ · V⊥ [U⊥ · U⊥ − (U 4 )2 ] > 0, (U 4 )2 

Remark 8.1 In view of the above result, we can state that every timelike 4-vector U defines infinitely many Lorentz frames with the origin at the event O. In fact, it is

242

8 Special Relativity in Minkowski Space

sufficient to consider the frame (O, e1 , e2 , e3 , U/|U|), where the mutually orthogonal unit vectors ei , i = 1, 2, 3, belong to  O,U . Proposition 8.2 If V is a spacelike 4-vector at O ∈ V4 , then it is possible to find at least a timelike 4-vector U such that U · V = 0. Proof Let (O, eα ) be a Lorentz frame at O. Then by adopting the same notation used in the preceding proposition, we have that 2

V⊥ · V⊥ − (V 4 ) > 0.

(8.1.7)

We must now prove that there exists at least a 4-vector U such that 2

U · U = U⊥ · U⊥ − (U 4 ) < 0,

(8.1.8)

U · V = U⊥ · V⊥ − U 4 V 4 = 0.

(8.1.9)

From (8.1.9), we derive that the 4-vector U is orthogonal to V if we choose arbitrarily the components U 1 , U 2 , U 3 , i.e., the spacelike 4-vector U ⊥ , provided that the component U 4 is given by U4 =

U⊥ · V⊥ . V4

For this choice of U 4 , condition (8.1.8) becomes U · U = U⊥ · U⊥ −

(U⊥ · V⊥ )2 . (V 4 )2

If we choose U⊥ = V⊥ , then we can write the preceding equation as

V⊥ · V⊥ , U · U = V⊥ · V⊥ 1 − (V 4 )2

(8.1.10)

so that in view of (8.1.7), the vector U = V⊥ + confirms (8.1.8) to (8.1.9).

V⊥ · V⊥ e4 (V 4 )2 

Remark 8.2 For every spacelike vector V at the event O, we can find infinitely many Lorentz frames (O, eα ) for which one of the unit 4-vectors ei is equal to V/|V |. In fact, it is sufficient to take one of the existing timelike 4-vectors e4 orthogonal to V and choose in the three-dimensional space  O,e4 , to which V belongs, two other unit vectors that are orthogonal to each other and orthogonal to V.

8.1 Minkowski Spacetime

243

Proposition 8.3 Let U ∈ C O+ be a timelike 4-vector at O ∈ V4 . Then V ∈ C O+ if and only if U · V < 0. (8.1.11) Proof In fact, U · U < 0, since U ∈ C O+ . But U · W is a continuous function of W ∈ C O+ and C O+ is a connected set. Consequently, if a 4-vector V ∈ C O+ existed for which U · V > 0, then there would exist a timelike 4-vector V ∈ C O+ such that U · V = 0. But this is impossible in view of Proposition 8.1. Conversely, owing to Remark 8.1, we can find a Lorentz frame at O with e4 = U/|U| as the timelike axis. In this frame one has U · U = −U 4 V 4 . On the other hand,  since U, V ∈ C O+ , we have U 4 > 0, V 4 > 0, and the proposition is proved. We denote by L + O the set of Lorentz frames (O, eα ) at O whose axes e4 belong to the positive cone C O+ . A Lorentz transformation between two Lorentz frames in L + O is said to be orthochronous. Proposition 8.4 Let (O, eα ) ∈ L + O be a Lorentz frame. Then the Lorentz frame (O, eα ) belongs to L + O if and only if A44 > 0.

(8.1.12)

Proof Since e4 = Aα4 eα , we have that e4 · e4 = −A44 . The statement is proved when we take into account Proposition 8.3. 

8.2 Physical Meaning of Minkowski Spacetime Now we want to attribute a physical meaning to some geometrical objects of Minkowski spacetime. Let I and I  be two inertial frames of reference and denote by u the translation velocity of I  with respect to I . The relation between the coordinates (x iA , t A ) and (x  iA , t A ) that the observers I and I  associate with the same event A is a Lorentz transformation. It is a very simple exercise to verify that the quadratic form s2 ≡

3 3



  (x iA − x Bi )2 − c2 (t A − t B )2 = (x iA − x Bi  )2 − c2 (t A − t B )2 , i=1

(8.2.1)

i=1

where (x Bi , t B ) and (x  iB , t B ) are the spacetime coordinates of another event B relative to I and I  respectively, is invariant under Lorentz transformations, and it becomes identical to (8.1.4) if we introduce the notation x 4 = ct. Further, from (7.2.42), we obtain that (8.2.2) A44 = γu > 0. Let e4 be a uniform timelike field in V4 . In view of Proposition 8.2 and Remark 8.2, we can define at every point O of V4 a Lorentz frame (O, eα ) ∈ L + O . We can introduce

244

8 Special Relativity in Minkowski Space

a one-to-one correspondence between the inertial frames I in physical space and the orthogonal or Lorentz frames of L + O in the following way. First, we associate a fixed  inertial frame I to an arbitrary Lorentz frame (O, eα ) in L + O . Let I be any inertial frame whose relation with I is expressed by the Lorentz transformations. Then we associate to I  the Lorentz frame (O  , eα ), whose transformation formulas (8.1.2) with respect to (O, eα ) are given by (see (8.1.2)) (x Oα  ) = (x Oi  , ct0 ), ⎛ ⎜ (Aαβ ) = ⎝

Q ij



u ju j j δ j + (γu − 1) 2 u j ui −γu Q i c

(8.2.3) −

γu Q ij γu

⎞ uj c ⎟ ⎠.

(8.2.4)

Since A44 = γu > 0,

(8.2.5)

the Lorentz frame (O  , eα ) belongs to L + O. + Conversely, let us assign the Lorentz frame (O  , eα ) ∈ L + O related to (O, eα ) ∈ L O 4 by (8.2.3)–(8.2.5), where A4 > 0. Then by (8.2.4), a new inertial frame I is defined. In particular, if I and I  are related by a special Lorentz transformation, then (8.2.3) and (8.2.4) reduce to the following formulas: (x Oα  ) = (0, 0), ⎛ (Aαβ )

γu

(8.2.6) u⎞ c⎟ 0 ⎟ ⎟. 0 ⎟ ⎠ γu

0 0 −γu

⎜ ⎜ 0 10 =⎜ ⎜ 0 01 ⎝ u −γu 0 0 c

(8.2.7)

The above considerations allow us to make clear the physical meaning of the definitions we have given in the preceding section. In fact, to any point P ∈ V4 with coordinates (x α ) in a Lorentz frame (O, eα ) ∈ L + O there corresponds the event that has coordinates (x i , x 4 /c) in the inertial frame I . In particular, all the events belonging to the light cone C O have coordinates satisfying (8.1.6). To these points of V4 correspond all the events whose coordinates in the inertial frame I satisfy the equation 3

(x i )2 − c2 t 2 ≡ r 2 − c2 t 2 = 0, (8.2.8) i=1

or equivalently, the two equations r − ct = 0 and r + ct = 0, which represent, respectively, a spherical light wave expanding from O and a spherical light wave contracting toward O.

8.2 Physical Meaning of Minkowski Spacetime

245

Moreover, due to Proposition 8.1 and Remark 8.1, for every point P ∈ C O+ it is possible to find a Lorentz frame in L + O and then an inertial frame I such that the event P has (0, 0, 0, t), t > 0, in I coordinates, meaning that it occurs in I after the event at the origin. If P ∈ C O− , then, the corresponding event occurs in I before the event at O. In other words, every event belonging to C O+ occurs after the event at O for some inertial frames, and every event in C O− occurs before the event at O for some inertial frames. Also, in view of Proposition 8.1 and Remark 8.1, an event belonging to the present of O occurs at the same time with respect to some inertial observer I . In Minkowski spacetime, time plays the role of a variable, in contrast to the role of a parameter played in the classical space and time description. In other words, let   I and I be two inertial frames and let u be the uniform velocity of I relative to I .  For Galileo’s transformations, the Cartesian coordinates of I are a function of the corresponding Cartesian coordinates of I , because time is just a parameter and does not transform. In contrast, Lorentz transformations combine the time coordinate with spatial coordinates. We conclude this section with a very important remark that is a consequence of the correspondence existing among the inertial frames and the Lorentz frames in V4 . If we succeed in formulating the physical laws by tensor relations in V4 , then they will be covariant with respect to Lorentz transformations.

8.3 Classification of Lorentz Transformations The set L of the Lorentz transformations is defined by affine transformations (8.1.2), where the matrix (Aαβ ) satisfies conditions (8.1.3). Therefore, we can state that L is a group, called the Poincaré group. More specifically, it is the group of all rotations and translations of V4 . Every element of L may be classified according to the structure of the matrix (Aαβ ) and the vector (x0α ). With respect to (x0α ), we have the following: • If (x0α ) = 0, then the transformation is called an inhomogeneous Lorentz transformation or Poincaré transformation, and (x0α ) represents a translation in spacetime. • If (x0α ) = 0, then the transformation is called a homogeneous Lorentz transformation. A homogeneous Lorentz transformations may be classified as follows: • If det (Aαβ ) = 1, then it is called a proper Lorentz transformation. • If det (Aαβ ) = −1, then it is called an improper Lorentz transformation. Further, a homogeneous Lorentz transformation may be a • general Lorentz transformation if the orthogonal matrix (Q ij ) in (8.2.4) is arbitrary; • Lorentz transformation without rotation if the orthogonal matrix (Q ij ) is equal to (δ ij ).

246

8 Special Relativity in Minkowski Space

• special Lorentz transformation if the matrix (Aαβ ) is given by (8.2.7); • boost transformation if it is a proper Lorentz transformation without rotation. For example, the matrix (Aαβ ) in (8.1.2) of a general homogeneous Lorentz transformation without rotation (a boost) is ⎛ i ⎜ δj

Aαβ = ⎝

ui u j + (γu − 1) 2 u ui −γu c

⎞ ui − γu ⎟ c ⎠, γu

(8.3.1)

and the matrix (Aαβ ) of a special boost with velocity u along the axis O x 1 is ⎛

γu ⎜ 0 Bx = ⎜ ⎝ 0 −γu u/c

0 1 0 0

⎞ 0 −γu u/c ⎟ 0 0 ⎟. ⎠ 1 0 0 γu

(8.3.2)

8.4 Four-Dimensional Equation of Motion Let I be an inertial frame and T ≡ (O, eα ) the corresponding Lorentz frame in V4 . The motion of a particle P with respect to I is given by the equations of motion x i = x i (t),

(8.4.1)

which are geometrically represented by a curve of V4 . In the Lorentz frame T , corresponding to I , the curve has equations x i = x i (t), x 4 = ct.

(8.4.2)

The curve is called a world trajectory or world line of P. Since the velocity of P in any inertial frame is less than the velocity c of light in vacuum, the tangent vector γ˙ = (x˙ i , c) to has a negative norm 3

(x˙ i )2 − c2 < 0.

(8.4.3)

i=1

In other words, the curve is timelike, and its tangent vector at every point P is contained in the nappe C O+ of the light cone at P. Let x i (t¯) be the position of P at the instant t¯ in the inertial frame I . The inertial frame that has the origin at x i (t¯) and moves with velocity v¯ with respect to I is called the rest frame or proper frame of P at the instant t¯, and will be denoted by I¯ (see Sect. 7.8). The corresponding Lorentz frame T ≡ ((x i (t¯), ct¯), eα ) has the time

8.4 Four-Dimensional Equation of Motion

247

axis e4 tangent to at the point (x i (t¯), ct¯), so that the vector e4 has components (0, 0, 0, c) in T . The infinitesimal distance between two events along is invariant in going from T to T , so that we have ds 2 =

 3 i=1

 (x˙ i )2 − c2 dt 2 = −c2 dτ 2 ,

(8.4.4)

where τ is the proper time, i.e., the time evaluated by the observer I (t) (see Sect. 7.11). From (8.4.4), we derive the relation dt = γv . dτ

(8.4.5)

If we adopt this time along , the parametric equations (8.4.2) become x α = x α (τ ).

(8.4.6)

The world velocity or 4-velocity is the 4-vector Uα =

dxα , dτ

(8.4.7)

which in the Lorentz frame T has components U α = (γv v, cγv ).

(8.4.8)

U α Uα = γv2 (v2 − c2 ) = −c2 < 0.

(8.4.9)

This is a timelike vector, since

Differentiating Eq. (8.4.7) with respect to τ , we have that the world velocity is orthogonal to the world acceleration Aμ = dU μ /dτ , U μ Aμ = 0.

(8.4.10)

Consider the relativistic equations of a particle (see Sect. 7.8), d (mv) = F, dt

(7.9.2)

d (mc2 ) = F · v, dt

(7.9.3)

 where m = m 0 / 1 − v 2 /c2 is the relativistic mass of P and m 0 is its rest mass. Introducing the 4-momentum and the 4-force respectively given by

248

8 Special Relativity in Minkowski Space

(P α ) = m 0 γv (v, c),

( α ) = γ

F·v , F, c

(8.4.11)

Equations (7.9.2) and (7.9.3) can be written in the following 4-dimensional form: d Pα dU α ≡ m0 = α . dτ dτ

(8.4.12)

8.5 Tensor Formulation of Electromagnetism in a Vacuum Let S be a continuous charge distribution moving with respect to the inertial frame I and denote by ρ and v the charge density and the velocity field of S, respectively. If S moves in a vacuum, then the continuity Eq. (7.5.3) becomes ∂ρ + ∇ · (ρv) = 0. ∂t

(8.5.1)

Consider a transformation (8.1.3) from a Lorentz frame I to another Lorentz frame I  and suppose that the quantities J α = (ρv, cρ) = (J, cρ) = ρ0 U α , ρ0 = ρ/γv ,

(8.5.2)

are the components of a 4-vector, which we call the 4-current. In other words, we are assuming that in going from I to I  , the quantities J α transform according to the formula (8.5.3) J α = Aαβ J β . Then (8.5.1) can be written as the divergence in V4 of the 4-current ∂ Jα = 0. ∂xα

(8.5.4)

The form of this equation is independent of the Lorentz frame, and consequently, it satisfies a principle of relativity. Exercise 8.1 Prove that from (8.5.3) to (8.2.7), we obtain again (7.6.10). Similarly, the first two Maxwell equations in vacuum (see (5.10.1)), ∇ × B = μ0 J + 0 μ0 ∇ ·E=

ρ , 0

∂E , ∂t

8.5 Tensor Formulation of Electromagnetism in a Vacuum

can be written in the form

∂ F αβ = μ0 J α , ∂xβ ⎞ 0 B 3 −B 2 −E 1 /c ⎜ −B 3 0 B 1 −E 2 /c⎟ ⎟. =⎜ 2 1 ⎝ B −B 0 −E 3 /c⎠ 0 E 1 /c E 2 /c E 3 /c

249

(8.5.5)

⎛

where F αβ

(8.5.6)

Consequently, if F αβ is supposed to comprise the components of a (skew-symmetric) 2-tensor, then (8.5.5) has the same form in every Lorentz frame, and the relativity principle is satisfied by Maxwell’s equations; F αβ is called the electromagnetic tensor. Introduce the adjoint tensor ∗Fαβ of F αβ (see Chap. 2), ∗ Fαβ =

1 λμαβ F λμ . 2

(8.5.7)

It is easy to verify that the matrix of covariant components of ∗F is obtained from the matrix F by the substitutions E/c → −B and B → −E/c: ⎞ 0 −E 3 /c E 2 /c B 1 ⎜ E 3 /c 0 −E 1 /c B 2 ⎟ ⎟. =⎜ 2 1 ⎝−E /c E /c 0 B3⎠ −B 1 −B 2 −B 3 0 ⎛

∗Fαβ

Consequently, the matrix of the contravariant components is ⎞ 0 −E 3 /c E 2 /c −B 1 ⎜ E 3 /c 0 −E 1 /c −B 2 ⎟ ⎟. =⎜ ⎝−E 2 /c E 1 /c 0 −B 3 ⎠ B1 B2 B3 0 ⎛

∗F αβ

Using the tensor ∗F αβ , the other two Maxwell’s equations (see (5.10.1)), ∇ ×E=− ∇ · B = 0, assume the covariant form

∂B , ∂t

∂ ∗F αβ = 0. ∂xβ

(8.5.8)

250

8 Special Relativity in Minkowski Space

Exercise 8.2 Verify that in terms of the tensor F αβ , Maxwell’s equations (8.5.8) can be written as follows: ∂ F αβ ∂ F βγ ∂ Fγα + + = 0. γ α ∂x ∂x ∂xβ

(8.5.9)

Hint: If two of the indices α, β, γ are equal, then the left-hand member of (8.5.9) is identically zero, since F αβ is skew-symmetric and the equation is trivial. The four possible cases that correspond to distinct values of α, β, and γ give Maxwell’s equations. Exercise 8.3 Verify that from F αβ = Aαλ Aβμ F λμ and (8.2.4), we obtain (7.7.1), (7.7.2). Exercise 8.4 The 4-vector F α = ρ0 F αβ Uβ = F αβ Jβ ,

(8.5.10)

which gives the 4-force acting on the charge contained in the unit volume, is called the Lorentz four-force. Prove that the components of (8.5.10) relative to an inertial frame are (E + v × B),

 E · v. c

8.6 Electromagnetic Potentials Maxwell’s equation ∇ · B = 0 implies the existence of a vector potential A0 for the magnetic induction B such that B = ∇ × A0 .

(8.6.1)

The vector potential is not uniquely defined, since A = A0 − ∇ϕ0 ,

(8.6.2)

where ϕ0 is an arbitrary function of the position and time, is also a vector potential for B: B = ∇ × A. (8.6.3)

8.6 Electromagnetic Potentials

251

Taking into account this result, Maxwell’s equation ∇ ×E=− gives

∂B ∂t



∂A ∂A0 = 0, ∇ × E + = 0, ∇× E+ ∂t ∂t

(8.6.4)

so that we can write E = −∇ϕ0 −

∂A ∂A0 , E = −∇ϕ − , ∂t ∂t

where ϕ = ϕ0 +

∂ψ , ∂t

(8.6.5)

(8.6.6)

ψ is an arbitrary function, and ϕ0 and ϕ are scalar potentials of the electromagnetic field. In conclusion, the pair (ϕ0 , A0 ) or any other pair (ϕ, A) that is related to (ϕ0 , A0 ) by (8.6.2) and (8.6.6) defines the same electromagnetic fields E, B. Introducing (8.6.3) and (8.6.5) into (5.10.2), (5.10.3), we obtain ∇ × (∇ × A) =μ0 ρv − 0 μ0 ∇ −∇ · ∇ϕ−

ρ ∂ ∇ ·A= . ∂t 0

∂ 2A ∂ϕ − 0 μ0 2 , ∂t ∂t

(8.6.7) (8.6.8)

In view of the arbitrariness of the scalar and vector potentials, we choose the pair ϕ, A satisfying the additional Lorentz condition ∇ · A + 0 μ0

∂ϕ = 0. ∂t

(8.6.9)

In view of this condition and the vector identity ∇ × (∇ × A) = ∇∇ · A − ∇ · ∇A,

(8.6.10)

we can give (8.6.7) and (8.6.8) the form ∂ 2A = μ0 ρv, ∂t 2 ∂ 2ϕ ρ

ϕ − 0 μ0 2 = , ∂t 0

A − 0 μ0

(8.6.11) (8.6.12)

252

8 Special Relativity in Minkowski Space

and we can state that the scalar and vector potentials are a solution of d’Alembert’s equation. We now suppose that α = (Ai , ϕ/c), α = (Ai , −ϕ/c)

(8.6.13)

are the contravariant and covariant components of a 4-vector of V4 . Then (8.6.3), (8.6.5), (8.6.9), (8.6.11), and (8.6.12) assume the tensor forms Fαβ =

∂β ∂α − , α ∂x ∂xβ

∂α = 0, ∂xα ∂ 2 α ηλβ λ β = μ0 J α , ∂x ∂x

(8.6.14) (8.6.15) (8.6.16)

when are invariant under Lorentz transformations.

8.7 The Electromagnetic Momentum–Energy Tensor in Vacuum The four-Lorentz force acting on the charge contained in a unit volume is (see (8.5.10)) (8.7.1) Fα = ρ0 Fαβ U β = Fαβ J β . In this section we prove the existence of a symmetric 4-tensor T αβ , called the electromagnetic momentum–energy tensor in vacuum, such that μ0 Fα = − where

∂ Tαβ , ∂xβ

1 T αβ = − F λμ Fλμ ηαβ + F αμ Fμβ . 4

(8.7.2)

(8.7.3)

In fact, from (8.7.1) to (8.5.5) we have μ0 Fα = Fαβ

∂ F βν ∂ ∂ Fαβ βν = ν (Fαβ F βν ) − F . ν ∂x ∂x ∂xν

(8.7.4)

On the other hand, in view of the skew-symmetry of F αβ and (8.5.9), we can write

8.7 The Electromagnetic Momentum–Energy Tensor in Vacuum

253

∂ Fαβ βν ∂ Fαν νβ ∂ Fνα βν F = F = F ν β ∂x ∂ x ∂xβ 1 ∂ Fαβ ∂ Fβν F νβ = + 2 ∂xν ∂xα 1 ∂ Fνβ νβ 1 ∂ =− F =− (Fνβ F νβ ), 2 ∂xα 4 ∂xα and (8.7.2), (8.7.3) are proved. Taking into account (8.6.6), the components of the tensor T αβ can be written as follows: 1 1 2 ij 1 i j (0 E 2 + B )δ − (0 E i E j + B B ), 2 μ0 μ0 1 = T 4i = (E × B)i , μ0 1 1 2 = (0 E 2 + B ). 2 μ0

T ij =

(8.7.5)

T i4

(8.7.6)

T 44

(8.7.7)

Here T i j is called the Maxwell stress tensor, T i4 is the Poynting vector, and T 44 is the electromagnetic energy density per unit volume.

8.8 Exterior Algebra and Maxwell’s Equations In this section we show that it is possible to formulate relativistic electromagnetism in terms of differential forms. First, denoting by Jα the covariant components of the 4-current vector (8.5.2), we define the 1-form (8.8.1) J = Jα d x α . The adjoint 3-form ∗J of J is the skew-symmetric tensor1

where

∗ J = ∗Jαβλ d x α ∧ d x β ∧ d x λ ,

(8.8.2)

∗ Jαβλ = μαβλ J μ , (α < β < λ).

(8.8.3)

We prove the following equivalence: ∂ Jα = 0 ⇐⇒ d ∗ J = 0. ∂xα

1∗

is called the Hodge operator.

(8.8.4)

254

8 Special Relativity in Minkowski Space

In fact, we have ∗J = ∗ J123 d x 1 ∧ d x 2 ∧ d x 3 + ∗J124 d x 1 ∧ d x 2 ∧ d x 4 + ∗ J134 d x 1 ∧ d x 3 ∧ d x 4 + ∗J234 d x 2 ∧ d x 3 ∧ d x 4 , so that ∂ ∗ J124 3 ∂ ∗ J123 4 dx ∧ dx1 ∧ dx2 ∧ dx3 + d x ∧ d x 1 ∧ d x 2 ∧ d x 4+ 4 ∂x ∂x3 ∂ ∗ J134 2 ∂ ∗ J234 1 dx ∧ dx1 ∧ dx3 ∧ dx4 + dx ∧ dx2 ∧ dx3 ∧ dx4 2 1 ∂ x ∂ x

∂ ∗ J234 ∂ ∗ J134 ∂ ∗ J124 ∂ ∗ J123 d x 1 ∧ d x 2 ∧ d x 3 ∧ d x 4. = − + − ∂x1 ∂x2 ∂x3 ∂x4

d ∗J =

On the other hand, from (8.8.3) we obtain ∗J234 = 1234 J 1 = J 1 , ∗J134 = 2134 J 2 = −J 2 , ∗J124 = 3124 J 3 = J 3 , ∗J123 = 4123 J 4 = −J 4 , and (8.8.4) is proved. Consider the following two-forms (see (8.6.13)): F = Fαβ d x α ∧ d x β , (α < β)  = α d x α .

(8.8.5)

F = d, d F = 0.

(8.8.6)

Then we have

The first of the above results shows that F is an exact 2-form, while the second result follows from the fact that an exact form is also a closed form. To verify (8.8.6), it is sufficient to evaluate the differential of : d =

∂β α 0,

(9.4.5)

(9.4.6)

We explicitly note that the variable τ has the meaning of proper time, provided that events belonging to the same curve of  are considered. In fact, along such a curve we have d x α = U α dτ and then

9.4 Intrinsic Deformation Gradient

279

Fig. 9.1 Space and time projection of the infinitesimal 4-vector dx

1 dτ = − Uα d x α . c

(9.4.7)

However, a proper time τ can be defined for the whole region W if and only if the differential form (9.4.7) is exact. Let us consider an event x on the world line ρ ∈  and let I(0) be the rest frame at this event. Since in view of (9.4.5), we have d x α = (∂x α /∂ y i )dy i + U α dτ , we can use (9.4.4) to obtain β

 α = Pαβ ∂x dy i ≡ F iα dy i , δx ∂ yi

(9.4.8)

iα ) is called the intrinsic deformation gradient. where ( F In order to recognize the physical meaning of (9.4.8), we consider the spatial vector dy ∈ X0 . If e0,i (0) is an orthonormal basis in X0 , we have dy = dy i e0,i (0). At the event x(τ ) ∈ ρ, corresponding to the value τ for the proper time, we consider the triad of the spatial vectors e0,i (τ ) obtained by e0,i (0) by a Fermi transport along ρ (see Fig. 9.2). Due to the properties of this transport, the three vectors e0,i (τ ) form an orthonormal basis in the three-dimensional space X (τ ) that is formed by the vectors orthogonal to the unit tangent vector e0,4 (τ ) to ρ at X(τ ). Moreover, the components dy i of dy remain constant during the transport (see Theorem 8.1) and dy i e0,i is the transported vector. If we refer (9.4.8) to the basis e0,i of X , then we have  i e0,i (τ ) = F ji dy j e0,i (τ ), δx

(9.4.9)

which defines an isomorphism between the three-dimensional spaces X0 and X . Introducing the notation i −1 )iα = ∂ y , (F (9.4.10) ∂x α

280

9 Continuous Systems in Special Relativity

Fig. 9.2 Evolution of the infinitesimal space 4-vector dx

we easily verify that jα Uα = 0, F

(9.4.11)

−1 )iα U α = 0, (F −1 )iα F jα = δ ij , (F

(9.4.12)

−1 )iα F iβ (F

(9.4.14)

=

(9.4.13)

Pβα .

In fact, from (9.4.4) and (9.4.8) we have iα Uα = F



∂x α 1 α ∂x β + U Uβ i ∂ yi c2 ∂y

 Uα = 0,

since U α Uα = −c2 . Moreover, −1 )iα U α = (F

∂ yi α ∂ yi = 0, U = ∂x α ∂τ

since y i and τ are independent variables. It is then very simple to verify (9.4.13) and (9.4.14). In what follows, we use the notation (9.4.4) to denote the projection on X of any 4-tensor. Now we prove the following theorem. Theorem 9.2 The following identities hold:  iρ dF

ρ iλ ∂U , =F dτ ∂x λ

(9.4.15)

9.4 Intrinsic Deformation Gradient

281

  ρ ∂U ρ −1 )iμ d Fi . = ( F ∂x μ dτ

(9.4.16)

Proof From (9.4.8) and (9.4.4), we have that   iα dF ∂ ∂x α 1 α ∂x β = + 2 U Uβ i dτ ∂τ ∂ y i c ∂y ∂x β Aα ∂x β Uα U α ∂U β ∂U α + 2 Uβ i + 2 A β i + 2 Uβ = i ∂y c ∂y c ∂y c ∂ yi ∂x μ α ∂U β 1 ∂x β = Pβ + 2 (Aα Uβ + U α Aβ ) i , i μ ∂y ∂x c ∂y

(9.4.17)

where Aα = dU α /dτ is the 4-acceleration. On the other hand, from (9.4.8) we obtain ν ∂x μ iμ − 1 U μ Uν ∂x , = F ∂ yi c2 ∂ yi

so that (9.4.16) becomes β μ ν β iα   β dF iμ Pβα ∂U + 1 Aα Uβ + U α Aβ ∂x − Pβα U Uν ∂x ∂U . =F μ 2 i 2 i μ dτ ∂x c ∂y c ∂ y ∂x

However, we also have Uμ

∂U β = Aβ , ∂x μ

Aα Uα = 0,

Pβα Aβ = Aα ,

iμ = P μ F λ F λ i ,

so that the above relation can be written as follows: β β iα dF iλ Pβα P μ ∂U + 1 U α Aβ ∂x . =F λ dτ ∂x μ c2 ∂ yi

(9.4.18)

Applying the projector Pαρ to both sides of (9.4.18) (see 8.9.5) and noting that ρ Pαρ Pβα = Pβ , Pαρ U α = 0, we finally obtain Pαρ

β iα dF iλ P ρ P μ ∂U , =F β λ μ dτ ∂x

and (9.4.15) is proved. The identity (9.4.16) follows from (9.4.15) and (9.4.14).  Remark 9.6 Equation (9.4.15) describes the evolution of the intrinsic deformation gradient with respect to every Lorentz frame I. If we denote by (O, eα ) the frame of reference in I and by (X(λ), ei , U/c) the family of frames that are obtained by the F-transport along the world trajectory ρ of a particle of the continuous system S, iα , we have then due the spatial character of F

282

9 Continuous Systems in Special Relativity j

Fi ej = Fiα eα .

(9.4.19)

On the other hand, since ei is obtained by an F-transport along ρ, we also have    j F d  j  i j 1 ej · dU U, ( Fi e j ) = i ej + F dτ dτ c2 dτ

(9.4.20)

and the projection of (9.4.18) on X(λ) coincides with (9.4.15). In conclusion, the Eq. (9.4.15) gives the spatial evolution of the intrinsic deformation gradient with respect to the proper observers that assume spatial axes in X(λ) that satisfy the Fermi transport along the world trajectory ρ. We conclude this section by defining the intrinsic Cauchy–Green tensor i j = ηαβ F iα F jβ . C

(9.4.21)

9.5 Relativistic Dissipation Inequality From the results of Sect. 9.3, we know that the relativistic balance equations for momentum and energy can be written as

where3

∂T αβ = f α, ∂x β

(9.5.1)

T αβ = U α U β − αβ + U α q β + q α U β

(9.5.2)

is the symmetric momentum–energy 4-tensor and   1 ( f α ) = f, (f · v + r) , c

f α Uα = −r,

(9.5.3)

is the 4-force acting per unit volume of the continuous system S. In (9.5.3), f is the force per unit volume with respect to an inertial frame, v is the velocity field of S, and r is the energy supply per unit volume. It is well known that the position 

ε =μ 1+ 2 c

 ≡ μχ

(9.5.4)

αβ is opposite to αβ in (9.3.12). Therefore, its spatial components in the proper frame are equal the components of Cauchy stress tensor.

3 Here

9.5 Relativistic Dissipation Inequality

283

defines the proper density of matter as well as the proper internal energy ε per unit proper mass. Moreover, in the absence of proper mass variation, μ satisfies the equation ∂ (μU α ) = 0. (9.5.5) ∂x α We also assume the following local form for the entropy inequality (see [38, 103, 112, 131]):  α ∂ r0 q α 2 ∂ + , (sU ) ≥ −c (9.5.6) ∂x α ∂x α θ θ where s is the proper entropy per unit volume, θ the proper absolute temperature, and r the proper energy supply per unit proper volume. If we define the proper specific entropy η as s = μ η, (9.5.7) then the above inequality assumes the form μ η˙ ≥ −c2

∂ ∂x α



qα θ



r + , θ

(9.5.8)

where the dot denotes the derivative with respect to the proper time τ along a world line of any particles of S. In order to derive an important inequality from (9.5.1) and (9.5.8), we begin by noting that (9.5.4) and (9.5.5) lead to ∂ ∂ (ρU α U β ) = (μχU α U β ) α ∂x ∂x α ∂ ∂ = χU α β (μU β ) + μU β α (χU α ) ∂x ∂x ε˙ d = μ (χU α ) = 2 U α + μχAα , dτ c

(9.5.9)

where Aα = dU α /dτ is the 4-acceleration. In view of (9.5.9), we can derive from (9.5.1) the following relation: μ

ε˙ α ∂αβ ∂ α U + μχA + + β (q α U β + q β U α ) = f α . c2 ∂x β ∂x

(9.5.10)

On the other hand, it is easy to verify that (9.3.5) and (9.3.11) imply that αβ Uβ = βα Uβ = 0, α

q Uα = 0.

(9.5.11) (9.5.12)

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9 Continuous Systems in Special Relativity

Moreover, from U α Uα = −c2 , we obtain Uα Aα = 0, ∂U α Uα β = 0. ∂x

(9.5.13) (9.5.14)

Finally, if we multiply (9.5.10) by Uα and consider (9.5.11)–(9.5.14), the condition Uα U α = −c2 , and (9.5.3), we get − μ ε˙ + Uα

α α ∂αβ β ∂q 2 ∂q + U U − c = −r . α ∂x β ∂x β ∂x α

(9.5.15)

Again taking into account (9.5.12) and (9.5.13), we can write the above equation in the form α ∂q α ∂U μ ε˙ = −αβ β − q α Aα − c2 α + r , (9.5.16) ∂x ∂x which, due to (9.4.16), becomes μ ε˙ = Sαi where

 iα dF ∂q α − q α Aα − c2 α + r , dτ ∂x

(9.5.17)

−1 )iβ . Sαi = −βα ( F

(9.5.18)

By eliminating r between (9.5.8) and (9.5.17) and noting that q α Uα = 0 (see (9.5.18), we obtain the inequality  iα c2 ˙ + Si d F − q α Aα − q α gα , − μ(ψ˙ + η θ) α dτ θ

(9.5.19)

ψ = ε − θη

(9.5.20)

 ∂θ ∂θ = Pαβ β . ∂x α ∂x

(9.5.21)

where is the specific free energy and  gα =

On the other hand, when we recall (9.4.10), we also find that 

∂θ ∂τ  gα = + β θ˙ i ∂y ∂x   −1 )iβ G i + τα θ˙ . ≡ Pαβ ( F Pαβ

−1 )iβ (F

 (9.5.22)

9.5 Relativistic Dissipation Inequality

285

Consequently, q α Pαβ = q β , since q α lies in the three-dimensional space orthogonal to U α , and so we have ˙ −1 )iα G i + τα θ. q α gα = q α ( F (9.5.23) Finally, the inequality (9.5.19), or its equivalent form

    iα dF q α τα ˙ 1 −1 )iα G i ≥ 0, (9.5.24) θ − Sαi −μ ψ˙ + η + c2 − q α Aα − q α ( F dτ θ θ is called the relativistic reduced dissipation inequality.

9.6 Thermoelastic Materials in Relativity As in the classical theory of continua, we call the pair of functions x α (y i , τ ), θ(y i , τ ), a thermokinetic process, while a thermodynamic process is a thermokinetic process together with the following functions: ψ = ψ(y i , τ ),

(9.6.1)

η = η(y , τ ),

(9.6.2)

αβ = αβ (y i , τ ),

(9.6.3)

i

α

α

q = q (y , τ ). i

(9.6.4)

In this section we analyze relativistic thermoelastic materials, i.e., materials that are described by the following constitutive equations: iα , Aα , θ, G i ), ψ0 = ψ( F iα , Aα , θ, G i ), η = η( F αβ



α

=

q =

iα , Aα , θ, G i ),  (F iα , Aα , θ, G i ). q α( F αβ

(9.6.5) (9.6.6) (9.6.7) (9.6.8)

It is reasonable to extend the dissipation principle (see Sect. 5.10) to special relativity by requiring that the constitutive equations (9.6.5)–(9.6.8) satisfy the reduced dissipation inequality (9.5.24) in every thermokinetic process. In order to derive the restrictions on the constitutive equations (9.6.5)–(9.6.8) resulting from this principle, we first prove the following theorem. Theorem 9.3 Let X = (y i , τ ) be an event on the world line of the particle (y i ) of the continuous system S. Then it is always possible to find, at least in a neighborhood of X in V4 , a motion x α (y i , τ ) such that the quantities

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9 Continuous Systems in Special Relativity

iα , F

αi d F ˙ G , G˙ , Aα , θ, θ, i i dτ

(9.6.9)

have arbitrary values at X. Proof The equations x i (y j , t) = a i (t) + F ji (t)y j

(9.6.10)

represent a motion of a continuous system S with respect to the inertial frame I if and only if the 3 × 3 matrix (F ji ) is nonsingular and the velocity of every point is less than c, provided less than c. The velocity u i (t) of the particle (y i ) = 0 will be  3 [(a˙ i (t))2 ] < c2 . that the arbitrary functions a i (t) are chosen in such a way that i=1 When this condition is satisfied, the squared modulus of the velocity (v i ) of the other particles of S, (9.6.11) v 2 = u 2 + 2u i F˙ ji y j + F˙ ji F˙hi y j y h , will be less than c2 , at least when the point (y i ) belongs to a suitable neighborhood I of 0. Under these conditions, we can define the congruence x i (y j , t) = a i (t) + F ji (t)y j , x = ct, 4

(9.6.12) (9.6.13)

of the world trajectories of the particles of S in the neighborhood I ×  of V4 . In this way, it is possible to arbitrarily define the 4-velocity and the 4-acceleration of any particle 0 of S. Now we prove that starting from (9.6.10), we can also arbitrarily iα /dτ ). In fact, let iα , (d  F assign the quantities F d xi =

∂x i i dy = F ji (t)dy j ∂y j

(9.6.14)

be the infinitesimal vector between the two events (0, t) and (dy i , t), which are simultaneous with respect to the observer I . In the Lorentz frame corresponding to x i , 0). Moreover, for the event X = (0, t), I , we can define the 4-vector (d x α ) = (d α we consider the 4-velocity U and the associate three-dimensional space X of the vectors that are orthogonal to U α . In this case, we have xβ, d x α = Pβα d so that

iβ . iα = Pβα F F

(9.6.15)

ji (t) corresponds an arbitrary This relation shows that to an arbitrary choice of F  α α . By differentiating with respect to time, we can also verify that d Fi choice of F β dτ

9.6 Thermoelastic Materials in Relativity

287

can be chosen in an arbitrary way starting from F ji (t). A similar line of reasoning can be applied to the proper field of temperature θ0 (y i , t) = a(t) + b j (t)y i .



In order to derive the restrictions on the constitutive equations due to the reduced dissipation inequality, we assume that this inequality is evaluated at an event X ∈ V4 , and we denote by U α the 4-velocity of the particle of the continuous system S at X and by X the three-dimensional space of the 4-vectors that are orthogonal to U α . If we differentiate (9.6.5) with respect to the proper time τ , iα ∂ψ ˙ ∂ψ d F ∂ψ ˙ ∂ψ ˙ α ∂ψ ¨ + ψ˙ = θ+ θ, Gi + A + α i dτ ∂G i ∂ Aα ∂F ∂θ ∂ θ˙

(9.6.16)

iα on X and U α by and we denote the projection of ∂ψ/∂ F   ∂ψ ∂ψ β = Pα , iα iα ∂F ∂F   ∂ψ ∂ψ 1 β = − 2 Uα U , iα iα c ∂F ∂F

(9.6.17) (9.6.18)



then the inequality (9.6.8) can also be written as follows: ⎡  ⎤ 

 α iα dF ∂ψ ⎦  q τ α − μ η + c2 θ˙ − ⎣ Sαi + μ iα dτ ∂F θ  ∂ψ ˙ α ) − μ ∂ψ G˙ − μ ∂ψ θ¨ − μ ∂ψ A˙ α  (F −μ i i α i ∂G i ∂ Aα ∂F ∂ θ˙

(9.6.19)



− qα A α − c 2

−1 )iα q α( F θ

G i ≥ 0.

Since this inequality must be satisfied for every choice of the quantities (9.6.9), we necessarily have that iα , θ), ψ = ψ( F η (e) = −

∂ψ ∂θ

,

(9.6.20) (9.6.21)

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9 Continuous Systems in Special Relativity

Sαi 

∂ψ iα ∂F



 ∂ψ iα , θ), = −μ = Sαi ( F iα ∂F

(9.6.22)

= 0,

(9.6.23)

||

μη (d) θ˙ − q α Aα − c2 where

−1 )iα q α( F θ

G i ≥ 0,

˙ G , Aα ) − η( F iα , θ) ≡ η( F iα , θ, θ, iα , θ, 0, 0, 0) η (e) ( F i

(9.6.24)

(9.6.25)

is the specific entropy at equilibrium and ˙ G , Aα ) = η( F ˙ G , Aα ) − η (e) ( F iα , θ, θ, iα , θ, θ, iα , θ) η (d) ( F i i

(9.6.26)

is the remaining part of the specific entropy. In particular, (9.6.24) leads to the following inequalities: α −1 i )α ˙ G , 0)θ˙ − c2 q ( F iα , θ, θ, η (d) ( F G i ≥ 0, i θ iα , θ, 0, 0, Aα )Aα ≤ 0. q α( F

(9.6.27) (9.6.28)

We have the following remarks. Remark 9.7 The dependence on Aα in the constitutive equations is essential. In fact, if we omitted it, then (9.6.28) would lead to q α = 0. Moreover, it implies the presence of heat conduction even in the absence of a temperature gradient. Remark 9.8 The inequality (9.6.28) is a relativistic extension of a result in [13] that implies that thermal waves have a finite propagation velocity. Remark 9.9 The condition (9.6.23) is a consequence of the objectivity principle in special relativity as formulated in [16, 17, 46, 90, 91, 103, 104, 157].4 It is possible to prove that from this form of the objectivity principle in special relativity, there follows i j , θ). ψ = ψ(C (9.6.29) Consequently, we have also lm ∂ψ ∂ψ ∂ C lβ ∂ψ , = = 2ηαβ F α α i lm ∂ F i li ∂F ∂C ∂C and (9.6.23) is proved when we take into account (9.4.11). 4 For a deeper analysis of the objectivity principle in special and general relativity, see [16,

17, 157].

9.6 Thermoelastic Materials in Relativity

289

We are now in a position to formulate an initial and boundary data problem to determine the evolution of a continuous system S. For simplicity, we limit our attention to a perfect fluid in the absence of thermal phenomena. In this case, the momentum–energy tensor is given by (9.3.16), and we have to find the fields U α , , provided that the constitutive equation p = p() is given. Since Uα U α = −c2 , we must determine four unknowns U i , i = 1, 2, 3, and ρ satisfying the four equations ∂β T αβ = 0. Let I be an inertial frame and denote by 0 the 3-dimensional region occupied by S at the instant t = 0 in I . In view of equation (9.2.11), we can suppose that 0 is occupied at the proper time τ = 0. Then we must assign the initial conditions U0α U0α = −c2 , U0α (x) = U α (x, 0), 0 (x) = (x, 0), ∀x ∈ 0 ,

(9.6.30) (9.6.31)

and the boundary condition Uα N α = 0, ∀t ∈ [0, T ],

(9.6.32)

where N α = (n, 0) is a space vector and n is the unit normal orthogonal to the boundary ∂(t) of the region (t) occupied by S at the instant t ∈ [0, T ].

9.7 On the Physical Meaning of Relative Quantities In the above sections we started from the assumption that the balance equations for momentum, energy, and angular momentum also hold during the evolution of a continuous system S in special relativity. However, we did not accept the classical expressions for the quantities that appear in the balance laws a priori; instead, we imposed the covariance of these equations under Lorentz transformations, proving that the above balance equations can be written as the divergence of the symmetric momentum–energy tensor T αβ of the spacetime V4 . As a consequence, the description of a continuous system in special relativity requires that we be able to express its components in terms of the thermodynamic process. Some examples of this tensor have been supplied together with the nature of the continua that they describe. In any case, the proposed momentum–energy tensors T αβ were defined by giving αβ their components T in the proper frame I , where their physical meaning was more evident. In this way, the whole description of S is realized in the spacetime, i.e., it is a geometric description in V4 of the evolution of S. It is evident that by resorting to the transformation formulas λμ (9.7.1) T αβ = Aαλ Aβμ T , λμ

it is possible to express the components T αβ in terms of the physical quantities T and then, using (9.3.11), to obtain the relative quantities g, t, h, p, which appear in

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9 Continuous Systems in Special Relativity

the balance equations, as functions of the proper components of T αβ . These formulas, which are proved in [46], have the following form:   

2 v · tv v·p v 2 2v g = γ h − γ(γ − 1) 2 + 2γ 2 − γ + 1 v c v 2 c2 p v·t −γ 2 +γ 2, c c   tv v · v·p 2 h =γ h− 2 +2 2 , c c v ⊗ (v · t) γ − 1 t · v + v2 γ v2 

(γ − 1)2 v · tv γ γ − 1 v · pv , + v ⊗ v + v ⊗ p − γ v4 c2 γ v2 

v · tv γ − 1 v(v · p) p = −γv · t + (γ − 1) 2 + γ p − . v γ v2

(9.7.2) (9.7.3)

t = t − (γ − 1)

(9.7.4) (9.7.5)

If the above expressions are introduced in the balance equations and the constitutive equations of the overlined quantities in the rest frame are provided, then we obtain a set of partial differential equations that at least in principle allow us to determine the evolution of the continuous system when they are equipped with suitable initial and boundary conditions. However, the balance equations relative to the inertial observer I involve only quantities that refer to the rest frame instead of quantities that relate to the observer I . In other words, it is not clear how we can define the stress, the specific energy, the heat current vector, etc., relative to I . It is evident that there are many possibilities to define these quantities by reasonable definitions supported by suitable experimental procedures to measure them. In the literature there are many different proposals to define the relative stress, the heat current vector, etc. In [46], the following definitions of the relative stress tensor t and the relative heat current vector s are proposed: α v ⊗ s, c2 p = v · t − s, t=t−

(9.7.6) (9.7.7)

where α is an unknown real number. Then, still in [46], homogeneous thermodynamic processes are considered to define the global energy U, the total work L, and the total heat Q that the system exchanges with the external world. Further, it is proved that for α = 0, the transformation formulas of U, L, Q in going from the rest frame to any inertial frame coincide with the transformation formulas proposed by Einstein, Planck, and von Laue (see [37, 112, 121]). For α = 1, the transformation formulas reduce to those proposed by Kibble and Møller: ([84, 112]). Finally, for α = c2 (γ −

9.7 On the Physical Meaning of Relative Quantities

291

1)/(γv 2 ), it is possible to derive the transformation formulas proposed by Landsberg [88].5 In conclusion, all the above choices and many others are acceptable, since each of them is associated with a particular arbitrary definition of the relative quantities, together with a corresponding measuring process.

5 See

[2].

Chapter 10

Electrodynamics in Moving Media

In Chap. 9 we analyzed the relativistic formulation of a continuous distribution of matter S in the presence of thermal phenomena. In this chapter we propose to extend the results of Chap. 9 to the case of a charged mass distribution. To achieve this goal we must answer the following questions: • How do we write Maxwell’s equations in matter? • How do we describe the interaction between electromagnetic fields and matter? We describe the most famous answers to the above questions (Minkowski, Abraham, Chu, Ampère, Boffi), showing their complete equivalence. In fact, we show that we can transform one model of matter proposed by one of the above proposals into another one by simply changing the fundamental variables by which we decide to describe the electromagnetic field. Consequently, the different models do not supply a microscopic physical description of matter. This is due to the fact that we cannot measure the electromagnetic fields inside a body, but we can evaluate only their exterior effects on matter.

10.1 Maxwell’s Equations in Matter In Chap. 9 we presented a relativistic theory of thermodynamics of moving continua. We now wish to account for the case of a moving continuum that carries charges and currents. To describe such a system we must determine: • the equations that govern the evolution of electromagnetic fields in matter; αβ • the momentum–energy tensor T(e) of the electromagnetic field. αβ

In particular, when the 4-tensor T(e) is known, we determine the momentum carried by the electromagnetic field and the 4-force acting on matter. The correct form of the momentum–energy tensor of an electromagnetic field has been debated for almost a century. Two different forms of the momentum–energy tensor were © Springer Nature Switzerland AG 2019 A. Romano and M. Mango Furnari, The Physical and Mathematical Foundations of the Theory of Relativity, https://doi.org/10.1007/978-3-030-27237-1_10

293

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10 Electrodynamics in Moving Media

originally proposed by Minkowski [109, 110] and Abraham [1], and many proposals for the form of this tensor have been added in later years. Since the literature about this subject is very extensive, we choose to cite only the following papers: [12, 14, 18, 44, 48, 68, 69, 71, 95, 98, 99, 103, 117, 122, 125, 152, 153], which cover the period 1908–2011. Why so many different forms of the momentum–energy tensor? The electrodynamics in vacuum in the presence of charges and currents is a well-established theory, and there is only one formulation for it. Consequently, there is one electromagnetic momentum–energy tensor and one electromagnetic 4-force (see Chap. 5 and Sect. 8.7). As regards electrodynamics in matter, many formulations have been proposed for the interaction between electromagnetic fields and moving matter. The first of them, which is due to Minkowski [109, 110], is purely phenomenological, and it assumes that Maxwell’s equations take the same form in moving bodies as in bodies at rest. In all the other formulations, matter is replaced by fictitious distributions of charges and currents that are derived using a microscopic model to describe magnetization and polarization in matter. The advantage of the latter approach is that it makes it possible to derive the Maxwell equations in moving media from the well-established equations in vacuum using the macroscopic charges and currents derived from the adopted microscopic model of matter as sources of electromagnetic fields. In spite of this considerable advantage, this approach shows some deficits. In fact, these models are obtained by starting from the microscopic structure of matter and introducing elementary charge and current distributions. Then classical physics instead of quantum physics is applied to describe the microscopic world. Further, the macroscopic quantities are obtained as mean values of the microscopic distributions by complex procedures that often have to be simplified to reach a result. α acting In any case, each model leads to a different expression of the 4-force f (e) αβ

on the matter, and thus to a different momentum–energy tensor T(e) for the electromagnetic field defined by the condition αβ

α =− f (e)

∂T(e) ∂x β

.

(10.1.1)

Further, the balance equations assume the form ∂T αβ = 0, ∂x β where

αβ

(10.1.2)

αβ

T αβ = T(m) + T(e) αβ

(10.1.3) αβ

is the total momentum–energy tensor, while T(m) and T(e) denote its mechanical part α leads to an and electromagnetic part, respectively. In some cases, the choice of f (e)

10.1 Maxwell’s Equations in Matter

295

αβ

electromagnetic tensor T(e) that is not symmetric together with the total momentum– energy tensor. The choice among the proposed models is a difficult task, and it appears rather a matter of preference. In many of the above-cited papers it is suggested that all these choices could be equivalent. The following considerations suggest the reasons for the equivalence of the many existing formulations of the electrodynamics of moving bodies. The evolution of a continuous system S that carries charges and currents is determined when the equations of motion x(X, t), the temperature field θ(X, t), and the electromagnetic fields E(X, t), H(X, t) are known. These fields can be found by solving the system of Maxwell’s equations (10.1.2), which involve the whole momentum energy tensor T αβ , provided that the constitutive equations of D, B, J, and T αβ are given. In contrast, by adopting a particular model, we obtain αβ the electromagnetic momentum–energy tensor T(e) for which we have to give the constitutive equations. But this tensor describes only a part of our system S, so that αβ to describe the whole system S, we have to assign the constitutive equations of T(m) . αβ

αβ

If our choice is such that T(m) + T(e) is equal to the previous choice of the total momentum–energy tensor T αβ , we obtain the same behavior of S.1

10.2 About the Equivalence of Formulations of the Electrodynamics of Moving Bodies Let S be a charged continuous system and let x be a particle of S that is moving with respect to the inertial observer I at the instant t. We denote by U α the 4-velocity at the point X = (x, t) ∈ V4 and the three-dimensional space of the spatial vectors that are orthogonal and parallel to U α at X by X and X , respectively. Finally, we denote by d α , eα , bα , h α the electric induction, electric field, magnetic induction, and magnetic field, respectively, in the proper frame I X . The electromagnetic field in matter is then described by the four fields d α , eα , bα , h α , and the two fields p α , m α , which represent the polarization and the magnetization in the rest frame I X of V4 at X. The following relations hold between these fields: 0 p α = d α − 0 eα , α

α

α

μ0 m = b − μ0 h ,

(10.2.1) (10.2.2)

where 0 is the dielectric constant and μ0 the magnetic permeability of a vacuum. In other words, only four of the six fields d α , eα , bα , h α , p α , m α are independent. On the other hand, there are only two independent Maxwell’s equations. Therefore, taking into account (10.2.1) and (10.2.2), we must give the constitutive equations for two of the above fields in terms of the remaining two fields. It is then possible to prove that 1A

complete comparison among the many proposals can be found in [76, 77].

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10 Electrodynamics in Moving Media

whichever of the four fields we decide to use to describe the electromagnetic fields in matter, there is a particular corresponding physical model.

Consequently, all the proposed models are fully equivalent, and they correspond to particular selections of the fundamental variables adopted in order to describe the interaction between matter and fields (see [141]). In a mechanical theory, the 4-vector c2 q α in (9.3.11) has been defined by the condition that its components in the rest frame coincide with the heat current vector. In the presence of electromagnetic fields, we assume that the vector c2 q α must be α α ) ∈ Xˆ , where the components of c2 σ(e) in substituted by the 4-vector c2 (q α + σ(e) the rest frame are (10.2.3) c2 σ α(e) = (E × H, 0). In other words, the energy current vector in the rest frame is obtained by adding the heat current vector and the Poynting vector. In every Lorentz frame we have that2 1 α c2 σ(e) = − αβλμ eβ h λ Uμ , c

(10.2.4)

where αβλμ is the Levi-Civita symbol, and (9.5.16) becomes μ ˙ = −αβ

α α α ∂σ(e) ∂U α α 2 ∂q 2 − (q + σ )A − c − c + r. α (e) ∂x β ∂x α ∂x α

(10.2.5)

α /∂x α in Minkowski’s forIn the next section we evaluate the 4-divergence ∂σ(e) mulation of the electrodynamics of moving bodies. We note that up to now, all our considerations have been based on the assumption (10.2.3), and we have not yet used either Maxwell’s equations or any model of interaction between matter and electromagnetic fields.

10.3 Minkowski’s Description We must adopt a description of the interaction between matter and electromagnetic fields if we want to evaluate the 4-divergence of σe in (10.2.5). Let S be a continuous system carrying electromagnetic fields. We start from Minkowski’s phenomenological model, which is based on the assumption that Maxwell’s equations in the form ∇ × HM = J + ∇ · DM = ρ f ,

∂D M , ∂t

(10.3.1) (10.3.2)

2 It is sufficient to verify that in the rest frame, (10.2.4) reduces to (10.2.3). In fact, in the rest frame,

(Uλ ) = (0, −c) and αβ4μ = 0 if one of the indices α, β, μ takes the value 4.

10.3 Minkowski’s Description

297

∇ × EM = − ∇ · B M = 0,

∂B M , ∂t

(10.3.3) (10.3.4)

hold in every inertial frame of reference.3 In (10.3.1)–(10.3.4), the subscript M recalls that we are referring to Minkowski’s approach. Further, E M , D M , H M , and B M denote the electric field, the electric induction, the magnetic field, and the magnetic induction, respectively, and they are the basic fields of Minkowski’s formulation. Finally, ρ f is free charge density and J the current flux. It is well known that the covariance of (10.3.1)–(10.3.4) under Lorentz transformations is proved if we succeed in giving them a tensor form. Introducing the skew-symmetric matrices ⎛

0 ⎜−H 3 αβ M (HM ) = ⎜ ⎝ HM2 cD 1M ⎛ 0 ⎜−E 3 ∗αβ M (E M ) = ⎜ 2 ⎝ EM 1 cB M and the matrix

⎞ HM3 −HM2 −cD 1M 0 HM1 −cD 2M ⎟ ⎟, 1 −HM 0 −cD 3M ⎠ cD 2M cD 3M 0 ⎞ 3 2 1 E M −E M −cB M 1 2 ⎟ 0 EM −cB M ⎟ 1 3 ⎠, −E M 0 −cB M 2 3 cB M cB M 0

JMα = (J, cρ f ),

(10.3.5)

(10.3.6)

(10.3.7)

equations can be written in the following form: αβ

∂ HM = JMα , ∂x β ∗αβ ∂ EM = 0, ∂x β

(10.3.8) (10.3.9)

which is a tensor form if we assume that (10.3.5), (10.3.6) define a skew-symmetric αβ αβ tensor and (10.3.7) a 4-vector of Spacetime V4 . Here HM and HM are respectively called the 4-tensor of magnetic field–electric induction and the 4-tensor of electric field–magnetic induction. Applying the decomposition (8.9.1) to the 4-current J M , we obtain J M + ρ f U. (10.3.10) JM = The spacelike vector J M is called the conductive current, the timelike vector ρ f U is the convective current, and ρ f is the charge density in the proper frame at X. To verify that ρ( f ) is the rest charge density, it is sufficient to note that from (8.9.2) it 3 At

least for low accelerations of the particles of S.

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follows that Jα = −

1 1 β Jβ U β U α = − 2 J β U U α = ρ f U α , 2 c c

since Jβ U β is invariant under Lorentz transformations. The hypothesis that H αβ and E ∗αβ are tensors of V4 determines the transformation character of electromagnetic fields under Lorentz transformations. In fact, from (10.3.5) and the transformation formulas αβ

λμ

HM = Aαλ Aβμ HM ,

∗αβ

EM

∗λμ

= Aαλ Aβμ E M ,

J α = Aαλ J λ

we easily derive B = B , E = E ,



B⊥ = γ B⊥ − c12 u × E⊥ ,  E⊥ = γ(E⊥ + u × B⊥ ).

B = B , E = E ,



B⊥ = γ B⊥ − c12 u × E⊥ ,  E⊥ = γ(E⊥ + u × B⊥ ).

H = H , D = D ,

 H⊥ = γ(H

⊥ − u1 × D⊥ ),   D⊥ = γ D⊥ + c2 u × H⊥ ,

J = γ(J  − ρ f U), U·J  . ρf = γ ρf − 2 c

 J⊥ = J⊥ ,

It is evident that these formulas must be experimentally verified in order to be accepted. We now propose to determine the electromagnetic momentum–energy tensor corresponding to the phenomenological Minkowski’s approach. First, we recall that formulas (8.9.23), (8.9.29), (8.9.30) hold for a skew-symmetric 4-tensor. Then they αβ ∗αβ αβ ∗αβ can be applied to HM and E M , and the tensors q α and sλ relative to HM and E M assume the following forms: Uλ λμ H , c2 M Uλ ∗λμ bα = −Pμα 2 E M , c 1 ρν h λ = λρνδ HM U δ , 2c 1 ∗ρν eλ = λρνδ E M U δ . 2c

d α = −Pμα

(10.3.11) (10.3.12) (10.3.13) (10.3.14)

The physical meaning of the space vectors (10.3.11)–(10.3.14) can be recognized by proving that their components in the rest frame T X are

10.3 Minkowski’s Description

299

(eα ) = (E M , 0), α

(d ) = (D M , 0), α

(h ) = (H M , 0), α

(b ) = (B M , 0), .

(10.3.15) (10.3.16) (10.3.17) (10.3.18)

We limit ourselves to verifying (10.3.15) and (10.3.16). From (10.3.14), we have 1 1 ∗ρν δ ∗ρν λρνδ E M U = λρν4 E M , 2c 2

eλ =

and (10.3.15) is proved. Further, from (10.3.11) we have     α α U Uμ 1 1 α U Uμ λμ α d = − 2 Uλ δ + HM = , δ + c c2 c μ c2 α

so that i

d =

1 4i i H = D M , e4 = 0. c M

In conclusion, we can write H αβ and E ∗αβ as follows: αβ

1 αβμλ  Uμ h λ + (U α d β − U β d α ), c 1 = αβμλ Uμ eλ + (U α bβ − U β bα ). c

HM = ∗αβ

EM

(10.3.19) (10.3.20)

α in (10.3.1), we prove the following Adopting the notation σ αM instead of σ(e) fundamental formula:

− c2 where

α ∂σ αM β ∂U = eα d˙ α + h α b˙ α −  Mα β + σ αM Aα − eα J α , α ∂x ∂x

(10.3.21)

αβ = −(eλ d λ + h λ bλ )η αβ + (eα d β + h α bβ ),

(10.3.22)

and J α is defined by (10.3.10). Using the notation ∂/∂x α = ∂α , from (10.2.4), we have 1 1 c2 ∂α σ αM = − ∂α (αβλμ Uμ eβ h λ ) = ∂α (αβμλ Uμ eβ h λ ) c c 1 1 αβμλ = eβ ∂α ( h λ Uμ ) + h λ ∂α (αβμλ eβ Uμ ) c c 1 αβμλ eβ h λ ∂α Uμ . −  c

(10.3.23)

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10 Electrodynamics in Moving Media

We now give the three terms on the right-hand side of (10.3.23) a more convenient form. In view of (10.3.19) and the condition eβ U β = 0 (see (10.3.14)), the first term becomes   1 eβ ∂α (αβμλ h λ Uμ ) = eβ J α − ∂α (U α d β ) + ∂α (U β d α ) c = eβ J β − d˙ β eβ − eβ d β ∂α U α + d α eβ ∂α Uβ .

(10.3.24)

For (10.3.15) and the condition h λ U λ = 0 (see (10.3.15)), the second term can be written as follows:   1 h λ ∂α (αλμβ eβ Uλ ) = h λ −∂α (U α bλ ) + ∂α (U λ bα ) c = −b˙ λ h λ − h λ bλ ∂α U α + h λ bα ∂α Uλ .

(10.3.25)

To transform the last term of (10.3.23) we need to apply the decomposition (8.9.19) to ∂α Uβ : 1 μ δ ∂α Uβ = ∂ α Uβ − 2 Pα Uβ U ∂μ Uδ c 1 1 − 2 Uα U μ Pβδ ∂μ Uδ + 4 Uα Uβ U μ U δ ∂μ Uδ . c c The second and fourth terms on the right-end side vanish, since U δ Uδ = −c2 implies U δ ∂μ Uδ = 0, and we have 1 α α ∂α Uβ = ∂ α U β − 2 U α A β , ∂α U = ∂α U , c

(10.3.26)

where Aα = dU α /dτ is the 4-acceleration. In view of (10.3.26), the third term can be transformed as follows: 1 1 1 − αβμλ eβ h λ ∂α Uμ = − αβμλ eβ h λ (∂ α Uβ − 2 Uα A μ ) c c c 1 1 = 3 αβμλ eβ h λ Uα Aμ = 3 μβλα eβ h λ Uα Aμ , c c since eβ h λ ∂ α Uβ is a space tensor. Finally, recalling (10.2.4), we obtain 1 μ − αβμλ eβ h λ ∂α Uμ = −σ M Aμ . c

(10.3.27)

Formulas (10.3.21), (10.3.22) are proved if we take into account (10.3.23)–(10.3.27). Minkowski’s electromagnetic specific energy ξ M in the rest frame is defined as

10.3 Minkowski’s Description

301

μ

dξ M = eα d˙ α + h α b˙ α , dτ

(10.3.28)

whereas Minkowski’s electromagnetic momentum–energy tensor is αβ

αβ

β

TM = μ0 ξ M U α U β +  M + σ αM U β + σ M U α .

(10.3.29)

αβ

We note that TM is not symmetric, since T [αβ] = [αβ] = e[α d β + h α bβ] .

(10.3.30)

Then on introducing (10.3.21) into (10.2.5), the balance of specific proper energy becomes μ

d β αβ α 2 α ( − ξ M ) = −(αβ −  M )∂ α Uβ − q Aα − c ∂α q − J eβ + r . (10.3.31) dτ

This equation could be interpreted as the balance of the “mechanical” part of the continuous system.

10.4 Ampère’s Model In this section we describe Ampère’s model [76, 122]. In view of what we stated in the preceding sections, this model can be obtained by choosing suitable fundamental variables to describe the electromagnetic field. More specifically, we use the fundamental variables bα , eα , p α , and m α , where the last two variables are given by constitutive equations that depend on at least bα , eα . We first express the electromagnetic momentum σ αM , now denoted by σ αA , in terms of the fields bα , eα , p α , and m α . Then from (10.2.4) and (10.2.1), we obtain σ αA = σ αM = − αβ

1 αβλμ  eβ c3



bλ − m λ Uμ . μ0

∗αβ

(10.4.1) ∗αβ

We also express HM and E M in terms of bα , eα , p α , and m α . It is evident that E M does not change, since it is already expressed in terms of these fields. Further, when we introduce the 4-tensors (see (10.3.19), (10.3.20)) αβ

1 αβμλ bλ  Uμ + (U α eβ − U β eα ), c μ0 1 ∗αβ = E M = αβμλ Uμ eλ + (U α bβ − U β bα ), c

HA = ∗αβ

EA

Maxwell’s equations assume the following form:

(10.4.2) (10.4.3)

302

10 Electrodynamics in Moving Media αβ

∂ HA = J Aα , ∂x β ∗αβ ∂EA = 0, ∂x β

(10.4.4) (10.4.5)

where the current vector J Aα is given by ∂ 1 ∂ J Aα = J α + αβλμ β (Uμ m λ ) − β (U α p β − U β p α ). c ∂x ∂x

(10.4.6)

Easy calculations lead to the conclusion that instead of the decomposition (10.3.10), we have J Aα = J Aα + ρ A U α , (10.4.7) with β β α J Aα = J Mα + Pλα p˙ λ + p α ∂ β U − p ∂β U 1 αβμλ + αβμλ Uμ ∂ Uμ m β A λ , β mλ − 2  c  A =  f − Pαλ ∂λ p α − λβνμ Uν m λ ∂ β Uμ .

(10.4.8) (10.4.9)

In order to derive the space form of equations (10.4.4), (10.4.5), we must define the Ampère electromagnetic fields E iA , H Ai , PAi , and M Ai with respect to a Lorentz frame I . From (10.4.5) and (10.3.6), it follows that E iA = E iM ,

i B iA = B M .

(10.4.10)

These identities supply the space form of the left-hand sides of (10.4.5). The space form of J Aα can be found by the definition of PAi and M Ai in terms of p α and m α . To this end, we observe that ⎞ ⎛ 1 i jh 1 i jh   (v m + c m ) v m h 4 h j h⎟ 1 αβμλ ⎜ c  (10.4.11) Uμ m λ = ⎝ c ⎠, 1 c − i j h v j m h 0 c where v is the velocity of a particle of the continuous system. Then we define the Ampère magnetization as follows: MA h ≡

1 γv (vh m 4 + c m h ), (M A h = m h ). c

(10.4.12)

It is evident that (10.4.12) defines the transformation properties of Ampère’s magnetization. Further, the identity

10.4 Ampère’s Model

303

m α U α = γv (m · v + c m 4 ) = 0 and (10.4.12) lead to the relations γv m i = M Ai + γv2

vi M A · v, c2

1 i jh 1 γv  v j m h = − (M A × v)i . c c

(10.4.13) (10.4.14)

In conclusion, (10.4.11) can be written as ⎞ 1 i 1 αβμλ ⎜  M A h − c (M A × v) ⎟  Uμ m λ = ⎝ 1 ⎠. c (M A × v)i 0 c ⎛

i jh

(10.4.15)

On the other hand, we have

γv (v i p 4 − c pi ) γv v [i p j] U p −U p = . 0 γv (v i p 4 − c pi ) α β

β

α

(10.4.16)

In view of (10.4.2), we define the Ampère polarization as follows: i

− c PAi ≡ γv (v i p 4 − c P i ), (P = pi ).

(10.4.17)

From the space character of p α , pα U α = γv (p · v − cp4 ) = 0,

(10.4.18)

we obtain 1 2 i γ v P A · v, c2 v γv v [i p j] = i j h (v × P A )h , γv pi = PAi +

(10.4.19) (10.4.20)

so that (10.4.16) becomes U α pβ − U β pα =



i j h (v × P A )h −c PAi . c PAi 0

(10.4.21)

Finally, by (10.4.4), (10.4.5), (10.4.15), (10.4.21), we can write Ampère’s equations in the following space form:

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∇×

∂P A ∂E A BA = + ∇ × (P A × v) + ∇ × M A − 0 μ0 ∂t ∂t 1 ∂ − 2 (M A × v) + J, c ∂t 1 0 ∇ · E A = −∇ · P A + 2 ∇ · (M A × v) + ρ f , c ∂B A , ∇ × EA = − ∂t ∇ · B A = 0.

(10.4.22) (10.4.23) (10.4.24) (10.4.25)

These equations are usually derived by adopting the models of electric dipoles for polarization and the current loops for magnetization. They were obtained here by merely substituting in Minkowski’s description the fields eα , d α , h α , and bα with the fields eα , bα , p α , and m α . Moreover, it can be proved that the transformation properties of M Ai and PAi deduced from (10.4.12) and (10.4.17) coincide with those obtained by Ampère’s model. The balance equation of energy (10.2.5) has to be expressed in terms of the fields eα , bα , p α , and m α to complete the Ampère scheme. In this way, the Ampère αβ momentum–energy tensor T A will be determined, and consequently the 4-force f Aα αβ is given by the relation f Aα = ∂β T A . In terms of the Ampère fields, the Poynting vector is given by (10.4.1), which we write as follows: bλ 1 1 c2 σ(e) = − αβλμ eβ Uμ + αβλμ eβ m λ Uμ ≡ c2 σ A + c2 σ M . c μ0 c

(10.4.26)

We notice that the terms on the right-hand side of (10.4.26) coincide with the righthand side of (10.2.4), provided that h α is replaced by bα /μ0 and −m α , respectively. In view of (10.3.21) and (10.3.22), we obtain

 bα b2 P αβ −c2 ∂α σ αA =0 eα e˙α + b˙ α + 0 e2 + μ0 μ0 (10.4.27)

 bα β α β α α 2 α  − 0 e e + b ∂α Uβ − eα J A + σ A Aα + c ∂α σ M . μ0 Denoting by ψ any function of X, τ and recalling (10.5.7), (10.3.26), we have 1 d μ 2 dτ

ψ 1 dψ 1 ψ dμ 1 dψ 1 = − = + ψ∂β U β μ 2 dτ 2 μ dτ 2 dτ 2 1 1 dψ + ψ P αβ ∂ = α Uβ . 2 dτ 2

This result allows us to write

10.4 Ampère’s Model

305

1 d 2 (e ) = 2 dτ 1 d 2 (b ) = bα b˙ α = 2 dτ eα e˙α =

1 d 2 1 (e ) + e2 P αβ ∂ α Uβ ; 2 dτ 2 1 d 2 1 (b ) + b2 P αβ ∂ α Uβ . 2 dτ 2

(10.4.28) (10.4.29)

In view of (10.4.28), (10.4.29), formula (10.4.27) becomes αβ α α α − c2 ∂α σ(e) = μ ξ˙ A −  A ∂ α Uβ − eα J A + σ A Aα + ∂α σ(m) ,

(10.4.30)

where

1 b2 2 , o e + ξA = 2μ μ0

1 b2 1 α β αβ 2 αβ α β P + 0 e e + b b , 0 e + A = − 2 μ0 μ0 1 αβλμ σA = −  eβ bλ Uμ , μ0 c3 1 σ(m) = − 3 αβλμ eβ m λ Uμ , c

(10.4.31) (10.4.32) (10.4.33) (10.4.34)

and the balance of energy (10.2.5) assumes the form μ

d αβ 2 α α (ξ − ξ A ) =(αβ −  A )∂ β Uα − c ∂α (σ − σ A ) dτ − (σ α − σ αA )Aα − eα J Aα .

(10.4.35)

The preceding results allow us to state that the momentum–energy tensor of Ampère’s formulation is αβ αβ β (10.4.36) TA = μ ξ A +  A + σαU β + σ A U α . In particular, in the rest frame T , we have

⎞ 1 i jl j 1 i j i j  E Bl ⎟ δ i j + 0 E E + B B αβ μ0 μ0 c TA ⎟ . 1 i jl j 1 2 2 ⎠ 1  E Bl 0 E + μ0 B μ0 c 2 (10.4.37) In (10.4.37), the subscript A has been omitted. In view of (10.4.2), (10.4.3), and (10.4.10), this tensor can also be written in the form ⎛

1 2 − ⎜ 2 0 E + =⎜ ⎝

2

B μ0



1 1 γδ αβ γβ T A = − (E αAγ H A + E A H Aγδ η αβ ). c 4

(10.4.38)

This statement can be easily verified by proving that in the rest frame, (10.4.38) and (10.4.37) are equal.

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10 Electrodynamics in Moving Media

It is possible to prove (see [141]) that the balance of energy (10.4.35) can be written in the form given in [68, 69]. In fact, from (10.4.8), we obtain the formulas α β β α eα J α =eα J Mα + eα p˙ α + eα p α ∂ β U − e p ∂β U 1 αβμλ + αβμλ Uμ eα ∂ Uμ eα m β Aλ , β Uα − 2  c

(10.4.39)

1 αβμλ α α  − σ(m) Aα = − αβμλ Uμ eα ∂ Uμ eα m β Aλ m ρ bν ∂ −c2 ∂α σ(m) β mλ + 2  ν uρ c β ˙α − m α bα ∂ β U − mαb , (10.4.40) from which, taking into account the identity (see (10.4.28), (10.4.29)) eα p˙ α = μ

d dτ



eα p α μ

β − eα p α ∂ βU ,

(10.4.41)

we obtain ∗

2 α α α ˙α α μξ˙ A = −t αβ ∂ β Uα − c ∂α q − q Aα − pα e˙ − m α b + eα J M ,

(10.4.42)

where ∗

1 eα p α (0 e2 + b2 /μ0 ) − , 2μ μ

1 b2 bα bβ P αβ + 0 eα eβ + 0 e2 + =αβ − 2 μ0 μ0

ξ A =ξ − t αβ

+ (eα p β − m α bβ ) + m λ bλ P αβ .

(10.4.43)

(10.4.44)

Equation (10.4.42) was determined in [68, 69]. Finally, we note that from (10.4.44), we have (10.4.45) t [αβ] = −( p [α eβ] + m [α bβ] ).

10.5 Boffi’s Formulation Boffi’s formulation adopts the same variables of Ampère’s formulation, namely eα , bα , p α , m α . Consequently, (10.4.4), (10.4.5) still represent the electromagnetic field equations, and (10.4.38) is the momentum–energy tensor. In other words, from a 4-dimensional point of view, there is no difference between the Ampère and Boffi formulations. However, they differ in respect to the spatial form of electromagnetic equations because the polarization PBi and magnetization M Bi of Boffi’s formulation have properties of transformation that differ from those of the corresponding vectors of Ampère’s formulation. In fact, starting from the adopted model, the following

10.5 Boffi’s Formulation

307

definitions of polarization and magnetization are assumed (see, for instance, [12, 122]), 1 M A × v, c2 MB = M A − v × PA, PB = P A −

(10.5.1) (10.5.2)

so that from (10.4.15) and (10.4.21) we have

i jh 1 αβμλ  M Bh c PBi ,  Uμ m λ − (U α p β − U β p α ) = −c PBi 0 c

(10.5.3)

and Minkowski’s equations can be written as follows: 1 ∂P B ∂E B + + ∇ × M B + J(e) , ∇ × B B = 0 μ0 ∂t ∂t 0 ∇ · E B = −∇ · P B + ρ(e) , ∂B B , ∇ × EB = − ∂t ∇ · B B = 0,

(10.5.4) (10.5.5) (10.5.6) (10.5.7)

where EB = E A,

BB = B A.

(10.5.8)

10.6 Chu’s Formulation Chu’s formulation is obtained by expressing Minkowski’s equations (10.3.8), (10.3.9) and the balance of energy (10.3.31) in terms of the 4-vectors eα , h α , p α , m α . We introduce the following two skew-symmetric 4-tensors: αβ

1 αβμλ  Uμ h λ + 0 (U α eβ − U β eα ), c 1 ≡ αβμλ Uμ eλ + μ0 (U α h β − U β h α ), c

HC ≡ αβ

EC

(10.6.1) (10.6.2)

depending only on eα and h α . Then using these 4-vectors, Minkowski’s equations assume the form αβ

∂β HC = JCα , αβ

∂β E C = Q Cα , where

(10.6.3) (10.6.4)

308

10 Electrodynamics in Moving Media α JCα = J(e) − ∂β (U α p β − U β p α ),

Q Cα

α

β

β

α

= μ0 ∂β (U m − U m ).

(10.6.5) (10.6.6)

By simple calculations it can be proved that instead of decomposition (10.3.10) we have JCα = J Cα + ρC U α , Cα + Q C U α , Q Cα = Q

(10.6.7) (10.6.8)

with β β α J Cα = J Mα + Pβα p˙ β ∂ β U − p ∂β U

ρC = ρ(e) − Pβα ∂α p β , Cα Q α

=

Q =

−μ0 Pβα m˙ β

(10.6.9) (10.6.10)

β − μ0 m ∂ βU + α

α  μ0 m β ∂ βU ,

μ0 Pβα ∂α m β .

(10.6.11) (10.6.12)

We conclude that the choice of variables eα , h α , p α , m α to describe the electromagnetic field leads to the electric dipole model for the electric field and to the magnetic dipole model for the magnetic field. In order to derive the spatial form of (10.6.3), (10.6.4) it is necessary to define the electromagnetic fields E Ci , HCi , PCi , MCi in a Lorentz frame I . Comparing (10.3.19), (10.3.20) with (10.6.1), (10.6.2) and taking into account (10.3.5), (10.3.6), we obtain αβ αβ the components in I of HC and E C . Further, proceeding as in Sect. 10.5, we verify that

i jh  (v × PC )h −c PCi α β β α (10.6.13) U p −U p = c PCi 0

i jh  (v × MC )h −cMCi U αmβ − U β mα = . (10.6.14) cMCi 0 Finally, the electromagnetic equations of Chu’s formulation are ∂EC ∂t 0 ∇ · EC ∂HC ∇ × EC + μ0 ∂t μ0 ∇ · HC ∇ × HC − 0

∂PC + ∇ × (PC × v) + J(e) , ∂t = −∇ · PC + ρ(e) , ∂MC = −μ0 ∇ × (MC × v), − = −μ0 ∇ · MC . =

(10.6.15) (10.6.16) (10.6.17) (10.6.18)

By operating as in the preceding sections, we can prove the following formulas:

10.6 Chu’s Formulation

309

−c2 ∂α σ αM = − c2 ∂α σCα    1 1 d 2 2 (0 e + μ0 h ) + (0 e2 + μ0 h 2 )P αβ =μ dτ 2μ 2  α α α − (0 eα eβ + μ0 h α h β ∂ α Uβ + eα JC + h α Q C + σC Aα , (10.6.19) d αβ 2 α α μ (ξ − ξ C ) = − (αβ − C )∂ β Uα − c ∂α (σ − σC ) dt Cα , − (σ α − σCα )Aα + eα J Cα + h α Q (10.6.20) where 1 (0 e2 + μ0 h 2 ), 2μ 1 αβ C = (0 e2 + μ0 h 2 )P αβ − (0 eα eβ + μ0 h α h β ), 2 1 α σC = − 3 αβμλ Uμ eβ h λ . c ξC =

(10.6.21) (10.6.22) (10.6.23)

In conclusion, the momentum–energy tensor of Chu’s formulation is αβ

αβ

β

TC = μξ C U α U β + C + σCα U β + σC U α ,

(10.6.24)

whose components in the proper frame are ⎞ 1 i jh 1 2 2 ij i j i j  E j Hh ⎟ ⎜ 2 (0 E + μ0 H )δ − (0 E E + μ0 H H ) c ⎟. =⎜ ⎠ ⎝ 1 i jh 2 2 1  E j Hh ( E + μ0 H ) 2 0 c (10.6.25) ⎛

αβ

TC

10.7 Final Remarks We have already noted that Maxwell’s equations in vacuum have been fully confirmed by many experimental data. In contrast, Maxwell’s equations in matter at once became a vexed question. It appeared evident that the only way to overcome the difficulties was to deduce these equations starting from Maxwell’s equations in empty space and proposing a model of matter that assumed suitable distributions of charges and currents as sources of electromagnetic fields. We have also noted that this procedure raises many doubts about the quantum behavior of elementary constituents of matter. Further, the average processes that allow us to deduce macroscopic quantities from the microscopic behavior of constituents of matter are difficult and some-

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times ambiguous. However, there was a flourishing of models to derive Maxwell’s equations in moving matter, and we cited some papers in which these models are discussed. We also referred to other papers in which their equivalence is proved starting from the following considerations. The behavior of the whole system is described by the total momentum–energy tensor. A particular electromagnetic momentum–energy tensor corresponds to the adopted model. But this tensor describes only a part of the continuous system, so that if the mechanical momentum–energy tensor is chosen in a convenient way, we can obtain the same total momentum–energy tensor for any model. In this chapter we have proved that all the models can be obtained by a simple change of the fundamental electromagnetic variables that we adopt to describe the electromagnetic field. In other words, we have verified that if one model is correct, then all the others are correct. This result leaves doubt about the physical consistency of the proposed models. It is impossible that two different descriptions of matter can give the same macroscopic results. Since we have verified just this result, this means that those models have no physical meaning. Perhaps the last reason for this result has to be found in the circumstance that the electromagnetic fields in moving matter are not observable, i.e., they are fictitious but auxiliary fields that lead to the same fields outside moving matter. Something like this happens in elasticity theory. We are unable to evaluate the tension state inside the matter. However, if we suppose that this state exists and that it is described by a Cauchy stress tensor, then we find the right deformation of the elastic body.

Part IV

General Relativity and Cosmology

Chapter 11

Introduction to General Relativity

The principle of special relativity states that all the optically isotropic frames of reference are equivalent for the description of physical phenomena. In other words, two optically isotropic observers I and I  register the same results when they carry out measurements by identical experimental devices in the same physical conditions. From a mathematical point of view this means that: the differential equations describing the phenomena that the observers are examining are the same for all of them. Consequently, when the observers integrate these differential equations by adopting the same initial and boundary conditions, they will obtain the same solution. If an observer adopts a noninertial frame of reference R, then he is compelled to introduce fictitious or inertial forces to obtain theoretical results in agreement with experiment. As a consequence, the evolutions of a phenomenon with respect to an inertial frame of reference I and a noninertial frame R produce different results, and the two frames are not equivalent, since they lead to different descriptions of the same phenomenon. In other words, the description of phenomena does not depend on the uniform translational motion of the inertial frame but depends on the acceleration of the frame of reference relative to an inertial frame. With the aim of overcoming this difference between inertial and noninertial frames of reference, Einstein proposed a new description of physical phenomena in the presence of gravitational fields that was much more innovative and revolutionary than the description adopted in special relativity. This new proposal is based on three milestones: the principle of general relativity, the equivalence principle, and the Mach principle. In this chapter we analyze these principles, and we show the heuristic role they had in suggesting to Einstein a deep modification of Minkowski’s spacetime to account for gravitational effects. Then we describe the new geometric structure of spacetime.

© Springer Nature Switzerland AG 2019 A. Romano and M. Mango Furnari, The Physical and Mathematical Foundations of the Theory of Relativity, https://doi.org/10.1007/978-3-030-27237-1_11

313

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11.1 Difficulties of Newtonian Gravitational Theory Newtonian gravitational theory, which was briefly recalled in Chap. 6, exhibited some fundamental experimental and theoretical difficulties in spite of its huge experimental success. The Newtonian law of gravitation was very successful, since it made possible to derive Kepler’s laws of planetary motion. However, more accurate astronomical observations showed that the planetary motions had a behavior much more complex than the behavior implied by Kepler’s laws. In other words, Kepler’s laws represented a first approximation of this motion, and consequently, Newton’s law did not describe the motion of the planets with sufficient accuracy. At this point, it was possible to assume two opposite positions: (1) Newton’s law must be modified; (2) Newton’s law was correct, and the discrepancies between experimental and theoretical results could be explained by taking into account that the motion of a planet depends not only on the solar action but also on the interaction among the planets themselves. We here sketch the second approach. The observed discrepancies were of two kinds: first, there were disturbances that have no cumulative effects, so they correct themselves after a fixed time. For this reason they were called periodic anomalies. Much more serious were those anomalies that proceeded in the same direction, always increasing in their departure from Keplerian motion. They were called secular anomalies. Before Laplace, the best known of them was the great anomaly of Jupiter and Saturn. Lagrange formulated the Newtonian equations of motion of a planet interacting with the Sun and the other planets in terms of their orbital characteristics. The righthand sides of these equations contain the first derivatives of the perturbation function, which is formulated in terms of the orbital parameters and takes into account the interactions among planets. This function can be developed as a Fourier series, and approximate solutions of the Lagrange equations can be evaluated. Laplace, starting from these equations, computed solutions to a higher order than his predecessors, and he found that in the final expression for the effect of Jupiter’s perturbing action on the mean motion of Saturn, the terms canceled. The same result held for the effect of any planet on the mean motion of any other: thus the mean motions of the planets cannot have any secular modification as a result of their mutual attractions. Laplace showed that the Jupiter and Saturn great anomaly is not a secular inequality, since it describes periodic anomalies of the motion of both the planets with the very long period of 929 years. A typical secular effect is represented by the anomalous motion of the perihelion of the planets. The above-mentioned approach allowed one to explain this secular effect, and the theoretical results were in agreement with the experimental ones except for Mercury, for which there was a difference of 42 seconds per century between the theoretical and experimental data. Now we recall some theoretical discrepancies. If the mass distribution in the universe fills the whole universe, then formula (6.3.1) implies an infinite value of the gravitational force at every point. In other words, finite forces are possible only

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in a bounded universe. But in view of the gravitational law, such a universe should collapse on itself. Further, the Newtonian gravitational force acting on a material point P at an instant t depends on the distances between P and the points interacting with P at the same instant. In other words, the velocity of propagation of the gravitational action is infinite. Laplace’s and Poisson’s equations, which follow from Newton’s law, are elliptic equations, and consequently, they do not admit wavelike solutions. According to the special theory of relativity, no physical action can propagate with velocity greater than the velocity of light, and then the Newtonian gravitational action cannot be accepted.

11.2 Attempts to Overcome the Difficulties of Newtonian Gravitational Theory (a) Newtonian Nonrelativistic Approaches. We here limit ourselves to recall only two classical attempts to match the theoretical and experimental data. First, if we suppose that the Sun, instead of being spherical, exhibits a polar flattening, then the solar action on a planet is described by a Newtonian central force and a dipole momentum (see Chap. 6). The presence of this momentum generates a planet’s perihelion advance. A polar flattening of the Sun producing Mercury’s perihelion advance of two seconds per century is already observable. However, this hypothesis must be rejected, since the strong polar flattening of the Sun that would be required to produce the delay of 42 seconds per century has not been observed. Another approach consists in changing the Newtonian gravitation law in such a way to justify Mercury’s observed perihelion advance without modifying the main results of the theory. For example, if we assume a force law such that F(R) = −h

M M vers R, Rn

it is possible to prove that the perihelion advance will be positive if n > 2 and negative if n < 2. Further, a value of n = 2.00000016 is sufficient to justify 42 seconds of Mercury’s perihelion advance. However, Brown and De Sitter (1905) proved that this correction implies unacceptable consequences on the motion of the Moon. In any case, all the classical approaches to modify Newton’s law are nonrelativistic and therefore unacceptable. (c) Relativistic approaches. From 1904 on, the Newtonian law of gravitation was examined in the context of the theory of relativity by Poincaré and Lorentz. Poincaré suggested modifications of the Newtonian formula, which were afterward discussed and further developed by Minkowski and Willem de Sitter. It was found that relativity theory would require secular motions of the planetary perihelia, which, however, are appreciable only for Mercury, and even for that planet they were too small to be observed.

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Another proposal to formulate a relativistic theory of gravitation is based on the fact that Poisson’s equation describes both the electrostatic field, when the source term is identified with the charge density , and the Newtonian gravitational field, when this term is identified with the mass density μ. This double interpretation of Poisson’s equation suggested the possibility to extend the relativistic formulation of electrodynamics (see Chap. 5) to the gravitational field. However, in the relativistic formulation of electrodynamics, the charge density , multiplied by the light velocity c in vacuum, is the fourth component of the 4vector J α = (J, c). Further, the electrostatic potential V is the fourth component of a 4-potential ϕμ that satisfies the following invariant equation in spacetime V4 :  ϕα =

4π α J , c

(11.2.1)

where  is the d’Alembert operator =

∂2 ∂2 1 ∂2 ∂2 + + − . c2 ∂t 2 ∂x12 ∂x22 ∂x32

By contrast, in special relativity the proper mass density μ¯ 0 times c2 is the component T¯ 00 in the proper frame I¯0 at a point of the continuous mass distribution of the symmetric momentum–energy tensor T αβ . Therefore, ρ and μ have different transformation properties, and the relativistic formalism of electrodynamics cannot be extended to gravitation. Gunnar Nordström’s gravitational theory is based on the assumption that in contrast to what holds for electromagnetic fields, the gravitational scalar potential V is supposed to be an invariant scalar, not the fourth component of a 4-vector. In order to preserve the covariance of Poisson’s equation under Lorentz’s transformation, the ¯ 2 , where mass density is supposed to be equal to the scalar term T = Tαα = μc ¯ αU β T αβ = μU λ

is the momentum–energy tensor of dust matter S, U λ = dU is the world speed of dτ the particles of S, and τ the proper time. Unfortunately, the relativistic equation so obtained, 4π ¯  V = − 2 hT = 4πhμ, c implies an incorrect evaluation of Mercury’s perihelion advance. Another attempt consists in retaining the tensor character of the source term of Poisson’s equation but substituting, in the proper frame of matter, the above equation with 4π (11.2.2) α¯ 44 = 2 hT¯44 , c

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so that the gravitational field is described by a symmetric tensor αλμ . Consequently, the relativistic formulation of (11.2.2) is  αλμ =

4π hTλμ . c2

(11.2.3)

Unfortunately, even this proposal cannot be accepted. In fact, adopting the momentum–energy tensor T αβ = μU ¯ α U β of dust matter and recalling the conservation equations ∂λ (μU λ ) = 0, ∂λ T λμ = 0, we obtain

so that

U λ ∂λ U μ = 0, d2xμ = 0, dτ 2

and the world trajectories of the particles of dust matter are straight lines. It is evident that this result can be accepted only in the absence of gravitation.

11.3 Principles of General Relativity and General Covariance Einstein spent more than ten years after his paper on special relativity (1905–1916) before proposing the general theory of relativity with the intent of realizing the following goals: • All the frames of reference can be adopted to describe physical phenomena. Therefore, the inertial frames of reference lose their privileged status. • Gravitational interaction is formulated in such a way as to satisfy the principle of relativity. In particular, its action propagates with finite velocity. Today everybody believes that the final formulation of the theory is correct, in view of the relevant amount of experimental confirmation, but there is no agreement about the logical foundations of the theory. On the other hand, this uncertainty in evaluating the foundations appears even in the papers that Einstein published before arriving at the final formulation of general relativity. In the years since 1916, a huge number of papers have been published about the interpretation of the foundations of the theory, and for every hypothesis that has been presented in one paper, the opposite hypothesis has been proposed in another. Consequently, the presentation of the theory as be found in textbooks on general relativity often differs from one book to another. In this gallery of opinions it seems that we can identify the following three fundamental principles that constitute the heuristic guidelines in formulating the theory:

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• The principle of general relativity: All of the systems of reference are equivalent with respect to the formulation of the fundamental laws of physics. • The equivalence principle: there are observers that do not experience gravitational effects. • Mach’s principle: the inertial forces are determined by the mass of the bodies distributed in the universe. We called these principles heuristic, since they have proved to be a valid support to the formulation of the theory, but they do not imply it uniquely. It must be noted that in the literature there are many different formulations even of these principles. Further, their role has changed over the years, with one or another of them assuming a dominant position with respect to the others at different times. We begin with a discussion of the principle of general relativity.1 The following question is quite natural: is it reasonable to postulate a principle of general relativity? In other words, is it possible to formulate the laws of nature for every observer? There is no doubt that special relativity has a deep consistency, but from a conceptual point of view, it is unsatisfactory, since it distinguishes the optically isotropic systems of reference from all other possible systems of reference. However, the extension of a relativity principle to an arbitrary frame of reference presents serious difficulties. Before discussing the difficulties linked to the formulation of the principle of general relativity, we highlight some aspects of the principle of special relativity. This principle states that all observers adopting optically isotropic systems of reference (inertial frames) to analyze physical reality are equivalent. It is important to underline that the stated equivalence of the inertial observers requires that for all of them, it is possible to adopt: 1. the same kind of spatial coordinates (for instance, orthogonal Cartesian coordinates); 2. rulers whose lengths are equal when they are compared at rest; 3. clocks whose behavior and global synchronization are based on the isotropic propagation of light. It has been proved that under these conditions, the principle of optical isotropy implies that the relation between space and time coordinates that two observers associate to the same event is given by the Lorentz transformations. Suppose that all the inertial observers adopt the above basic criteria to analyze physical phenomena. Then under these conditions, the principle of special relativity states that all inertial observers obtain the same experimental results if each of them carries out the same experiments. In other words, when all the above conditions are satisfied, all the inertial observers formulate the laws of physics in the same way, so that they are completely equivalent in describing the evolution of physical phenomena. 1A

broad analysis of the debate about the principle of general relativity can be found in [115]. See also the interesting discussion in Fock [58].

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319

For instance, let I and I  be two inertial observers. If I analyzes the electric field produced by a charge q at rest with respect to I , and I  analyzes another equal charge q at rest with respect to I  , then they will find the same electrostatic field E0 . Further, if I analyzes the electromagnetic field produced by the charge q of I  , he will experience the same electromagnetic field that I  experiences when he observes the charge q of I . In this way, they can determine, at least in principle, the transformation formulas of electric and magnetic fields when the inertial frame is changed. To analyze a more general situation, consider two inertial observers I and I  that analyze a class of phenomena characterized by a set of scalar, vector, tensor fields F = (F1 , F2 , . . .) for I and F  = (F1 , F2 , . . .) for I  . Suppose that conditions (1), (2), and (3) are satisfied in I and I  . Then the observers in I and I  will determine the transformation formulas F → F  for the measures of these fields in going from I to I  . If E is the set of basic differential equations describing the class of phenomena that I is examining, then the equations E relative to the observer I  are obtained by applying to E both the transformation formulas F → F  and the Lorentz transformations. Then the special principle of relativity states that the equations E are identical to E. Suppose that I  adopts another set of spatial coordinates (x  , y  , z  ) (for instance, spherical coordinates) and denote by ϕ : (x  , y  , z  ) → (x  , y  , z  ) the coordinate transformation. Then the equations E relative to I  are obtained by applying to E the transformation ϕ. Equations E differ from E, but if vector notation is used, then the basic physical equations in I and I  have the same vector form. It is well known that all the above considerations can be formulated by resorting to a suitable mathematical framework, i.e., by introducing Minkowski spacetime V4 . In fact, as we already explained in Chap. 8, we begin by associating a system of orthogonal axes O x 1 x 2 x 3 x 4 of V4 to an inertial frame of reference I . Only after this association does every point of V4 become an event, that is, the physical occurrence that is characterized in the frame I by the Cartesian coordinates (x 1 , x 2 , x 3 ) and the time t = x 4 /c. Further, every other system of orthogonal axes O x 1 x 2 x 3 x 4 of V4 can be associated with an inertial frame I  , and the Lorentz transformation between I and I  can be regarded as the orthogonal transformation between the axes O x 1 x 2 x 3 x 4 and O x 1 x 2 x 3 x 4 . Finally, we conclude these remarks by noting that: (a) Physical laws are invariant under Lorentz transformations if they can be written as tensor equations of V4 . (b) Well-defined criteria allow us to obtain physical quantities and physical laws when their tensor formulation in V4 is given. For instance, a pair of Maxwell’s equations in V4 assumes the form ∂ F αβ = J α, ∂x β

(11.3.1)

where F αβ is the skew-symmetric electromagnetic 4-tensor and J α is the 4-current. However, the procedure to derive the standard spatial form of Maxwell’s equations relative to any inertial observer I from (11.3.1) is well known. Consequently, in

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special relativity we can go from the physical formulation to the 4-dimensional formulation and vice versa. Now we wish to overcome the limitation due to the use of only inertial frames. In other words, we wish to analyze the consequences deriving from assuming a general relativity principle. We here state this principle in the following form: Principle 11.1 (Principle of general relativity) All observers have the same right to describe physical reality. In this form, the principle of general relativity does not state the equivalence of all frames of reference: it is a statement of democracy for all observers. Everybody can adopt a frame of reference in which the coordinates of spatial points are arbitrarily assigned by well-defined procedures with rulers and clocks arbitrarily chosen. In particular, the observer could adopt rulers and clocks that have, when they are at rest in an inertial frame I , the same length and behavior as the rulers and clocks of I , provided that in the new theory of gravitation the inertial frames still exist. At this point it is quite natural to ask whether it is possible to reach an absolute formulation of physical laws if the measures that the observer carries out are relative to that observer. Any knowledge of physical reality starts from measures that are subjective and conventional. However, the knowledge of physical reality does not reduce to these subjective data; it is not given by the numbers obtained by any laboratory. On the contrary, it must be identified with the relations that link the data of different laboratories. We attain universal knowledge if we know how the data obtained by analyzing a phenomenon in one laboratory are related to the data about the same phenomenon obtained by another laboratory. For instance, consider a classical observer at rest in a noninertial frame of reference R, namely a rigid reference solid equipped with the same rulers and clocks of an inertial frame I but moving relative to I with an arbitrary rigid motion. It is well known that to justify the results of mechanical experiments, the observer in R is compelled to introduce fictitious or inertial forces, that is, forces to which he is not able to attribute a physical origin. In particular, this implies that the evolution of a phenomenon with respect to an inertial frame of reference I and a noninertial frame R are different. In other words, the two frames are not equivalent, since they lead to different descriptions of the same phenomenon. However, forgetting our inability to provide an explanation of the appearance of inertial forces, we can adopt the arbitrary rigid frame R, since we can relate to each other the descriptions relative to I and R. In conclusion, if we found a physical motivation of fictitious forces, then we could state that inertial and noninertial frames can be adopted, since the observers can compare their results and thereby decide whether they are analyzing the same phenomenon even if their descriptions of the same phenomenon are different. From the above considerations it follows that we can accept a principle of general relativity if we are able to supply the relations between the experimental data obtained by arbitrary observers. More precisely, even if we accept the formulation of the

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321

general principle of relativity of this section instead of the principle formulated in Sect. 11.3, we should be able to solve the following problems: • Define a general frame of reference R and a procedure to introduce at least either one physical system of coordinates (x α ) ≡ (x i , x 4 ) in R by rulers and clocks or an arbitrary criterion that allows one to associate spacetime coordinates uniquely to an event in R. • Determine the coordinate transformations (x α ) → (x α ) from the frame R to another frame R  . • Write the differential equations E of fundamental laws in any frame R. • Determine the transformation laws of the fields involved in the equations E. To the best of our knowledge, an answer to the above questions does not exist, but in Chap. 17, we discuss these problems in more detail. At this point, we can attempt to satisfy the principle of general relativity by following a different approach. Suppose in addition that in the new theory of gravitation, the set of all events is still represented by a 4-dimensional manifold V4 , possibly with a geometric structure different from the geometric structure of Minkowski spacetime. Then we suppose that the following principle holds. Principle 11.2 (Principle of general covariance) The basic laws of physics can be formulated in tensor form in V4 . This principle requires the natural extension of the tensor form of physical laws to an arbitrary system of coordinates of V4 . At the end of this section we briefly discuss some of the objections to this principle that can be found in the literature. For the present, we wish to analyze the relation between the general covariance and the principle of general relativity. Einstein supposed that: • an arbitrary frame of reference will be represented in V4 by a local or global system of coordinates; • a transformation of coordinates in V4 corresponds to a change of frame of reference. Consequently, if the basic laws of physics are formulated in tensor form in V4 , they conserve their form in every system of coordinates of V4 and then in every frame of reference. For the above reasons, this principle is often considered the mathematical formulation of the principle of general relativity in the spacetime V4 . However, this identification is subject to criticism. First, the relation between an arbitrary frame of reference R and the corresponding coordinates (x α ) of V4 is not defined. In other words, how can we identify the frame of reference R corresponding to a given system of coordinates (x α ) in V4 ? In particular, is it possible to assign a physical meaning to an arbitrary system of coordinates? If not, which are the coordinates in V4 to which a physical meaning can be assigned?2 Further, if we know the tensor form of a physical 2 In the Lagrangian or Hamiltonian formalism of classical mechanics, arbitrary Lagrangian or sym-

plectic coordinates are used to simplify the solution of a problem. However, the solution thus obtained is meaningless if we don’t know how the coordinates in which we solved the problem are related to the coordinates of which the physical meaning is known. For instance, the

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law in the coordinates (x α ), how can we obtain its form relative to the observer R? In particular, what is the rule that allows us to obtain the physical fields from a tensor field of V4 ? For instance, how can we obtain the electric and magnetic fields relative to the observer R when the components Fαβ of the electromagnetic tensor are given in the coordinates (x α )? In conclusion, the identification of the principle of general relativity with the principle of general covariance is meaningless unless we answer the above questions. After Einstein’s formulation of the principle of general covariance, Erich Kretschmann [87] proved that every theory can be written in a covariant way by adding a suitable set of fields and equations to the fields defining the theory. In other words, the principle of general covariance has no physical meaning, and it is unable to limit the form of physical laws. More recently, Ryoyu Utiyama [173] proved that the principle of general covariance selects all the equations depending on a fixed set of fields and satisfying given symmetry properties. In this way, many equations are discarded, and this result seems to contradict Kretschmann’s results. In effect, the two authors resort to different hypotheses to state their results.

11.4 Principle of Equivalence In order to introduce the principle of equivalence, we begin with some considerations based on classical mechanics. Let I be a classical inertial frame of reference and denote by R a frame of reference in translational nonuniform motion with respect to I . The equation describing the motion of an isolated point P with respect to R is ma = −mar ,

(11.4.1)

where m is the inertial mass of P and ar is the drag acceleration. According to (11.4.1), all the material points will have the same acceleration in R independently of their inertial mass. Then an observer carrying out mechanical experiments in R can state that R is an inertial frame and the material points are acted on by a gravitational field G = −ar . We consider now a different situation. Let I be a frame of reference with the origin at the center of the Earth and axes directed toward fixed stars. It is well known that I is a good approximation of an inertial frame of reference. Furthermore, let R be a local free-falling frame of reference with respect to I . In other words, the motion of R relative to I is a translational uniformly accelerated motion whose acceleration is equal to the terrestrial gravitational field g in the small region occupied by R. If we denote by m and m g the inertial and gravitational masses of a material point P, respectively, then the equation of motion of P relative to R is Arnold–Liouville theorem states the existence of coordinates in the phase space in which the solution of a completely integrable system is trivial but the relation of these coordinates to the physical ones is not given. To overcome this problem, the angle-action variables are introduced.

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323

ma = m g g − mar .

(11.4.2)

Many accurate experiments lead to the conclusion that m g = m. Then (11.4.2) assumes the form ma = mg − mg = 0. (11.4.3) On this basis, the observer carrying out mechanical experiments in R may state that R is an inertial frame without any contradiction. In other words, under the assumption that inertial and gravitational mass are equal, the inertial force due to the accelerated motion of R relative to I is balanced by the terrestrial gravitational field. In conclusion, the inertial fields may be globally eliminated by an appropriate choice of a frame of reference, but the gravitational fields can be only locally eliminated. In other words, the two kinds of fields are at least locally equivalent from a classical point of view. All the above observations suggested to Einstein the following principle: Principle 11.3 ([Strong] principle of equivalence) Consider a small spacetime region W in which the spatial and temporal gravitational changes are negligible. Then in W , there always exists a local frame of reference I in which there is no gravitational effect on any physical phenomena. The equivalence stated by this principle requires a common cause for real and fictitious fields. Mach’s principle states that the inertia of a body is due to nonuniform motion with respect to the fixed stars. In other words, in an empty universe, test particles have no inertial property. At the beginning of his research into general relativity, Einstein thought that Mach’s principle should play an important role in the formulation of a new theory of gravitation. In fact, the inertial forces are due to the nonuniform motion relative to far masses, and these forces are locally equivalent to real gravitational fields. Since in Einstein’s view, gravitation is described by the geometry of spacetime V4 , the geometric properties of V4 must be determined by the masses in the universe (field equations). However, Einstein’s final field equations disregard this principle. For instance, Alexander Friedmann’s solution corresponds to an empty universe that is not flat. Further, Karl Schwarzschild’s solution is obtained by supposing that at infinity there is no mass and space is asymptotically flat.

11.5 The Spacetime of General Relativity The spacetime V4 of special relativity is assumed to be a Euclidean space with signature (+, +, +, −). In his paper of 1916, Einstein proposes the following model of spacetime. Postulate 11.1 Spacetime is a C 2 Riemannian manifold V4 with hyperbolic signature (+, +, +, −). This assumption implies that the tangent space T p (V4 ) at every point p ∈ V4 is Euclidean and that bases can be found at p in which the metric assumes the form

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ds 2 = (d x 1 )2 + (d x 2 )2 + (d x 3 )2 − (d x 4 )2 .

(11.5.1)

By choosing such a base at every point p ∈ V4 , we obtain a field of bases. However, these are not holonomic bases, even locally. All the point properties of Minkowski space hold also in this more general spacetime. In fact, the metric g p at p ∈ V4 allows us to classify the vectors of T p (V4 ) into timelike, spacelike, and null-like vectors. A similar classification can be extended to curves and surfaces of V4 . Recall that at every point p0 of V4 , it is always possible to adopt locally geodesic coordinates (cf. Theorem 4.6), that is, coordinates (x¯ α ) in which g¯αβ (x¯0α ) = ηαβ

g¯αβ,γ (x¯0α ) = 0,

(11.5.2)

where x¯0α are the coordinates of the point p0 . Note that the vectors e¯ i tangent to the coordinate curves at p0 are spacelike for i = 1, 2, 3, whereas e¯ 4 is timelike. From now on, the holonomic frame { p0 , (¯eα )} associated with the locally geodesic coordinates will be called either a geodesic frame or a Lorentz frame at p0 . Furthermore, in a neighborhood of p, the frames { p, (¯eα )} are Lorentz frames up to terms of order greater than (x¯ α − x0α ). The next postulate is a geometric version of the principle of equivalence, and it also supplies a first physical interpretation of the geometric objects of V4 . Postulate 11.2 A geodesic frame { p0 , (¯eα )} in V4 is the geometric representation of a physical local frame of reference in which the physical laws of special relativity hold. Such a frame of reference will also be called a local inertial frame of reference. In other words, there is no action of a gravitational field on any physical phenomenon inside the regions of V4 with reference to coordinates in which the metric satisfies (11.5.2). Since V4 is a general Riemannian manifold, the gravitational field can be eliminated only in locally geodesic frames, and gravitation is described by the metric coefficients gαβ . In view of Postulate 11.2, all the laws of special relativity hold in a geodesic frame at the point p of V4 . Consider now a free-falling particle P. In a geodesic frame of reference, the equation of motion of P is dU α = 0, dτ

(11.5.3)

where (U α ) is the four-velocity of P and τ the proper time evaluated along the world line of P. Then in a general system of coordinates, (11.5.3) assumes the form dU α α + βλ U β U λ = 0, dτ

(11.5.4)

so that the world line of a free-falling particle is a spacetime geodesic. Similarly, in general relativity a continuum is still described by the momentum– energy tensor T αβ , and the evolution equations can be written

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325

∇β T αβ = 0.

(11.5.5)

Finally, in general coordinates, Maxwell’s equations have the form Jα

0 = 0.

∇β F αβ = ∇β F

∗αβ

(11.5.6)

11.6 Einstein’s Gravitational Equations In this section we analyze the central problem of Einstein’s theory of gravitation. It consists in deriving the equations that determine the gravitational potentials, that is, the metric of spacetime. In view of the principle of general covariance, these equations must be written in a covariant form, for instance as tensor relations. Postulate 11.3 The gravitational field equations have the form χ Tαβ , G αβ = −

(11.6.1)

where  χ is a constant, G αβ is a tensor depending on gαβ , their first derivatives gαβ,γ , and linearly on the second derivatives gαβ,γδ . Finally, Tαβ is the momentum–energy tensor of matter and energy distribution. It is possible to prove (see Cartan [23]) that one and only one tensor G αβ exists satisfying these properties and the conditions ∇β G αβ = 0

(11.6.2)

that follow from the identification of Tαβ with the momentum–energy tensor. The tensor G αβ is called the Einstein tensor, and it is written as G αβ

    R  +  gαβ , = h Rαβ − 2

(11.6.3)

γ

where  h and  are arbitrary constants, Rαβ = Rα βγ is the Ricci tensor, and R = Rαα (see Chap. 4) is the scalar of curvature. Introducing (11.6.3) in (11.6.1), we obtain Einstein’s gravitational equations  Rαβ −

 R +  gαβ = −χTαβ , 2

where χ=

 χ .  h

(11.6.4)

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Contracting the indices α and β in (11.6.3), we obtain R + 4 = χTαα ≡ χT,

χ= χ/ h,

and Einstein’s equation can be written in the equivalent form  Rαβ + gαβ = −χ Tαβ

 T − gαβ . 2

(11.6.5)

We conclude this section with the following remarks: (a) We show later that the constants χ and  can be determined by requiring that Eq. (11.6.5) reduce to Poisson’s equation when the gravitational fields are weak and the velocities of the points of material distribution producing the gravitational field are small compared with the velocity of light in vacuum. In this way, we obtain the following values for χ and : χ=

8πh ,  = 0. c4

Einstein suggested that one should assume  = 0 for cosmological problems. (b) Equation (11.6.5) contain the evolution equations of the matter–energy system that generates the gravitational field, since ∇β G αβ = ∇β T αβ = 0. This aspect is quite new with respect to all the other physical theories. For example, the equation of Newtonian mechanics is not included in Poisson’s equation. Further, Maxwell’s equations do not include either the force acting on charges and currents or the equation of motion. In other words, Einstein’s equations show that matter determines the geometry of spacetime, and in turn, this geometry determines the motion of matter. (c) For the sake of simplicity, if we consider a region  of V4 in which there is no matter and energy, so that Tαβ = 0, and suppose  = 0, then (11.6.5) becomes Rαβ = 0.

(11.6.6)

Since Rαβ is symmetric, (11.6.6) is a system of ten second-order quasilinear partial differential equations in the ten unknowns gαβ . Due to the identities ∇β G αβ = ∇β Rαβ = 0, only six equations of (11.6.6) are independent, in full agreement with the principle of general covariance. Indeed, if (11.6.6) were independent, then it would be possible

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327

to determine the gravitational potential gαβ (x γ ) in a neighborhood U of the point p, provided that the values of gαβ and their derivatives gαβ,γ were assigned over a surface  containing p (Cauchy’s problem). Choosing in U a new coordinate system (x  α ), α α (11.6.7) x  = x  (x β ), and considering the covariance of (11.6.6), the metric coefficients

should satisfy the equations

 = Aγα Aδβ gγδ gαβ

(11.6.8)

 = 0, Rαβ

(11.6.9)

    depend on gαβ , gαβ,γ , gαβ,γδ as Rαβ depend on the gαβ , gαβ,γ , gαβ,γδ . Then where Rαβ  (x  γ ) assigning the reasoning as we did for (11.6.6), we should determine the gαβ values of these functions and their first derivatives on . Choosing the (x α ) → (x  α ) transformation in a such way that it is the identity transformation in a neighborhood of  and arbitrary elsewhere, we should obtain  gαβ = gαβ

 gαβ,γ = gαβ,γ

 on , and gαβ (x  γ ) should be equal to gαβ (x  γ ). This result contradicts (11.6.8),  at points sufficiently far from . which states that gαβ differ from gαβ In conclusion, the principle of general relativity implies that Einstein’s equations cannot be independent. Their solutions must contain four arbitrary functions corresponding to the four arbitrary functions (11.6.7). Note that in view of (11.6.6), it is always possible to fix arbitrarily four of gαβ so that the other six independent Eq. (11.2.2) allow us to determine the remaining gravitational potentials. Taking into account the results of Sect. 4.8, Einstein’s equations in the empty space can also be written in the form

  1 R αβ = − g γδ (g αβ ),γδ + H gγλ , ∂gγδ /∂xμ = 0, 2

(11.6.10)

so that they represent a hyperbolic system whose characteristic surfaces are solutions of the equation (11.6.11) g αβ f ,α f ,β = 0. These surfaces are isotropic, since (11.6.11) shows that the normal vector to these surfaces is isotropic. This remark shows that gravitational waves are possible and that their speed of propagation is equal to the speed of light in empty space.

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11.7 Experimental Determination of gαβ We highlight the deep change of perspective in going from special relativity to general relativity. In fact, in special relativity the procedures allowing inertial observers to measure space and time are specified before one formulates any physical law. Furthermore, these procedures allow one to identify three-dimensional spaces in Minkowski spacetime as well as one-dimensional spaces that are the geometric representations of space and time relative to an inertial observer. In general relativity, the metric of spacetime is dynamic, i.e., it is determined by the evolution of matter and energy by the field equations. This means that we cannot speak about local or global space and time before solving Einstein’s equations and the conservation laws. In other words, before knowing the metric, we cannot speak about space, time, geodesics, etc. In particular, the definitions of local and global space and time will depend on the form of the metric. Furthermore, the elegant and revolutionary formulation of physical phenomena we have outlined in the previous two sections places us in a 4-dimensional Riemannian manifold V4 without attributing a physical meaning to many of the geometric structures with which V4 is equipped. Only the equivalence principle introduces a correspondence between V4 and the physical world, associating the proper frames with the geodesic coordinates in V4 . This correspondence represents the only way to go from V4 to the real world; however, it allows us to attribute only locally a physical meaning to some mathematical structures. To make explicit the problem with which we are faced, suppose that we succeed in determining: • A solution gαβ of Einstein’s equation in a given system of coordinates (W, x α ), where W is an open region of V4 ; • The parametric equations x α (τ ) of the geodesics of metric coefficients gαβ . Then we know the 4-velocity U α = d x α /dτ of a test particle P acted on by the gravitational field gαβ . It is evident that experiments are carried out in the physical space by adopting a frame of reference R, where space and time are separated. This frame must be supposed to be equipped with suitable devices to measure physical quantities. Then to verify whether the above mathematical results are in agreement with the experimental data, it is necessary to determine: • A (local) physical frame of reference R, if it exists, with well-defined coordinates (x α ), that can be considered the physical counterpart of the mathematical chart (W, x α ) in V4 . In this way, the points of the region W of V4 can be interpreted as events. • An experimental procedure to measure the metric coefficients gαβ . • How to obtain the velocity v of a test particle P relative to R from the 4-velocity (U α ). No comparison is possible between mathematical results and experimental data before solving the above problems. This connection will be widely analyzed in

11.7 Experimental Determination of gαβ

329

Chap. 17, but for the present, we limit ourselves to recalling, in an equivalent form, a criterion proposed by Einstein to determine the metric coefficients gαβ through measures carried out in a frame of reference R. In general relativity, a frame of reference R must be intended as a fluid of reference S at every point of which are distributed rulers and clocks with arbitrary behavior. We denote by y i , i = 1, 2, 3, space coordinates in S and by t the coordinate time measured by the clock at the point P ∈ S. The world line σ P of each point P ∈ S is a timelike curve of V4 , and the collection of the ∞3 world lines on varying P ∈ S defines a timelike congruence  in a region W of V4 . Adopting in W the coordinates (y i , ct), the curves of  will have equations y i = const and ct = var. Coordinates (y i , ct) with this property are said to be adapted to the congruence . Let (W, x i ) be an arbitrary chart of V4 related to (y i , t) by known transformation functions x α = x α (y i , ct).

(11.7.1)

These coordinates, which can have no physical meaning, can be used both in the region W of V4 and in the frame of reference R, since in view of the invertible functions (11.7.1), they define a point P = (y i ) of S and a coordinate time t at P. In other words, x = (x α ) defines the event given by a point P ∈ S and a time t. In the sequel, we use the notation x = (y i , ct). Let x ∈ W be an event and denote by γ = (U α /c) the unit vector tangent at x to the unique curve of  containing x. It is well known that geodesic coordinates (x α ) exist in a neighborhood of x ∈ V4 such that the corresponding holonomic base at x is a Lorentz base, whose fourth axis is γ. These coordinates are the geometric representation of the proper inertial frame I x (the freely gravitating cabin) of the particle P of R at the event x. In this frame, in view of the equivalence principle, there is no gravitation. For the observer I x , special relativity locally holds, so that he knows how to obtain the physical quantities from 4-vectors and 4-tensors. Therefore, if x + dx = (y i + dy i , t + dt) is an event close to x and the decomposition (8.9.1) is applied to dx, one has + dx , (11.7.2) dx = dx is the space vector between the points (y i ) and (y i + dy i ) and dx is c where dx times the time interval dT between the events x and x + dx for the inertial frame I x . In view of (8.9.4), the components of these two vectors in I x are given by α = (gαβ + γα γβ )d x β ≡ γαβ d x β , dx 1 dTs = − γα d x α , c

(11.7.3) (11.7.4)

where dTs is called the standard time interval. In adapted coordinates (y i , ct), the curves of  have equations y i = const and ct = var. Then since these curves are timelike, the square of the elementary distance between two events belonging to the same curve of  satisfies the condition ds 2 = g44 (cdt)2 < 0, so that in adapted coordinates, one has

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11 Introduction to General Relativity

g44 < 0.

(11.7.5)

On the other hand, the unit vector γ tangent to the curves of  in adapted coordinates has components (0, 1) and gαβ γ α γ β = −1. Therefore, we have that γ i = 0, γ 4 = √

1 gα4 , γα = √ . −g44 −g44

(11.7.6)

The fundamental role of I x in evaluating elementary space and time intervals relative to the general frame of reference R is expressed by the following assumption (see [4, 6, 7, 112]): Assumption 11.1 The observer in the frame of reference R adopts at x the measures of space and time intervals obtained by the observer in I x after expressing them in terms of his arbitrary coordinates. In other words, the observer in R entrusts the observer in I x with the task of measuring physical quantities in the absence of gravitation with the aim to adopt the results of I x after expressing them in terms of his arbitrary coordinates. This assumption is more or less explicitly accepted in all the papers about this subject; moreover, in [112], it is justified by assuming that the measuring rods have a length independent of their acceleration relative to an inertial frame. In any case, it is used to define the measures of elementary lengths and time intervals. In particular, the above assumption makes it possible to define the elementary spatial distance dσ between two particles of the frame R by the formula obtained by (11.7.2), (11.7.7) dσ 2 = γαβ d x α d x β . It is evident that measures of elementary space distances dσ and elementary time intervals dT allow one to obtain γαβ and γα and then gαβ . Finally, in view of (11.7.3), (11.7.4), we can write the metric of V4 in the form ds 2 = dσ 2 − c2 dT 2 .

(11.7.8)

11.8 The Rotating Frame In [40], Einstein introduces a frame of reference R rotating with respect to an inertial frame I with constant angular velocity ω. The frame R is supposed to be equipped with rulers and clocks identical to those of I but at rest relative to R. Denoting by O x 1 x 2 x 3 the Cartesian axes in I , we suppose that the space of R rotates about the axis O x 3 . Then every point p ∈ R describes a circumference in a plane orthogonal to O x 3 with the center belonging to this axis. Special relativity requires that the space of R be a cylinder C whose radius rc satisfies in the frame I the condition

11.8 The Rotating Frame

331

ω rc < c,

(11.8.1)

where c is the velocity of light in vacuum. During the motion, every section orthogonal to O x 3 of C is a disk rotating about O x 3 with constant angular velocity ω. Consequently, the kinematics of all the disks are the same, and we refer to the rotating disk D corresponding to the section x 3 = 0. In [40], Einstein remarks that the rulers along the radii of D do not contract, whereas the rulers along the circumference dc of D undergo a Lorentz contraction. Therefore, if rc is the radius of D, then the measure l of dc for an observer on D is greater than 2πrc : l > 2πrc .

(11.8.2)

In conclusion, the geometry on the disk D is non-Euclidean. For Einstein, this remark is very important, since it suggests that the inertial forces acting inside the disk D, which are locally quite equivalent to real gravitational fields, modify the space geometry on D. In other words, Einstein’s approach to the problem of the spatial geometry of the rotating disk had the purpose to show that the gravitational fields modify both the spacetime geometry and the space geometry. So Einstein’s treatment of the rotating disk indicates the way to go from the special theory of relativity toward a relativistic theory of gravitation based on the idea that the geometry of the familiar Minkowski spacetime of the special theory must be deformed by a gravitational field. Now we analyze mathematically the problem of the rotating disk, starting from Assumption 11.1 of the previous section (see, for instance, [86, 134], and references therein). Let I be an inertial frame equipped with rigid rulers and synchronized clocks in which Cartesian coordinates (X i ) are introduced. Denote by I the corresponding Lorentz frame in the Minkowski space V4 . Introduce inside I a system of cylindrical coordinates (R, , X 3 ) related to the Cartesian coordinates (X i ) by the relations X 1 = R cos ,

X 2 = R sin ,

X 3 = X 3.

(11.8.3)

Now we define a new frame of reference R in V4 by the coordinate transformation r = R, θ =  − ωT, x 3 = X 3, t = T.

(11.8.4) (11.8.5) (11.8.6) (11.8.7)

The procedure to measure the cylindrical coordinates of a point P ∈ I is carried out with the rulers of R. In V4 , these coordinates are adapted to the congruence  of the straight lines X i = const or (R, , X 3 ) constant and T = var, where T is the time evaluated by the clocks in I , i.e., the standard time (see Sect. 11.7).

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11 Introduction to General Relativity

In the chart (r, θ, x 3 , t), the time t is equal to the time T of the inertial frame I . In this way, we label each event in the frame R using the time of a clock at rest in I . Assigning to the rotating frame R the coordinate T of the inertial frame I is the only way to synchronize the clocks in R, since the proper times of these clocks cannot be synchronized by Einstein’s convention (Assumption 11.1 of the previous section), as we show in the sequel of this section. Of course, if we are not interested in the global synchronization on the disk, a different choice of the chart can be made, which has a direct operational meaning for an observer on the disk, since it consists in substituting the coordinate time t with the proper time of the clocks at rest in R. From (11.8.4)–(11.8.7) we obtain R = r,

(11.8.8)

ω  = θ + x 4, c X 3 = x 3,

(11.8.10)

X =x ,

(11.8.11)

4

4

where x 4 = X 4 = cT , and the Jacobian matrix of this transformation is ⎛ ⎞ 1000 ⎜0 1 0 0⎟ ⎟ (Aλμ ) = ⎜ ⎝0 0 1 0⎠. 0000

(11.8.9)

(11.8.12)

In the coordinates (R, , X 3 , X 4 ) of the Lorentz frame I, the metric of V4 has the form (11.8.13) ds 2 = d R 2 + R 2 d2 + (d X 3 )2 − (d X 4 )2 , and the components G αβ of the metric tensor are ⎛

1 ⎜0 (G λμ ) = ⎜ ⎝0 0

0 R2 0 0

0 0 1 0

⎞ 0 ω/c ⎟ ⎟. 0 ⎠ −1

(11.8.14)

On the other hand, the components gαβ of the metric in the coordinates (r, θ, x 3 , x 4 ) are related to the components G λμ by the formulas μ

gαβ = Aλα Aβ G λμ

(11.8.15)

11.8 The Rotating Frame

333

so that we obtain

⎛

1 0 0

0 ⎜ r 2ω 2 ⎜0 r 0 ⎜ c (gλμ ) = ⎜ 0 ⎜0 0 1 ⎝ r 2ω r 2 ω2 0 −1 + 2 0 c c

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎠

(11.8.16)

and   r 2ω r 2 ω2 4 ds = dr + r dθ + dz + 2 (d x 4 )2 . dθd x − 1 − 2 c c 2

2

2

2

2

(11.8.17)

Since the coordinates (r, θ, x 3 , x 4 ) are adapted to the congruence  of the world lines of the particles of R, we can apply (11.7.3) and (11.7.6) to obtain γ i = 0, γ 4 = 

γ1 = 0, γ2 = √ and

1 r 2 ω2 1− 2 c 2 r ω

c2 − r 2 ω 2 ⎛

,

(11.8.18) 

, γ3 = 0, γ4 = − 1 −

1

0 2 ⎜ r ⎜0 (γλμ ) = ⎜ ⎜ 1 − r 2 ω 2 /c2 ⎝0 0 0 0

00

r 2 ω2 , (11.8.19) c2

⎞

⎟ 0 0⎟ ⎟. ⎟ 1 0⎠ 00

(11.8.20)

Let W be the region of V4 covered by the congruence . We now introduce the following equivalence relation Q: Two events of W are equivalent if they belong to the same curve of . Then we define space V3 ≡ W/Q as the three-dimensional space of the rotating frame of reference R (see [134]). The considerations of Sect. 11.7 suggest that we regard V3 as a Riemannian manifold with the metric dσ 2 = dr 2 +

r2 dθ2 . 1 − r 2 ω 2 /c2

(11.8.21)

In conclusion, if we accept Assumption 11.1 of Sect. 11.7, the space of the rotating frame of reference is not Euclidean. In particular, consider the circumference c given by the points of V3 for which r is constant x 3 = 0 and θ ∈ [0, 2π]. Then the measure of the radius rc of c is obtained by integrating (11.8.21) for dθ = 0, 

rc

rc =

dr, 0

(11.8.22)

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11 Introduction to General Relativity

while the measure lc of the circumference is given by  lc = 0

so that

2π

rc rc  dθ = 2π  , 2 2 2 1 − rc ω /c 1 − rc2 ω 2 /c2

(11.8.23)

lc 2π = > 2π. rc 1 − rc2 ω 2 /c2

(11.8.24)

Further, the elementary time interval evaluated by standard clocks in the frame R is given by (see (11.8.4), (11.8.19)) r 2ω 1 1 dTs = − √ dθ + 1 − r 2 ω 2 /c2 d x 4 . c c2 − r 2 ω 2 c

(11.8.25)

Finally, we evaluate the proper time taken by light to travel along a circumference of radius rc in the rotational direction and in the opposite one. To obtain this proper time we must integrate (11.8.25) along the curve r = rc , θ ∈ [0, 2π], z = 0, and x 4 = x 4 (θ) and along the curve r = rc , θ ∈ [0, −2π], z = 0, and x 4 = x 4 (θ). In the first integration we have 2πωrc2 , T1 = A −  c c2 − ω 2 rc2 whereas the second integration leads to the result 2πωrc2 T2 = A +  , c c2 − ω 2 rc2 where A is the integral of the second term on the right-hand side of (11.8.25). In conclusion, the first ray of light arrives at the starting point after the second ray with a delay 4πωrc2 4ωS =  , (11.8.26) T = T2 − T1 =  2 2 2 2 c c − ω rc c c − ω 2 rc2 where S is the area of the circle of radius rc . When the condition rc ω c is satisfied (nonrelativistic rotational velocity of the disk), then we obtain the formula T =

4ωS , c2

(11.8.27)

which has been verified in the experiments of Sagnac and Pogany [126, 150, 151]. It must be noted that the rotating frame of reference in which all the above results have been deduced is an ideal frame of reference. In fact, it is formed by axes that

11.8 The Rotating Frame

335

remain rectilinear during the rotation relative to the inertial frame. It is not clear whether these results can be applied to a real rotating disk. We conclude by noticing that an exact theory of rotating disks should begin with a Riemannian spacetime equipped with a metric ds 2 = gαβ d x α d x β whose coefficients are the solution of the Einstein equations in which the momentum– energy tensor describes the mass distribution of the disk.

11.9 Variational Formulation of Gravitation In the previous section we showed how Einstein derived the gravitational field equations from some physical and mathematical assumptions. In this section we show how to obtain these equations by the variational principle proposed by David Hilbert. In order to formulate a variational principle, we must search for a scalar Lagrangian L G . √ Since the expression −g d 4 x is an invariant, the action integral for the gravitational field should have the form of the integral 

√ 

−g L G d 4 x,

(11.9.1)

where the integration is carried out over the spacetime region . First, we notice that the Einstein equations contain derivatives of the metric tensor gμν no higher than the second. Since the Lagrange equations contain the derivatives of the Lagrangian, the function L G should be only a function of the metric tensor and its first derivatives. In other words, L G should include the metric tensor gμν α and the Christoffel symbols μν . A scalar depending on these variables is the scalar curvature R. However, R depends also on the second derivatives of the metric tensor, but the second derivatives occur linearly in R. Consequently, the terms containing the second derivatives do not contribute to the field equations if we adopt R as the Lagrangian of our variational principle. For all these reasons, the action integral for the gravitational field will be taken as follows:  √ −g Rd 4 x. (11.9.2) 

It is evident that we must add to this integral another one taking into account all the other nongravitational fields that describe our physical system. Therefore, we can generally give the following form to the action integral of gravitation and fields:  I =



√ −g (R − 2χL F )d 4 x,

(11.9.3)

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11 Introduction to General Relativity

where L F is the Lagrangian relative to the nongravitational fields. The variational principle consists in requiring that the first variation of (11.9.3) vanish, δ I = 0,

(11.9.4)

for every variation of the metric coefficient vanishing on ∂. In (11.9.4), χ = 8πh/c4 , h is the gravitational constant, and c is the speed of light in empty space. The variation of the first term of the integral (11.9.4) gives  δ 

√

 √ −g Rd 4 x = δ −g g μν Rμν d 4 x     √ √ = −g g μν δ Rμν d 4 x + Rμν δ −g g μν d 4 x. 

(11.9.5)



Furthermore, the variation of the Ricci tensor δ Rμν in geodesic coordinates is  δ Rμν = δ  =δ =

ρ ∂μν

∂x μ ρ ∂μν

∂x ρ ρ ) ∂(δμν

− −

ρ ∂μρ

+ ∂x ν  ρ ∂μρ

σ ρ μν ρσ

−

σ ρ μρ νσ



∂x ν ρ ) ∂(δμρ

(11.9.6)

− ρ ∂x ∂xν  ρ   ρ . = ∇ρ δμν − ∇ν δμρ It is important to notice that (11.9.6) is a tensor equation, so that it must be valid in every system of coordinates and at every point of spacetime, not only in geodesic coordinates. Therefore, the function under the first integral on the right-hand side of (11.9.5) can be written in the form   √ √ ρ ρ −g g μν δ Rμν = −g g μν ∇ρ (δμν ) − ∇ν (δμρ )   √ ρ ρ ) · ∇ν (g μν δμρ ) = −g ∇ρ (g μν δμν (11.9.7)   √ μν α μα ρ = −g ∇α (g δμν ) − ∇α (g δμρ ) . Hence we can write

where

√ √ −g g μν δ Rμν = −g ∇α V α ,

(11.9.8)

α ρ − g μα δμρ V α = g μν δμν

(11.9.9)

is a contravariant vector. Then since the divergence of a vector V μ is  ∂ √ 1 −gV μ , ∇μ V μ = √ −g ∂x μ

11.9 Variational Formulation of Gravitation

337

we can give the first integral on the right-hand side of (11.9.5) the following form:  

√



μν

gg δ Rμν d x =

∂

4

∂

 √ −g V α 4 d x. ∂x α

(11.9.10)

The integral on the right-hand side of (11.9.10) vanishes, since on applying Gauss’s theorem, it is equal to the surface integral of √ α ρ −g V α = g μν δμν − g μα δμρ , which is equal to zero, since the variations of the Christoffel symbols vanish on ∂. Finally, we obtain the result 

√

−g g μν δ Rμν d x 4 = 0.

(11.9.11)

The second integral on the right-hand side of (11.9.5) gives  

Rμν δ

√

 −g g μν d 4 x = =



√



√



−g Rμν δg μν d 4 x + μν 4

−g Rμν δg d x +

  

√ Rμν g μν δ −g d 4 x √ Rδ −g d 4 x. (11.9.12)

On the other hand, we have √ 1√ 1 1 δg = − −g gμν δg μν , δ −g = − √ 2 −g 2

(11.9.13)

and on inserting this result into (11.9.12), we obtain  

 √ Rμν δ −g g μν d 4 x =

 

√ −g

 Rμν

 1 − gμν R δg μν d 4 x. 2

(11.9.14)

From (11.9.11) and (11.9.14) we have the final form  δ 

√ −g Rd 4 x =



√ 

 −g

 1 Rμν − gμν R δg μν d 4 x 2

(11.9.15)

of the variation of the gravitational part of the action integral for every variation δgμν that vanishes on ∂. The first variation of the second term of the action integral (11.9.3) is  δ

√

  √   √ ∂ −g L F ∂ −g L F μν 4 μν −g L F d x = δg + δg,α d x. μν ∂g μν ∂g,α 4

(11.9.16)

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11 Introduction to General Relativity

On the other hand, we have  √   √ √ ∂ −g L F ∂ −g L F μν ∂ −g L F μν δg,α = δg − δg μν . μν μν μν ∂g,α ∂g,α ∂g,α ,α When we apply Gauss’s theorem to the first term of the above equation and recalling that the variations δg μν on ∂ vanish, (11.9.16) becomes  √      √ ∂ −g L F ∂ −g L F ∂ δ − α δg μν d 4 x. μν ∂g μν ∂x ∂g,α (11.9.17) If we define the momentum–energy tensor as 

√ −g L F d 4 x =

Tμν

2 = −√ −g 

then we obtain δ

 √  √    ∂ −g L F ∂ −g L F ∂ − α , μν ∂g μν ∂x ∂g,α √

−g L F d 4 x = −

1 2



√

−g Tμν δg μν d 4 x

(11.9.18)

(11.9.19)

for the variation of the nongravitational part of the action integral (11.9.3). Using (11.9.15) and (11.9.19) in (11.9.3) and (11.9.4), we finally get  δI =

√ −g

 Rμν

 1 − gμν R + χTμν δg μν d 4 x. 2

(11.9.20)

Since this equation is supposed to be valid for an arbitrary variation δg μν that vanishes on ∂, we conclude that at every point of  the following Einstein equations hold: 1 Rμν − gμν R = −χTμν . 2

(11.9.21)

In conclusion, we have derived Einstein’s equations from a variational principle.

11.10 Palatini’s Variational Principle Einstein’s field equations can also be derived from Palatini’s variational principle [119, 169]. The fundamental idea of this approach consists in using a variational principle whose fundamental fields are the metric coefficients gαβ and the coefficients μ αβ of a torsion-free affine connection. In other words, the connection coefficients are no longer determined by the metric but are assumed to be independent field variables together with gαβ . The action integral is the same as (11.9.3), with the μ proviso that the Ricci tensor is a function of gαβ and αβ , which are now supposed

11.10 Palatini’s Variational Principle

339

to be independent variables, that is,  I P [g, ] =

 √  μν −g g Rμν () + 2χL F (g) d 4 x,

(11.10.1)

where λ λ λ σ λ σ Rαβ = R λ αλβ = ∂μ αβ − ∂β αμ + σμ αβ − σβ αμ , μ

(11.10.2)

μ

and αβ = βα . We suppose that the Lagrange density L F depends not only on fields such as the mass density in the proper frame ρ0 and the 4-velocity, but also on the metric g μν . Since we are searching for an extremum of the integral I , we evaluate the first μ variation of I on varying both gαβ and βα and then we equate it to zero:  0 = δ IP = δ

√

  −g g αβ Rαβ − 2χL F d 4 x.

(11.10.3)

Consider now the variations of the individual terms of (11.10.3) inside the parentheses. The variation of the first factor is just δ g αβ . In the variation of Rαβ , changes λ is of g αβ play no part; only changes in the ’s appear. Moreover, the variation δαβ λ a tensor, even though αβ itself is not. λ Note that the variation δ Rαμβ of the curvature consists of two terms of the form λ δαβ,μ and four terms of the form δ. Since on dealing with a tensor, one coordinate system is as good as another, we may choose a coordinate system in which all the ’s vanish at the point under study. Thus the terms δ drop out. In this coordinate system, the variation of the curvature is expressed in terms of the first derivatives λ . One then need only replace the ordinary derivatives by of quantities such as δαβ covariant derivatives to obtain a formula correct in any coordinate system, λ λ λ = δαβ;μ − δαμ;β , δ Rαμβ

(11.10.4)

from which by contraction we have λ λ − δαλ;β . δ Rαβ = δαβ;λ

(11.10.5)

√ Finally, the variation of the factor −g that appears in the variational principle is given by (11.9.13). In order for an extremum to exist, the following expression must vanish:  

   λ  √ 1 αβ αβ λ Rαβ − gαβ R δg + g δαβ;λ − δαβ;λ −g d 4 x + 2    √ δL F 1 − gαβ L F δg αβ −g d 4 x = 0. αβ δg 2

(11.10.6)

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11 Introduction to General Relativity

We now focus attention on the term in (11.10.6) containing the variations of , 

 λ √ λ − δαβ;λ −g d 4 x, g αβ δαβ;λ

and integrate by parts to eliminate the derivatives of the δ. After some computations [111], we obtain ∂gβγ σ σ − gγσ βλ − gβσ γλ = 0. ∂x λ Solving these equations for the , which up to now have been considered independent of the gβγ , we obtain the standard equations for the connection coefficients, ρ μν =

1 ρσ g (gμσ,ν + gσν,μ − gμν,σ ), 2

(11.10.7)

as is required in Riemannian geometry with a metric connection (see Sect. 4.8). Similarly, equating to zero the coefficients of δg αβ in (11.10.6), we obtain Einstein’s field equations in the form G αβ

  δL F = gαβ L F − 2 αβ . δg

(11.10.8)

One of the simplest metric variations is g  μν = gμν + δgμν = gμν + ξμ;ν + ξν;μ ,

(11.10.9)

generated by the infinitesimal coordinate transformation μ

x  = x μ − ξμ.

(11.10.10)

We highlight that the action integral I is a scalar invariant, a number whose value depends on the physics, not on the adopted coordinates. This invariance holds for both parts of the action principle. Therefore, neither part will be affected in value by the variation (11.10.9). In other words, the quantity 

 √ G αβ ξ α;β + ξ β;α −g d 4 x  √ = −2χ G αβ ;β ξ α −g d 4 x

δ IG ≡ χ

(11.10.11)

must vanish for every vector field ξ α . In this way, one sees from a new point of view the contracted Bianchi identities, G αβ ;β = 0.

(11.10.12)

11.10 Palatini’s Variational Principle

341

The combination of the two constraints induced by the Palatini variation once again leads to the dynamics (Einstein’s) deduced by the Hilbert action. The “neutrality” of the action principle with respect to a coordinate transformation (11.10.9) shows once again that the variational principle—and with it Einstein’s equations— cannot determine the coordinates or the metric, but only the 4-geometry itself. The equivalence of the two approaches, however, is by no means always the case in all theories of gravity; see, for example, [21].

11.11 Conclusions and Perspectives Einstein, starting from heuristic considerations, proposed a generalization of Minkowski spacetime. In fact, he stated that in the presence of matter and energy, the mathematical representation of the collection of events is a Riemannian manifold V4 with hyperbolic signature. The metric g of V4 is a solution of Einstein’s equations (11.6.4),   R +  gαβ = −χTαβ , (11.11.1) Rαβ − 2 where Rαβ is the Ricci tensor, R the scalar curvature,  the cosmological constant, and Tαβ the momentum–energy tensor satisfying the balance or conservation laws ∇β Tαβ = 0.

(11.11.2)

The momentum–energy tensor Tαβ for a perfect fluid S has the form  p Tαβ = 0 + 2 Uα Uβ + pgαβ , c

(11.11.3)

where 0 is the proper mass density, p = p(0 ) the proper pressure, and Uα the 4-velocity of the particles of S. In view of (11.11.2), only six of the ten Einstein’s equations are independent. Consequently, they determine the components of the metric up to four of them, which can be assigned by choosing a convenient coordinate transformation. Then the first problem we face is a suitable formulation of Cauchy’s problem for the equations (11.11.1), (11.11.2). In other words, we must prove local existence and uniqueness theorems for the following problem: given the values of gαβ , their first derivatives, Uα , and 0 on a convenient three-dimensional surface , prove the local existence a unique solution of (11.11.1), (11.11.2). This topic will be analyzed in Chap. 13. For a fluid S, the world lines of the particles form a timelike congruence that occupies an unknown region T of V4 called a universe tube. Since Einstein’s equations are partial differential equations, they admit infinitely many solutions. If we want to determine a unique solution, we must assign initial and boundary data. In conclusion, (i) , 0 , Uα ) of (11.11.1), (11.11.2) (interior problem) we have to find a solution (gαβ

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11 Introduction to General Relativity

(e) together with a solution gαβ of Einstein’s equations in vacuum,

Rαβ = 0

(11.11.4)

(exterior problem), both satisfying the initial and boundary data. Further, on the boundary of T the interior and exterior solutions must together equal their first derivatives (matching conditions). The spacetime V4 equipped with such a solution is called a model of the universe. Determining a model of the universe is a very difficult problem, since we ask for a global solution of Einstein’s equations. However, if physical considerations allow us to suppose that our solution satisfies particular symmetries, then the number of unknowns to be determined is reduced drastically, and the problem can be solved. Schwarzschild’s solution (Chap. 14) and the homogeneous and isotropic models of the universe (Chap. 16) are very important examples of particularly symmetric global solutions.

Chapter 12

Linearized Einstein’s Equations

Einstein’s equations are nonlinear, and this makes it very difficult to exhibit their explicit solutions. In order to obtain some solutions, we can adopt one of two points of view: search for approximate solutions or impose symmetries on spacetime that can be justified by physical motivations. In this chapter we analyze approximate solutions obtained by linearizing Einstein’s equations. Although the linear formulation of Einstein’s equations is based on very drastic approximation, we derive very important results. In fact, we show that Einstein’s theory reduces to Newtonian gravitation for weak and static fields. If the gravitational fields are weak but depend on time, then the linear equations foresee the existence of gravitational radiation. The linearizing process is based on the assumptions that the mass density of the matter generating the gravitational field and the velocity of its particles assume very low values. For this reason, the linearizing process is called weak-field approximation of Einstein’s equations. In the next sections we set up the mathematical apparatus leading to the linearized Einstein’s equations, and we illustrate the physical meaning of weak fields. We also derive the values of the constants χ and  appearing in Einstein’s equations.

12.1 Quasi-Minkowskian Spacetime In the presence of weak gravitational fields, the spacetime V4 should be close to a Minkowski spacetime. The following definition makes clear this idea. Definition 12.1 (Quasi-Minkowskian spacetime) A spacetime V4 is called quasiMinkowskian if it is homeomorphic to 4 and admits at least a chart (4 , x α ) in which the metric coefficients assume the form gαβ = ηαβ + h αβ , © Springer Nature Switzerland AG 2019 A. Romano and M. Mango Furnari, The Physical and Mathematical Foundations of the Theory of Relativity, https://doi.org/10.1007/978-3-030-27237-1_12

(12.1.1) 343

344

12 Linearized Einstein’s Equations

where ηαβ is the Minkowski metric tensor, the quantities h αβ satisfy the conditions |h αβ |  1,

|h αβ,β |  1,

lim h αβ = 0,

r →∞

(12.1.2)

 3 α 2 and r = α=1 (x ) . α Coordinates (x ) satisfying the above conditions are called quasi-Minkowskian coordinates. We notice that in our approximation, the mixed and contravariant components of h αβ are obtained by resorting to the Minkowski tensor ηαβ as follows: h αβ ≡ η αγ h γβ and h αβ ≡ η αγ η βδ h γδ .

(12.1.3)

It is important to remark that if a system of quasi-Minkowskian coordinates exists, then there are infinitely many systems of quasi-Minkowskian coordinates. In fact, starting from quasi-Minkowskian coordinates (x α ), every other system of coordinates obtained by a Lorentz transformation x α = Aαβ x β

(12.1.4)

is quasi-Minkowskian, since in the new coordinates, we have μ

 = Aλα Aβ (ηλμ + h αβ ) ≡ ηαβ + h αβ . gαβ

(12.1.5)

We now analyze another procedure to generate quasi-Minkowskian coordinates. Consider the coordinate transformation called gauge transformation, x α = x α + ξ α ,

(12.1.6)

where the functions ξ α satisfy the conditions |ξ α |  1,

|ξ α ,β |  1,

lim ξ α = 0.

r →∞

(12.1.7)

The Jacobian matrix of (12.1.6) is Aαβ ≡

∂x α = δβα + ξ α ,β , ∂x β

while the inverse matrix is given by (A−1 )αβ =

∂x α  δβα − ξ α ,β . ∂x β

(12.1.8)

12.1 Quasi-Minkowskian Spacetime

345

Therefore, the metric coefficients in the coordinates (x α ) are ∂x λ ∂x μ (ηλμ + h λμ ) ∂x α ∂x β μ = (δαλ − ξ λ ,α )(δβ − ξ μ ,β )(ηλμ + h λμ )

 gαβ =

μ

μ

 (δαλ δβ − δαλ ξ μ ,β −ξ λ ,α δβ )(ηλμ + h λμ )  ηαβ + h αβ − ξα,β − ξβ,α , and we conclude that the new components h αβ of h αβ in the coordinates (x α ) are h αβ = h αβ − (ξα,β + ξβ,α ).

(12.1.9)

We now prove that by a suitable choice of the functions ξ α , it is possible to determine harmonic quasi-Minkowskian coordinates (x α ), i.e., quasi-Minkowskian coordinates satisfying the conditions (see Chap. 4) α α  η λμ λμ , α = 1, . . . , 4. x α ≡ g λμ λμ

(12.1.10)

The Christoffel symbols corresponding to the metric ηαβ + h αβ can be written in the form   1 α (12.1.11) = η αβ h μβ,λ + h λβ,μ − h λμ,β  , λμ 2 where we have used the notation a,λ = ∂a/∂x λ . Therefore, conditions (12.1.10) become (12.1.12) η αβ η λμ (h μβ,λ + h λβ,μ − h λμ,β  ) = 0. On the other hand, taking into account (12.1.8) and supposing that a is a first-order quantity, we have ∂x μ μ (12.1.13) a,λ = a,μ λ  a,μ δλ = a,λ . ∂x From this result and (12.1.9), conditions (12.1.12) can also be written as  η αβ η μλ h μβ,λ − (ξμ,βλ + ξβ,μλ )+ h λβ,μ − (ξλ,βμ + ξβ,λμ )−  h λμ,β + (ξλ,μβ + ξμ,λβ ) = 0,

(12.1.14)

so that we have η αβ η μλ (h μβ,λ + h λβ,μ − h λμ,β − 2ξβ,λμ ) =h λα ,λ +h μα ,μ −η αβ h λλ,β − 2η λμ ξ α ,μλ =2h

λα

,λ −η αβ h λλ,β

λμ α

− 2η ξ ,μλ = 0.

(12.1.15)

346

12 Linearized Einstein’s Equations

In conclusion, the coordinates x α = x α + ξ α are harmonic if in the coordinates (x ), the functions ξ α satisfy conditions (12.1.7) and the wave equation α

1 ξ α ≡ f α ≡ h αλ ,λ − η αλ h,λ , 2

(12.1.16)

where ξ α = η λμ ξ α ,λμ is the d’Alembert operator and h = h λλ . In other words, the fields h αβ determine the functions f λ in the coordinates (x α ). Then every solution ξ α of (12.1.16) determines harmonic quasi-Minkowskian coordinates by (12.1.6). We prove that it is also possible to characterize harmonic coordinates (x α ) by the components of the tensor field h αβ . We begin by introducing the auxiliary tensor field that in coordinates (x α ) has components 1 1 λ h αβ = h αβ − ηαβ h, h α = h λα − ηαλ h, 2 2

(12.1.17)

h = h λλ = η λμ h λμ .

(12.1.18)

where

Although ηαβ + h αβ is not a metric tensor, there is a one-to-one correspondence between h αβ and h αβ . In fact, from (12.1.18) we obtain α

so that

h ≡ h α = −h,

(12.1.19)

1 h αβ = h αβ − ηαβ h. 2

(12.1.20)



Now we evaluate the components h αβ of h αβ in the new quasi-Minkowskian coordinates (x α ) given by (12.1.6). Then in view of (12.1.9), we obtain h αβ = h αβ − (ξα,β + ξβ,α ),

(12.1.21)

h λ λ

(12.1.22)

=

h λλ

λ

− 2ξ ,λ ,

so that we also have 1  h αβ = h αβ − ηαβ h  2 1 = h αβ − (ξα,β + ξβ,α ) − ηαβ (h − 2ξ λ ,λ ) 2 = h αβ − (ξα,β + ξβ,α ) + ηαβ ξ λ ,λ .

12.1 Quasi-Minkowskian Spacetime

347

By (12.1.13), if we differentiate (12.1.23) and contract two indices in the result, we obtain α

α

h β,α = h β,α − ξ α ,βα −η αμ ξβ,αμ + η αμ ηαβ ξ λ ,λμ α

= h β,α − ξ α ,βα −η αμ ξβ,αμ + ξ λ ,λβ 1 = h αβ,α − ηβα h,α −η αμ ξβ,αμ . 2 That is,

α

α

h β,α = h β,α − ξβ .

(12.1.23)

Comparing this result with (12.1.16) and (12.1.17), we can state that the quasiMinkowskian coordinates (x α ) are harmonic if and only if the Lorentz gauge conditions are satisfied: α h β,α = 0. (12.1.24) In particular, if both the coordinates (x α ) and (x α ) are harmonic, then (12.1.23) becomes (12.1.25) ξβ = 0. In other words, if the coordinates (x α ) are harmonic, then every solution ξ α of (12.1.25) leads to new harmonic coordinates x α = x α + ξ α .

12.2 Einstein’s Linearized Equations For the Ricci tensor μ

μ

μ

ν ν μ νβ − αβ νμ , Rαβ = αμ,β − αβ,μ + αμ

(12.2.1)

we have that in the linear approximation, the terms  can be neglected, and we can write it as follows: μ

μ

 1 μλ  η h λα,μ + h λμ,α − h αμ,λ ,β 2  1 μλ  − η h λα,β + h λβ,α − h αβ,λ ,μ 2   1 = η μλ h λμ,αβ − h αμ,λβ − h λβ,αμ + h αβ,λμ , 2

Rαβ  αμ,β − αβ,μ =

(12.2.2)

which through a simple calculation may be transformed into the form Rαβ =

1 μ h,αβ −h λα,λβ − h β,αμ + h αβ . 2

(12.2.3)

348

12 Linearized Einstein’s Equations

On the other hand, from (12.1.17) we obtain 1 1 λ h α,βλ = h λα,βλ − ηαλ h,λβ = h λα,βλ − h,αβ , 2 2 so that (12.2.3), when we recall that we have adopted harmonic quasi-Minkowskian coordinates (see (12.1.24)), can also be written as follows: Rαβ =

1 1 λ μ h αβ − h α,λβ − h β,αμ = h αβ . 2 2

(12.2.4)

Consequently, the scalar of curvature is R

1 h, 2

(12.2.5)

and Einstein’s tensor assumes the form 1 G αβ  Rαβ − Rηαβ

2  1 1 1 h αβ − ηαβ h = h αβ . = 2 2 2

(12.2.6)

Finally, in harmonic quasi-Minkowskian coordinates, the linearized Einstein’s equations can be written in the following equivalent forms:

 1  h αβ − ηαβ h = −2χTαβ 2 h αβ = −2χTαβ .

(12.2.7) (12.2.8)

It is evident that both the systems (12.2.7), (12.2.8) are hyperbolic, and furthermore, the second system is formed by ten d’Alembert equations with separated unknowns. It is possible to give the same form also to the first system. In fact, on contracting the indices α and β in (12.2.8), we obtain h = 2χTλλ ,

(12.2.9)

and this result allow us to give (12.2.7) the form 

1 h αβ = −2χ Tαβ − ηαβ Tλλ . 2

(12.2.10)

In Chap. 11 we remarked that only six of the ten Einstein’s equations are independent, owing to the conservation laws, which in the linear approximation become T αβ ,β = 0.

(12.2.11)

12.2 Einstein’s Linearized Equations

349

Consider the momentum–energy tensor Tαβ of a perfect fluid. This tensor depends on the world velocity U α , the rest mass density 0 , and the rest pressure p0 = f ( 0 ). Therefore, (12.2.11) is a system of four equations in the four unknowns 0 and U α (since Uα U α = −c2 ), which in principle, can be determined by (12.2.11). In other words, Eq. (12.2.10) show that in the linear approximation, the matter distribution modifies the geometry but the geometry has no action on the matter distribution. It is evident that the field equations (12.2.10) determine the unknowns h αβ , provided that we take into account the four conditions (12.1.21) defining the harmonic coordinates.

12.3 Momentum–Energy Tensor for Weak Fields It is evident that a spacetime V4 equipped with a quasi-Minkowskian Riemannian metric, being close to Minkowski spacetime, is incompatible with an arbitrary energy–matter distribution, i.e., with any tensor Tαβ . We now suppose that a momentum–energy tensor is reasonably a source of a quasi-Minkowskian metric if: 1. Tαβ is different from zero in a spatially bounded region C, that is, Tαβ (x α , x 4 ) = 0,

3 (x α )2 < k,

(12.3.1)

i=1

where k is a finite positive real number. 2. The source of the gravitational field is dust matter, Tαβ = 0 Uα Uβ , 0  1,

(12.3.2)

where 0 is the mass density in the proper frame of reference and U α is the four-velocity of particles. 3. In harmonic and quasi-Minkowskian coordinates, the spatial velocities (v α ), i = 1, 2, 3, of material particles are nonrelativistic, 3 1 α 2 (v )  1, c2 u=1

(12.3.3)

so that γ = (1 − v 2 /c2 )−1/2  1. In view of hypotheses (1)–(3), we obtain Uα = γ(vi , −c) = γ(v i , −c)  (v i , −c), Ti j = 0 vi v j ,

(12.3.4) (12.3.5)

350

12 Linearized Einstein’s Equations

Ti4 = − 0 vi c,

(12.3.6)

T44 = 0 c . 2

T ≡ = ηαβ T

(12.3.7) αβ

= 0 (v − c )  − 0 c , 2

2

2

(12.3.8)

and Einstein’s equations (12.2.7) become  h i j = −χ 0 δi j c2 ,

 h i4 = 2χc 0 vi ,

 h 44 = −χc2 0 .

(12.3.9)

Recalling hypothesis (1) and Kirchhoff’s retarded potential theory, we can state that a solution of system (12.3.8) is given by the following functions:

 0 1 χδi j dc, (12.3.10) hi j = 4π C r

  0 vi 1 h i4 = − cχ dc, (12.3.11) 2π r

c  0 1 χc2 dc, (12.3.12) h 44 = 4π C r where (x α ) is the point at which h αβ are evaluated, (x  α ) is the variable point over the bounded region C, and we have adopted the notation  = 0 (x i , t − r/c), 0

vi = vi (x  , t − r/c),   3  r =  (x i − x  i )2 . i

(12.3.13)

i=1

In view of (12.3.10)–(12.3.12), we can write the metric of V4 as follows: ds 2 = (1 + h 44 )

3

(d x i )2 + h i4 d x i d x 4 − (1 − h 44 )(d x 4 )2 .

(12.3.14)

i=1

Finally, it is easy to verify that in our approximation, the conservation laws (12.3.8) can be written in the form ∂ 0 + ∇ · ( 0 v) = 0, ∂t ∂ 0 v + ∇ · ( 0 v ⊗ v) = 0. ∂t

(12.3.15) (12.3.16)

This system, in principle, allows one to determine the unknowns 0 and v when the initial and boundary conditions are given. It is very important to remark that in the

12.3 Momentum–Energy Tensor for Weak Fields

351

linear approximation, the moving dust matter determines the gravitational potentials h αβ , but there is no influence of the gravitational field on the motion of matter. This influence appears at a higher order of approximation.

12.4 Static Matter Distribution In all the formulas of the previous sections, the value of the constant χ is undetermined. To attribute a value to this constant, we consider the case of a static matter distribution, that is, we suppose v i = 0 and 0 independent of time. Under these conditions, (12.3.10)–(12.3.12) become h i j = δi j h 44 , h i4 = 0,

0 1 χc2 dc. h 44 = 4π C r

(12.4.1) (12.4.2) (12.4.3)

Furthermore, the linearized Einstein’s equations (12.3.9) reduce to the single equation h 44 = −χc2 0 ,

(12.4.4)

and the metric of spacetime becomes ds 2 = (1 + h 44 )[(d x 1 )2 + (d x 2 )2 + (d x 3 )2 ] − (1 − h 44 )(d x 4 )2 .

(12.4.5)

In other words, in the static case, the ten gravitational potentials h αβ reduce to h 44 . Furthermore, (12.4.4) has the same structure of Poisson’s equation. In order to find the relation between h 44 and Newtonian gravitational energy, we consider the motion of a test particle P acted on by the gravitational field associated with the metric (12.4.5). In this case, the world line of P is a geodesic of (12.4.5): δ d2xi i dx +  δλ dτ 2 dτ 2 4 δ d x 4 dx + δλ 2 dτ dτ

dxλ = 0, dτ dxλ = 0. dτ

(12.4.6) (12.4.7)

In the linear approximation, the Christoffel symbols are first-order quantities. α Since the velocity is of the same order, it is possible to neglect the terms iλ (d x i /dτ ) λ (d x /dτ ) in Eqs. (12.4.6), (12.4.7). Further, the coefficients h αβ are independent of time, so that we have 4 = 44

1 44 η (g44,4 + g44,4 − g44,4 ) = 0 2

(12.4.8)

352

12 Linearized Einstein’s Equations

and (see (12.4.5)) i 44 =

1 ij 1 1 η (g j4,4 + g j4,4 − g44, j ) = − g i j g44, j = − η i j h 44, j . 2 2 2

(12.4.9)

Finally, (12.4.6), (12.4.7) reduce to d2xi 1 − η i j c2 h 44, j = 0, dτ 2 2 d 2t = 0. dτ 2

(12.4.10) (12.4.11)

The second of the above equations allows us to identify the proper time τ with the coordinate time t; consequently, the first equation becomes d2xi 1 = c2 η i j h 44, j , dt 2 2

(12.4.12)

and it can be identified with Newton’s law of motion, provided that the Newtonian gravitational energy U is defined as follows: 1 U = − c2 h 44 . 2

(12.4.13)

On introducing this expression into (12.4.4), we obtain U =

1 4 χc 0 , 2

so that a comparison with Poisson’s equation U = 4πh 0 , where h denotes the gravitational constant, leads to the following value of χ: χ=

8πh . c4

(12.4.14)

Finally, we can write the metric (12.4.5) in the form ds 2 = (1 −

 2U 2U  ) (d x 1 )2 + (d x 2 )2 + (d x 3 )2 − (1 + 2 )(d x 4 )2 . 2 c c

(12.4.15)

Taking into account (12.4.14), we have that adopting quasi-Euclidean and harmonic coordinates allows us to write Einstein’s equations as follows:

12.4 Static Matter Distribution

353

⎧ 8πG ⎪  h i j = − 2 0 δi j , ⎪ ⎪ ⎪ c ⎪ ⎨ 16πG  h i4 = 0 vi , ⎪ c3 ⎪ ⎪ ⎪ ⎪ ⎩  h = − 8πG . 44 0 c2

2U δi j , c2 2 h i4 = − 2 Ai , c 2U h 44 = 2 , c 3 h ii − h 44 = h= hi j =

(12.4.16)

i=1

(12.4.17) (12.4.18) (12.4.19) 4U , c2 (12.4.20)

where

U =G C

 0 dc r

(12.4.21)

is the Newtonian potential, and

G Ai = dc, c C  3 



2U 4 2U ds 2 = 1 + 2 (d x i )2 − 2 Ai d x i d x 4 − 1 − 2 (d x 2 )2 c c c i=1 1 1 ¯ h¯ αβ = h αβ − ηαβ h, h αβ = h¯ αβ + ηαβ h, 2 2 16πG  h¯ αβ = − 2 Tαβ c 2U 2U h¯ ii = 2 − 2 = 0, c c 2 h¯ i4 = h i4 = − 2 Ai , c 4U 1 h¯ 44 = h 44 + h = 2 . 2 c

(12.4.22) (12.4.23) (12.4.24) (12.4.25) (12.4.26) (12.4.27) (12.4.28)

12.5 Plane Waves The linearized Einstein equations in vacuum are obtained from (12.2.7) with Tαβ = 0. If (12.2.9) is taken into account, then we have that in empty space, Einstein’s equations can equivalently be written h αβ = h αβ = 0.

(12.5.1)

354

12 Linearized Einstein’s Equations

Since the above equations describe the gravitational field in regions that are very far from the sources, we search for a solution representing a plane wave. Then we put h αβ = aαβ sin (k · x − ωt) ≡ aαβ sin K μ x μ ,

(12.5.2)

where the polarization tensor of the gravitational wave is a constant symmetric tensor (12.5.3) aαβ = aβα , and K μ = (ki , −ω/c)

(12.5.4)

is the 4-wave vector. Introducing (12.5.2) into (12.5.1), we obtain aαβ K 2 cos(K ν x ν ) = 0, so that the wave vector K α is isotropic, Kα K α = k2 −

ω2 = 0, c2

(12.5.5)

and the gravitational waves propagate with the velocity of light. Furthermore, Eq. (12.5.1) hold only in harmonic coordinates (x α ), i.e., in coordinates satisfying the Lorentz gauge condition (12.1.25). We search for a new set (x α ) of harmonic coordinates in which the tensor (aαβ ) has a simpler form. In view of (12.1.24) and (12.1.25), the transformation (x α ) → (x α ) is such that α

h β,α = 0, ξα = 0.

(12.5.6)

As a solution of the equation ξα = 0, we choose ξα = bα cos (l · x − ωt) ≡ bα cos L μ x μ ,

(12.5.7)

where L is an isotropic vector (L α L α = 0). Then we determine the field (ξα ) requiring that four components of h αβ in the new coordinates have given values. Precisely, we determine (ξα ) in such a way that in the new harmonic coordinates x α = x α + ξ α , we have (see (12.1.21)) h α4 = aα4 − (ξα,4 + ξ4,α ) = 0.

(12.5.8)

Recalling (12.5.2) and (12.5.7), the above equation gives bα L 4 + b4 L α = −aα4 .

(12.5.9)

12.5 Plane Waves

355

It is a simple exercise to determine (bα ) in terms of aα4 and L α solving system (12.5.9). The coordinate transformation determined by this vector (bα ) is called a TT gauge. We adopt now the new harmonic coordinates (xα ) and omit the prime in all the formula. Then in these coordinates we have, aαβ K β = 0,

(12.5.10)

aα4 = a4α = 0.

(12.5.11)

For α = 4, the condition (12.5.10) is satisfied in view of (12.5.11). Therefore, (12.5.10) reduces to three independent conditions. It is simple to verify that using the arbitrariness of the isotropic vector L α , it is possible to impose the condition that in the new coordinates, aαβ is traceless. In conclusion, in the new harmonic coordinates, the components aαβ satisfy the eight conditions aiβ K β = 0, aα4 = a4α = 0,

(12.5.12) (12.5.13)

η αβ aαβ = 0,

(12.5.14)

so that only two of them are independent. In order to find these components, we consider a plane wave propagating along the direction x 1 , h αβ = aαβ sin(K 1 x 1 + K 4 x 4 ) = aαβ sin(k1 x 1 − ωt).

(12.5.15)

Introducing (12.5.15) into (12.5.12)–(12.5.14), we obtain ai1 K 1 = 0,

a11 + a22 + a33 = 0,

(12.5.16)

so that ai1 = 0, a23 = 0, a33 = −a22 . In conclusion, we have

⎛

0 ⎜0 (aαβ ) = ⎜ ⎝0 0

0 0 a+ a× a× −a+ 0 0

⎞ 0 0⎟ ⎟, 0⎠ 0

(12.5.17)

(12.5.18)

where we have introduced the notation a22 = a+ , a23 = a× to denote the two polarization states of the gravitational waves in the linear approximation. Therefore, the two polarization states can be taken to be

h+ αβ

⎛ 0 ⎜0 = a+ ⎜ ⎝0 0

0 1 0 0

0 0 −1 0

⎞ 0 0⎟ ⎟ sin(k1 x 1 − ωt), 0⎠ 0

(12.5.19)

356

12 Linearized Einstein’s Equations

h× αβ

⎛ 0 ⎜ × ⎜0 =a ⎝ 0 0

0 0 1 0

0 1 0 0

⎞ 0 0⎟ ⎟ sin(k1 x 1 − ωt), 0⎠ 0

(12.5.20)

where a+ and a× are the “plus” and “cross” amplitudes, respectively.

12.6 Gravitational Wave Detection The choice of coordinates has a fundamental role in deriving the results of the above sections. In contrast, any attempt to compare the theoretical results with possible experimental data should be independent of the choice of coordinates. The coordinate-independent feature of any gravitational field is its tidal effect. Thus the detection of gravitational waves involves the recording of minute changes in the relative positions of a set of test particles. In this section, we shall first deduce the oscillatory pattern of such displacements, then briefly describe the principle of a gravitational wave interferometer. The effect of a gravitational wave on a test particle P is determined by the geodesic equation dU α α U λ U μ = 0. (12.6.1) + Fλμ dτ In the linear approximation, the velocity v of P is such that |v|  c. Therefore, (12.6.1) reduces to the equation dU α α = −c2 44 , dτ

(12.6.2)

where the Christoffel symbols α 44 =

 1 αβ  η h β4,4 + h 4β,4 − h 44,β 2

vanish because the metric perturbation h αβ has, in the TT gauge, polarization components a4α = aα4 = a44 = 0. Therefore, (12.6.2) becomes dU α = 0, dτ

(12.6.3)

and we can state that if initially P is at rest, then it remains at rest at the initial position. In other words, the particle is stationary with respect to the chosen harmonic coordinates, i.e., the TT gauge coordinate labels are attached to the particle at every instant. Thus it is impossible to discover any gravitational effect on a single particle. This is in agreement with the equivalence principle, stating that gravity can always

12.6 Gravitational Wave Detection

357

Fig. 12.1 Effects of the tidal forces produced by waves h + on a circle of test particles

be transformed away at a point by an appropriate choice of coordinates. We need to examine the relative motion of at least two particles in order to detect the effects of a gravitational wave. Consider the effect of a gravitational wave with “plus-polarization” h + αβ propagating along the axis x 1 and acting on two test particles at rest. One of these particles is supposed to be located at the origin and the other at an infinitesimal distance δ along the x 2 -axis. Then the variation of coordinates of the two particles at the same instant is (d x α = (0, δ, 0, 0)), and the proper distance

 1 √ ds = g22 δ  1 + h 22 δ 2 

1 = 1 + a+ sin(k1 x 1 − ωt) δ 2

(12.6.4)

changes with time. Similarly, for two particles separated along the x 3 -axis one has d x α = (0, 0, δ, 0), and the gravitational wave modifies the separation according to 

1 √ ds = g33 δ  1 + h 33 δ 2

 1 = 1 − a+ sin(k1 x 1 − ωt) δ. 2

(12.6.5)

Thus the separation in the x 2 direction is elongated, while along the x 3 direction, it is compressed. It is easy to recognize that there is no separation along the direction x 1 (see Fig. 12.1). In other words, like electromagnetic waves, gravitational radiation propagates by transverse waves. To show the pattern of relative displacements, we do not refer to two particles but to a set of particles located on a circle of radius δ. Figure 12.2 shows the configuration of these particles during a cycle of oscillation. The effect of a wave with cross-polarization h × αβ on two particles with a dif√ α ferential interval of d x =  (0, 1, ±1, 0)δ/ 2 alters the proper separation as ds   1 ± 21 a × sin(k1 x 1 − ωt) δ. The configurations of a circle of particles during a cycle of oscillation is shown in Fig. 12.1. While the two independent polarization directions of an electromagnetic wave are at 90◦ from each other, those of a gravity wave are at 45◦ .

358

12 Linearized Einstein’s Equations

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

12.7 Gravitoelectromagnetism The general theory of relativity postulates that the momentum–energy distribution, described by the momentum–energy tensor Tαβ , determines the geometry of spacetime by Einstein’s equations. In turn, the geometry tells matter how it must move. In the previous sections we have determined the first-order effects of momentum– energy on the geometry, and we have noticed that at this order, there is no influence of the geometry on the motion. In this section we show that without modifying the results, we can adopt a different point of view. We suppose that the spacetime is the Minkowski spacetime V4 even in the presence of weak gravitational fields. Consequently, this manifold is still Euclidean, and (12.4.15) is not considered to be the metric induced by the momentum–energy distribution. In this space, we can adopt rectilinear coordinates or arbitrary curvilinear coordinates. Therefore, h αβ is no longer a metric tensor but a tensor field describing the gravitation in V4 (not the geometry of V4 ). In other words, we are considering a Euclidean gravitational theory in which the effects of matter on the geometry can be neglected but not the forces that these small effects generate. The quantities U and A introduced in Sect. 12.4 are called scalar and vector potentials, respectively, of the gravitational field. We recall that the coordinates (x α ) are supposed to be harmonic. In view of (12.4.27) and (12.4.28), then, (12.1.25), for i = 4, gives

∇·

 1 1 A + U,i = 0, 2 c

(12.7.1)

which is formally identical to the gauge invariance that in electrodynamics holds between the electric scalar and vector potentials, up to a factor 1/2 that multiplies the vector potential. Introducing, as in electrodynamics, the gravitoelectric vector and the gravitomagnetic vector, 1 ∂ EG = −∇U − c ∂t BG = ∇ × A,

 A , 2

(12.7.2) (12.7.3)

12.7 Gravitoelectromagnetism

359

Einstein’s equations (12.2.10) reduce to ∇ · EG = 4πh 0 , ∇ · BG = 0, 1 ∂BG , ∇ × EG = − 2c ∂t 8π F 2 ∂EG ∇ × BG = 0 v + . c c ∂t

(12.7.4) (12.7.5) (12.7.6) (12.7.7)

These equations are called Maxwell’s equations for gravitoelectromagnetism. It is also possible to prove that under the hypothesis of static fields and low velocities, the geodetic equation assumes the form 2 x¨ = EG + x˙ × BG , c

(12.7.8)

where x¨ = (x¨ α ) and x˙ = (x˙ α ) denote respectively the material point acceleration and the velocity under the action of a gravitoelectromagnetic field.

Chapter 13

Cauchy’s Problem for Einstein’s Equations

There are two complementary ways to use Einstein’s equations: G αβ = −χ Tαβ . After choosing a momentum–energy tensor on the basis of some physical assumptions, we can try to determine the solutions of Einstein’s equations corresponding to that momentum–energy tensor. For example, if the momentum–energy tensor of a perfect fluid is chosen and a spherically symmetric solution is searched for, then a reasonable stellar model is obtained. Alternatively, if the geometric structure of spacetime is assigned, then Einstein’s equations allow one to determine the matter– energy distribution generating that geometry. This is the approach we will adopt in Chap. 15 to obtain cosmological models. In this chapter, instead of finding explicit solutions of Einstein’s equations, we analyze the Cauchy problem relative to Einstein’s equations in empty space Rαβ = 0.

13.1 Cauchy’s Problem and First Considerations In view (4.9.1) and (4.8.9), in every system of coordinates (x α ) of V4 , Einstein’s equations in empty space have the form Rαβ =

 1 ρσ  g gσα,ρβ + gρβ,ασ − gαβ,ρσ − gρσ,αβ + K αβ = 0, 2

© Springer Nature Switzerland AG 2019 A. Romano and M. Mango Furnari, The Physical and Mathematical Foundations of the Theory of Relativity, https://doi.org/10.1007/978-3-030-27237-1_13

(13.1.1)

361

362

13 Cauchy’s Problem for Einstein’s Equations

where f,αβ = ∂ 2 f /∂x α ∂x β and the functions K αβ depend only on the metric coefficients gαβ and their first derivatives gαβ,σ . Let S be a three-dimensional surface of V4 and suppose that in the coordinates (x α ), it is represented by the equation x 4 = 0. Then the Cauchy problem for (13.1.1) can be formulated as follows [92]: Having assigned the metric coefficients on S together with their first derivatives, prove that in a neighborhood of S, a unique solution gαβ of (13.1.1) exists that on S assumes the given values. A positive answer to this problem means that Einstein’s equations are deterministic, since if the values of gαβ and their first derivatives gαβ,σ are known on S (x 4 = 0), then the evolution of the metric coefficients is known for x 4 > 0, at least in a neighborhood of S. The classical procedure to solve this problem was proposed by Cauchy for a special case and Kovalevskaya for the full result. In this approach, the data and the coefficients of the equations are supposed to be analytic, and an analytic solution is searched for. Then it is sufficient to prove that the derivatives of gαβ at every order on S can be determined by the data on S and that the power series whose coefficients are these derivatives is convergent in a neighborhood of S. In this section we limit ourselves to proving that under suitable conditions, it is possible to determine all the derivatives of gαβ on S starting from the Cauchy data (see, for instance, [92]). With simple manipulations of (13.1.1) we obtain the equations 1 Ri j = − g 44 gi j,44 + Fi j (gαβ , gαβ,λ , gαβ,iλ ) = 0, 2 1 j4 Ri4 = g gi j,44 + i4 (gαβ , gαβ,λ , gαβ,iλ ) = 0, 2 1 R44 = − g i j gi j,44 + i4 (gαβ , gαβ,λ , gαβ,iλ ) = 0, 2

(13.1.2) (13.1.3) (13.1.4)

which on S become 1 Ri j = − g 44 gi j,44 + Fi j (C.D.) = 0, 2 1 j4 Ri4 = g gi j,44 + i4 (C.D.) = 0, 2 1 R44 = − g i j gi j,44 + i4 (C.D.) = 0, 2

(13.1.5) (13.1.6) (13.1.7)

since the variables gαβ , gαβ,λ , gαβ,iλ on S can be expressed in terms of the Cauchy data (C.D.). Regarding system (13.1.5)–(13.1.7), we notice that: • It should allow one to evaluate all the second derivatives gαβ,44 on S, but the derivatives gα4,44 do not appear in the above system. • The six unknown derivatives gi j,44 must satisfy ten equations. • the derivatives gi j,44 can be obtained from (13.1.5), provided that g 44 = 0.

(13.1.8)

13.1 Cauchy’s Problem and First Considerations

363

If S is a spacelike surface, its normal vector (Nα ) is time-oriented at every point g αβ Nα Nβ < 0. On the other hand, in the coordinates (x α ), the equation of S is x 4 = 0, and the above condition reduces to g 44 < 0, so that (13.1.8) is satisfied. This result shows that if S is a lightlike surface, then the Cauchy problem is ill posed. Consequently, the characteristic surfaces of (13.1.1) are the lightlike surfaces defined by the equation g αβ f,α f,β = 0. With the aim of overcoming the difficulties highlighted in the first two items, we show now that (13.1.6), (13.1.7) are equivalent to two other conditions that contain only the Cauchy data. In fact, the Einstein tensor in the absence of the cosmological constant is 1 (13.1.9) G βα = Rαβ − Rgαβ , 2 so that G i4 = g 44 Ri4 + g 4 j Ri j ,  1  44 g R44 − g i j Ri j . G 44 = 2

(13.1.10) (13.1.11)

This system shows that given the components Ri j , there is a one-to-one correspondence between the pairs (Ri4 , R44 ) and (G i4 , G 44 ). Further, on the surface S, when we take into account (13.1.6) and (13.1.7), we have G i4 (C.D.) = g 44 i4 + g j4 Fi j = 0,  1  44 g  − g i j Fi j = 0. G 44 (C.D.) = 2

(13.1.12) (13.1.13)

In conclusion, Eqs. (13.1.6), (13.1.7), or the equivalent equations (13.1.12) and (13.1.13), represent compatibility conditions of the Cauchy data. In other words, these data must be assigned in such a way to satisfy (13.1.12) and (13.1.13). Thus, we can equivalently analyze the system Ri j = 0, G 4α = 0, provided that the Cauchy data satisfy (13.1.12), (13.1.13). We omit proving1 that if (13.1.12), (13.1.13) are satisfied on S, then G 4α = 0 in a neighborhood of S. Finally, the metric coefficients gi j can be determined by (13.1.5). Differentiating (13.1.5) 1 See

[92].

364

13 Cauchy’s Problem for Einstein’s Equations

with respect x 4 , we obtain the derivatives gi j,444 , and so on. In the next section we analyze the consequences of the absence of derivatives gα4,44 in (13.1.5).

13.2 About the Uniqueness of the Solution of Cauchy’s Problem Let gαβ be a solution of a Cauchy problem relative to (13.1.1) and to a surface S that in the coordinates (x α ) has equation x 4 = 0. Equation (13.1.1) have the same form in every other system of coordinates (x α ). In other words, the components  of the Ricci tensor in the coordinates (x α ) depend on the metric coefficients Rαβ  gαβ as the components Rαβ depend on gαβ . We show that it is possible to make a  ) that reduces to the identity only on the surface change of coordinates (x α ) → (xαβ S. Consequently, in the new coordinates, the surface S still has equation x 4 = 0, and the Cauchy data are not modified by the coordinate transformation. In other words, this coordinate transformation does not modify the Cauchy problem. However, the  transformation differs from the identity outside S, and the metric coefficients gαβ differ from gαβ evaluated at points in a neighborhood of S. This means that if gαβ is a solution of the Cauchy problem, we obtain infinitely many solutions by applying to gαβ a coordinate transformation that does not modify S and the Cauchy data. All these solutions are said to be physically equivalent, because it is reasonable to think that they describe the same gravitational field. In conclusion, the solution of a Cauchy problem depends on four arbitrary functions. This situation can also be explained by noticing that Einstein’s equations are not independent, since they satisfy four conservation laws. Now we analyze the coordinate transformation 1 x λ = x λ + (x 4 )3 ϕλ (x i ), 6

(13.2.1)

satisfying the above-listed properties, and where ϕλ (x i ) are arbitrary functions defined on S. First, when x 4 = 0, we have also x 4 = 0. Further, introducing the notation ∂x λ , Aλμ = ∂x μ it is easy to verify that on S we have (Aλμ ) S = δμλ , (Aλ4,i4 ) S

=

λ (Ai,44 )S

= 0,

(Aλ4,μ ) S = (Aλμ,4 ) S = 0,

(13.2.2)

(Aλ4,44 ) S

(13.2.3)

λ

= ϕ = 0.

On the other hand, under this coordinate change, the metric coefficients and their first derivatives transform as follows:

13.2 About the Uniqueness of the Solution of Cauchy’s Problem

365

 gλμ = Aαλ Aβμ gαβ ,

gλμ,4 =

(13.2.4)

ρ  Aαλ Aβμ A4 gαβ,ρ 

 + Aαλ,4 Aβμ gαβ +

β  Aμ,4 Aαλ gαβ ,

(13.2.5)

  ρ where gαβ,ρ  = ∂gαβ /∂x . In conclusion, under the coordinate change (13.2.1) on the surface S, we have that   ) S , (gλμ,4 ) S = (gλμ,4 (gλμ ) S = (gλμ  )S .

(13.2.6)

It remains to evaluate the effect of the coordinate change on the second derivatives with respect to x 4 . It is ρ

β

 α β  α  gλμ,44 = Aαλ Aβμ A4 Aσ4 gαβ,4  4 + A λ,44 A μ gαβ + A μ,44 A λ gαβ + T,

where T denotes terms containing the first derivatives of Aαβ that vanish on S (see (13.2.2), (13.2.3)). Therefore on S, we have gi j,44 = gi j,4 4 ,  ρ 4 ρ  4 gλ4,44 = gλ4,4  4 + ϕ gλρ + δλ ϕ gρ4 = gλ4,4 4 + ϕλ + δλ ϕ4 .

(13.2.7) (13.2.8)

In conclusion, under this coordinate change, the second derivatives gi j,44 are not modified on the surface S, whereas the second derivatives gλ4,44 can assume arbitrary values. It is possible to prove that with a convenient choice of the functions ϕλ , these derivatives become continuous across S. All the considerations of Sects. 13.1 and 13.2 are based on the classical approach of Cauchy–Kovalevskaya, which requires that the Cauchy data and the solutions be analytic. This hypothesis is too strong for general relativity, where these functions are supposed to be of class C 2 . Therefore, Yvonne Choquet-Bruhat proved the existence and uniqueness theorems for the Cauchy problem under the weaker hypothesis of class C 2 of solutions and data, resorting to Leray’s theorems relative to hyperbolic systems. This approach is discussed in the following sections.

13.3 Mathematical Preliminaries The main tools to prove existence and uniqueness theorems for the Cauchy problem relative to Eq. (13.1.1), under the hypothesis of differentiability of solutions and Cauchy data, is represented by Leray’s theorem (see [93]). Some definitions and results are necessary before we can state it. Let Vn be a C k differential manifold with k sufficiently large and denote by a(x, ∂) a differential linear operator defined on the set of differentiable functions on Vn . For instance, a(x, ∂) could be the operator

366

13 Cauchy’s Problem for Einstein’s Equations

a(x, ∂) = bhk (x)

∂2 ∂ + ch (x) h . ∂x h ∂x k ∂x

Let Tx∗ be the covector space at x ∈ Vn . If m is the order of a, then a(x, ξ), ξ ∈ Tx∗ is an mth-degree real polynomial in ξ. Denoting by h(x, ξ) the principal part2 of a(x, ξ), from h(x, ξ) = 0 we have h(x, λξ) = 0 for every λ ∈ . Thus the set Vx (h) = {x ∈ Vn , ξ ∈ Tx∗ : h(x, ξ) = 0} is the projective conic surface in Tx∗ defined by h(x, ξ) = 0, where x is a fixed point of Vn . Definition 13.1 (Strictly Hyperbolic operator) The operator a is said to be strictly hyperbolic at x if there are elements ξ ∈ Tx∗ such that each straight line issuing from ξ and not containing the vertex of the cone Vx (h) intersects this cone in m distinct real points. Under these conditions, the set of these points ξ is the interior of two opposite convex nappes x+ and x− of a cone whose boundary belongs to Vx (h). Consider now a diagonal matrix A(x, ∂) sufficiently differentiable at x: ⎛ a1 (x, ∂) · · · ⎜ .. .. A(x, ∂) = ⎝ . . 0

0 .. .

⎞ ⎟ ⎠

· · · a p (x, ∂)

Definition 13.2 (Strictly hyperbolic matrix) The matrix A(x, ∂) is said to be strictly hyperbolic at x if (a) the aτ , τ = 1, . . . , p, are strictly hyperbolic operators at x; (b) The two opposite convex nappes of the cone, x+ (A) = ∩τ x+ (aτ ),

x− (A) = ∩τ x− (aτ ),

have nonempty interior. We introduce in the vector space Tx tangent to Vn at x the convex nappe Cm+ (A), the dual of m+ (A). With the same meaning, we introduce the opposite nappe C x− (A), the dual of x− (A), and the set C x+ ∪ C x− (A). A path of Vn is said to be timelike with respect to A if the half-tangent at each point x belongs to C x+ (A). A regular hypersurface  of Vn is said to be spacelike with respect to A if the vector subspace tangent to  at x is exterior to C x (A). Definition 13.3 Let  be a domain (a connected open set) of Vn . The matrix A(x, ∂) is said to be strictly hyperbolic in  if the following conditions are satisfied: polynomial P of degree m is called homogeneous if P(λx1 , . . . , λxn ) = λm P(x1 , . . . , xn ) for all λ ∈ C. Every polynomial P can be written as P = P0 + P1 , where P0 is homogeneous of degree m and the degree of P1 is less than m. The polynomial P0 is called the principal part of P.

2A

13.3 Mathematical Preliminaries

367

(1) A(x, ∂) is strictly hyperbolic at each point x ∈ . (2) The set of timelike paths, with respect to A, joining in  two points x, x  ∈  is either compact or empty. If A is strictly hyperbolic at x, then it is possible to prove that there exists a domain  containing x and homeomorphic to the interior of a sphere of n such as A is strictly hyperbolic in .

13.4 Leray’s Theorem On a differentiable manifold Vn , we consider the quasilinear differential systems in the unknowns σ = 1, . . . , p, U (x) = (u σ (x)), defined by A (x, U, ∂) U = B(x, U ),

(13.4.1)

where A is a diagonal matrix ⎛ a1 (x, U, ∂) · · · ⎜ .. .. A(x, U, ∂) = ⎝ . . 0 and B(x, U ) is

B(x, U ) = (bτ (x, U )),

0 .. .

⎞ ⎟ ⎠

· · · a p (x, U, ∂) τ = 1, . . . , p.

Denote by m(τ ) the order of the differential operator aτ (x, U, ∂). To make clear the assumptions about aτ (x, U, ∂) and bτ (x, U ), it is convenient to associate an integer index s(σ) ≥ 1 with each unknown u σ , and an integer index t (τ ) ≥ 1 with each equation τ of the system such as m(τ ) = s(τ ) − t (τ ) + 1. We suppose that the bτ and the coefficients of the differential operator aτ are sufficiently regular functions of x, u σ , and the derivatives of u σ of order s(σ) − t (τ ). When s(σ) − t (τ ) < 0, the terms bτ and the coefficients of aτ are independent of u σ . The structure of the system (13.4.1) defines the indices s(σ) and τ up to an additive constant. We now define Cauchy’s problem for the differential system (13.4.1). Definition 13.4 (Cauchy’s Problem) Let  be a regular hypersurface contained in  ⊂ Vn . Introduce in a neighborhood of  the auxiliary functions

368

13 Cauchy’s Problem for Einstein’s Equations

V (x) = (vσ (x))

σ = 1, . . . , p,

where vσ has locally square integrable derivatives of order ≤ s(σ) + 1. Furthermore, suppose that 1. the matrix A(x, V, ∂) is strictly hyperbolic in  and the hypersurface  is spacelike with respect to A; 2. the differences aτ (x, V, ∂)vτ − bτ (x, V ) and their derivatives of order < t (τ ) vanish on . This is the formulation of a Cauchy’s problem for system (13.4.1). A solution of a Cauchy problem is a set of functions U = (u σ (x)), locally square integrable together with their derivatives, that are a solution of the system (13.1.1) and satisfy the condition that the differences u σ (x) − vσ (x) and their derivatives of order ≤ s(σ) vanish on . Definition 13.5 (Leray systems) A system of partial differential equations (13.4.1) in which bv (x) and aτ satisfy the above hypotheses is called a Leray system. If it is possible to choose V such that A(x, V, ∂) is strictly hyperbolic, then we say that (13.4.1) is a strictly hyperbolic Leray system, provided that the Cauchy data are compatible and sufficiently regular. For this problem, Leray proved some existence and uniqueness theorems; see [30]. We state without proof the following theorem. Theorem 13.1 (Leray’s theorem) If x ∈ , then Cauchy’s problem for the system (13.1.1) under our hypothesis admits a solution in a neighborhood of x. If u σ and (u¯ σ ) have locally square summable derivatives of order ≤ s(σ) + 1, these solutions coincide.

13.5 Harmonic Coordinates Let V4 be a spacetime manifold, i.e., a Riemannian manifold whose metric has signature +, +, +, −, and let F k (Vn ) be, for k ≥ 2, the set of C k functions on V4 . We recall from Chap. 4 the following definition. Definition 13.6 (Harmonic functions) A function f ∈ F(V4 ) is said to be harmonic with respect to the metric of V4 if

f = g αβ ∇α ∂β f = 0. The level surfaces ( f = const) are called isothermal surfaces.

(13.5.1)

13.5 Harmonic Coordinates

369

In particular, a local system of coordinates (U, x α ) is harmonic with respect to the metric of V4 if the coordinates x α satisfy in U Laplace’s equation

x α = 0. The four families of local surfaces x α = const are isothermal surfaces. We introduce the quantities α , F α = x α = g ρσ ρσ

(13.5.2)

depending on the gravitational potentials g ρσ and linearly on their first derivatives α depend linearly on these derivatives. Then the local since the Christoffel symbols ρσ α coordinates (x ) are harmonic with respect to the metric of V4 if the gravitational potentials satisfy the equations (13.5.3) F α = 0. Recall that in every coordinate system (x α ), the components of the Ricci tensor are (see (13.1.1)) Rαβ =

 1 ρσ  g gσα,ρβ + gρβ,ασ − gαβ,ρσ − gρσ,αβ + K αβ = 0. 2

(13.5.4)

Now we want to determine a decomposition of the Ricci tensor that differs from (13.5.4). Notice that we have

= = = =

 1 ρσ  g gσα,ρβ + gρβ,ασ − gρσ,αβ 2  1 ρσ  g gασ,βρ + gαρ,βσ − gρσ,αβ − gρσ,αβ + gβσ,αρ + gβρ,ασ 4  1 ρσ   g [ρσ, α] ,β + [ρβ, σ] ,α − K αβ (gσρ , gσρ,λ ) 2   1 1 λ ρσ λ σρ  gαλ ρσ g ,β + gαλ σρ g ,α −K αβ 2 2    1  gαλ F λ ,β +gβλ F λ ,α − K αβ gρσ , gρσ,λ . 2

Substituting this expression into (13.5.4), we obtain (i) + L αβ , Rαβ = Rαβ

(13.5.5)

where we have introduced the notation   1 (i) Rαβ = − g ρσ gαβ,ρσ + f αβ gρσ , gρσ,λ , 2  1 L αβ = gαλ F λ ,β −gβλ F λ ,α . 2

(13.5.6) (13.5.7)

370

13 Cauchy’s Problem for Einstein’s Equations

The decomposition (13.5.5) holds in every system of coordinates (x α ). In harmonic coordinates, from (13.5.3) we have L αβ = 0, and (13.5.5) may be written as 1 (i) Rαβ = Rαβ = − g ρσ gαβ,σρ + f αβ (gσρ , gσρ,λ ). 2

(13.5.8)

Introducing the quantities (i) , R (i) = g σρ Rσρ σρ

from (13.5.5), we have

(13.5.9)

L = g L σρ

(13.5.10)

R = R (i) + L .

(13.5.11)

(i) and L αβ are not tensors, and so R and L are not scalars. It can be proved that Rαβ

13.6 Einstein’s Equations in Harmonic Coordinates For the exterior case, when the cosmological constant is neglected, Einstein’s equations assume the form (13.6.1) G αβ = Rαβ = 0, and taking into account (13.5.5), we have (i) + L αβ = 0. Rαβ = Rαβ

(13.6.2)

In particular, if the coordinates (x α ) are harmonic, the Eq. (13.6.2) reduce to (i) = 0. Rαβ = Rαβ

(13.6.3)

We now reformulate Cauchy’s problem for Einstein’s equations (13.6.3). Let V4 be a C 2 manifold, and denote by (U, x α ) the harmonic coordinates. Due to the character of the coordinates, in U the following conditions must be satisfied: Rαβ = 0,

F α = 0.

(13.6.4)

We again show that Cauchy’s problem cannot be formulated in the following way: (a) Let S be a space-oriented hypersurface with equation x 4 = 0 in U . Given the Cauchy data (C.D.) (13.6.5) gαβ (x i ), gαβ ,4 (x i ), i = 1, 2, 3,

13.6 Einstein’s Equations in Harmonic Coordinates

371

on S, determine the gravitational potentials gαβ satisfying system (13.6.4) outside of S and Cauchy data on S. Indeed, on S we have F α = 0, and from (13.5.2), we have that the quantities α F are functions of gαβ and their first derivatives. Since all these derivatives can be obtained by Cauchy data (13.6.5), we conclude that on S these data must satisfy the following compatibility conditions: F α (x σ ) = 0.

(13.6.6)

Further, in view of (13.1.11), (13.1.12), on S the Cauchy data must satisfy the additional conditions (13.6.7) Rα4 (C.D.) = 0. Concluding, in harmonic coordinates, Cauchy’s problem becomes the following: (b) Given on the spaceoriented surface S(x 4 = 0) ⊂ U the Cauchy data gαβ , gαβ ,4 ,

(13.6.8)

confirming on S the conditions Rα4 (C.D.) = 0,

F α (C.D.) = 0,

(13.6.9)

determine the gravitational potentials, in a neighborhood of S, solutions of the equations (13.6.10) Rαβ = 0, F α = 0. Finally, we show that the formulation (b) of Cauchy’s problem is equivalent to the following: (c) Given on the space oriented surface S with equation x 4 = 0 the Cauchy data confirming conditions (13.6.9), determine the gravitational potentials gαβ that solve the equations (13.6.11) Ri(i) j =0 in a neighborhood of S. It is evident that every solution of (b) is a solution of (c). To prove that every solution of (c) satisfies (b), we begin by recalling that if (13.6.7) are satisfied by the Cauchy data on S, then they hold also outside of S. It is possible to prove that a similar conclusion holds also for the equations F α = 0 (see [93]). Finally, we have i Ri j = Ri(i) j = 0, because (x ) are harmonic coordinates, and every solution of (c) is also a solution of (b). It is simple to verify that (13.6.11) is a Leray system, and an existence and uniqueness theorem for Cauchy’s problem relative to Einstein’s exterior equations is proved in harmonic coordinates.

Chapter 14

Schwarzschild’s Universe

In this chapter we analyze a model of the universe proposed by Karl Schwarzschild. In the literature, this model is presented in many equivalent ways. Here we adopt an approach that in our opinion is less formal. However, in Sect. 14.4 we introduce a definition that allows a comparison with another usual presentation of Schwarzschild’s model. Schwarzschild’s solution of Einstein’s equations describes the gravitational field inside and outside a nonrotating spherically symmetric mass distribution S. Since the Sun and the planets are assumed to be slowly rotating and approximately spherically symmetric, Schwarzschild’s solution should supply a good description of Einstein’s gravitational field produced by these celestial bodies. A spacetime equipped with Schwarzschild’s solution is called a Schwarzschild spacetime. Denoting by Tαβ the momentum–energy tensor, Schwarzschild’s metric satisfies equations Rαβ = 0 outside the body S and equations G αβ = −χTαβ inside S, and it is continuous with its first derivatives across the boundary of S. It is important to remark that this global solution of Einstein’s equations can be obtained only if the radius of the sphere occupied by S is greater than the Schwarzschild radius r = 2hM/c2 , where M is the total mass of S. In Sects. 14.9–14.10, we discuss the physical meaning of the mathematical quantities, and we describe the applications of Schwarzschild’s model to planetary orbits, light paths, and the behavior of clocks. The exterior Schwarzschild solution admits an analytic extension to the region contained in the Schwarzschild radius except for the origin. In this way, a model of spacetime is obtained that could describe the gravitational field of a black hole. We postpone the analysis of the properties of this model to Chap. 15.

14.1 Gaussian Coordinates Let V4 be a spacetime, i.e., a C 2 four-dimensional hyperbolic Riemannian manifold whose metric has signature (+, +, +, −). We denote by  a time-oriented threedimensional surface of V4 such that the unit normal vector V(y) to  at every point © Springer Nature Switzerland AG 2019 A. Romano and M. Mango Furnari, The Physical and Mathematical Foundations of the Theory of Relativity, https://doi.org/10.1007/978-3-030-27237-1_14

373

374

14 Schwarzschild’s Universe

y ∈  is spacelike: V(y) · V(y) = 1, for all y ∈ S. Let  be the family of geodesics starting from the points y ∈  and having at these points the tangent vector V(y). We suppose that an interval (a, b) of the affine parameter s along the geodesics exists such that the corresponding geodesic arcs do not intersect each other. In other words, we suppose that in a neighborhood of , the family of geodesics  is a local congruence. Since the scalar product is invariant under parallel transport, the tangent vector field V(x) to the geodesics is a spacelike unit vector field, and  is formed by spacelike curves. In conclusion, in an arbitrary system of coordinates of V4 , the vector field V(x) satisfies the conditions V α ∇α Vβ = 0, V α ∇β Vα = 0,

(14.1.1)

so that we have also V α (∇α Vβ − ∇β Vα ) = V α (Vβ,α − Vα,β ) ≡ V α ωαβ = 0,

(14.1.2)

where f,α = ∂ f /∂x α . Denote by (x i ), i = 2, 3, 4, arbitrary local coordinates on . If x 1 is the affine parameter along the curves of , then (x 1 , x i ) becomes a coordinate system in a neighborhood of . In these coordinates, x 1 = 0 is the local coordinate representation of , the geodesics of  have equations x i = const, and x 1 = var, and V i = 0. From V i = 0 and (14.1.2) we have that (14.1.3) ω1i = 0. On the other hand, we have ω, ω ≡ ωαβ d x α ∧ d x β = d(Vα d x α ) ≡ d

(14.1.4)

 = 0. dω = d 2 ω

(14.1.5)

so that

Since ω =ω12 d x 1 ∧ d x 2 + ω13 d x 1 ∧ d x 3 + ω14 d x 1 ∧ d x 4 + ω23 d x 2 ∧ d x 3 + ω24 d x 2 ∧ d x 4 + ω34 d x 3 ∧ d x 4 , Equation (14.1.5) can be written explicitly as follows: dω =(ω23,1 − ω13,2 + ω12,3 )d x 1 ∧ d x 2 ∧ d x 3 + (ω24,1 − ω14,2 + ω12,4 )d x 1 ∧ d x 2 ∧ d x 4 + (ω34,1 − ω14,3 + ω13,4 )d x 1 ∧ d x 3 ∧ d x 4 + (ω34,2 − ω24,3 + ω23,4 )d x 2 ∧ d x 3 ∧ d x 4 = 0.

(14.1.6)

14.1 Gaussian Coordinates

375

In view of (14.1.4), the above condition gives ω23,1 = ω24,1 = ω34,1 = 0

(14.1.7)

in a neighborhood of . On the other hand, on  (x 1 = 0) the covariant components Vi of V vanish identically, since V is orthogonal to . Then on S we have ωi j = 0, i, j = 2, 3, 4,

(14.1.8)

so that (14.1.8), (14.1.7), and (14.1.3) lead to the result ωαβ = 0,

(14.1.9)

which, in turn, implies the integrability of the differential form Vα d x α . In conclusion, a differentiable function f exists such that Vα = f,α ,

(14.1.10)

and we can state that the geodesics  are orthogonal to the family of surfaces f = const. Further, on  one has Vi = f,i = 0, and the function f is constant on . Choosing f = 0 on , the variables (x α ) with x 1 = f become local coordinates in a neighborhood of  that are called Gaussian coordinates associated with . In these coordinates, the metric assumes the form ds 2 = (d x 1 )2 + gi j d x i d x j ,

(14.1.11)

g11 = V · V = 1.

(14.1.12)

since It is not difficult to verify that if in a neighborhood of a surface  a system of coordinates (x α ) exists in which the metric has the form (14.1.11), then the coordinates (x α ) are Gaussian coordinates for  (see [92], p. 60).

14.2 Matching Conditions Let S be a mass–energy distribution in a region V of the three-dimensional space of a frame of reference R. In the spacetime V4 , to the matter–energy S there corresponds a congruence  of timelike curves representing the world lines of the points of S. This congruence fills a region  of V4 that is called the universe tube. It is evident that outside the region , the metric of V4 satisfies Einstein’s equations relative to the empty space, whereas inside , the metric is a solution of Einstein’s equations whose right-hand sides contain the momentum–energy tensor Tαβ of the mass–energy distribution S.

376

14 Schwarzschild’s Universe

From now on, we suppose that on the boundary ∂ of the universe tube  the following condition is satisfied: Matching conditions—For every point x ∈ ∂, there exists a system of coordinates compatible with the atlas of V4 such that the metric coefficients and their first derivatives are continuous across ∂. We omit the proof of the following theorem1 : Theorem 14.1 Every system of Gaussian coordinates (x α ) relative to ∂ is compatible with the atlas of V4 . Further, the metric coefficients and their first derivatives are continuous across ∂. The above theorem allows to state that the continuity conditions can be formulated as follows: • In a system of Gaussian coordinates relative to the surface ∂, the metric coefficients and their first derivatives are continuous across ∂. • The Gaussian coordinates on both sides of ∂ are compatible with the atlas of V4 . In concrete cases, this procedure may be not easy to apply. Then we can resort to the following theorem (see [92]), which holds when the mass distribution generating the gravitational field is a perfect fluid. Theorem 14.2 Let  be a bounded region of V4 occupied by a perfect fluid F. Then the matching conditions are satisfied in a Gaussian system of coordinates relative to ∂ if: • The boundary ∂ is generated by the world lines of the particles of F. • The pressure vanishes on ∂.

14.3 Static and Stationary Spacetime In general relativity, a model of the universe is a four-dimensional manifold V4 such that: • It is equipped with an atlas A of class C 2 . • It possesses a Riemannian metric g with hyperbolic signature (+, +, +, −) of class C 2 (V4 ) that is a solution of Einstein’s equations corresponding to a given momentum–energy tensor Tαβ . • The metric coefficients and their first derivatives are continuous across the surface of the universe tube in which Tαβ = 0.

1 See

[92], p. 61.

14.3 Static and Stationary Spacetime

377

Definition 14.1 (Static spacetime) Let V4 be a four-dimensional hyperbolic Riemannian manifold and denote by V3 a three-dimensional manifold. Suppose that a diffeomorphism f : V3 ×  → V4 exists such that: 1. The curves f p : { p} ×  → V4 , for all p ∈ V3 , satisfy the following properties: • They  are timelike and diffeomorphic to ; • f ( p) f (q) = ∅, for all p, q ∈ V4 , p = q. We denote by  the timelike congruence of these curves. 2. The hypersurfaces f t : V3 × {t} → V4 , for all t ∈ , are spacelike surfaces diffeomorphic to V3 and orthogonal to the curves of . We denote by F the family of the surfaces f t . If (U, x i ), i = 1, 2, 3, and (, x 4 ) are coordinates for V3 and  respectively, then (U × , (x i , x 4 )) is a chart on V4 in which the curves of  have parametric equations x i = const, x 4 = var, while the surfaces of F have the equation x 4 = const. These coordinates are said to be adapted to the congruence . They are defined up to a coordinate transformation such that x = x (x 1 , x 2 , x 3 ), i

i

x = x (x 1 , x 2 , x 3 , x 4 ). 4

4

(14.3.1)

3. In adapted coordinates, the metric coefficients satisfy the conditions gi4 = 0

(14.3.2)

gαβ,4 = 0,

(14.3.3)

i.e., the curves of  are orthogonal to the surfaces of F, and the metric coefficients are independent of the coordinate x 4 . A hyperbolic manifold V4 satisfying the above properties is said to be a static spacetime. The frame of reference R defined by the congruence  (see Sect. 11.7) has the following property: the unit tangent vector γ to the curve of  at the point p ∈ V4 is orthogonal to the surfaces S p ∈ F containing p. Consequently, the three-dimensional subspace  p orthogonal to  at p is tangent to S p at the point p. Further, (11.7.8) becomes (14.3.4) ds 2 = dσ 2 − c2 dT 2 ,

378

with

14 Schwarzschild’s Universe

dσ 2 = gi j d x i d x j , 1√ dT = − −g44 d x 4 , c

i, j = 1, 2, 3.

(14.3.5)

From now on, we use the notation V3 ×  to denote a static spacetime. When all the above conditions are satisfied except for (14.3.2), the spacetime V4 is said to be stationary.

14.4 Isometries and Killing’s Vector Fields In order to define a static spacetime in a different way, we introduce some mathematical considerations. Let V4 be a Riemannian manifold, X a smooth vector field on V4 , and g the metric tensor. Then the following formula holds: L X g = (∇α X β + ∇β X α )d x α ⊗ d x β , where (x α ) are arbitrary coordinates of V4 . In fact, from Sect. 2.3 we recall that   ∂Xν ∂Xν ν ∂gαβ L Xg = X + gνβ α + gαν β d x α ⊗ d x β . ∂x ν ∂x ∂x

(14.4.1)

(14.4.2)

On the other hand, we have ∂ Xβ ∂ Xα ν ν − αβ Xν + − βα Xν α ∂x ∂x β  ∂  ∂ ν ν = α gβν X ν − αβ X ν + β (gαν X ν ) − βα Xν ∂x ∂x ∂gβν ∂Xν ∂Xν ν ∂gαν ν =X ν + g + X + g − 2αβ Xν βν αν ∂x α ∂x α ∂x β ∂x β ν ν ∂gβν ∂X ∂gαν ∂X =X ν + gβν α + X ν + gαν β − α β ∂x ∂x ∂x ∂x   ∂gλβ ∂gαβ νλ ∂gλα Xν g + − ∂x β ∂x α ∂x λ ∂gαβ ∂Xν ∂Xν =X ν + g + g , νβ αν ∂x ν ∂x α ∂x β

∇α X β + ∇β X α =

and formula (14.4.2) is proved. We now introduce new definitions of stationary and static spacetimes, and then we prove that they are essentially equivalent to the definitions of Sect. 14.2.

14.4 Isometries and Killing’s Vector Fields

379

Definition 14.2 (Stationary spacetime) A spacetime V4 with Riemannian metric g is said to be stationary if a one-parameter group of transformations φ :  × V4 → V4

(14.4.3)

without fixed points exists satisfying the following properties: • The family  of orbits of φ is a timelike congruence. • Every transformation φt , t ∈ , is an isometry, that is (see (3.2.17)), (φt )φt (x) (g) = g(x), ∀x ∈ V4 .

(14.4.4)

Let us introduce in V4 the following equivalence relation R: two points of V4 are equivalent if they belong to the same orbit of the transformation group φ. Since  is a congruence, the manifold V3 ≡ V4 /R is three-dimensional, and V4 = V3 × . Further, since φ is an isometry, condition (14.4.4) is equivalent to the condition L X g = 0,

(14.4.5)

where the infinitesimal generator X of φ is a timelike vector field on V4 that, in view of (14.4.1) and (14.4.5), satisfies the condition ∇α X β + ∇β X α = 0.

(14.4.6)

Such a vector field is called a Killing vector field.2 Finally, we introduce a coordinate system (x i , x 4 ) on V4 , where (x i ), i = 1, 2, 3, are coordinates in V3 , and x 4 a coordinate on . Then these coordinates are adapted to the congruence , so that the infinitesimal generator X of φ has contravariant components (0, 0, 0, 1).

14.5 Three-Dimensional Spherically Symmetric Manifolds Definition 14.3 (Manifold with Spherical Symmetry) A properly Riemannian manifold V3 is said to have spherical symmetry about a point q ∈ V3 if a diffeomorphism ψ : 3 → V3 exists such that: 1. ψ(0) = q, where 0 is the origin of 3 ; 2. the coefficients γi j of the metric of V3 , dσ 2 = γi j d x i d x j ,

(14.5.1)

2 It is evident that if a Killing vector fields exists on a spacetime V , that is, the infinitesimal generator 4

of a one-parameter group of global transformations, then V4 has all the properties listed in Definition 14.1.

380

14 Schwarzschild’s Universe

are invariant under a change of orthogonal Cartesian coordinates of 3 with origin 0 ∈ 3 , or equivalently, under an orthogonal transformation of E 3 . Theorem 14.3 The manifold V3 diffeomorphic to 3 is symmetric about the point q ∈ V3 if and only if the coefficients γi j of the properly Riemannian metric (14.5.1) have the form 3  (xi )2 , (14.5.2) γi j = a(r )δi j + b(r )xi x j , r = i=1

in every chart (V3 , xi ), where (xi ) are Cartesian coordinates of 3 with the origin at ψ(q) and a(r ), b(r ) are positive functions. Proof We begin by proving that (14.5.2) is sufficient for spherical symmetry about j j the origin. Let xi = Q i x j be a change of orthogonal Cartesian coordinates, where Q i is an orthogonal matrix. If (14.5.2) are the coefficients of the metric in the coordinates (x i ), then in the orthogonal Cartesian coordinates (x i ), we have γi j = Q ih Q kj (a(r )δhk + b(r )x h xk ) = a(r )δi j + b(r )xi x j , since Q ij is orthogonal, i.e., Q ih Q hj = δi j , and r is invariant under the transformation (xi ) → (xi ). Conversely, if the metric is invariant under an arbitrary orthogonal transformation xi = Q ih x h of 3 , then we have γi j (r ) = Q ih Q kj γhk (r), where r is the position vector of 3 with components (xi ), and r is the transformed vector with components (xi ). In matrix form, the above condition can also be written γ(Q(r)) = Q(r)γ(r)QT , and we conclude that the function γ(r) is isotropic. From general theorems of algebra, we obtain (14.5.2).  We now adopt in the Euclidean space 3 spherical coordinates (r, θ, ϕ) related to the orthogonal Cartesian ones by the formulas x1 = r sin θ cos ϕ, x2 = r sin θ sin ϕ, x3 = r cos θ,

(14.5.3)

where θ ∈ [0, π] is the colatitude and ϕ ∈ [0, 2π] the longitude. In view of the diffeomorphism ψ : 3 → V3 , (r, θ, ϕ) are also coordinates in V3 . With simple but tedious calculations, it is possible to verify that in the spherical coordinates, the metric dσ 2 assumes the form

14.5 Three-Dimensional Spherically Symmetric Manifolds

381

    dσ 2 = a(r ) + b(r )r 2 dr 2 + a(r )r 2 dθ2 + sin2 θdϕ2 .

(14.5.4)

To further simplify (14.5.4), we consider the two-dimensional submanifold  of V3 corresponding to the point at which r assumes a constant value. On this surface, the induced metric is   dσ2 = a(r )r 2 dθ2 + sin2 θdϕ2 . Consequently, the determinant g of the metric coefficients and the area A of  are respectively given by g = a2 r 4 sin2 θ √ 2 A= g dθdϕ = ar 

π 0

sin θdθ

2π

dϕ = 4πar 2 .

0

This last formula shows that the area A of  is equal to the area of a Euclidean sphere of radius r if and only if we choose a = 1. When we adopt this choice, the coordinate r is called area distance, and the metric of V3 assumes the form     dσ 2 = 1 + b(r )r 2 dr 2 + r 2 dθ2 + sin2 θdϕ2 .

(14.5.5)

In order to make clearer the above definitions, we consider in 3 , in reference to Cartesian coordinates x, y, and z, a surface S diffeomorphic to 2 (or to an open set of 2 ) with a symmetry of revolution about the z-axis (see Fig. 14.1). We consider the diffeomorphism ψ : 2 → S such that at every point (x, y) ∈ 2 , it associates the intersection point of the straight line orthogonal to 2 at (x, y) with S. Notice that the Cartesian coordinates in 2 are curvilinear coordinates on S. The equations of S are x = x, y = y, z = h(x 2 + y 2 ) = h(r 2 ), so that from the third equation we have dz 2 = 2xh d x + 2yh dy, and the metric is dσ 2 = d x 2 + dy 2 + dz 2 = (1 + 4x 2 h )d x 2 + 8x yh d xd y + (1 + 4y 2 h )dy 2 . (14.5.6) It is evident that the metric coefficients have the form (14.5.2), where 2

a(r ) = 1,

2

b(r ) = 4h 2 .

2

382

14 Schwarzschild’s Universe

Fig. 14.1 A surface with a symmetry of revolution about the axis O x3

When we adopt polar coordinates in 2 , x = r cos ϕ y = r sin ϕ, the metric becomes

dσ 2 = (1 + 4h 2 r 2 )dr 2 + r 2 dθ.

In particular, if S is a hemisphere of radius R, then h(x 2 + y 2 ) = r 2 /(R 2 − r 2 ), and the metric in polar coordinates is 

4r 4 dσ = 1 + 2 R − r2 2

√

R 2 − r 2 , h 2 =

 dr 2 + r 2 dθ2 .

Similar considerations can be repeated for a three-dimensional surface S of 4 with a symmetry of revolution about the axis x 4 of a Cartesian system of coordinates of 4 : xi = xi ,

i = 1, 2, 3,

x4 = h(r ), 2

14.5 Three-Dimensional Spherically Symmetric Manifolds

where r = have

3

i=1 (x i )

2

383

. With notation already adopted in the previous example, we

d x4 = 2xi h d xi , dσ = 2

3 

d xi2 + d x42 = (δi j + 4h xi x j )d xi d x j .

i=1

If we adopt spherical coordinates in the subspace (x1 , x2 , x3 ) of 4 , x1 = r sin θ cos ϕ, x2 = r sin θ sin ϕ, x3 = r cos θ, by similar computations we obtain dσ 2 = (1 + 4h 2r 2 )dr 2 + r 2 (dθ2 + sin2 θdϕ2 ).

(14.5.7)

More specifically, if S is a sphere, then (14.5.7) may be written as follows:  dσ 2 = 1 +

4r 4 2 R − r2

 dr 2 + r 2 (dθ2 + sin2 θdϕ2 ).

(14.5.8)

14.6 Schwarzschild’s Exterior Solution Definition 14.4 (Static spacetime with spatial symmetry about a point) The static spacetime V4 = V3 ×  is said to have spatial spherical symmetry if (a) V3 has spatial spherical symmetry with respect to a point q ∈ V3 ; (b) its metric in adapted coordinates (r, θ, φ, t) assumes the form (see (14.5.5)) ds 2 = a(r )dr 2 + r 2 (dθ2 + sin2 dϕ2 ) − b(r )c2 dt 2 ,

(14.6.1)

where the functions a(r ), b(r ) are arbitrary except for the qualitative conditions a(r ) > 0, b(r ) > 0. The first condition implies that the spatial distance between two points is a positive definite quadratic form, while the second condition ensures that the curves of the congruence  of the world lines of the reference fluid are timelike. To simplify the next calculations, we prefer to give (14.6.1) the following equivalent form: (14.6.2) ds 2 = eλ(r ) dr 2 + r 2 (dθ2 + sin2 dϕ2 ) − eν(r ) c2 dt 2 ,

384

14 Schwarzschild’s Universe

where

a(r ) = eλ(r ) , b(r ) = eν(r ) .

(14.6.3)

Now we want to prove that the metric tensor (14.6.2) becomes a solution of Einstein’s equations in empty space with an appropriate choice of the functions λ(r ), ν(r ). We begin by noticing that in the coordinates (x α ) = (r, θ, ϕ, t), we have3 g11 = eλ(r ) , g22 = r 2 , g33 = r 2 sin2 θ, g44 = −c2 eν(r ) ,

(14.6.4)

since the other metric components vanish. Further, we have 1 = e−λ(r ) , g11 1 1 = = 2 2 , g33 r sin θ

1 1 = 2, g22 r 1 e−ν(r ) = =− 2 . g44 c

g 11 =

g 22 =

g 33

g 44

(14.6.5)

After easy but boring computations, the nonvanishing Christoffel symbols may be written as λ (r ) 1 1 = −r e−λ(r ) , 33 = −r e−λ(r ) sin2 θ, , 22 2 1 1 2 2 = eν(r )−λ(r ) ν (r ), 12 = = − sin θ cos θ, 33 2 r 1 1 3 4 23 = , = cot θ, 14 = ν (r ), r 2

1 = 11 1 44 3 13

(14.6.6)

where the prime denotes differentiation with respect to r . It is also easy to show that Einstein’s equations for the external case will produce only three independent equations in the unknowns λ(r ) and ν(r ),

R22 R44



 1 1 + ν (r ) − = 0, r r

2 1 ν (r ) 1 − λ (r )ν (r ) + (ν (r ) − λ (r ) = 0, = ν

(r ) + 2 2 r   1 1 −λ(r )

λ (r ) − − = 0, = −e r r

R11 = e−λ(r )

(14.6.7)

while the remaining equations are either identically satisfied or dependent on Eqs. (14.6.7). From the third of the above equations we have

3 All

the following computations can be carried out with the notebook Geometry.nb written in Mathematica.

14.6 Schwarzschild’s Exterior Solution

385

e−λ(r ) (λ (r )r − 1) = 1. After putting the above equation in the form (e−λ(r )r ) = 1, we integrate to obtain

e−λ(r ) r = r − 2m,

where 2m is an integration constant. In conclusion, we have eλ(r ) =

1 . 2m 1− r

(14.6.8)

Subtracting the third equation from the first of (14.6.7), we obtain the condition λ (r ) + ν (r ) = 0, which implies λ(r ) + ν(r ) = C,

(14.6.9)

where C is an integration constant. In order to find the integration constants m and C, we first note that the metric must reduce to the Minkowski metric when r → ∞. Therefore, we have lim λ(r ) = lim ν(r ) = 0, r →∞

r →∞

and from (14.6.9) we have that C = 0 and ν(r ) = −λ(r ). In view of (14.6.8), we finally obtain eν(r ) = 1 −

2m , r

(14.6.10)

and the exterior metric (14.6.2) assumes the form   1 2m 2 2 2 2 2 2 2 ds = c dt . dr + r (dθ + sin dϕ ) − 1 − 2m r 1− r 2

(14.6.11)

To find the value of m, we notice that for large values of r , we have the approximate formula 2m , eλ(r )  1 + r

386

14 Schwarzschild’s Universe

so that for large values of r , we get 

2m ds = 1 + r



2

  2m 2 2 dr + r (dθ + sin dϕ ) − 1 − c dt . r 2

2

2

2

2

Comparing this metric with (12.4.15), which holds in the linear approximation, we can state that 2m/r plays the role of h 44 in the metric (12.4.15). Then from (12.4.13), we get U 2hM 2m = 2 = 2 , r c c r i.e., m=

hM , c2

(14.6.12)

where M is the mass of the body generating the gravitational field. Taking into account (14.6.11), (14.6.12), (14.6.2), we finally obtain the exterior solution of Einstein’s equations in a static spacetime V4 = V3 × , where V3 is spherically symmetric with respect to the point r = 0: ds 2 =

  dr 2 2hM c2 dt 2 . + r 2 (dθ2 + sin2 θdϕ2 ) − 1 − 2 2hM c r 1− 2 c r

(14.6.13)

With regard to (14.6.13), we notice the following: (a) The spatial manifold V3 , which is assumed to be diffeomorphic to 3 , is a Riemannian manifold with metric dσ 2 =

dr 2 + r 2 (dθ2 + sin2 θdϕ2 ). 2hM 1− 2 c r

(14.6.14)

(b) The conditions eλ(r ) > 0, eν(r ) > 0 are satisfied by the metric (14.6.13) only if 1−

2hM > 0, c2 r

(14.6.15)

2hM . c2

(14.6.16)

that is, when r> Furthermore, the metric is singular at rS =

2hM = 2m. c2

14.6 Schwarzschild’s Exterior Solution

387

This value of r is called the Schwarzschild radius. For the Sun, the Schwarzschild radius has the value 2.956 · 105 cm, while the solar radius is 6.963 · 1010 cm. For the Earth, the Schwarzschild radius is 9 mm. We conclude by observing that a static spacetime V4 , with the properties listed in Sect. 14.3 and equipped with the metric (14.6.13), cannot be taken as a model of the universe. In fact, this metric is not regular at r = 0 and when r is equal to the Schwarzschild radius. To build a model of the universe we have to assign the momentum–energy tensor in a region of V4 , integrate the corresponding Einstein equations in this region, and show that the interior and exterior solutions can be connected in such a way to obtain a C 1 (V4 ) global solution of Einstein’s equations.

14.7 Schwarzschild’s Interior Solution In this section we consider a static perfect fluid F filling a sphere S of V3 with spherical symmetry and denote by  the universe tube of V4 formed by the world lines of the particles of F. Then we integrate the corresponding Einstein’s equations in  and impose that the coefficients of the metrics corresponding to the interior and exterior Schwarzschild solutions assume, with their first derivatives, the same values on ∂, i.e., that they satisfy the matching condition of Sect. 14.2. The first step in achieving these results consists in searching for the functions λ(r ) and ν(r ) in (14.6.2) such that they are a solution in  of Einstein’s equations G βα = Rαβ −

1 β Rg = −χTαβ , 2 α

(14.7.1)

where Tαβ is the momentum–energy tensor of a perfect fluid, Tαβ = Uα U β + pgαβ +

p Uα U β , c2

(14.7.2)

the pressure p and the density  evaluated in the proper frame are related by the state equation p = f (), (14.7.3) and the fields ρ, p, and U α depend only on the variable r . Before writing Einstein’s equations, we notice that in the frame of reference (r, θ, φ, ct) of V4 , the fluid is at rest. Consequently, we have U α = (0, 0, 0, U 4 ).

(14.7.4)

√ √ From gαβ U α U β = g44 (U 4 )2 = −c2 it follows that U 4 = c/ −g44 = c/ b, where 4 2 we used (14.6.5) √ and (14.6.3). On the other hand, we also have U4 U = −c and thus U4 = −c b.

388

14 Schwarzschild’s Universe

Tedious computations lead to the following Einstein’s equations4 : 

 1 1 ν (r ) − 2 = χ p(r ), =e + r2 r r   e−λ(r ) ν 2 λ ν

ν − λ

2

G2 = ν + = χ p, − + 2 2 2 r 

 λ (r ) 1 1 − 2 − 2 = −c2 χ(r ). G 44 = −e−λ(r ) r r r G 11

−λ(r )

(14.7.5) (14.7.6) (14.7.7)

αβ

Further, in our case, the conservation laws T,β = 0 reduce to the single equation 1 p = − ν (c2 + p). 2

(14.7.8)

Since (14.7.8) is a consequence of Einstein’s equations, it can be used in place of one of the three equations (14.7.5). In conclusion, we must determine four functions λ(r ), ν(r ), ρ(r ), and p(r ) that solve system (14.7.3), (14.7.5), (14.7.7), (14.7.8). We limit ourselves to solving this system by supposing that  = const.

(14.7.9)

When we accept the Schwarzschild hypothesis (14.7.9), we have no constitutive Eq. (14.7.3), and we have only to solve system (14.7.5), (14.7.7), (14.7.8) in the unknowns λ(r ), ν(r ), and p(r ). It is important to notice that (14.7.9) does not lead to a good stellar model, since a constant mass density is a first approximation only for small stars in which the pressure is not too large. We begin by writing (14.7.7) in the form −e−λ(r ) (λ (r )r − 1) − 1 = −c2 χr 2 , so that

−(e−λ(r ) r ) + 1 = c2 χρr 2 .

Integrating this equation yields r −e

−λ(r )



1 2 r3 r =2 c χ 2 3

and finally we can write

4 The

reader may refer to the notebook Geometry.nb.

 ≡ 2m(r ),

14.7 Schwarzschild’s Interior Solution

eλ(r ) =

389

1 1 , ≡ 2 2 1 − Ar 2 c χr 1− 3

where A=

(14.7.10)

χc2 ρ . 3

Introducing (14.7.10) into (14.7.5), we have the condition ν (r ) =

χ p(r )r 3 + 2m(r ) , r (r − 2m(r ))

(14.7.11)

which allows us to deduce from the conservation law (14.7.8) the equation p (r ) = (c2  + p(r ))

χ p(r )r 3 + 2m(r ) , 2r (2m(r ) − r )

(14.7.12)

called the Tolman–Hoppenheimer–Volkoff equation. Instead of solving this equation, in the case  = const, we can use (14.7.8) in the form 1 ( p + c2 ) = − ν (r )( p + c2 ), 2 whose integration, by separated variables, gives p + c2  = Be−ν/2 ,

(14.7.13)

where B is an integration constant. It remains to determine the function ν(r ). To find this function we choose the equation obtained by subtracting (14.7.7) from (14.7.5) and taking into account (14.7.13): (14.7.14) χ( p + c2 ) = e−λ (λ + ν ) = χBe−ν/2 . After substituting (14.7.10) into this equation and noticing that λ (r ) = −d(log(1 − Ar 2 ))/d x, we have (14.7.15) eν/2 (2 A − Ar ν + ν /r ) = χB. Equation (14.7.15) is equivalent to 

2(1 − Ar 2 )3/2 eν/2 (1 − Ar 2 )−1/2 = χBr, whose integration gives eν/2 =

χB − D 1 − Ar 2 , 2A

(14.7.16)

390

14 Schwarzschild’s Universe

where D is a new integration constant. In conclusion, the interior metric for ρ = const has the form  

dr 2 χB 2 2 2 2 2 c2 dt 2 , + r (dθ + sin θdϕ ) − 1 − Ar ds 2 = − D 1 − Ar 2 2A (14.7.17) where B and D are constants to be determined. It is important to remark that this solution is meaningless when r2 >

3 1 = 2 . A χc ρ

(14.7.18)

14.8 Matching Interior and Exterior Solutions We can determine the constants A, B, and D that appear in the interior Schwarzschild metric (14.7.17) by imposing the matching conditions, that is, the continuity of metric coefficients and their first derivatives across the boundary ∂ of the universe tube  = S ×  containing the static fluid F that generates the gravitational field. More explicitly, we want to determine these constants in terms of the constant mass density  and the radius r1 of the sphere S containing the fluid. From (14.6.11) to (14.7.17) we obtain the equality 1 1 = , 2m 1 − Ar12 1− r1 which leads to 2m = Ar13 .

(14.8.1)

However, it is an easy exercise to verify that at r = r1 , the derivatives with respect to r of the first coefficients of the interior and exterior metrics are not equal. Consequently, we cannot satisfy the matching conditions on the surface ∂. This result tells us that the coordinates (r, θ, ϕ, t) we adopted in the spacetime V4 are incompatible with the atlas that defines the differential class of V4 . Taking into account the results of Sect. 14.2, we must impose the matching conditions in a Gaussian system of coordinates relative to ∂. This can be obtained by operating two different coordinate transformations inside and outside  that lead to the same system of Gaussian coordinates relative to ∂. Since this approach is very tedious to apply, we adopt an equivalent way to match the interior and exterior solutions, resorting to Theorem 14.3. To apply this theorem we first notice that the surface ∂ is generated by the world lines of the particles of F and that from (14.7.13), (14.7.16) we obtain

14.8 Matching Interior and Exterior Solutions

p = Be

−ν/2

391

√ 1 3AD 1 − Ar 2 − χB/2 − ρc = . √ χ χB/2 A − D 1 − Ar 2 2

(14.8.2)

We determine the integration constants A, B, and D by imposing the continuity across ∂ of eλ(r ) , eν(r ) and the condition p = 0 on ∂: 2m 1 − Ar12 = 1 − , r1  2  2m 2 χB/2 A − D 1 − Ar1 = 1 − , r1  3AD 1 − Ar12 = χB/2.

(14.8.3) (14.8.4) (14.8.5)

The solution of this system is hM 1 3 Ar = 2 , 2 1 c   1 χB = 3A 1 − Ar12 = χ 1 − χc2 r12 , 3 1 D= . 2 m=

(14.8.6) (14.8.7) (14.8.8)

In conclusion, the exterior and interior Schwarzschild solutions are respectively given by the following formulas:   dr 2 2m 2 2 c dt , + r 2 (dθ2 + sin2 θdϕ2 ) − 1 − 2m r 1− r 2 dr ds 2 = + r 2 (dθ2 + sin2 θdϕ2 )− 1 − Ar 2    3 1 1 − Ar12 − 1 − Ar 2 c2 dt 2 , 2 2 ds 2 =

where  = const,

2m = Ar13 ,

A=

1 χc2 . 3

(14.8.9)

(14.8.10)

(14.8.11)

Finally, the pressure inside the fluid is √ √ 2 2 A 1 − Ar 2 − 1 − Ar1  p= , χ 3 1 − Ar 2 − √1 − Ar 2 1

and the interior space metric is

(14.8.12)

392

14 Schwarzschild’s Universe

dσ 2 =

dr 2 + r 2 (dθ2 + sin2 θdϕ2 ). 1 − Ar 2

(14.8.13)

The Schwarzschild universe model we have obtained can be applied to the Sun, since for this star we have r1 = 6.95 × 108 m and

rS  10−6 < 1. r1

Recalling the values of the gravitational constant h, the velocity of light in vacuum, and the solar mass density ρ, h = 6.67 10−11

m3 m Kg , c = 3 108 ,  = 1.4 103 3 , 2 Kg s s m

from (12.4.14), (14.7.10), (14.8.1) we obtain 2 8πh −43 s , = 2.07 10 c2 Kg m 1 χc2  = 8.7 10−24 2 , A= 3 m 2m = Ar13  2.86 103 m.

χ=

(14.8.14) (14.8.15) (14.8.16)

We are now in a position to evaluate how the solar mass modifies the geometry of space. To this end, we consider the coefficients grr of the metrics (14.8.9) and (14.8.10) that determine the deviation between Schwarschild’s space geometry and Euclidean geometry. First, we write these coefficients in the form 1 1 = , 2m 2m 1− 1− r r1 (r/r 1) 1 1 = , grr ≡ 2 1 − Ar r2 1 − (Ar12 ) 2 r1 grr ≡

where r1 is the solar radius. Then Fig. 14.2 shows the difference grr − 1 versus r/r1 inside and outside the Sun. Note that if we adopt the variable r , the metric coefficient grr is continuous across the solar surface, but its derivative ∂grr /∂r is discontinuous, in agreement with the result we obtained at the beginning of this section.

14.9 Physical Remarks About Schwarzschild’s Solution

393

Fig. 14.2 grr − 1 versus r/r1

14.9 Physical Remarks About Schwarzschild’s Solution All the results of the previous sections were obtained by adopting in the spacetime V4 the coordinates (r, θ, ϕ, t). In this section we try to understand whether these coordinates have a physical meaning, that is, whether there is an experimental procedure to associate the value of these coordinates to any event of V4 . We recall that the coordinates (r, θ, ϕ) have a physical meaning only in the Euclidean space 3 , to which the space V3 of the product manifold V4 = V3 ×  is diffeomorphic. As a consequence of this diffeomorphism, they are still coordinates in V3 . However, for an observer in this space, the coordinates (r, θ, ϕ) do not retain the same physical meaning they have inside 3 . We wish to discover how these coordinates can be determined by the observer in V3 . The variable r does not represent the distance of a generic point (r, θ, ϕ) from the origin of spherical coordinates as happens in 3 . In this regard, we have already noted in Sect. 14.5 that r is a variable such that 4πr 2 denotes the area A of a sphere of radius r . It is often claimed that this property allows one to determine r experimentally by the measure of the area of the spherical surface S of radius r . But how can we single out the points of S before knowing a way to measure r ? It is important to emphasize that in general relativity we can use arbitrary coordinates, provided that we know how they are related to other coordinates with a physical meaning. With the intent to determine new physical coordinates and their relationship to (r, θ, ϕ), consider the curve r = ξ, with 0 < ξ < r , θ = const, ϕ = const, where r ≤ r1 , and r1 is the value of r corresponding to the points of the sphere S containing the fluid F in V3 . If we adopt in S the space metric (14.8.13), then the arc length l(r ) of this curve is given by

394

14 Schwarzschild’s Universe

Fig. 14.3 (l − r )/r1 versus l/r1 inside S

l(r ) =

r



r

dσ =

0

0

1 1 − Aξ 2

dξ =

√ arcsin( Ar ) , √ A

(14.9.1)

where l(r ) is measured with the rulers of the observer operating inside S. From (14.9.1) we easily obtain the function √ sin Al , r (l) = √ A

(14.9.2)

whose nondimensional form is shown in Fig. 14.3. Similar results can be obtained with more difficult computations starting from the space metric (14.6.14). It remains to show that the coordinates (θ, ϕ) can also be determined experimentally for every point of V3 . Before doing so, we wish to highlight an important property of the coordinates (l, θ, ϕ). In fact, on adopting them, the metrics (14.6.13) and (14.6.14) assume the following form:   2m c2 dt 2 , ds = dl + r (l)(dθ + sin θdϕ ) − 1 − r (l)

(14.9.3)

dσ 2 = dl 2 + r 2 (l)(dθ2 + sin2 θdϕ2 ).

(14.9.4)

2

2

2

2

2

2

Comparing the preceding metrics with (14.1.11), we can state that: • The coordinates (l, θ, ϕ, ct) are Gaussian coordinates relative to the boundary ∂ of the universe tube  containing the world lines of fluid F. Further, the curve l = var and θ, ϕ, t constant are geodesics of the metric (14.9.3) that are orthogonal to ∂.

14.9 Physical Remarks About Schwarzschild’s Solution

395

• The coordinates (l, θ, ϕ) are Gaussian coordinates relative to the spherical surface S containing the fluid F. Further, the curve l = var and θ, ϕ constant are geodesics of the metric (14.9.4) that are orthogonal to ∂ S. Further, they are the space projections of the geodesics orthogonal to ∂. Denote by l a sphere of V3 with it center at the center O of S and radius l ∈ [0, ∞). The metric on l is obtained from (14.9.4) with dl = 0: dσ 2 = r 2 (l)(dθ2 + sin2 θdϕ2 ).

(14.9.5)

Consequently, the metric tensor on l is   2 r 0 , g= 0 r 2 sin2 θ and the area a(l) of ∂r is given by

π

a(l) = r 2 (l) 0



2π

sin θdθ

dϕ = 4πr 2 (l).

0

Consider the 2-dimensional submanifolds θ,0 and 0,ϕ of V3 with parametric equations ϕ = 0 and θ = 0, respectively. Then θ,0 intersects the sphere l along a circumference γθ,0 with parametric equations l = const, θ ∈ [0, π], and ϕ = 0. The length σ of an arc of γθ,0 , with θ ∈ [0, θ], is given by the formula σ = r (l)θ,

(14.9.6)

which shows how we can evaluate θ by measuring lengths. Similar considerations hold also for the angle ϕ. We now analyze the behavior of the clocks measuring the coordinate time t. From (14.6.4) to (14.6.6) it follows that the equations of a geodesic, for which θ and ϕ are constant, reduce to   2m c2 dt 2 m 1 − mr 2 (s) r (s)  + r

(s) = 0, + (14.9.7) − 2m 2 r 2 (s) 1− r (s) r 2mr (s)

− (14.9.8) t (s) + t

(s) = 0, r (2m − r ) where s is a canonical parameter along the geodesic. If we require that this geodesic also be lightlike, then the following condition must be added:

396

14 Schwarzschild’s Universe

 1−

2m r



1 r 2 (s) = 0. 2m 1− r (s)

c2 t 2 (s) −

From (14.9.7) to (14.9.9) we get

r

(s) = 0,

(14.9.9)

(14.9.10)

so that r can be adopted as a canonical parameter. In conclusion, the lightlike geodesics along which θ and ϕ are constant are given by the equations r = var,

    2m 2 dt 2 c − 1− r dr

1 = 0. 2m 1− r

(14.9.11)

We are now in a position to recognize the physical meaning of the time coordinate t. Denote by R the radius beyond which the gravitational effects on the clock’s behavior are no longer appreciable. All the clocks in the region W , with r > R, can be synchronized by the procedure of special relativity. Suppose that we want to synchronize a clock a at the point A of the region of V3 ata distance ra < R from the origin O. Let b be a clock at a point B belonging to W γ, where γ is the curve r = var, θ, ϕ = const, containing the points O and A. Consider a light ray starting from B, when the clock at this point measures the time t B = 0, and moving along the curve γ. The propagation of this ray satisfies the differential condition (14.9.11)2 , which is equivalent to the following one: dt =

1 c

dr . 2m 1− r

(14.9.12)

In conclusion, the clock at A is synchronous with the clock at B if a light pulse starting at t B = 0 from B along γ arrives at A at the coordinate time t, t A = tB +

1 c



rA

rB

dr . 2m 1− r

(14.9.13)

In view of the spherical symmetry, this procedure does not depend on θ and ϕ.

14.10 Planetary Orbits in a Schwarzschild Field In this section we analyze the motion of a planet acted on by the gravitational field produced by the Sun, assuming that this field is described by Schwarzschild metric coefficients. The solar center is assumed to be coincident with the center of spherical

14.10 Planetary Orbits in a Schwarzschild Field

397

coordinates introduced in Sect. 14.2. Further, the planet P is considered a material point whose mass is negligible compared to the solar mass. Under these conditions, the contribution of P to the solar gravitational field can be neglected. From the axioms of general relativity it follows that the possible world lines of P are spacelike geodesics of the Schwarzschild metric. In view of (14.6.6), the equations of these geodesics are d 2r 1 + 11 dτ 2



dr dτ



2 1 + 22

dθ dτ



2 1 + 33

dϕ dτ



2 1 + 44

dt dτ

2 = 0,

 2 dϕ d 2θ 2 dr dθ 2 +  + 2 = 0, 12 33 2 dτ dτ dτ dτ d 2ϕ 3 dr dϕ 3 dϕ dθ + 213 + 223 = 0, dτ 2 dτ dτ dτ dτ dt 2 4 dr dt = 0. + 214 2 dτ dτ dτ

(14.10.1)

On the other hand, since eλ(r ) =

1 2m , , eν(r ) = 1 − r 1 − 2m r

(14.10.2)

we have also 

1 λ(r ) = log 1 − 2m r



  2m , , ν(r ) = log 1 − r

(14.10.3)

and the Christoffel symbols assume the following form: m , r (r − 2m) m = 3 (r − 2m), r 1 = , r

1 11 =−

1 22 = −r (r − 2m),

1 44

2 12 =

3 13

1 , r

3 23 = cot θ,

1 33 = −(r − 2m) sin2 θ, 2 33 = − sin θ cos θ, 4 14 =

Finally, the equations of the geodesics (14.10.1) become

m . r (r − 2m)

(14.10.4)

398

14 Schwarzschild’s Universe

 2 dr d 2r m − 2 dτ r (r − 2m) dτ       2 dθ 2 dϕ mc2 dt 2 2 − (r − 2m) + sin θ − 3 = 0, dτ dτ r dτ  2 dϕ 2 dr dθ d 2θ + = 0, − sin θ cos θ dτ 2 r dτ dτ dτ d 2 ϕ 2 dr dϕ dθ dϕ + + 2 cot θ = 0, 2 dτ r dτ dτ dτ dτ dr dt 2m d 2t = 0. + dτ 2 r (r − 2m) dτ dτ

(14.10.5)

Recalling that the 4-velocity U α = (d x α /dτ ) is tangent to the geodesic described by 4

the planet P and satisfies the condition gαβ U α U β = gαβ (U α )2 = −c2 , we obtain i=1

the first integral r r − 2m



dr dτ

2



dθ dτ

2



dϕ +r + r sin θ dτ  2 r − 2m 2 dt c − = −c2 . r dτ 2

2

2

2

(14.10.6)

Suppose now that initially planet P belongs to the submanifold θ = π/2 and has initial velocity tangent to this submanifold. Then at the initial instant we have dθ/dτ = 0, and we at once verify that (14.10.5)2 admits the solution θ = π/2. In other words, if the planet is initially moving in the plane that satisfies these initial conditions, then its motion will continue in such a plane. When we set θ = π/2 in (14.10.5) and take into account (14.10.6), instead of (14.10.5)1 , we obtain r r − 2m



 2   2 dϕ dr 2 dt c2 + r2 − (r − 2m) = −c2 , dτ dτ r dτ d 2 ϕ 2 dr dϕ = 0, + dτ 2 r dτ dτ d 2t dr dt 2m = 0. + 2 dτ r (r − 2m) dτ dτ

(14.10.7)

On introducing the notation w = dϕ/dt, v = dt/dτ , we see that (14.10.7)2,3 assumes the form

14.10 Planetary Orbits in a Schwarzschild Field

399

dw 2 dr + w = 0, dτ r dτ dv 2m dr + v = 0, dτ r (r − 2m) dτ so that we have d  2  r w = 0, dτ

d dτ



 r − 2m v = 0. r

From these equations we derive α dϕ = 2, dτ r βr dt = , v= dτ r − 2m

w=

(14.10.8)

where α and β are integration constants. Introducing (14.10.8) into (14.10.5)1 , we have  2 dr r − 2m 2 2mc2 2 2 2 . (14.10.9) + α − c β = −c + dτ r3 r Further, from (14.10.8)1 it follows that d d dϕ α d = = 2 , dτ dϕ dτ r dϕ and (14.10.9) becomes 

α dr r 2 dϕ

2 +

2mα2 α2 2mc2 2 2 2 + = −c + c β + . r2 r r3

Putting u = 1/r , this equation can also be written as follows: 

du dϕ

2 + u2 = −

c2 c2 β 2 2mc2 + + u + 2mu 3 . α2 α2 α2

Finally, we give this equation the form 

du dϕ

2 = A + Bu − u 2 + 2mu 3 ,

(14.10.10)

where the identification of the constants A and B is evident. It is easy to recognize that the last term on the right-hand side of (14.10.10) is much smaller than the others. Neglecting this term, we obtain the Newtonian equation

400

14 Schwarzschild’s Universe



du dϕ

2 = A + Bu − u 2 ≥ 0.

(14.10.11)

The maximum and minimum values of u, corresponding to the perihelion and aphelion respectively (since u = 1/r ), are determined by computing the roots of the equation (14.10.12) A + Bu − u 2 = 0, which admits two positive real roots when the orbit of P is bounded. Denoting by u 1 < u 2 the two roots, Eq. (14.10.11) can also be written as 

du dϕ

2 = −(u − u 1 )(u − u 2 ),

(14.10.13)

whose right-hand side is positive only if u 1 < u < u 2 , in agreement with the fact that the orbit is bounded. Further, the variation of ϕ as u varies inside the interval (u 1 , u 2 ) is given by the integral [ϕ] =

⎤u 2 1 (u u − + u ) 1 2 du ⎥ ⎢ 2 = ⎣arcsin √ ⎦ = π. (14.10.14) 1 (u − u 1 )(u 2 − u) (u 2 − u 1 ) 2 u1 ⎡

u2 u1

Therefore, the variation of ϕ between two successive perihelia is equal to 2π, and the orbit is closed. If in (14.10.10) we don’t neglect the 2mu 3 term but but still suppose that 2mu is smaller than 1, then the third equation, A + Bu − u 2 + 2mu 3 = 0,

(14.10.15)

admits three roots u 1 , u 2 , u 3 , two of which, for example u 1 and u 2 , have values very close to the values of the roots of (14.10.11). In such a case, since the sum of the roots satisfy the equality 1 , (14.10.16) u1 + u2 + u3 = 2m u 3 must also be real. Writing (14.10.9) as 

du dϕ

2 = 2m(u − u 1 )(u − u 2 )(u − u 3 ) > 0,

(14.10.17)

we can state that the right-hand side of (14.10.17) is positive for u ∈ [u 1 , u 2 ], provided that u 3 > u 2 . From (14.10.17) to (14.10.15) it follows that

14.10 Planetary Orbits in a Schwarzschild Field



du dϕ

401

2 = (u − u 1 )(u − u 2 )(2mu − 2mu 3 )  = (u − u 1 )(u 2 − u) [1 − 2m(u 1 + u 2 )] · 1 −

 2mu , 1 − 2m(u 1 + u 2 )

and separating variables yields du 1 dϕ = √  √ (u − u 1 )(u 2 − u) 1 + 2m(u 1 + u 2 ) 1−

1 2mu 1 − 2(u 1 + u 2 )

.

Substituting the second and third factors on the right-hand side with the first-order terms of their Taylor expansions, we obtain the following approximate equation: dϕ = √

   du 2mu 2m(u 1 + u 2 ) 1+ . 1+ 2 2 (u − u 1 )(u 2 − u)

Integrating between u 1 and u 2 and considering the variation of ϕ between two successive perihelia, we obtain

1 + 2mu/2 du = (u − u 1 )(u 2 − u) u1   2m = [2 + 2m(u 1 + u 2 )] 1 + (u 1 + u 2 ) 4 6πm  2π + (u 1 + u 2 ). 2

2[ϕ] = [2 + 2m(u 1 + u 2 )]

u2

√

(14.10.18)

If we denote by a the semimajor axis of the Newtonian elliptic orbit and by e its eccentricity, then we have r1 =

1 1 = a(1 + ), r2 = = a(1 − ), u1 u2

and (14.10.18) can also be written as follows: ϕ =

6πm . a(1 − 2 )

(14.10.19)

When this formula is applied to Mercury, we get an advance of the perihelion equal to 42.9 seconds per century, which is in good agreement with the observed difference between the experimental value and the value obtained by Newton’s theory of gravitation, after taking into account the effect on the motion of P due to the presence of the other planets (Hamiltonian perturbation theory) (Fig. 14.4).

402

14 Schwarzschild’s Universe

Fig. 14.4 Precession of the orbit

14.11 Gravitational Deflection of Light In this section we analyze the path of a light ray R in the solar gravitational field, assuming again that this field is described by Schwarzschild metric coefficients. The solar center is assumed to be coincident with the center of spherical coordinates introduced in Sect. 14.2. From the axioms of general relativity it follows that the possible world lines of R are lightlike geodesics of the Schwarzschild metric. Owing to the character of these geodesics, we must use an affine parameter λ along the geodesics instead of the proper time τ . Then in view of (14.6.5), the equations of the geodesics are  2 dr m d 2r − 2 dλ r (r − 2m) dλ   2   2 2  dθ 2 dt dϕ mc − (r − 2m) + sin2 θ − 3 = 0, dλ dλ r dλ  2 dϕ 2 dr dθ d 2θ − sin θ cos θ + = 0, 2 dλ r dλ dλ dλ d 2 ϕ 2 dr dϕ dθ dϕ + 2c tan θ = 0, + 2 dλ r dλ dλ dλ dλ d 2t dr dt 2m + = 0. dλ2 r (r − 2m) dλ dλ

(14.11.1)

To select the lightlike geodesics, we must impose the condition r r − 2m



dr dλ

2



dθ dλ

2



dϕ +r + r sin θ dλ  2 r − 2m 2 dt c − = 0. r dλ 2

2

2

2

(14.11.2)

14.11 Gravitational Deflection of Light

403

Suppose now that a point P of a light ray belongs to the submanifold θ = π/2 and its tangent vector at P is tangent to this submanifold M. Then at the initial instant, we have dθ/dλ = 0, and we at once verify that (14.11.1)2 admits the solution θ = π/2. In other words, if the light ray lies initially on M, then the whole ray lies on M. When we set θ = π/2 in (14.11.1) and take into account (14.11.2), instead of (14.11.1)1 , we obtain r r − 2m



 2   2 dϕ dr 2 dt c2 + r2 − (r − 2m) = 0, dλ dλ r dλ d 2 ϕ 2 dr dϕ = 0, + dλ2 r dλ dλ d 2t dr dt 2m = 0. + 2 dλ r (r − 2m) dλ dλ

(14.11.3)

On introducing the notation w = dϕ/dλ, v = dt/dλ, Eqs. (14.11.3)2,3 assume the form dw 2 dr + w = 0, dλ r dλ

dr dv 2m + v = 0, dλ r (r − 2m) dλ

so that d  2  r w = 0, dλ

d dλ



 r − 2m v = 0. r

From these equations we derive α dϕ = 2, dλ r βr dt = , v= dλ r − 2m

w=

(14.11.4)

where α and β are integration constants. Introducing (14.11.4) into (14.11.3)1 , we have  2 dr r − 2m 2 + α − c2 β 2 = 0. (14.11.5) dλ r3 Further, from (14.11.4)1 it follows that d d dϕ α d = = 2 , dλ dϕ dλ r dϕ and (14.11.5) becomes

404

14 Schwarzschild’s Universe

Fig. 14.5 Gravitational deflection of light



α dr r 2 dϕ

2 +

α2 2mα2 = c2 β 2 + . 2 r r3

Putting u = 1/r , this equation can also be written as follows: 

du dϕ

2 + u2 =

c2 β 2 + 2mu 3 . α2

Finally, we give this equation the form 

du dϕ

2 = A − u 2 + 2mu 3 ,

where A=

(14.11.6)

c2 β 2 . α2

Neglecting the last term on the right-hand side of (14.11.5), since it is much smaller than the others,5 we obtain the equation 

du dϕ

2 = A − u 2 ≥ 0,

(14.11.7)

√ √ so that u ∈ [− A, A]. Now consider a light ray coming from infinity (u = 0) along the direction ϕ = 0. Integrating (14.11.7) and adopting the notation (Fig. 14.5) 1 D=√ , A

we obtain

u

ϕ=

∓

0

i.e., u= 5 In

Ddξ 1 − (Dξ)2

(14.11.8)

= arcsin(Du),

1 D sin ϕ, r = . D sin ϕ

(14.11.9)

other words, we are considering the propagation of light in the absence of gravitation.

14.11 Gravitational Deflection of Light

405

We can state that in the adopted coordinates the path of the light ray is a straight line that passes the solar origin at a distance D for ϕ = π/2 and goes at infinity for ϕ = π. In order to obtain an approximate solution of (14.11.6), we give it the form du 1 1 − σ2 , = dϕ D

(14.11.10)

where we have introduced the new variable √ σ = Du 1 − 2mu.

(14.11.11)

Since 2mu is a small quantity, to first order we have σ = Du (1 − mu) , Finally, we have

 mσ  . Du = σ (1 + mu) = σ 1 + D

(14.11.12)

  2mσ Ddu = dσ 1 + , D

and from (14.11.10) we obtain

σ D dξ 1 + 2mξ/D

ϕ= = dξ √ 2 1−σ 1 − ξ2 0 0 2m 2m . 1 − σ2 + = arcsin σ − D D u

(14.11.13)

In view of (14.11.10), the maximum value u M of u, that is, the value for closest approach of the ray to the Sun, corresponds to σ = 1. Then the corresponding value ϕ M of the angle ϕ is π 2m . (14.11.14) ϕM = + 2 D We can conclude that the curve (14.11.13) is represented by a slightly curved line with a total deflection  π  4m = . (14.11.15) ψ = 2 ϕM − 2 D This represents the deflection of a light ray in the solar field as noted by an observer on Earth, which can be regarded as infinitely far from the Sun.

406

14 Schwarzschild’s Universe

14.12 Gravitational Shift of Spectral Lines One of the most crucial confirmations of general relativity is the shift of spectral lines. This effect is a direct consequence of the equivalence principle. Consider a light source s at rest at a point q1 belonging to the three-dimensional space V3 of Schwarzschild’s universe V3 ×  and denote by ν0 the proper frequency of the light pulses emitted by s, that is, the number of light waves emitted in the unit proper time τ 0 at q1 . The frequency ν for the coordinated clock associated to q1 , according to the formula

(14.12.1) dτ 0 = −g44 (q1 )dt, is

√ ν1 = ν0 / −g44 .

(14.12.2)

Since the metric coefficients are independent of time, the number of waves that reach an observer located at q2 ∈ V3 , in the coordinate time scale t, is independent of time and equal to the number of emitted waves. The frequency ν20 measured with the proper clock at q2 is  ν20 = ν0

g44 (q1 ) , g44 (q2 )

(14.12.3)

since a√ time interval of a proper clock at q2 corresponds to a coordinate time interval  t = g44 (q2 ). From (14.12.1), (14.12.2) we have  ν = ν0

g44 (q1 ) , g44 (q2 )

(14.12.4)

and the observed frequency differs from the proper frequency ν 0 by the amount  ν = ν − ν0,  ν g44 (q1 ) − 1. (14.12.5) = ν0 g44 (q1) The right-hand side of (14.12.5) may be written as  2m      1− 1 2m 1 1 2m 1 r1 1+ − 1  −m , (14.12.6) −1 1− −  2 r1 2 r2 r1 r2 2m 1− r2 because in the exterior Schwarzschild metric, g44 = − (1 − 2m/r ). Taking into account that m = hM/c2 , (14.12.5) becomes

14.12 Gravitational Shift of Spectral Lines

ν hM =− 2 ν0 c

407



1 1 − r1 r2

 .

(14.12.7)

If in (14.12.4), r1 is the solar radius and r2 the Sun–Earth distance, then we obtain the shift of a spectral line emitted by an atom on the solar surface with respect to an atom on the terrestrial surface. From a rigorous point of view, these values of r are not equal to the distance values computed with ordinary astronomical methods, which are based on the assumptions that the space is Euclidean and the light rays propagate along straight lines. However, using the astronomical values instead of the correct ones, the differences are sufficiently small, and the errors are of greater order with respect to the effect (14.12.4) to be observed. Substituting numerical values into (14.12.7), we obtain ν = −2.12 10−6 . ν0 Since the minus sign means that the shift is toward the red, it follows that the observed spectral line should be displaced toward the red. This effect is in a good agreement with experimental results.

Chapter 15

Schwarzschild’s Solution and Black Holes

In Chap. 14 we showed that the exterior Schwarzschild solution of Einstein’s equation describes the gravitational field outside a static spherical mass with constant mass density. This solution has the peculiar characteristic that some components of the metric tensor became infinite at the Schwarzschild radius rs = 2m and at r = 0. The nature and meaning of this behavior was not quite understood at the time of its discovery (1916), although the bending of light and the redshift of clocks in a gravitational field were phenomena already acquired. At first it was suspected that the strange features of the Schwarzschild solution were pathological artifacts due to the imposed symmetry conditions. In 1924, Arthur Eddington showed that the mathematical singularity at rs disappeared after a change of coordinates. Then in 1933 Georges Lemaître realized that the metric singularity at the Schwarzschild radius had no physical meaning [75]. Passing to coordinates that eliminate this pathology, we have that a light ray that is inside a sphere of radius rs cannot escape and that any object that falls inside this sphere is constrained to fall until diverging tidal forces split it apart. This chapter is devoted to the study of these singularities and to the analysis of the physical object described by the Schwarzschild exterior solution when we extend it to the whole space except the origin r = 0. Then the Kerr exterior solution of Einstein’s equations is presented, showing that it is a candidate to describe the gravitational field outside a rotating black hole. Finally, some more recent results about black holes are recalled.

15.1 On the Singularity of Schwarzschild’s Exterior Solution In the attempt to extend the Newtonian star evolution theory to relativity, both the self-gravitating collapsing body and the need to take into account quantum effects emerged. For example, Subrahmanyan Chandrasekhar [174] proved that a nonrotat© Springer Nature Switzerland AG 2019 A. Romano and M. Mango Furnari, The Physical and Mathematical Foundations of the Theory of Relativity, https://doi.org/10.1007/978-3-030-27237-1_15

409

410

15 Schwarzschild’s Solution and Black Holes

ing body of electron-degenerate matter above the limit mass of 1.4 M (called the Chandrasekhar limit) has no stable solution. Chandrasekhar’s arguments were criticized by many of his contemporaries, like Eddington and Landau, because a white dwarf slightly more massive than the Chandrasekhar limit will collapse into a neutron star, which is itself stable. Further, in 1939, Robert Oppenheimer [118] predicted that neutron stars with a mass above some limit1 (the Tolman–Oppenheimer–Volkoff limit) would collapse further. Oppenheimer and his coauthors interpreted the singular spherical surface of radius rs as indicating the boundary of a bubble in which time stopped. Oppenheimer concluded that no physical law was able to stop at least some stars from collapsing to black holes [148]. In 1958, David Finkelstein identified the Schwarzschild surface as an event horizon, “a perfect unidirectional membrane: causal influences can cross it in only one direction” [56]. This did not strictly contradict Oppenheimer’s results, but extended them to include the point of view of infalling observers. Finkelstein’s solution extended the Schwarzschild solution to describe the future of observers falling into a black hole. In spite of this invisible interior region, the presence of a black hole can be inferred through its interaction with exterior matter and electromagnetic radiation such as visible light. Matter that falls into a black hole can form an external accretion disk heated by friction, forming some of the brightest objects in the universe. If there are other stars orbiting around a black hole, then their orbits can be used to determine the black hole’s mass and its location. Such observations can be used to exclude possible alternatives such as neutron stars. In this way, astronomers have identified numerous stellar black holes that are possible candidates in binary systems, and established that the core of our Milky Way contains a supermassive black hole of about 4.3 million solar masses. In Sect. 14.5 it was shown that the static, spherically symmetric Schwarzschild exterior metric   1 2m 2 2 2 2 2 2 2 2 c dt (14.6.11) dr + r (dθ + sin dϕ ) − 1 − ds = 2m r 1− r is a solution of Einstein’s equations, at least for r > 2m. In order to obtain a regular solution of Einstein’s equations, we supposed that the static, uniform, and spherically symmetric matter–energy distribution generating this external field was contained in a region of the space V3 of “radius” r1 > 2m. We have from (14.6.11) that the metric coefficients become infinite at r = 0 and r = 2m ≡ rs , an evident sign that something is going wrong. The metric coefficients, of course, are coordinate-dependent quantities, so we should not make too much of their values. It is certainly possible to have a coordinate singularity that results from a breakdown of a specific coordinate system rather than from the underlying manifold. 1 Their

original calculations, based on the Pauli exclusion principle, gave it as 0.7 M ; subsequent consideration of the strong force due to neutron–neutron repulsion raised the estimate to approximately 1.5 M to 3.0 M .

15.1 On the Singularity of Schwarzschild’s Exterior Solution

411

An example occurs at the origin of polar coordinates in the plane, where the metric ds 2 = dr 2 + r 2 d2 becomes degenerate and the component g θθ = r −2 of the inverse metric blows up, even if that point of the plane is no different from any other. Thus it is quite natural to ask whether the singularity at rs = 2m of the exterior solution is linked to the coordinates we are employing or whether it refers to a property of spacetime. In other words, it seems reasonable to pose the question, is it possible to extend the exterior solution inside the radius r ≤ 2m, and if so, what is the nature of spacetime in that region? To answer this question we first note that if r = 2m, then instead of gtt < 0, we have gtt = 0 and lim grr = ∞ (see (14.6.11)). Further, for r = 0, the funcr →2m

tions grr and gtt are undefined, so that the metric exhibits a singularity both at r = 0 and r = 2m. Finally, for 0 < r < 2m, the physical admissibility requires that t and r be intended as time and spatial coordinates, respectively. On eliminating the curve r = 0 and t = var from the spacetime V4 and the surface r = 2m, we obtain a manifold with two disjoint components U1 and U2 , where U1 is defined by (r, t) ∈ (0, 2m) × (−∞, ∞), and U2 by (r, t) ∈ (2m, ∞) × (−∞, ∞). Since the space must be connected, since otherwise, no information can be exchanged between the two components, it is necessary to take in account only one of the above two components, and the obvious choice consists in retaining U2 , equipped with the metric (14.6.11). It would be of interest to find a coordinate transformation such that the metric in these coordinates is everywhere regular except for the origin O, so that V4 − {O}, equipped with this metric, could represent a model of the universe. Equivalently, we can say that a manifold V4 and a regular isometry i : V4 → U2 exist such that U2 is embedded in V4 and this manifold is the extension of U2 . Suppose c = 1 and consider in V4 the coordinate transformation (Eddington– Finkelstein) ⎧ r = r , θ = θ , ϕ = ϕ , ⎪ ⎪  r ⎪ ⎨ −1 , r > 2m, t = t  − 2m log (15.1.1) 2m ⎪  ⎪ r ⎪ ⎩ t = t  − 2m log 1 − , r < 2m, 2m whose inverse transformation is ⎧  r = r, θ = θ, ϕ = ϕ, ⎪ ⎪  ⎪ ⎨  r −1 , r > 2m, t = t + 2m log 2m ⎪  ⎪ ⎪ ⎩ t  = t + 2m log 1 − r , r < 2m. 2m

(15.1.2)

The above transformations are defined for every value of r = 2m, and for r > 2m, the transformation (15.1.2) defines a map i : U2 → V4 . Further, the coordinates r  , θ , and ϕ have the same physical meaning that we highlighted in Sect. 14.6, while

412

15 Schwarzschild’s Solution and Black Holes

coordinate t  is defined in terms of t and r , whose measure procedure was described in the same section. To determine the form of Schwarzschild’s exterior metric (14.6.11) under the transformation (15.1.1), we begin by differentiating (15.1.1)2,3 to obtain, both for r > 2m and r < 2m, 2m dt = dt  + dr, (15.1.3) 2m − r so that ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

4m 2 4m dt 2 = dt 2 + dr dt  , dr 2 + 2 (2m − r ) 2m − r     2m 2m 4m 2 4m dt 2 = − 1 − dt 2 + dr 2 + dr dt  , − 1− r r r (2m − r ) r ⎪ ⎪ ⎪ ⎪ 1 2m 4m 2 ⎪ ⎪ =1+ . + ⎪ ⎪ ⎪ r (2m − r ) r ⎩ 1 − 2m r (15.1.4) Introducing (15.1.4) into (14.6.11), we obtain the form of Schwarzschild’s metric ds 2 =

 

 4m  2m dr 2 + r 2 dθ2 + sin2 θdϕ2 + 1+ dt dr r r   2m dt 2 − 1− r

(15.1.5)

in the coordinates (r, θ, ϕ, t  ), which we assume to be the metric of the whole spacetime V4 . Thus the metric coefficients of V4 different from zero are   2m g11 = 1 + , g22 = r 2 , g33 = r 2 sin2 θ, r   (15.1.6) 2m 2m , g14 = g41 = . g44 = − 1 − r r Furthermore, this metric is everywhere regular throughout V4 (for r = 0). Resorting to the notebook Geometry.nb, it is possible to verify that Einstein’s exterior equations are satisfied by the metric (15.1.5). Now we want to analyze the lightlike geodesics of the metric (15.1.5). Due to the symmetry of the space metric, we limit the following analysis to the submanifold V2 defined by θ =const, ϕ =const, that is, to the submanifold (r, t  ) ∈ (0, ∞) × (−∞, ∞) equipped with the metric 

2m ds = 1 + r 2



  4m  2m dr + dt dr − 1 − dt 2 . r r 2

(15.1.7)

15.1 On the Singularity of Schwarzschild’s Exterior Solution

413

Again using the notebook Geometry.nb, we obtain the equations of the geodesics of V2 relative to the metric (15.1.7): ⎧   2  dt ⎪ 2 dt ⎪ ⎪ = 0, ⎨ m(2m + r ) + 4m dr + m(2m − r ) dr   2 ⎪ dt d 2t  dt  ⎪ ⎪ ⎩ 2m(m + r ) + 2m(2m + r ) + r 3 2 = 0. + 2m 2 dr dr dr

(15.1.8)

To select the lightlike geodesics among all the geodesics (15.1.8), we must impose the condition ds 2 = 0, which can also be written as follows:      2  4m dt  2m dt 2m + − 1− 1+ = 0, r r dr r dr

(15.1.9)

where we are adopting r as an affine parameter. Noting that (15.1.9) is a seconddegree algebraic equation in the unknown dt  /dr , it is easy to verify that we have dt  = −1 dr

dt  r + 2m = . dr r − 2m

(15.1.10)

The families of lightlike geodesics of V2 can be obtained by integrating (15.1.10). The first family is formed by the straight lines t  = −r + C,

(15.1.11)

where C is an integration constant. The second family is obtained by integrating (15.1.10)2 : r > 2m, t  = r + 4m log(r − 2m) + C, (15.1.12) t  = r + 4m log(2m − r ) + C, r < 2m, where C denotes an arbitrary constant. It is evident that the straight line r = 2m is an asymptote for all the curves (15.1.12). Introducing (15.1.10) into (15.1.8), we can verify that they are identically satisfied. In conclusion, all the lightlike curves are lightlike geodesics of V2 , that is, possible trajectories of photons. Recall that material particles move along timelike geodesics, that is, along world lines whose tangent vectors at every point are interior or tangent to the light cone at that point. Figure 15.1 shows that a photon that at a given instant is inside the surface r = 2m cannot cross this surface and reach the exterior region r > 2m. Furthermore, it will reach the line r = 0 in a finite coordinate (or proper) time. The fact that no particle can cross the surface r = 2m toward the region r > 2m means that no observer P located in this region can receive information about the events happening in the interior region r < 2m. For this reason, we say that the surface r = 2m represents the event horizon for all the observers in the region r > 2m.

414

15 Schwarzschild’s Solution and Black Holes

Fig. 15.1 Trajectories of photons near the event horizon

The event horizon is characterized by the following simple geometric property: the vectors orthogonal to this surface have norm equal to zero. In fact, a normal to the surface r = rs is defined by the equation f (x α ) = r − 2m = 0, so that it is proportional to the vector n α = f ,α = (1, 0, 0, 0). On the other hand, the contravariant components of the metric tensor are 2m , r 1 = 2 2 , r sin θ

g 11 = 1 − g 33

2m g 14 = g 41 = − , r   2m , g 44 = − 1 − r

g 22 =

1 , r2

while all the other components vanish. Consequently, g αβ n α n β = g 11 , g 11 = 0 when r = 2m, and the norm of the vector n α vanishes at r = 2m. Concluding this section, we observe that Schwarzschild’s solution exhibits a mathematical singularity only at r = 0, while the singularity at rs must be ascribed to the coordinates we adopted.

15.2 Physical Interpretation of the Event Horizon In order to assign a physical interpretation to the singularities that the metric exhibits at rs , we analyze two different scenarios. First, consider an observer attached to a particle collapsing into the black hole. Consider an observer I freely falling along

15.2 Physical Interpretation of the Event Horizon

415

a spacelike radial geodesic. In view of (14.10.8) and (14.10.9), the equation of this geodesic is dr = ∓ A2 − c2 (1 − 2m/r ), (15.2.1) dτ where A2 = c2 β 2

(15.2.2)

and β is a constant. Then the distance L covered by the observer I in going from the position r0 > 2m to the position 2m is given by

L=

2m

√

r0

Recalling (14.10.8)2 ,

dr . 1 − 2m/r

(15.2.3)

A dt = , dτ c (1 − 2m/r )

(15.2.4)

we find that the coordinate time t spent by I to cover this distance is infinite: t =

A c

2m

r0



dr

(1 − 2m/r ) A2 − c2 (1 − 2m/r )

→ ∞.

(15.2.5)

In other words, the observer I never reaches the surface r = 2m if the process of free fall is evaluated by adopting the coordinate time in the space V3 . Consider now the same scenario but from the point of view of the observer I that during free fall uses clocks that measure the proper time τ . In other words, it observes this phenomenon from far away. Then by (15.2.1), the time τ required to go from r0 to 2m is the finite proper time interval

τ =

2m



r0

dr A2

−

c2 (1

− 2m/r )

.

(15.2.6)

Consider now a photon going from r0 to 2m. In this case, along the radial lightlike geodesic we have 1 dr 2 = c2 (1 − 2m/r ) dt 2 , (15.2.7) 1 − 2m/r so that the coordinate time spent by the photon to cover the distance L is again infinite, since it is given by t =

1 c

2m

r0

dr 1 − 2m/r

→ ∞,

(15.2.8)

416

15 Schwarzschild’s Solution and Black Holes

since the function under the integral sign is not integrable in the interval (0, 2m). We cannot describe the process from the point of view of the photon, since the proper time is meaningless. Thus we have that the description of physics in the presence of the event horizon depends on the choice taken by the observer. In other words, if the observer decides to follow the collapsing matter during its fall up to the event horizon, then he will see it falling into matter of infinitely high density. No restraining force whatsoever has the power to hold him away from this catastrophe once he crosses the critical surface that we called the event horizon. The final collapse occurs in finite time after this surface is crossed, but it is inevitable. At the center of event horizon, as described by general relativity, lies a gravitational singularity, a region where the spacetime curvature becomes infinite. The region delimited by the horizon is called a Schwarzschild black hole. It can be shown that the singular region has zero volume and contains all the mass of the black hole, so that its density is infinite. Observers falling into a nonrotating and uncharged Schwarzschild black hole cannot avoid being carried into the singularity once they cross the horizon. What kind of coordinate-independent signal should we look for as a warning that something about the geometry is out of control? This turns out to be a difficult question to answer. One simple criterion for understanding that something is going wrong is to recognize that the curvature becomes infinite. The curvature is measured by the Riemann tensor, and it is hard to say when a tensor becomes infinite, since its components are coordinate-dependent. But from the curvature we can construct various scalar quantities, and since scalars are coordinate-independent, it is meaningful to say that they become infinite. The simplest such scalar is the Ricci scalar R = g μν Rμν , but we can also construct higher-order scalars such as R μν Rμν , R μνρσ Rμνρσ , Rμνρσ R ρσλτ Rλτ μν , and so on. If any of these scalars (but not necessarily all of them) goes to infinity as we approach some point, we regard that point as a singularity of the curvature. We should also check that the point is not infinitely far away, that is, that it can be reached by traveling a finite distance along a curve. We therefore have a sufficient condition for considering a point to be singular. For our purposes, we will control whether the geodesics are well behaved at the point in question, and if so, we will consider the point nonsingular. In the case of the Schwarzschild metric (14.6.11), direct computation reveals that R μνρσ Rμνρσ =

48m 2 . r6

(15.2.9)

This is enough to convince us that r = 0 represents an honest singularity. In contrast, at rs = 2m, we can suppose that the spacetime has no effective singularity, since at rs = 2m the determinant of the metric coefficients is equal to r 4 sin2 θ, and the invariant associated with the curvature tensor is given by (15.2.9), which is singular only at r = 0.

15.3 Gravitational Collapse

417

15.3 Gravitational Collapse Let S be a spherically symmetric self-gravitating body filled by an ideal gas. A gravitational collapse begins if at a given instant, the gravitational force in S exceeds the pressure, and S contracts until the gravitational force and pressure balance each other. Recall that in Sect. 6.9, we analyzed some proposals to describe the evolution of a spherically symmetric self-gravitating ideal gas S in the context of Newtonian gravitation. Before describing the gravitational collapse in the context of general relativity, we prove that the Schwarzschild metric is also the unique vacuum solution for timedependent matter distribution, provided that spherical symmetry is preserved at each instant. Theorem 15.1 (Birkhoff) Every spherically symmetric vacuum solution of Einstein’s equations is independent of time. Proof A general spherically symmetric metric has the form ds 2 = eλ(r,t) dr 2 + r 2 (dθ2 + sin2 θdϕ2 ) + eν(r,t) c2 dt 2 .

(15.3.1)

Introducing the above metric into Einstein’s equations in vacuum yields (see the notebook Geometry.nb)   ⎧ Rtt = ∂t2 λ + (∂t λ)2 − ∂t ν∂t λ + ⎪ ⎪   ⎪ ⎪ ⎪ 2 ⎪ 2(νλ) 2 2 ⎪ e ∂r ν + (∂r ν) − ∂r ν∂r λ + ∂r ν = 0, ⎪ ⎪ r ⎪ ⎪ ⎪   ⎪ ⎪ 2 ⎪ 2 2 ⎪ ⎪ ⎨ Rrr = − ∂r ν + (∂r ν) − ∂r ν∂r λ − r ∂r λ +   . ⎪ e2(λ−ν) ∂t2 λ + (∂t λ)2 − ∂t ν∂t λ = 0, ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ Rtr = ∂t λ = 0, ⎪ ⎪ r ⎪ ⎪ ⎪ −2λ ⎪ ⎪ R = e [r (∂r λ − ∂r ν) − 1] = 0, θθ ⎪ ⎪ ⎩ Rφφ = Rθθ sin2 θ.

(15.3.2)

Taking the time derivative of Rθθ = 0 and using the result ∂t λ = 0 (see (15.3.2)3 ), we get ∂t ∂r ν = 0, so that we can write λ = λ(r ),

ν(r, t) = f (r ) + g(t).

(15.3.3)

Then the time term in the metric (15.3.1) is −e f (r ) eg(t) dt 2 . But we can always redefine the time coordinate by the substitution dt → e−g(t)/2 dt. In other words, we are free to choose t such that g(t) = 0. Consequently, ν(t, r ) = f (r ), and we have ds 2 = e2λ(r ) dr 2 + r 2 d2 − e2ν(r ) dt 2 ,

(15.3.4)

418

15 Schwarzschild’s Solution and Black Holes

and all the metric components are independent of the coordinate t. The metric (15.3.4) is equal to the Schwarzschild exterior metric we derived in Sect. 14.6,     2G M 2G M −1 2 dt 2 + 1 − ds 2 = − 1 − dr + r 2 d2 . r r and Birkhoff’ s theorem is proved.

(15.3.5) 

An important consequence of Birkhoff’s theorem is that if a spherically symmetric mass distribution, such as a star, evolves while remaining spherically symmetric, then no disturbance can propagate into the surrounding space. This means that a pulsating spherically symmetric star cannot emit gravitational waves. If a spherically symmetric source is restricted to the region r ≤ 2m, then the solution for r > 2m must be the exterior Schwarzschild solution. However, the converse is not true, since a source that gives rise to an exterior Schwarzschild solution is not necessarily spherically symmetric. In general relativity, the stress state inside the star is a source of a further gravitational field, and this leads to a situation that is completely different from the corresponding classical situation. In fact, a detailed discussion of this problem shows that a positive value a f of the final radius after the collapse exists only if the total mass M of the star is smaller than a certain value M0 . For M > M0 , no positive value a f exists for any state equation. Thus in general relativity, a star of mass M > M0 will collapse until all the stellar material is concentrated at the center of symmetry. This situation is called gravitational collapse, and this phenomenon is described by the metric (15.3.5). When the stellar radius a(t) is greater than 2m, the contraction rate will depend on the state equation, and this contraction may eventually stop when a > 2m. Conversely, if a value less than 2m is reached, then we will only have a contraction whose rate cannot be smaller than a certain minimum value determined from the shape of the light cones. However, if the surface of the star reaches the spherical surface r = 2m, then the collapse will be completed in a finite time. Recalling that no material particle or photon crosses the surface r = 2m outward, we note that an observer located in the region r > 2m cannot receive information about what happens in the region a < 2m. We have already said that the collapsed body represents a black hole. General relativity forecasts that a star with enough mass always collapses into a black hole. To describe the gravitational collapse inside the event horizon (black hole), we should first recall the static configurations inside spherically symmetric stars. In Sect. 14.7 we studied the Schwarzschild interior solution for a static perfect fluid filling a sphere S of V3 under the assumption that = const. In this case, we need no constitutive equation. Furthermore, we proved the Tolman–Hoppenheimer–Volkoff equation

15.3 Gravitational Collapse

419

p  (r ) = (c2 + p(r ))

χ p(r )r 3 + 2m(r ) , 2r (2m(r ) − r )

(14.7.12)

where m(r ) is related to (r ) as follows: dm = 4πr 2 dr

⇒

m(r ) =

4 π r 3 . 3

(15.3.6)

Integrating (14.7.12), we obtain the pressure inside the fluid √ √ 2c2 1 − Ar 2 − 1 − A R 2 p= , √ √ 3 3 1 − A R 2 − 1 − Ar 2

(14.8.12)

with

= const, 2m = A R 3 ,

A=

1 8πh χ c2 , χ = 2 . 3 c

(14.8.11)

The pressure increases near the core of the star, as one would expect. Indeed, for a star of fixed radius R, the central pressure p(0) will need to be greater than infinity if the mass exceeds 4 Mmax = R. (15.3.7) h We derived this result from the rather strong assumption that the density is constant, but it remains valid also when this assumption is considerably weakened. Of course, this still doesn’t mean that realistic astrophysical objects will always ultimately collapse to black holes. An ordinary planet, supported by material pressures, will persist essentially forever. The pressure supporting a star comes from the heat produced by fusion of light nuclei into heavier ones. When the nuclear fuel is used up, the temperature declines, and the star begins to shrink under the influence of gravity. The collapse may eventually be halted by Fermi degeneracy pressure: electrons are pushed so close together that they resist further compression simply on the basis of the Pauli exclusion principle. A stellar remnant supported by electron degeneracy pressure is called a white dwarf; a typical white dwarf is comparable in size to the Earth. White dwarfs are the end state for most stars, and they are extremely common throughout the universe. If the total mass is sufficiently high, however, the star will reach the Chandrasekhar limit, where even the electron degeneracy pressure is not enough to resist the pull of gravity. Calculations put the Chandrasekhar limit at about 1.4 M , where M = 2 × 1033 g is the mass of the Sun. When it is reached, the star is forced to collapse to an even smaller radius. At this point, electrons combine with protons to make neutrons and neutrinos (inverse beta decay), and the neutrinos simply fly away. The result is a neutron star, with a typical radius of about 10 km. Neutron stars have a low total luminosity, but they possess strong magnetic fields and are often rapidly spinning. This combination gives rise to pulsars, discovered by Jocelyn Bell in 1967. Since

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15 Schwarzschild’s Solution and Black Holes

the conditions at the center of a neutron star are very different from those on Earth, we do not have a perfect understanding of the equation of state. Nevertheless, we believe that a sufficiently massive neutron star will itself be unable to resist the pull of gravity and will continue to collapse; current estimates of the maximum possible neutron-star mass are around 3 to 4 M , the Oppenheimer–Volkoff limit. Since a fluid of neutrons is the densest material we know, it is believed that the outcome of such a collapse is a black hole. It was long questioned whether such black holes could actually exist in nature or whether they were merely pathological solutions to Einstein’s equations. Einstein himself wrongly thought that black holes could not form, because he held that the angular momentum of collapsing particles would stabilize their motion at some radius [41]. Once a black hole has formed, it can continue to grow by absorbing additional matter from its surroundings, such as gas and interstellar dust.

15.4 Summary of Schwarzschild Spacetime and Black Holes We wrap up this section by summarizing what this brief examination of the features of the Schwarzschild spacetime have revealed: • In general relativity, the Schwarzschild spacetime represents a monopolar “gravitational field.” • This solution describes the exterior spacetime of every spherically symmetric body, even one that is time-dependent (as long as the time-dependence preserves spherical symmetry). • The spacetime contains an event horizon: a spherical surface at rs = 2m that causally disconnects all events at r < rs from those at r > rs . Things can go into the horizon (from r > rs to r < rs ), but they cannot get out; once inside, all causal trajectories (timelike or null) tend inexorably to the classical singularity at r = 0. • From the perspective of distant observers, dynamics near the horizon appears weird—such observers never actually see anything cross it and fall inside. This is largely a consequence of extreme redshifting—clocks slow down so much relative to distant observers that physical processes appear to come to a stop. At any rate, this weirdness is hidden from distant observers due to the extreme redshifting and deflection of the photons that they would use to observe such processes. For all these reasons, the Schwarzschild solution is known as a black hole: a region of spacetime that is completely dark and whose interior is completely cut off from the rest of the universe. The simplest static black holes have mass but neither electric charge nor angular momentum, and are often referred to as Schwarzschild black holes. According to Birkhoff’s theorem, the Schwarzschild solution is the only exterior spherically symmetric solution of Einstein’s equations. This means that there is no observable

15.4 Summary of Schwarzschild Spacetime and Black Holes

421

difference between the gravitational field of such a black hole and that of any other spherical object of the same mass. The popular notion of a black hole “sucking in everything” in its surroundings is therefore correct only near a black hole’s event horizon; far away, the external gravitational field is identical to that of any other body of the same mass. The no-hair conjecture postulates that once it achieves a stable condition after formation, a black hole has only three independent physical properties: mass, charge, and angular momentum; the black hole is otherwise featureless. If the conjecture is true, any two black holes that share the same values for these properties, or parameters, are indistinguishable from each other. The degree to which the conjecture is true for real black holes, under the laws of modern physics, is currently an unsolved problem [74]. Solutions describing more general black holes also exist. Nonrotating charged black holes are described by the Reissner–Nordström metric, while the Kerr metric describes a noncharged rotating black hole. The most general stationary black hole solution is the Kerr–Newman metric, which describes a black hole with both charge and angular momentum [72]. Because a black hole eventually achieves a stable state with only three parameters, there is no way to avoid losing information about the initial conditions: the gravitational and electric fields of a black hole give very little information about what went in. The information that is lost includes every quantity that cannot be measured far away from the black hole’s event horizon, including approximately conserved quantum numbers such as the total baryon number and lepton number. This behavior is so puzzling that it has been called the black hole information loss paradox [128]. Black holes are commonly classified according to their mass; see Table 15.1. The size of a black hole, as determined by the radius of the event horizon, or Schwarzschild radius, is roughly proportional to the mass M, through rs =

2hM M ≈ 2.95 km, 2 c M

where rs is the Schwarzschild radius and M is the mass of the Sun.

Table 15.1 Black hole classifications Class Approx. mass Supermassive black hole Intermediate-mass black hole Stellar black hole Micro black hole

105 −1010

MSun 103 MSun 10 MSun up to MMoon

Approx. size 0.001–400 AU 103 km ≈ REarth 30 km up to 0.1 mm

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15 Schwarzschild’s Solution and Black Holes

15.5 Heuristic Derivation of the Kerr Metric Karl Schwarzschild solved Einstein’s field equations, which determine the exact spacetime geometry of a nonrotating spherical body. It was relatively quickly realized, via Birkhoff’s uniqueness theorem [34, 133], that the spacetime geometry in vacuum, outside any localized spherically symmetric source, is equivalent to a portion of the Schwarzschild geometry, and so of direct physical interest to modeling the spacetime geometry exterior to idealized nonrotating spherical stars and planets. From astrophysics, we know that stars rotate, and if a rotating star were to undergo gravitational collapse, then the resulting black hole would be expected to retain at least some fraction of its initial angular momentum, suggesting on physical grounds that somehow there should be an extension of the Schwarzschild geometry to the situation in which the central body carries angular momentum. Physicists and mathematicians looked for such a solution for many years until Roy Kerr discovered a solution in 1963 [83]. The original Kerr derivation [83] is complicated and unintuitive. It strongly relies on the Petrov algebraic classification and Cartan calculus [82]. Chandrasekhar, in [29], gave a derivation of the Kerr metric based on symmetry arguments. He introduced five unknown functions to be found by solving Einstein’s equations. The main goal of this section is to collect the efforts to find an intuitive derivation, backed up with physical arguments, in order to show that a simple physical reasoning and observation of some features of the Schwarzschild and Kerr solutions can lead to the derivation of the Kerr metric. For a review, see [162]. The main tools in deriving the Kerr metric consist in combining the ellipsoid coordinate transformation and the procedure to obtain the Schwarzschild metric according to these steps: 1. transforming the coordinates (x, y, z, t) of the Minkowski spacetime into a spacetime with ellipsoidal symmetry, i.e., with reference to coordinates (r, θ, φ, t); 2. transforming the ellipsoidal coordinates (r, θ, φ, t) into a new coordinate system (r, θ, , T ) in order to eliminate the major difference in metric components between the Kerr and Schwarzschild metrics, i.e., there are no off-diagonal terms, and the product g44 g11 becomes −1; 3. solving Einstein’s equations in vacuum using this ansatz metric tensor; 4. applying the limit method to calculate the Ricci curvature tensor and finally deduce the Kerr metric. In the next section we exhibit the Kerr metric tensor expressed in the Boyer– Lindquist coordinates and summarize its main properties. In Sect. 15.6 we describe how to extend the Schwarzschild heuristic strategy to derive the Kerr metric. In Sect. 15.7 we introduce the ellipsoid coordinate transformation necessary to derive the Kerr metric. Finally, in Sect. 15.8 we use the previous transformation to solve Einstein’s equations and thereby derive the complete form of the Kerr metric.

15.6 Kerr Metric and Its Properties

423

15.6 Kerr Metric and Its Properties The Kerr line element expressed in the Boyer–Lindquist coordinates (ρ, θ, φ, t) takes the form    1 2mr 4mra sin2 θ ρ dr 2 + ρ2 dθ2 dtdφ + ds 2 = − 1 − 2 dt 2 − ρ ρ2  (15.6.1)   2mra 2 sin2 θ 2 2 2 2 sin θdφ , + r +a + ρ2 where m and a denote mass and angular momentum for a unit mass, respectively, ρ2 = r 2 + a 2 cos2 θ, and  = r 2 − 2mr + a 2 . The Boyer–Lindquist coordinates are particularly useful, since they minimize the number of off-diagonal components of the metric: there is now only one off-diagonal component. So the Kerr metric is not orthogonal, but it has the same symmetry as the ellipsoidal spacetime metric, which is orthogonal. Another particularly useful feature is that the asymptotic (r → ∞) behavior in the Boyer–Lindquist coordinates is       1 1 4ma sin2 θ 2m 2 +O + O − dt dφdt, ds 2 = − 1 − r r3 r r3 (15.6.2)   

2  1 2m 2 2 2 2 dr +O . + r (dθ + sin θdφ + 1+ r r2 A comparison with the formulas of Chap. 14 allows us to confirm that m is the mass, and J = ma the angular momentum. If a → 0, the Boyer–Lindquist line element reproduces the Schwarzschild line element in standard Schwarzschild coordinates. If m → 0, the Boyer–Lindquist line element reduces to r 2 + a 2 cos θ2 2 dr , ds 2 → − dt 2 + r 2 + a2

2  

+ r + a 2 cos θ2 dθ2 + r 2 + a 2 sin2 θdφ2 ,

(15.6.3)

i.e., to the flat Minkowski metric in so-called oblate spheroidal coordinates. We can relate these coordinates to the usual Cartesian coordinates of Euclidean 3-space by the formulas (15.6.4) x = r 2 + a 2 sin θ cos φ, 2 2 y = r + a sin θ sin φ; (15.6.5) z = r cos θ. (15.6.6)

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15 Schwarzschild’s Solution and Black Holes

15.7 The Schwarzschild and Kerr Solutions In this section we delineate a heuristic strategy to derive the Kerr metric, imitating the strategy adopted to derive the Schwarzschild solution. In Minkowski spacetime we may adopt global coordinates, combining the three Euclidean space coordinates and time, obtaining the metric ds 2 = d x 2 + dy 2 + dz 2 − dt 2 ,

(15.7.1)

which in polar coordinate assumes the form ds 2 = dr 2 + r 2 dθ2 + r 2 sin2 θdφ2 − dt 2 .

(15.7.2)

To find the metric for a static spherically symmetric object, Schwarzschild employed polar coordinates in which two unknown functions ν(r ), λ(r ), appear: ds 2 = e2λ(r ) dr 2 + r 2 dθ2 + r 2 sin2 θdφ2 − e2ν(r ) dt 2 .

(15.7.3)

The corresponding metric was used as an ansatz to solve the Einstein field equations. More specifically, Schwarzschild used the vacuum condition, Rμν = 0, calculating Ricci tensor from (15.7.2), and he obtained the first exact solution of the Einstein field equation [154]:     2M −1 2 2M 2 2 2 2 2 dt 2 . dr + r dθ + r sin θdφ − 1 − ds = 1 − r r 2

(15.7.4)

However, the Schwarzschild metric cannot be used to describe rotating, axially symmetric, and charged bodies. From an examination of the metric tensor gμν in the Schwarzschild metric, one obtains the following components:   2M −1 grr = 1 − , r gφφ = r sin θ, 2

2

gθθ = r 2 ,   2M . gtt = − 1 − r

(15.7.5)

Differences in the metric tensor gμν between the Schwarzschild metric (15.7.3) and Minkowski spacetime (15.7.2) are to be found in the terms (gtt ) and (grr ). The Kerr metric is the second exact solution of the Einstein equations, which can be used to describe spacetime geometry in a vacuum near a rotating, axially symmetric body [83]. It is a generalized form of the Schwarzschild metric. The Kerr metric in the Boyer–Lindquist coordinates can be expressed as follows:

15.7 The Schwarzschild and Kerr Solutions

425

  ρ2 2 2Mra 2 sin2 θ 2 2 2 2 sin2 θdφ2 ds = dr + ρ dθ + r + a +  ρ2   4Mra sin2 θ 2Mr dt 2 . − dtdφ − 1 − 2 ρ2 ρ 2

(15.7.6)

By examining the components of the metric tensor gμν in (15.7.6), one obtains g11 =

ρ2 , 

g22 = −ρ2 ,

g44 = 1 −

2Mr , ρ2

2Mra sin2 θ g43 = g34 = , ρ2   2Mra 2 sin2 θ sin2 θ. g33 = r 2 + a 2 + ρ2

(15.7.7)

Comparing the Schwarzschild and Kerr components of the metric tensor, we observe that: 1. Both off-diagonal terms g43 (gtφ ) and g34 (gφt ) of the Kerr metric are not present in the Schwarzschild metric, apparently due to rotation. If the rotation parameter a is equal to 0, these two terms vanish. 2. In the Schwarzschild metric, we have g00 g11 = gtt grr = −1, but not in the Kerr metric. 3. When the spin parameter a vanishes, the Kerr metric turns into the Schwarzschild metric, and therefore it represents a generalized form of the Schwarzschild metric.

15.8 Transformation of Ellipsoid Symmetric Orthogonal Coordinate The Kerr metric cannot be derived simply using the spherical symmetry method used for the derivation of the Schwarzschild metric. In [179], Chou proposes to modify the starting point of the Schwarzschild method, first expressing the metric tensor in ellipsoidal coordinates, and then to use Schwarzschild’s method for the Kerr metric to solve the Einstein equations. In other words, the following change to ellipsoidal coordinates was applied to (15.7.1): x → (r 2 + a 2 )1/2 sin θ cos φ, y → (r 2 + a 2 )1/2 sin θ sin φ, z → r cos θ, t → t,

(15.8.1)

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15 Schwarzschild’s Solution and Black Holes

where a is the coordinate transformation parameter. In the new coordinates, the metric becomes ds 2 = ρ2 dr 2 + ρ2 dθ2 + (r 2 + a 2 ) sin2 θdφ2 − dt 2 .

(15.8.2)

Equation (15.8.2) represents the metric in the coordinates with ellipsoidal symmetry in vacuum; it can also be obtained by assigning m = 0 to the Kerr metric (15.8.2). Bijan [11] started from (15.8.2) and derived the following Schwarzschild-like solution for ellipsoidal objects:     2M 2m −1 ρ2 2 2 2 2 2 dt 2 , + ρ dθ − (r + a ) sin θdφ 1 − ds = 1 − r r 2 + a2 r (15.8.3) which can be transformed into Schwarzschild’s solution (15.7.3) when the parameter a is equal to 0. In order to eliminate the difference between the Kerr metric and the Schwarzschild-like metric (15.8.3), more specifically to eliminate the off-diagonal term, we may apply the following transformation: 2

dT ≡ dt − pdφ,

d ≡ dφ − qdt.

(15.8.4)

In these new coordinates we can rewrite the Kerr metric as ds 2 = G 11 dr 2 + G 22 dθ2 + G 33 d2 + G 44 dT 2 ,

(15.8.5)

G 44 G 11 = −1.

(15.8.6)

and we obtain

By comparing the coefficients, we obtain 2Mra sin2 θ , ρ2 2Mr G 44 + G 33 q 2 = 1 − 2 , ρ   2Mra 2 sin2 θ  2  2 2 sin2 θ, G 44 p + G 33 = − r + a + ρ2

G 44 p + G 33 q = −

(15.8.7) (15.8.8) (15.8.9)

G 22 = −ρ2 ,

(15.8.10)

−ρ . 

(15.8.11)

G 11 =

2

By determining the six variables G 44 , G 11 , G 22 , G 33 , p, q in the six equations (15.8.6)–(15.8.11), we obtain

15.8 Transformation of Ellipsoid Symmetric Orthogonal Coordinate

427

ρ2 ,  2 = −ρ ,

G 11 = − p = ± a sin2 θ, a , q=± 2 r + a2

take the positive result

G 22

take the positive result

G 33 = − G 44

(r 2 + a 2 )2 sin2 θ , ρ2  = 2. ρ (15.8.12)

Putting these into Eq. (15.8.1), we have  2 a ρ2 2 (r 2 + a 2 )2 sin2 θ  dφ − 2 ds = dr + dt − 2 (dt − a sin2 θdφ)2 . 2 2  ρ r +a ρ (15.8.13) As a consequence of the coordinate transformation there is no off-diagonal term, and g44 g11 = −1. Finally, making the ellipsoidal coordinate transformation 2

dT ≡ dt − a sin2 θdφ, a dt d ≡ d − 2 (r + a 2 )

(15.8.14)

transforms (15.8.13) into ds 2 =

ρ2 (r 2 + a 2 )2 sin2 θ r 2 + a2 2 2 2 2 2 dr + ρ dθ + d − dT . r 2 + a2 ρ2 ρ2

(15.8.15)

We apply the Schwarzschild procedure by introducing two functions e2ν(r,θ) , e2λ(r,θ) , so that we have ds 2 = +e2ν(r,θ) dr 2 + ρ2 dθ2 +

(r 2 + a 2 ) sin2 θ d2 − e2ν(r,θ) dT 2 , ρ2

(15.8.16)

where the parameters ρ2 and h are given by ρ2 ≡ r 2 + a 2 cos2 θ, h ≡ r 2 + a2.

(15.8.17)

15.9 A Solution of Einstein’s Equations in Vacuum The last step consists in solving the vacuum Einstein’s equations. We don’t give the details of these computations, which can be found in [179]. We begin with the equations Rμν = 0, R = 0. Then combining R00 and R11 , we get (15.9.1), and solving this equation yields (15.9.2)–(15.9.4):

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15 Schwarzschild’s Solution and Black Holes

2r (∂1 ν + ∂1 λ) = 0, h ∂1 ν + ∂1 λ = ∂1 (ν + λ) = 0,

e−2(ν−λ) R00 + R11 =

ν = −λ + c, ν(r, θ) = −λ(r, θ) + c, ν

e =e

−λ

.

(15.9.1) (15.9.2) (15.9.3) (15.9.4)

To solve this partial differential equation, one has to keep in mind that when a → 0, the Kerr metric (15.7.6) turns into the Schwarzschild metric (15.7.3). Then we obtain the following equations: lim h = r 2 ,

a→0

lim ρ = r,

a→0

(15.9.5)

lim R22 = e−λ (r (∂1 λ − ∂1 ν) − 1) + 1,

(15.9.6)

lim R33 = sin2 θe−2λ (r (∂1 λ − ∂1 ν) − 1 + 1 = sin2 θ R22 ,

(15.9.7)

lim R22 = 0,

(15.9.8)

a→0

a→0 a→0

e2ν = 1 +

C , r

let C = −2M.

(15.9.9)

When the angular momentum approaches zero (a → 0), the equations can be solved, giving r 2 − 2Mr 2M = lim e2ν = 1 − a→0 r r2 (15.9.10) −1  2M r2 2λ . lim e = 1 − = 2 a→0 r r − 2Mr Setting another limit condition on flat spacetime, when the mass approaches zero (M → 0), from (15.8.11), we have r 2 + a2 r 2 + a2 = 2 2 ρ r + a 2 cos2 θ ρ2 r 2 + a 2 cos2 θ = 2 = . 2 r +a r 2 + a2

lim e2ν =

M→0

lim e2λ

M→0

(15.9.11)

From the above conditions (15.9.10)–(15.9.11), Einstein’s equations can be solved, giving r 2 − 2Mr 2 2 a r + a 2 cos2 θ, e2ν = + (15.9.12) r 2 + a 2 cos2 θ 2λ . e = 2 r − 2Mr + a 2 Finally, the Kerr metric is obtained in the form

15.9 A Solution of Einstein’s Equations in Vacuum

429

 2 a ρ2 (r 2 + a 2 )2 sin2 θ 2 2 2 dφ + 2 ds = 2 dr + ρ dθ + dt , r − 2Mr + a 2 ρ2 r + a2 r 2 − 2Mr + a 2 (dt − a sin2 θdφ)2 , − ρ2 (15.9.13) where ρ2 ≡ r 2 + a 2 cos2 θ and  ≡ r 2 − 2Mr + a 2 , m is the mass of the rotational material, a is the spin parameter or specific angular momentum, and J = Ma. 2

15.10 Consequences of Kerr’s Solutions The Kerr solutions possibly describe exterior gravitational fields of stationary rotating axially symmetric sources. While it is expected that the exterior region of the Kerr solution is stable and that all rotating black holes will eventually approach a Kerr metric, the interior region of the solution appears to be unstable, much like a pencil balanced on its point [175]. However, no satisfactory interior solutions are known. Although the Kerr solution appears to be singular at the roots of  = 0, these are actually coordinate singularities, and with an appropriate choice of new coordinates, the Kerr solution can be smoothly extended through the values of r corresponding to these roots. A black hole in general is surrounded by a surface situated at the Schwarzschild radius for a nonrotating black hole, called the event horizon. Within this surface, no observer/particle can maintain itself at a constant radius. A rotating black hole has the same static limit at its event horizon, but there is an additional surface outside the event horizon that can be intuitively characterized as the sphere where the rotational velocity of the surrounding space is dragged along with the velocity of light. Within this sphere, the dragging is greater than the speed of light, and any observer/particle is forced to corotate. The region outside the event horizon but inside the surface where the rotational velocity is the speed of light is called the ergosphere. Particles falling within the ergosphere are forced to rotate faster and thereby gain energy. Because they are still outside the event horizon, they may escape the black hole. The net process is that the rotating black hole emits energetic particles at the cost of its own total energy. The possibility of extracting spin energy from a rotating black hole was first proposed by the mathematician Roger Penrose in 1969 and is thus called the Penrose process. Rotating black holes in astrophysics are a potential source of large amounts of energy and are used to explain energetic phenomena, such as gamma ray bursts.

Chapter 16

Elements of Cosmology

In Chap. 14, some models of the universe that can be considered reasonable representations of a region of the universe have been described. In this chapter, general relativity is applied to cosmology. Any model of the universe that is adopted in cosmology is based on a spacetime V4 with the following properties: • A global or cosmic time t exists with respect to which the evolution of the universe is evaluated. Further, a procedure is defined to measure this time by synchronized clocks. • The 3 dimensional submanifolds t = const are spacelike and represent the whole physical space at the instant t. This means that V4 is the union of the “slices” t = const. • Every slice t = const is homogeneous and isotropic, in agreement with the cosmological principle. • The momentum–energy tensor Tαβ is different from zero at every point of V4 . The cosmological principle and the above properties drastically reduce the number of metric coefficients gαβ . The remaining unknown metric coefficients are determined by requiring that they satisfy Einstein’s equations. In particular, we analyze the Friedmann, Einstein, and de Sitter universes.

16.1 Global Properties of Spacetime Einstein was the first to discuss cosmological problems within the framework of general relativity, but his conclusions were not experimentally confirmed. Satisfactory results were obtained by other authors by adopting more general hypotheses. We begin by formulating the mathematical properties of the spacetime V4 that lead to a good description of the whole universe. © Springer Nature Switzerland AG 2019 A. Romano and M. Mango Furnari, The Physical and Mathematical Foundations of the Theory of Relativity, https://doi.org/10.1007/978-3-030-27237-1_16

431

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16 Elements of Cosmology

Definition 16.1 Let V4 be a four-dimensional hyperbolic Riemannian manifold and denote by V3 a three-dimensional manifold (see Sect. 14.1). Suppose that a diffeomorphism f : V3 ×  → V4 exists such that: 1. The family  of the curves f p : { p} ×  → V4 , ∀ p ∈ V3 , satisfies the following property: • It is a congruence of timelike geodesics diffeomorphic to an interval ]a, b[⊂  (Weyl’s hypothesis), where a and b could be finite or infinite; • lim f p (t) = x,∀ p ∈ V3 , where x ∈ V4 . t→a

In other words,  is a congruence of timelike geodesics starting from the same point x. 2. The hypersurfaces f t : V3 × {t} → V4 , for all t ∈ , are spacelike surfaces diffeomorphic to V3 and orthogonal to the curves of . We denote by F the family of the surfaces f t . 3. The space sections f t are homogeneous and isotropic (cosmological principle). 4. The geodesics of  are the timelike world lines of the particles of a perfect fluid S that fills V4 and is described by the momentum–energy tensor  p Tαβ =  + 2 U α U β + pgαβ , c

(x) = 0,

∀x ∈ V4 ,

(16.1.1)

where  = h/c2 is the proper density and h the proper matter–energy density of S (the fluid S is freely gravitating). Essentially, we modify Definition 14.1 by supposing that the congruence  is formed by timelike geodesics and requiring that the geometry of space sections f t be homogeneous and isotropic. If (U, x i ), i = 1, 2, 3, and (, x 4 ) are coordinates for V3 and  respectively, then (U × , (x i , x 4 )) is a chart on V4 in which the curves of  have parametric equations x i = const, x 4 = var, while the surfaces of F have equation x 4 = const. These coordinates, which are said to be adapted to the congruence  or comoving coordinates, are the Lagrangian coordinates of the particles of the reference fluid S. They are defined up to a coordinate transformation such that 

x  = x  (x 1 , x 2 , x 3 ), i

i

x  = x  (x 1 , x 2 , x 3 , x 4 ). 4

4

(16.1.2)

Since the curves of  are geodesics orthogonal to the surfaces f t , the adapted coordinates are Gaussian coordinates for every surface f t , provided that the coordinate transformation has the form (16.1.3) x 4 = x 4 .

16.1 Global Properties of Spacetime

433

In these coordinates, the metric coefficients satisfy the conditions gi4 = 0, g44 = 1,

(16.1.4)

and the metric of V4 assumes the form ds 2 = gi j d x i d x j − (d x 4 )2

i, j = 1, 2, 3.

(16.1.5)

αβ = ∇α Uβ − ∇β Uα = Uβ,α − Uα,β = 0.

(16.1.6)

Finally (see Sect. 14.1), we have

On introducing the 1-form ω = Uα d x α , we see that the above condition is equivalent to requiring that ω be closed: dω = 0. Since V4 is supposed to be linearly connected, this property is equivalent to the existence of a differentiable function f : V4 →  such that ω = df

⇒

Uα = f ,α .

(16.1.7)

This function f is just the function whose existence is supposed in item 1.

16.2 On the Geometry of Space Sections In this section we analyze the implications of the cosmological principle on the geometry of space sections. According to this principle, the sections f t must be homogeneous and isotropic. We recall that a Riemannian manifold Vn is isotropic at the point p ∈ Vn if for every pair of unit tangent vectors X, Y at p, an isometry  : Vn → Vn exists such that ∗ (X) = Y. Further, Vn is homogeneous if for every pair of points p and q of Vn , an isometry  : Vn → Vn exists such that ( p) = q. It is possible to prove that if Vn is isotropic at a point p and is homogeneous, then it is isotropic at every point of Vn . It is fundamental for our purpose to determine the metric (16.1.5) of the spacetime V4 for which the space sections f t are homogeneous and isotropic. This difficult problem was solved in [60, 61]. The results there obtained were applied to cosmology in [135, 136] (see also the references cited in [136]) and allow one to write the metric (16.1.5) as follows: (16.2.1) ds 2 = a 2 (t)dσ 2 − c2 dt 2 , where a(t) is an arbitrary function of the global time t, and the metric

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16 Elements of Cosmology

dσ 2 = γi j (x 1 , x 2 , x 3 )d x i d x j ,

i, j = 1, 2, 3,

(16.2.2)

refers to a three-dimensional properly Riemannian space with constant scalar curvature K : (3) R = K. (16.2.3) It remains to determine the most general metric of a three-dimensional Riemannian manifold with constant curvature scalar (3) R. In differential geometry it is proved that the most general three-dimensional Riemannian manifold satisfying this condition admits a global coordinate system (x i ) in which the metric has the form dσ 2 =

3  2 (xi d x i )2 (d x i )2 + , 1 − r 2 i=1

r=

3 i=1

(x i )2

 21

,

(16.2.4)

where  = −1, 0, 1. Resorting to an appropriate embedding into a Euclidean space E 4 , it is possible to supply a geometric representation of a 3-dimensional manifold V3 with metric (16.2.4). In fact, consider Euclidean spaces E 4 admitting global z) in which the metric assumes the following forms: coordinates ( x i , ds 2 = ds 2 =

3 

1 z )2 , (d x i )2 + (d  i=1

3 

(d x i )2 + (d z )2 ,

 = ±1,

(16.2.5)

 = 0.

(16.2.6)

i=1

In other words, E 4 is the usual 4-dimensional Euclidean space when  = 0, 1 and a Minkowski’s space when  = −1. For  = 0, 1, we consider the hypersurface V3 embedded in E 4 , equipped with the metric (16.2.5), whose points have coordinates satisfying the equation 



( x i )2 + z2 = K 2,

(16.2.7)

i

where K > 0. It is evident that for  = 0, Eq. (16.2.7) defines two 3-planes whose distance from the 3-plane O z = 0 is ±K . When  = 1, (16.2.7) defines a hypersphere in the usual Euclidean space E 4 . In contrast, for  = −1, the hypersurface V3 , embedded in E 4 with the metric (16.2.6), whose points have coordinates satisfying the equation 3  1 z)2 = −K 2 , (d x i )2 + (d  i=1

(16.2.8)

16.2 On the Geometry of Space Sections

435

is a pseudosphere, i.e., the locus of points at the same distance from the origin of Minkowski space. x i /K and z =  z/K , If we introduce the nondimensional coordinates x i =  Eqs. (16.2.7), (16.2.8) assume the following form: 

 (x i )2 + z 2 = 1.

(16.2.9)

i

Let (x i , z) and (x i + d x i , z + dz) be two points of the hypersurface V3 defined by (16.2.9). Then the distance induced by the metric of E 4 between these points is given by 3  1 dσ 2 = (d x i )2 + (dz)2 ,  = ±1, (16.2.10)  i=1 where in view of (16.2.9), we have zdz = −



xi dxi

⇒

i

dz 2 =

2 i (xi d x i )2 . 1 −  i (x i )2

These relations allow us to transform (16.2.10) into (16.2.4), so that the geometric meaning of the metric (16.2.4) becomes evident. If we introduce into the manifold V3 the spherical coordinates r, θ, ϕ by the relations x 1 = r sin θ cos ϕ, 0 < θ < π, x 2 = r sin θ sin ϕ,

0 < ϕ < 2π,

(16.2.11)

x = r cos θ, 3

then (16.2.4) can be written as dσ 2 =

dr 2 + r 2 dθ2 + r 2 sin2 θdϕ2 , 1 − r 2

and the metric (16.2.1) of spacetime V4 becomes the Robertson–Walker metric:  dr 2 2 2 2 2 2 + r dθ + r sin θdϕ − c2 dt 2 . ds = a (t) 1 − r 2

2

2

(16.2.12)

It is important to highlight that the coordinates (x 1 , x 2 , x 3 ) and (r, θ, ϕ) were introduced in the subspace z = 0 of E 4 . Consequently, they have no physical meaning for observers located at the points of the cosmic fluid S. However, we can adopt the following measure procedure to assign a physical meaning to these coordinates. If  = 1, then r ∈ [0, 1], and we can introduce the new coordinate

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ξ = arcsin r, r = sin ξ,

(16.2.13)

so that (16.2.12) can be written as follows:

ds 2 = a 2 (t) dξ 2 + sin2 ξ dθ2 + sin2 θdϕ2 − c2 dt 2 .

(16.2.14)

If we denote by  the measure at the instant t of the arc of the space curve u ∈ [0, ξ], θ and ϕ constant, then from (16.2.14) we obtain ξ=

 , a(t)

(16.2.15)

and ξ is determined by a space measure at the instant t. Finally, since r is related to ξ by (16.2.13), we can also use r as a coordinate. If  = −1, then r ∈ [0, ∞), and introducing the coordinate η = arc sinh r, r = sinh η,

(16.2.16)

Equation (16.2.14) becomes

ds 2 = a 2 (t) dη 2 + sinh2 η dθ2 + sin2 θdϕ2 − c2 dt 2 ,

(16.2.17)

and we obtain for η a formula like (16.2.15). Finally, we remark that the coordinate time t of an observer at a point O of the cosmic fluid is equal to the proper time τ of O. In fact, the proper time is evaluated along the world line of O (η, θ, and ϕ are constant), so that ds 2 = −c2 dτ 2 = −c2 dt 2 . The clock at O can be synchronized with the clock at a point P( ξ, θ, ϕ) ∈ S, sending at the instant t = 0 for O a light signal toward P along the arc ξ ∈ [0,  ξ] connecting O and P. During the propagation of the signal, we have the condition a 2 (t)dξ 2 − c2 dt 2 = 0, so that by integrating we obtain  t  du ξ . = 2 c 0 a(u)

(16.2.18)

Therefore, when the signal arrives at P, the clock at this point must mark the time t satisfying condition (16.2.18), and this time interval [0, t] is just the proper time interval for the clock at O. The clocks at O and P remain synchronous, since their evolution does not depend on the point of the space section (homogeneity hypothesis). A similar procedure can be applied by starting from the metric (16.2.17).

16.2 On the Geometry of Space Sections

437

We conclude by noticing that using the notebook Geometry.nb, we can obtain the scalar curvature of the space section t = const: K =

 a 2 (t)

.

(16.2.19)

We have now deduced all the consequences of the cosmological principle and the hypotheses about the geometric properties of V4 . It remains to determine the function a(t) and the mass density (r, t) using Einstein’s equations.

16.3 Conservation Laws We begin with the conservation laws that are a consequence of Einstein’s equations. Using the notebook Geometry.nb, we obtain G 2 ≡ ∇β T 2β = 0, G 3 ≡ ∇β T 3β = 0,

(16.3.1)

as well as

1 ∂p 1 − r 2  = 0, 2 a ∂r 3ca˙ ∂ 3 pa˙ + +c = 0. = ca a ∂t

G 1 ≡ ∇β T 1β =

(16.3.2)

G 4 ≡ ∇β T 4β

(16.3.3)

Equation (16.3.2) shows that p depends only on t, i.e., p = p(t). Since p is a function of  through a state equation p = p(), we have that  = (t), and (16.3.3) gives 3 pa˙ 3ca˙ + + c˙ = 0. ca a

(16.3.4)

By simple algebraic manipulations, (16.3.4) becomes ˙ 3a˙ =− 2 , p + c2  c a

(16.3.5)

or equivalently, c ˙ 2 a 3 + 3c2 a 2 a˙ = −3 pa 2 a. ˙ This last equation can be written in the form da 3 d 2 3 c a = − p , dt dt

(16.3.6)

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which can evidently be considered the balance of energy, since it shows that the variation of relativistic energy is equal, up to sign, to the power of pressure for the instantaneous variation of the volume. In other words, it corresponds to the first principle of thermodynamics for adiabatic transformations. On the other hand, all the transformations of the whole universe must be adiabatic, since it is isolated. The state equation p = p(t) is usually written as p = wc2 ,

(16.3.7)

where w is a parameter that characterizes the material content of the considered system. For instance, for nonrelativistic matter, the pressure is very small compared with the rest energy of the particles of the cosmic fluid, so that it can be neglected and we can choose w = 0. For the radiation, w = 1/3, and so on. When the pressure can be neglected, (16.3.6) gives (t)a 3 (t) = (0)a 3 (0) = (0),

(16.3.8)

since a(0) = 1.

16.4 Friedmann’s Equations In the previous section we found the consequences of the identities (16.3.1)–(16.3.3) when the Robertson–Walker metric is adopted. Up to now, a model of the universe based on this metric requires the determination of the functions (t) and a(t), since the function p(t) can be obtained by the constitutive equation (16.3.7). Equation (16.3.5) supplies the mass density (t), provided that a(t) is known. In this section we show that Einstein’s equations lead to a differential equation in the unknown a(t). Using again the notebook Geometry.nb, we obtain that if the Robertson–Walker metric is adopted, then Einstein’s equations reduce to a˙ 2 c2  8πh 2a¨ + 2 + 2 − c2 = − 2 p, a a a c a˙ 2 c2  1 2 8πh , + 2 − c = a2 a 3 3

(16.4.1) (16.4.2)

where  is the cosmological constant. Equation (16.4.2) can also be written as a˙ 2 =

8πh 2 1 2 2 a + c a − c2 , 3 3

while on subtracting (16.4.2) from (16.4.1), we obtain the equation

(16.4.3)

16.4 Friedmann’s Equations

439

a¨ = −

4πh a 3



 3p 1 +  + c2 a. c2 3

(16.4.4)

Equations (16.4.1), (16.4.2) or the equivalent equations (16.4.3), (16.4.4) are called Friedmann’s equations. It seems that we have two equations in the unknown a(t). We now prove that (16.4.3) is a first integral of (16.4.4), so that it is satisfied for every solution of (16.4.4). First, we have that

3p 3 2 a 3p c +  = + 3 − 2 = + p − 2 = − ˙ − 2, c2 c2 c2 a˙ where in the last equality we have taken into account (16.3.5). Introducing this result in (16.4.4), we obtain  1 4πh  a a ˙ + 2 + c2 a. 3 a˙ 3

a¨ =

(16.4.5)

Multiplying (16.4.5) by a, ˙ we have a¨ a˙ = or

4πh 1 ˙ + c2 a a, ˙ (a ˙ 2 + 2a a) 3 3

4πh d da 2 1 1 d a˙ 2 = (a 2 ) + c2 . 2 dt 3 dt 6 dt

In other words, the quantity a˙ 2 −

8πh 2 1 2 2 a − c a = const 3 3

is constant, and (16.4.3) is obtained, provided that we identify the integration constant with −c2 . We have thus proved that (16.4.3) is a first integral of (16.4.3). Before analyzing the behavior of the function a(t), we want to verify whether a static solution a(t) = const is possible. From (16.4.3), (16.4.4) we have 8πh 2 1 2 2 a + c a − c2  = 0, 3

3 1 4πh 3p +  + c2 a = 0. − a 3 c2 3

(16.4.6) (16.4.7)

Solving these equations with respect to  and  yields =

4πh(3 p + c2 ) 4πha 2 ( p + c2 ) , = , 4 c c4

(16.4.8)

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so that  and  are positive quantities and the universe is closed. This result holds if  = 0, since if  = 0, then from (16.4.8) we obtain the absurd result p=−

c2  < 0. 3

Since p  c2 and  = 1, we can neglect p in (16.4.8), so that we obtain an approximate evaluation of the radius of Einstein’s universe: a=√

c . 4πh

(16.4.9)

It is possible to prove that Einstein’s universe is static but unstable.

16.5 Models of Universe for  = p = 0 If  plays an important role when large parts of the universe are taken into account, then the consequences of Friedmann’s equations hold for small regions of the universe when the hypothesis  = 0 is accepted. When  = p = 0, Eq. (16.4.3) and (16.4.4) become 8πh 2 a − c2 , 3 4πh a¨ = − a. 3

a˙ 2 =

(16.5.1) (16.5.2)

Taking into account (16.3.8), we can write the above equations as follows: 8πh 0 − c2 , 3a 4πh0 a¨ = − . 3a 2

a˙ 2 =

(16.5.3) (16.5.4)

We now introduce some constant quantities that allow us to write the above equations in a more useful form. We define the Hubble constant H0 and the critical mass density c by the formulas a(t ˙ 0) 3H02 = a(t ˙ 0 ), , (16.5.5) H0 = c = a(t0 ) 8πh where t0 is the present time, and we have introduced the notation 0 =

0 . c

(16.5.6)

16.5 Models of Universe for  = p = 0

Then we have 8πh =

441

3H02 , c

0 = 0 c ,

(16.5.7)

and (16.5.4), (16.5.3) become 0 H02 , 2a 2 0 H02 − c2 . a˙ 2 = a a¨ = −

(16.5.8) (16.5.9)

Evaluating (16.5.9) at t = t0 and recalling (16.5.5), we obtain the result H02 = 0 H02 − c2 , from which we derive that  = (0 − 1)

H02 . c2

(16.5.10)

(16.5.11)

Finally, we can write (16.5.8), (16.5.9) in the form 0 H02 , 2a 2 1 1 2 0 H02 a˙ − = − (0 − 1)H02 ≡ C. 2 2a 2 a¨ = −

(16.5.12) (16.5.13)

This system has the following structure: a¨ = f (a),

(16.5.14)

1 a˙ + W (a) = const, 2

(16.5.15)



where W (a) = −

f (a) da.

This particular form of the system allows us to derive the qualitative behavior of solutions by Weierstrass’s method. The results of this analysis will be shown in the next section.

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16.6 Qualitative Analysis of Friedmann’s Equations for  = p = 0 To analyze the behavior of solutions of the system (16.5.12), (16.5.13) by Weierstrass’s method, we refer to Fig. 16.1, in which the “potential energy” −0 H02 /2a versus a is shown together with two dashed lines corresponding to two values of the constant C. The line C > 0 corresponds to a value of 0 < 1, while the line C < 0 corresponds to a value of 0 > 1. Taking into account (16.5.13), from Fig. 16.1 we see that if C > 0 and a(t ˙ 0 ) > 0, then the “velocity” a˙ increases up to the limit value lim a˙ =

t→∞

 √ 2C = H0 1 − 0 .

(16.6.1)

This value reduces to zero when C = 0. In contrast, if C < 0 and a(t ˙ 0 ) > 0, then a˙ decreases, becoming zero when a = ae . We now evaluate: • the value of ae ; • the time interval [0, T ] to reach the value ae ; • the evolution of a(t) for t > T . The value of ae is obtained by equating to zero a˙ in (16.5.13), ae =

0 . 0 − 1

Further, separating the variables in (16.5.13), we obtain the result √ a = dt, √ H0 a(1 − 0 ) + 0

Fig. 16.1 Plot of −0 H02 /2a

(16.6.2)

16.6 Qualitative Analysis of Friedmann’s Equationsfor  = p = 0

443

Fig. 16.2 Solutions a(t) for different values of 0

from which on integrating we obtain 

0 0 −1

0

√ a da = T. √ H0 a(1 − 0 ) + 0

(16.6.3)

In Chap. 11 of the notebook Geometry.nb, it is proved that T =

π0 . 2H0 (0 − 1)

(16.6.4)

To determine the behavior of a(t) for t > T , we notice that a˙ has at t = T a simple zero. Further, at t = T we have a(T ) = ae , a(T ˙ ) = 0, and a(T ¨ ) < 0 (see (16.5.13)). Consequently, a(t) decreases and reaches the value 0 after time 2T . In conclusion, if 0 < 1, the function a(t) starts from zero at t = 0; it then increases up to the value ae (big bang) and decreases until it assumes again the value 0 at t = 2T (big crunch). In Fig. 16.2 are shown solutions a(t) of (16.5.12) corresponding to initial data a(0) = 0, a(0) ˙ = 1. The model for 0 = 1 separates the open models from the closed ones, that is, it separates the expanding models from the collapsing models. It is called the Einstein– de Sitter model. In view of (16.6.1), in this model the expansion velocity tends to zero as t goes to infinity. Further, (16.5.13) becomes a˙ =

H0 , a 1/2

and integrating between the instants t0 and t, we obtain

(16.6.5)

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2 3/2 a 3

t = H0 (t − t0 ).

(16.6.6)

t0

Since a(0) = 0 and a(t0 ) = 1, we obtain the age of the universe T = and the solution

a(t) =

2H0 3 2t 3H0

2/3 .

(16.6.7)

16.7 Models of the Universe for p = 0 and   = 0 In the absence of pressure, Eqs. (16.4.3), (16.4.4) become a¨ = − a˙ 2 =

1 4πh a + c2 a, 3 3 8πh 2 1 2 2 a + c a − c2 . 3 3

(16.7.1) (16.7.2)

We notice that on the right-hand side of (16.7.1), the term containing the cosmological constant represents an expansive force, since it has opposite sign to the gravitational term −4πha/3. There is no classical interpretation of , but quantum theory foresees fluctuations of quantum fields in empty space that could be described by the cosmological term. Taking into account (16.3.2), we have a¨ = − a˙ 2 =

4πh 0 + 3a 2 8πh 0 + 3a

1 2 c a, 3 1 2 2 c a − c2 , 3

(16.7.3) (16.7.4)

where 0 = (t0 ). If we adopt (16.5.6), (16.5.7) and introduce the notation  =

c2 , 3H02

(16.7.5)

the above equations assume the following form: a¨ = −

0 H02 +  H02 a, 2a 2

(16.7.6)

16.7 Models of the Universe for p = 0 and  = 0

a˙ 2 =

0 H02 +  H02 a 2 − c2 . a

445

(16.7.7)

Evaluating (16.7.7) at t = t0 and recalling that a(t0 ) = 1, a(t ˙ 0 ) = H0 , we obtain H02 = 0 H02 +  H02 − c2 , so that c2  = (0 +  − 1)H02 ,

(16.7.8)

a˙ 2 + W (a) = 1 − (0 +  ), H02

(16.7.9)

and (16.7.7) becomes

where



 0 2 W (a) = − +  a . a

(16.7.10)

Equation (16.7.9), which is a first integral of (16.7.6), shows that given 0 and  , the admissible values of a must satisfy the following inequality: 1 − (0 +  ) ≥ W (a).

(16.7.11)

Before analyzing condition (16.7.11), we summarize the meaning of 0 and  in the following list: 0 ≥ 1 ↔ ρ0 ≥ ρc , 0 < 1 ↔ ρ0 < ρc ,

(16.7.12) (16.7.13)

 ≥ 0 ↔  ≥ 0,  < 0 ↔  < 0,

(16.7.14) (16.7.15)

0 +  > 1 ↔  = 1, 0 +  = 1 ↔  = 0,

(16.7.16) (16.7.17)

0 +  < 1 ↔  = −1.

(16.7.18)

When  = 0, (16.7.9) reduces to (16.5.13). Further, the consequences of the restriction (16.7.11) depend on which of (16.7.12)–(16.7.18) we choose. We refer only to the case  = 0. If  = 0 and  = 0, then in Sect. 16.6 we proved that ρ0 = ρc , and we obtained an expanding universe. Now, if  > 0, condition (16.7.17) holds, and (16.7.11) gives − W (a) =

0 +  a 2 ≥ 0. a

(16.7.19)

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This condition is satisfied for all a ∈ [0, ∞), provided that 0 < 1, i.e., ρ0 < ρc . In contrast, if  < 0, the admissible values of a are obtained from inequality the − W (a) =

0 − | | a 2 ≥ 0, a

(16.7.20)

where in view of (16.7.17), 0 − |λ | = 1. In conclusion, we have

0 < ac ≤

0 |λ |

1/3 ,

(16.7.21)

and the radius of the universe oscillates between 0 and ac . Similar considerations can be extended to the other cases.

Chapter 17

Relative Formulation of Physical Laws

In Chap. 7 we described the procedures that in special relativity allow us to derive the physical laws relative to an inertial frame starting from their 4-dimensional formulation and vice versa. Determining analogous procedures in general relativity is a more difficult task. In this chapter, we analyze reasonable criteria leading to the relative formulation of physical laws starting from their 4-dimensional formulation. By this approach we determine the momentum equation and the energy equation of a material point relative to an arbitrary frame of reference starting from the fourdimensional momentum–energy equation. Then we compare the obtained equation with similar equations proposed in the literature. Finally, the above procedure is applied to derive the relative charge conservation, Maxwell’s equations, and the balance equations of an incoherent fluid.1

17.1 Introduction In Chap. 7 we analyzed how Einstein derived the Lorentz transformations from the optical isotropy principle. Furthermore, as a consequence of the relativity principle, we showed that physical laws must be invariant in form under Lorentz transformations. After verifying the covariance of Maxwell’s equations under these transformations, Einstein determined the equation governing the dynamics of a particle in special relativity. In Chap. 8 we described the geometric formulation of special relativity proposed by Minkowski, who adopted a pseudo-Euclidean 4-dimensional space V4 as a model of spacetime. The hyperbolic nature of the metrics allows one to define at every point x ∈ V4 a cone Cx , called the light cone at x. In this model, every inertial frame I is represented in V4 by an orthogonal frame of reference R = O x 1 x 2 x 3 x 4 with the fourth axis O x 4 contained in the light cone C O . As a consequence, the Lorentz 1 For

the content of this chapter, see the review paper [142].

© Springer Nature Switzerland AG 2019 A. Romano and M. Mango Furnari, The Physical and Mathematical Foundations of the Theory of Relativity, https://doi.org/10.1007/978-3-030-27237-1_17

447

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17 Relative Formulation of Physical Laws

transformation between two inertial frames I and I  is regarded as an orthogonal transformation between the orthogonal frames R and R  , corresponding in V4 to the inertial frames I and I  , respectively. Then the principle of relativity is satisfied by those physical laws that can be written as tensor equations of V4 . In conclusion, in this formulation of special relativity, the correspondence between physical quantities and geometric objects of V4 is well defined, since the rule is given to go from the physical description of a phenomenon in an inertial frame to its geometric formulation in the corresponding orthonormal frame of V4 . Then we showed that to overcome the contradictions contained in Newton’s theory of gravitation, Einstein adopted a hyperbolic Riemannian manifold as a model of spacetime V4 , assigning to the metric coefficients the role of gravitational potentials (see Chap. 11). Furthermore, assuming that the metrics of V4 are determined from the distribution of mass and energy, Einstein formulated the celebrated equations relating the gravitational field to the curvature of spacetime. This elegant formulation of physics places us in a 4-dimensional Riemannian manifold V4 without attributing a physical meaning to the many geometric structures contained in V4 . Only the equivalence principle introduces a partial correspondence between V4 and the physical world, associating local inertial frames with geodesic coordinates at a point of V4 . It is well known that a mathematical model can be considered a formal description of a class of phenomena if the correspondence between observable quantities and elements of the model is given. For instance, the Schwarschild model is a particular model of spacetime obtained by integrating Einstein’s equations under reasonable symmetry hypotheses. Then the correspondence between the mathematical objects and physical quantities is defined. A similar procedure can be adopted in cosmological models. Is it possible to extend this approach to any model of spacetime? Can we derive the form relative to an observer of a physical law if its tensor form is known? Conversely, can a physical observer go back to the structure of spacetime by suitable measures carried out in his frame of reference? All the papers devoted to the problem of deducing the physical laws, i.e., their relative form, from the corresponding 4-dimensional laws have the same starting point: the natural decomposition of any tensor quantity of V4 in space and time tensors in a geodesic frame. By an extensive use of this decomposition of tensors of V4 , in some papers (see, for instance, [4, 5, 7, 25, 27, 28, 52–55, 112]) are derived relative laws that are justified by their similarity with the corresponding classical laws in accelerated frames of reference. In other papers (see, for instance, [97, 100– 102, 130]), many affine connections are introduced in V4 that are compatible with the Riemannian metrics of V4 . Then it is shown that only one of these connections satisfies the Fermi–Walker transport (see [50, 176]).2 It is just this connection that is used to obtain the relative laws. Finally, in other papers (see, for instance, [47, 161]), fields of Lorentz frames are introduced on V4 (tetrad fields) together with affine connections in the fiber bundle of these tetrad fields to relate one Lorentz frame to another. Once again, the connection related to the Fermi–Walker transport is chosen 2 See

Sect. 8.11.

17.1 Introduction

449

to obtain reasonable relative laws. In the cited papers some geometric structures are introduced a priori in V4 , and then their physical meaning is investigated. The same procedure is adopted in [137, 149]. In this chapter, we propose reasonable physical assumptions to derive the relative form of physical laws, and we analyze the corresponding geometric structures that they introduce in V4 . From now on, we say that a physical law is expressed in relative form when it refers to an arbitrary physical frame of reference R, and in absolute form when it is written in arbitrary coordinates of V4 . Then from the above physical assumptions, we derive the relative equations of momentum and energy of a moving particle, already obtained by different approaches. Finally, this analysis is extended to charge conservation, Maxwell’s equations, and continuum mechanics. In such a way, we find how the observer in the frame of reference R describes the action of the gravitational field on a moving particle, an electromagnetic field, or a continuous system. In Sect. 17.2 we adopt the usual definition of a frame of reference R as a continuum of ∞3 particles. Its representation in V4 is given by the set of the world lines of these particles, which are supposed to form a timelike congruence , covering a region W of V4 . The frame R will be local if W ⊂ V4 , global if W coincides with V4 . Let x ∈ W be an event and denote by γ the unit vector tangent at x to the unique curve of  containing x. It is well known that (see Chap. 4) geodesic coordinates (x α ) exist in a neighborhood of x ∈ V4 such that the corresponding holonomic basis at x is a Lorentz basis, whose fourth axis is γ. These coordinates give the geometric representation of the proper inertial frame I x (the freely gravitating cabin) of the particle of R, at the event x. In this frame, in view of the equivalence principle, there is no gravitation. For the observer I x , special relativity holds locally, so that he knows how to obtain the physical quantities from 4-vectors and 4-tensors. The fundamental role of I x in formulating physical laws relative to R is stated by the following assumption (see [4–7, 112]): Assumption 17.1 The observer in the frame of reference R adopts at x the measures of the physical quantities carried out by the observer in I x , after expressing them in terms of his arbitrary coordinates. In other words, the observer in R entrusts the observer in I x with the task of measuring physical quantities in the absence of gravitation with the aim to adopt the results of I x , after expressing them in terms of his arbitrary coordinates. This assumption is more or less explicitly accepted in all the papers about this topic. Moreover, in [112] it is justified by assuming that the measuring rods have a length independent of their acceleration relative to an inertial frame. In any case, it is used to define the measures of elementary lengths and time intervals. Geometrically, this procedure corresponds to the decomposition of the absolute quantities along γ(x) and the three-dimensional vector space x of the 4-vectors orthogonal to γ(x) (see [25–28, 47, 51, 53–55, 65, 97, 100–102, 129, 130, 149, 176, 178]).3 3 See

Sect. 8.9.

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17 Relative Formulation of Physical Laws

In particular, Assumption 17.1 makes it possible to define the elementary spatial distance dσ between two particles of the frame R and the elementary standard time interval dT relative to R between two events. Furthermore, for a moving particle P, we can introduce kinematic and dynamic quantities relative to R as the standard velocity, standard momentum, and so on, at any event of the time world line x(τ ) of P, where τ is the proper time (Sect. 17.2). It is also important to notice that Assumption 17.1, although it introduces a simple criterion to measure physical quantities, generates a “tower of Babel effect,” since all the quantities used by R during the evolution of P are supplied by different inertial observers I x(τ ) that are unrelated to each other. Consequently, it is impossible to describe the dynamic evolution of P if there is no possibility to compare the different pieces of information supplied by the observers I x(τ ) . The second assumption, which introduces such a criterion, makes the following assumption (see, for instance, [24, 47, 50, 97, 100–102, 129, 130, 138, 149, 176, 178]): Assumption 17.2 Along the timelike world line of P, the spatial axes of the observer I x(τ +dτ ) must be related to the spatial axes of I x(τ ) by a Lorentz transformation without rotation. In other words, the spatial axes of the different frames I x(τ ) undergo a Fermi– Walker transport along the world line x(τ ) of P (Sect. 8.11). Although the literature on the Fermi–Walker transport is very extensive (see, for instance, [47, 50, 97, 100–102, 129, 130, 149, 176, 178]), in Sects. 17.2–17.3 we analyze the derivative d F V/dT along x(τ ) of a 4-vector V, introduced by Fermi–Walker transport (F–W transport). That is done with the aim to put in evidence the physical meaning of the F–W transport on the arbitrary observer rather than the geometric structures that can be defined in V4 by such a transport. Using the elementary spatial distance dσ between two particles of R and the Fermi transport, it is possible to determine the evolution equation of dσ along the world line of an arbitrary particle of R (see [50]). This equation allows us to describe all the kinematic characteristics of the frame R (Sect. 17.7). A first formulation of the relative equations of momentum and energy of a single particle P moving relative to R was proposed in [26–28] by a formal approach, which can be summarized as follows. Consider a material particle P with a rest mass m 0 and denote by x(τ ) its world line. Then the absolute equation of motion is d Pα = α , dτ

(17.1.1)

where d/dτ is the absolute derivative along x(τ ), P α = m 0 U α is the 4-momentum of P, U α = d x α /dτ its 4-velocity, and α the 4-force acting on P. Let R be a frame of reference defined by the timelike congruence  and denote by γ(x) the unit vector field of the tangent vectors to  and by x(τ ) the three-dimensional space orthogonal to γ(x(τ )). After the standard momentum  p and the standard energy E have been defined, the relative equations of momentum and energy are obtained by projecting

17.1 Introduction

451

(17.1.1) onto the three-dimensional space x(τ ) and along γ, respectively, and introducing without a physical motivation in the projection along x(τ ) a covariant space derivative of the standard momentum  p of P along its trajectory. v= Denote by P the linear operator projecting a 4-vector on x and let  P (dU α /dT ) be the standard velocity (Sects. 8.10, 17.2). Then in [65, 66], these equations are obtained from the following assumptions: dF of the standard • the force  F relative to R equates the Fermi–Walker derivative dT momentum  p of the particle P, and the power of  F equates the derivative of the energy E: d F p  = F, dT dE  = F · v. dT

(17.1.2) (17.1.3)

• The Eqs. (17.1.2), (17.1.3) are obtained by projecting the Eq. (17.1.1) on x(τ ) and γ, respectively. In [54, 55], some inconsistencies of the equations determined in [26–28] are pointed  of  out. Furthermore, some terms G F are identified with the gravitational force, since they have the same form of the inertial forces of classical mechanics. Before going on, we remark that the gravitational force acting on the material particle P during its motion relative to the frame of reference R is given by the vector  α dP p  = d F − P . (17.1.4) G dT dT Furthermore, in special relativity the two terms on the right-hand side of (17.1.4) are  vanishes. Finally, the first term has the same form equal, and the gravitational force G as in special relativity, except for the substitution d/dT → d F /dT . In conclusion, we can obtain (17.1.2) and (17.1.3) under the following assumption.  = dτ P (α ) evaluated by Assumption 17.3 The relative quantities  p, E,  dT Assumption 17.1 from P α and α (i.e., in the absence of gravitation) satisfy the equations obtained from the corresponding equations of special relativity, with the substitution d/dT → d F /dT and the addition of the gravitational force (17.1.4) to  the applied force . The dynamics of a single material point P describes the evolution of the 4momentum along the world line of P. Therefore, the Fermi–Walker transport is sufficient to determine the corresponding relative equations of momentum and energy. On the other hand, the Maxwell equations and the balance equations of a relativistic continuum are differential relations among vectors and tensor fields in regions of spacetime. Consequently, to obtain the relative form of these laws, in Sects. 17.5 and 17.6

452

17 Relative Formulation of Physical Laws F

we define the Fermi–Walker covariant derivative ∇, which naturally follows from the F–W transport in the fiber bundle x∈V4 x . It remains to generalize Assumption 17.3 to obtain the relative form of the physical laws starting from their absolute form. Let A be a set of tensor fields satisfying a system B of differential equations in an arbitrary inertial frame I of special relativity. Denote by A the tensor fields representing the fields A in the spacetime V4 of general relativity. Furthermore, let B be the covariant formulation in V4 of the equations B. For instance, for the density of current j and the charge density , we have that A = ( j, c ), B reduces to the equation ∇ ·j+

∂ = 0, ∂t

(17.1.5)

A is the current 4-vector (J α ), and B is given by the covariant equation ∇α J α = 0.

(17.1.6)

 the physical quantities evaluated by the observer R according Finally, we denote by A to Assumption 17.1, i.e., in the absence of gravitation, starting from the tensor fields A. Then we extend Assumption 17.3 as follows.  satisfy the equations A that are obtained Assumption 17.4 The physical quantities B from B, that is, from the equations that the quantities A satisfy in special relativity, by the following correspondence rules: dF d → , dt dT

∂ ∂F → , ∂t ∂T

F

∇ → ∇,

(17.1.7)

∂F is the F–W derivative along a curve of the frame R. Furthermore, the ∂T difference between the space and time projections of the absolute equations B and   and the gravitational field. B give the interaction between the fields A

where

For instance, in a frame R, instead of 17.1.5, we obtain F  ∂F  = φ, ∇ · j+ ∂T

where

  F  ∂F  φ = ∇ · − P (∇α J α ). j+ ∂T

(17.1.8)

(17.1.9)

In this way, in Sect. 17.10 we determine the relative form of the continuity equation of charge, while in Sects. 17.11 and 17.12 we find the relative forms of both Maxwell’s equations and the balance equations of an incoherent fluid. On the right-hand side of all these relative equations there appear terms that describe the interactions of charges and currents with the gravitational field.

17.2 Timelike Congruences

453

17.2 Timelike Congruences In this section we recall some definitions and results of previous chapters for completeness. Let V4 be a 4-dimensional Riemannian manifold whose metric tensor g has hyperbolic signature (+, +, +, −) and let Tx V4 be the vector tangent space to V4 at the point x ∈ V4 . Then every vector v ∈ Tx V4 is said to be, respectively, a spacelike vector, a lightlike vector, or a timelike vector according to whether it verifies the first, second, or third of the following conditions: g(v, v) > 0,

g(v, v) = 0,

g(v, v) < 0.

Definition 17.1 The element (x, eα ) of the tangent bundle T V4 , α = 1, . . . , 4, is a Lorentz frame at x if g(eα , eβ ) = ηαβ , where ηαβ = 0 if α = β, and ηii = 1, i = 1, 2, 3, η44 = −1. Definition 17.2 A curve γ of V4 is timelike if its tangent vector is timelike at every point of γ. It is evident how to define a lightlike curve and a spacelike curve of V4 . Definition 17.3 A timelike congruence  on the region W ⊂ V4 is a family of timelike curves such that every point x ∈ W belongs to one and only one curve of . Definition 17.4 A system of coordinates (y α ), α = 1, . . . , 4, is said to be adapted to the congruence  if the curves y 4 = var locally coincide with the curves of . If (eα ) denotes the natural basis of these coordinates, then the vector field e4 is tangent to the curves of . In adapted coordinates, the unit vector field γ = e4 /(e4 · e4 ) has, respectively, the following contravariant and covariant components: 1 ), γ α = (0, 0, 0, √ −g44 gα4 γα = gα4 γ 4 = √ . −g44

(17.2.1) (17.2.2)

A change of adapted coordinates is expressed by the functions y i = y i (y 1 , y 2 , y 3 ), 4

4

i = 1, 2, 3,

y = y (y , y , y , y ). 1

2

3

4

(17.2.3) (17.2.4)

Theorem 17.1 Let  be a timelike congruence in the region W of V4 and let γ(x) be the unit timelike vector field tangent to the curves of . Then in a neighborhood of every x ∈ W , there exist local coordinates (x α ), called geodesic coordinates, with the following properties:

454

17 Relative Formulation of Physical Laws

• The coordinates of x vanish. • γ(x) is the tangent vector at x to the coordinate curve x i = 0, i = 1, 2, 3, x 4 = var. • The curves x i = const, i = 1, 2, 3, x 4 = var are timelike in a neighborhood of x. Therefore, they locally define a timelike congruence to which they are adapted. • This leads to the result g αβ (x) = ηαβ ,

α

 βμ (x) = 0.

(17.2.5)

Then the covariant derivative at x coincides with the ordinary derivative. Furthermore, x and the natural base (ei , γ(x)) form a Lorentz base. • The world line of the origin of Ix , in a neighborhood of x 4 = 0, is an arc of a geodesic. Proof The proof of the above theorem is given in Sect. 4.11.



Let  be the timelike congruence in the region W ⊂ V4 and denote by γ(x) the unit vector field tangent to the curves of . For all x ∈ W , we consider the onedimensional vector space x (γ) of all timelike vectors such that x (γ) = {u = λγ, λ ∈ },

(17.2.6)

as well as the three-dimensional space v ∈ Tx , v · γ = 0} x (γ) = {

(17.2.7)

of all spacelike vectors  v that are orthogonal to γ at x. For every choice of the orthonormal triad (ai ) ∈ x (γ), we obtain a Lorentz frame (x, ai , γ(x)) at x. Let Tx be the tangent space to the spacetime V4 at the point x ∈ W . Then every v ∈ x (γ) 4-vector v ∈ Tx can be written as the sum of a spacelike vector P (v) =  and a timelike vector Pθ (v) ∈ x (γ): v = P (v) + P (v).

(17.2.8)

If we put P (v) = μγ, then the scalar product of (17.2.8) and γ, since γ · γ = −1, gives us the value of μ: μ = −γ · v. (17.2.9) Substituting this expression into (17.2.8), we finally get P (v) = (g + γ ⊗ γ)v, P (v) = −(γ · v)γ.

(17.2.10) (17.2.11)

Let (x i ) be an arbitrary system of coordinates in a neighborhood of x and let (gi j ) be the components of the metric tensor in these coordinates. Then the components of the tensor G + γ ⊗ γ are

17.2 Timelike Congruences

455

γαβ = gαβ + γα γβ .

(17.2.12)

For the extension of the decomposition (17.2.8) to any tensor, see Sect. 8.10, [4, 6, 27]. From now on, we define space vectors and time vectors as the vectors belonging to x (γ) and x (γ), respectively. Remark 17.1 For every space tensor we can use both the tensors gαβ and γαβ to relate contravariant and covariant components. In other words, γαβ defines a scalar product in every three-dimensional subspace x (γ). The pseudotensor

 ηαβμ = ηαβμδ γ δ ≡

√

−g αβμδ γ δ ,

(17.2.13)

where g = det(gαβ ) and αβμδ is the Levi-Civita symbol, is a space pseudotensor, since from the symmetry of γ α γ δ and the skew-symmetry of αβμδ , it follows that  ηαβμ γ α = ηαβμδ γ δ γ α = 0,

(17.2.14)

and the same result holds for the other indices. In coordinates (y α ), adapted to the congruence , in view of (17.2.11), we can write (17.2.13) as  ηαβμ =

√ 1 −g αβμ4 √ , −g44

and all the components of  ηαβμ with index equal to 4 vanish. Furthermore, we have g44 γ = g, where γ = det(γαβ ).4 Consequently, we have that  ηαβμ =

√

γ αβμ4 ≡

√

γ

αβμ .

(17.2.15)

u The space tensor  ηαβμ allows us to define the cross product of two space vectors  and  v by the formula √

αβμ u β vμ. (17.2.16) ( u × v)α = γ  αβ : α∗ of a given space 2-tensor T Similarly, we can define the adjoint space vector T βμ . α∗ = 1 √γ 

αβμ T T 2

(17.2.17)

We conclude this section by recalling a theorem proved in Sect. 8.11. Let σ be any curve of V4 defined by the equation x = x(λ), λ ∈ [0, a]. We denote by γ(x), x ∈ W , the unit timelike vector field γ · γ = −1, 4 See

[112], Appendix n.8.

(17.2.18)

456

17 Relative Formulation of Physical Laws

Fig. 17.1 Fermi–Walker transport

which is tangent to the integral curves of a timelike congruence  at every point. Finally, we introduce at the initial point x(0) of σ three unit space vectors ai0 , i = 1, 2, 3, which are orthogonal to each other and to γ(0). Consequently, (x(0), ai0 , γ(0)) is a Lorentz frame in V4 , with the origin at the point x(0) of σ (see Fig. 17.1). Theorem 17.2 ([50]) It is possible to determine at every point x(λ) of the curve σ three unit space vectors ai (λ), i = 1, 2, 3, such that for all λ ∈ [0, a]: • L Rx(λ) ≡ (x(λ), ai (λ), γ(λ)) is a Lorentz frame. • The Lorentz frame L Rx(λ) is related to the Lorentz frame L Rx(λ+dλ) by an infinitesimal Lorentz transformation without rotation. • The space vectors ai (λ) are the solution of the following initial value problem:   dai dγ = ai · γ, dλ dλ ai (0) = ai0 .

(17.2.19) (17.2.20)

Definition 17.5 The solution aα (λ) of the initial value problem (17.2.19), (17.2.20) is called the Fermi–Walker transport of the Lorentz frame (ei0 , γ(0)) along the curve σ.

17.3 The Fermi–Walker Derivative Let  be a timelike congruence in the region W ⊂ V4 and denote by γ(x) the unit vector field tangent to the curves of . Let (ai (λ), γ(λ)) be the Lorentz frames obtained by the Fermi–Walker transport along the curve x(λ) of a Lorentz frame (ai (0), γ(0)) (see Theorem 17.1). Let us consider a C 1 vector function v(λ) assigned α along x(λ). If V (λ) denote the components of v(λ) relative to the base (ai (λ), γ(λ)), then we have i 4 v + v, (17.3.1) v = V ai + V γ ≡ 

17.3 The Fermi–Walker Derivative

457

where (see (17.2.10), (17.2.11)) i

 v = V ai ,

4

 v = V γ = −(v · γ)γ.

(17.3.2)

Definition 17.6 The Fermi–Walker derivative (F–W derivative) of v(λ) along the curve x(λ) is the vector function along x(λ) given by α

dV dF v = aα , dλ dλ

(a4 = γ).

(17.3.3)

In other words, the F–W derivative of v(λ) is evaluated with respect to the family of Lorentz frames (ai (λ), γ(λ)), where the vectors ai (λ), which are transported according to (17.2.19), are supposed to be constant. In Sect. 8.11 we proved the following theorem. Theorem 17.3 The Fermi–Walker derivative along a curve x(λ) is related to the absolute derivative in V4 by the formula   dF v dv dγ dγ = − v· γ + (v · γ) . dλ dλ dλ dλ

(17.3.4)

In particular, the F–W derivative along x(λ) of a space vector field v(λ)∈x(λ) (γ(λ)) is equal to the space projection of the absolute derivative of  v(λ) along x(λ): d F v = dλ



d F v dλ

 

 =

d v dλ



 

,

v d F dλ

 

= 0,

(17.3.5)

where we have used the more compact notation =x(λ) (γ(λ)) and =x(λ) (γ(λ)). The following definition and theorem extend (17.3.3), (17.3.4), and (17.3.5) to a 2-tensor. Definition 17.7 Let (aα (λ)) ≡ (ai (λ), γ(λ)) be the Lorentz frames obtained by the Fermi–Walker transport along the curve x(λ) of a Lorentz frame (ai (0), γ(0)). αβ Denote by T(λ) a 2-tensor of C 1 class along x(λ). If T (λ) are the components of T(λ) relative to the base (aα (λ)), then we have that T=T

αβ

aα ⊗ aβ .

(17.3.6)

The Fermi–Walker derivative of T(λ) along the curve x(λ) is αβ

dT dF T = aα ⊗ aβ , dλ dλ

(17.3.7)

that is, the derivative relative to the base (aα ), which undergoes an F–W transport along the curve x(λ).

458

17 Relative Formulation of Physical Laws

Theorem 17.4 The Fermi–Walker derivative of a 2-tensor T(λ) along the timelike curve x(λ) is dF T dT dγ dγ = −γ⊗ ·T+ ⊗γ·T dλ dλ dλ dλ (17.3.8) dγ dγ ⊗γ+T·γ⊗ . −T· dλ dλ Furthermore, if T(λ) ≡  T(λ) ∈  ⊗ , then we obtain dF T = dλ



dF T dλ



 

=

dT dλ

 

.

(17.3.9)

Proof Taking into account (17.3.6) and (17.3.7), we see that the absolute derivative of T(λ) is given by dF T dT daβ αβ daα αβ = +T ⊗ aβ + T aα ⊗ . dλ dλ dλ dλ

(17.3.10)

Introducing (8.11.2) in (17.3.10), we obtain   dT d F T dγ dγ αβ αβ aα · = +T γ ⊗ aβ − T (aα · γ) ⊗ aβ dλ dλ dλ dλ     dγ dγ αβ αβ γ − T aα ⊗ aβ · γ + T aα ⊗ aβ · dλ dλ dγ dF T dγ αβ αβ +γ⊗ · T aα ⊗ aβ − ⊗ γ · T aα ⊗ aβ = dλ dλ dλ dγ dγ αβ αβ ⊗ γ − T aα ⊗ aβ · γ ⊗ . + T aα ⊗ aβ · dλ dλ Recalling (17.3.7), we conclude that formula (17.3.8) is proved. If T is a 2-tensor belonging to  ⊗ , then T · γ = γ · T = 0, γ is orthogonal to , and the theorem is proved. 

17.4 The Fermi–Walker Covariant Derivative Let x be a vector field of class C ∞ in V4 . Denoting by F(V4 ) the set of C ∞ functions on V4 , by x(λ) an arbitrary integral curve of the field x, and by X(V4 ) the F(V4 )module of the vector fields of class C ∞ on V4 , we give the name Fermi–Walker covariant derivative (F–W covariant derivative) to the derivation operator F

∇ : X(V4 ) → X(V4 ),

17.4 The Fermi–Walker Covariant Derivative

which is defined as follows:

459

F dF v = x · ∇v. dλ

(17.4.1)

By introducing (17.3.4) into (17.4.1) and using the arbitrariness of x, we have that F

∇v = ∇v − (∇γ · v) ⊗ γ + (v · γ)∇γ.

(17.4.2)

Note that the map F

v ∈ X(V4 ) → x · ∇v ∈ X(V4 )

(17.4.3)

is F(V4 )-linear in x. Furthermore, if we define the F–W derivative along the integral curve x(λ) of x of the function f ∈ F(V4 ) as df dF f = , dλ dλ

(17.4.4)

then the map (17.4.1) satisfies the property F

F

F

∇ x ( f v) = f ∇ x v + v · ∇ x f.

(17.4.5)

In conclusion, the F–W covariant derivative is a derivation in the tangent fiber bundles T V4 of the vector fields on V4 . Finally, we remark that from (17.4.2), it follows that the covariant and contravariant components of the F–W derivative are related by the metric tensor g of V4 as in the ordinary covariant derivative. In components, (17.4.2) can be written as follows: F

∂V α α + βλ V λ + (γλ ∇β γ α − ∇β γλ γ α )V λ ∂x β F ∂V α α λ ≡ +  βλ V , ∂x β

∇β V α =

(17.4.6)

where we have introduced the quantities, which we call F–W symbols, F   α  αβλ = βλ + γλ ∇β γ α − ∇β γλ γ α .

(17.4.7)

Relations (17.4.7) show that the connection introduced in V4 by the F–W transport is not symmetric and that it depends on both the metrics of V4 and the frame of reference R.  Let us consider the fiber bundle x∈V4 x of the space vector fields of V4 corresponding to a given choice of the timelike congruence . Let f (x) be a function of x be a space vector field. Then from (17.4.4), we obtain the following F(V4 ) and let  expression for the F–W gradient of f :

460

17 Relative Formulation of Physical Laws

dF f = x · ∇ f = x · P (∇ f ), dλ

∀ x∈



x ,

(17.4.8)

x∈V4

which, owing to the arbitrariness of  x, leads us to introduce the operator F

∇ f = P (∇ f ),

(17.4.9)

whose components are F

∇ β f = γαμ

∂f = ∂xμ



∂ ∂ + γβ γ μ ∂xβ ∂xμ



β f. f ≡∂

(17.4.10)

Remark 17.2 Equality (17.4.10) shows that the F–W gradient of the function f β f introduced by Carlo Cattaneo in [4, 6, coincides with the transverse derivative ∂ 27]. However, (17.4.10) makes clear its meaning relating the transverse derivative to the F–W derivative. Theorem 17.5 Let  bea timelike congruence on a region W ⊂ V4 and consider the fiber bundle V4 ≡ x∈V4 x of all three-dimensional subspaces that at every F

point x ∈ V4 are orthogonal to γ(x). Then the restriction of ∇ to V4 is a covariant derivation in V4 such that F

∇ v = (∇ v) . Proof If  x, v∈

 x∈V4

(17.4.11)

x , then from (17.3.2), (17.3.3), we have that

i dv i v d F j ∂v = ai = X ai . (17.4.12) dλ dλ ∂x j  v/dλ ∈ x∈V4 x , but also, the F–W derivaConsequently, not only do we have d F tive becomes a space operator, which in geodesic coordinates has the following components: F ∂v i ∇ v= ai ⊗ a j . (17.4.13) ∂x j

Owing to the space character of  x and  v, from (17.4.2) we obtain F

∇ v = P (∇ v − (∇γ ·  v) ⊗ γ) = P (∇ v + (∇ v · γ) ⊗ γ) ,

(17.4.14)

where the space projection refers to the operator ∇, that is, to the derivation index. The above equality can also be written as F

∇ v = P (∇ v) − P ((∇ v · γ) ⊗ γ) + P ((∇ v · γ) ⊗ γ) ,

17.4 The Fermi–Walker Covariant Derivative

461



and (17.4.11) is proved. Remark 17.3 In general coordinates, (17.4.11) can be written as F

∇ β v α = γμα γβν ∇ν vμ,

(17.4.15)

and in [4, 6, 27] it is shown that the right-hand side of (17.4.15) can also be put in the form F α μ β β ∇ β vα = ∂ v α + βμ  v ≡∇ vα , (17.4.16) where α βμ =

 1 αν  μ γνβ − ∂ ν γβμ . ∂β γμν + ∂ γ 2

(17.4.17)

β Then the covariant derivative ∇ v α introduced by Cattaneo coincides with the F–W covariant derivative. In other words, the coefficients of F–W connection are obtained by the metric tensor γαβ by substituting the ordinary spatial derivative with the transverse one.

17.5 F–W Derivation of 2-Tensors The results of the above section can be easily extended to any tensor. In particular, the F–W covariant derivative of the 2-tensor T is given by (see (17.3.8)) F

∇T = ∇T − γ ⊗ ∇γ · T + ∇γ ⊗ γ · T − T · ∇γ ⊗ γ + T · γ ⊗ ∇γ,

(17.5.1)

which in components can be written as follows: F

αβ

∇Tλ = ∇λ T αβ − γ α ∇λ γμ T μβ + ∇λ γ α γμ T μβ − T αμ ∇λ γμ γ β + T αμ γμ ∇λ γ β .

(17.5.2)

When  T is a space tensor, instead of (17.4.11), we have that F   ∇ T = ∇ T  .

(17.5.3)

Theorem 17.6 The following results hold: F

∇ λ gαβ = 0,

F

∇ λ γαβ = 0.

(17.5.4)

462

17 Relative Formulation of Physical Laws

Proof Since ∇λ gαβ = 0, from (17.5.2) we obtain that F

∇ λ = −γα ∇λ γ μ gμβ + ∇λ γα γ μ gμβ − gαμ ∇λ γ μ γβ + gαμ γ μ ∇λ γβ = −γα ∇λ γβ + γβ ∇λ γα − γβ ∇λ γα + γα ∇λ γβ = 0. On the other hand (see (17.2.12)), we have γαβ = gαβ + γα γβ , so that in view of (17.5.4) and (17.5.2), we have F

F

∇ λ γαβ = ∇ λ (γα γβ ) = ∇λ (γα γβ ) − γα ∇λ γμ γ μ γβ + ∇λ γα γ μ γμ γβ − γα γ μ ∇λ γμ γβ + γα γ μ γμ ∇λ γβ . But γ μ γμ = −1 implies γ μ ∇λ γμ = γμ ∇λ γ μ = 0, and from the above equation there  follows (17.5.4)2 . Remark 17.4 In view of (17.5.4), both the tensors gαβ and γαβ commute with the F

derivation operator ∇ applied to a space tensor. Remark 17.5 In components, (17.5.3) can be written as F

μ

νρ . ∇ λ Tαβ = γλ γαν γβρ ∇μ T As for (17.4.15), in [4, 6, 27], it is proved that F

α μβ λ T αβ + λμ αμ . ∇ λ T αβ = ∂ T + λμ T β

17.6 Frames of Reference Minkowski’s formulation is obtained by starting from a physical approach to special relativity. In other words, its geometric objects, i.e., spacetime V4 , events, metrics, 4-momentum, electromagnetic tensor, etc., are derived from physical quantities. Consequently, there is no ambiguity about the physical meaning of the geometric objects defined on V4 . In particular, in V4 only rectilinear coordinates are used as a geometric representation of the inertial frames. A quite different situation occurs in general relativity. For instance, general coordinates in V4 represent only an arbitrary way to label the events of the spacetime, so they can have no physical meaning. Consequently, there is no correspondence between coordinates of V4 and physical frames of reference R. Even if such a correspondence is introduced, can the observer in R evaluate the space distances and time intervals in R? More generally, can he deduce the physical quantities from the

17.6 Frames of Reference

463

geometric objects? All these questions require an answer if we wish to return from the spacetime to the observers and their physical description of the world. Up to now, we have equipped the spacetime V4 with many geometric structures. From now on, we make clear their physical meaning. Definition 17.8 A frame of reference R is formed by a fluid of reference S every particle p of which is equipped with a local time variable t. A timelike congruence  corresponds in V4 to the frame R. The curves of  represent the world lines of the ∞3 particles of R. Definition 17.9 A system of coordinates (y α ) is said to be adapted to the frame of reference R if it is adapted to the congruence defining R. Remark 17.6 A change of coordinates adapted to the same frame R does not modify the frame R. Therefore, every physical quantity relative to a frame R must behave like a tensor under a change of adapted coordinates. Let x ∈  be an event relative to a particle p of the fluid S of the frame of reference R. The event x is a point of the curve γ ∈  representing the world line of p. At x we consider the unit vector γ(x) tangent to  at x and the vector spaces x (γ), x (γ) defined by (17.2.6) and (17.2.7). Definition 17.10 The geodesic rest frame I x relative to the event x is the local geodesic frame (see Theorem 17.5) around x having as natural base at x the Lorentz axes (ai , γ), where (ai ), i = 1, 2, 3, is an orthogonal base in x (γ). This frame has to be identified with the local frame in which gravitation is absent and the physical laws of special relativity hold. Remark 17.7 Let R be a frame of reference represented by the congruence  and denote by x(λ) the world line of a particle moving relative to R. Then the above definition allows us to say that F–W transport along x(λ) of a Lorentz base (ai (λ), γ(λ)) defines a family of geodesic rest frames along x(λ) with the property that the space axes of (ai (λ))) and (ai (λ + dλ)) are related to each other by a Lorentz transformation without rotation. The first fundamental assumption about the measures in the general frame R can be formulated as follows: Assumption 17.1 The measures of physical quantities at an arbitrary event x ∈ W are carried out by an observer in the geodesic frame I x and then adopted by R after being expressed in terms of arbitrary coordinates of V4 . The next assumption makes clear how to evaluate the variations of a space vector with respect to a frame R: Assumption 17.2 The variations of a space vector  v along a curve x(λ) must be evaluated with respect to the family of local frames (ai (λ), γ(λ)) that are F–W transported along x(λ).

464

17 Relative Formulation of Physical Laws

In view of the first of the above assumptions, the observer R at x adopts the space and time measures carried out by the observer in the Lorentz frame Ix ≡ (x, ai , γ(x)). Consequently, the space distance dσ and the time interval dT between two events x and x + dx are obtained as follows. In the base (ai , γ), we have that i

+ P (dx). dx = d

x ai + cdT γ = P (dx) + P (dx) ≡ dx

(17.6.1)

Then from (17.6.1), (17.2.10), (17.2.11), and (17.2.12), in the coordinates (x α ), we can write

· dx

= γαβ d x α d x β , dσ 2 = dx 1 dT = − γα d x α . c

(17.6.2) (17.6.3)

Finally, from (17.2.12), (17.6.2), and (17.6.3) we obtain ds 2 = dσ 2 − c2 dT 2 ,

(17.6.4)

where ds is the spacetime distance between x and x + dx.

17.7 Kinematic Characteristics of a Frame of Reference Let R be a frame of reference defined by the timelike congruence . We denote by (y α ) coordinates adapted to , by (x α ) general coordinates, and by x α = x α (y β ),

(17.7.1)

the equations of  in the coordinates (x α ). A curve σ ∈  is obtained from (17.7.1) by assigning y 1 , y 2 , y 3 and varying y 4 . In view of (17.6.3) and (17.1.1), the elementary time interval dT along σ becomes dT =

1√ −g44 dy 4 , c

so that on σ, we can introduce the variable  1 √ T = −g44 dy 4 , c σ

(17.7.2)

(17.7.3)

whose variations along σ coincide with the elementary interval dT . If we adopt this new parameter along the curves of , then we can write (17.7.1) in the form x α = x α (y i , T ).

(17.7.4)

17.7 Kinematic Characteristics of a Frame of Reference

465

Differentiating (17.7.4), δx α =

∂x α μ δ y , (y 4 = T ), ∂ yμ

(17.7.5)

and considering the projection of (17.7.5) on x , we obtain the space vector between the two particles (y i ) and (y i + δ y i ) of the frame of reference R: δ x = P (δx).

(17.7.6)

Assumption 17.2 introduces a criterion to evaluate the variations along the curve x(λ) ∈  of the space vector d x with respect to the frame R. In fact, we must consider the family of geodesic frames I (T ) along x(λ), i.e., the proper geodesic frames of the particle of R having x(λ) as world trajectory. Then after choosing a space base ai (0) in the geodesic frame I (0), we introduce in any geodesic frame I (T ) a space base ai (T ) obtained by a Fermi–Walker transport of ai (0) along x(λ). Consequently, the space vector d x is given by δ x = δx i ai (T ).

(17.7.7)

We assume that the variations of δ x with respect to R are expressed by the F–W derivative (17.3.3), δx i dF δ x= ai (T ), (17.7.8) dT dT which, in view of (17.3.5), can also be written in the equivalent form dF δ x = P dT



 d δ x . dT

(17.7.9)

Since it is possible to prove that  P

 d δ x = δ x · P (∇γ), dT

(17.7.10)

Equation (17.7.9) assumes the following form: dF d x = δ x · P (∇γ). dT

(17.7.11)

If we introduce the space tensors  1 αλ βμ  γ γ ∇λ γμ − ∇μ γλ , 2   1 = γ αλ γ βμ ∇λ γμ + ∇μ γλ , 2

αβ = 

(17.7.12)

αβ K

(17.7.13)

466

17 Relative Formulation of Physical Laws

which are, respectively, skew-symmetric and symmetric, then (17.7.11) becomes dF  + K).  d x = δ x · ( dT

(17.7.14)

We can recognize the physical meaning of the above two space tensors by noting that in the proper geodesic frame I (T ) they have only spatial components, given by 1 i j = 2 Kij

1 = 2



∂γ j ∂ Xi ∂γ j ∂ Xi

− +

∂γi ∂X j ∂γi

,

(17.7.15)

,

(17.7.16)

∂X j

 the space curly tensor and K  the space stretching so that it is justified to call  tensor. From (17.6.2) and (17.7.14) we obtain the relation dF d d δσ 2 = (δx · δx) = δ x· d x dT dT dT  + K)  · δ  · δ = δ x · ( x = δ x·K x,  is skew-symmetric and K  is symmetric. The above equation allows us to since   = 0 (Born rigidity). state that a frame R is rigid if and only if K We conclude by proving the following theorem: Theorem 17.7

αβ − γα Cβ , αβ + K ∇α γβ = 

where Cα =

∇γ α , C α γα = 0 ds

(17.7.17)

(17.7.18)

is the curvature 4-vector of the world lines of the particle of the frame R. Proof First, we note that the tangent space Tx at the point x ∈ V4 can be decomposed into the following direct sum of its vector subspaces: Tx = x ⊗ x + x ⊗ x + x ⊗ x + x ⊗ x .

(17.7.19)

Consequently, the components ∇α γβ are obtained by adding the components of the following tensors belonging to the subspaces appearing in (17.7.19):

17.7 Kinematic Characteristics of a Frame of Reference

467

αβ , αβ + K γ αλ γ βμ ∇λ γμ =  −γ αλ γ β γ μ ∇λ γμ = 0, −γ α γ λ γ βμ ∇λ γμ = −γ α Cβ ,

(17.7.20)

γ α γ λ γ β γ μ ∇λ γμ = 0. Equalities (17.7.20) can easily be proved. In fact, (17.7.20)1 follows from (17.7.12), (17.7.13). Furthermore, from the condition

it follows that

γμ γ μ = −1,

(17.7.21)

γ μ ∇λ γμ = 0,

(17.7.22)

and (17.7.20)2 is proved. Since Cα =

∇γα = γ μ ∇μ γα ds

(17.7.23)

and the condition (17.7.22) implies that γ α Cα = γ α

∇γα = γ α γ μ ∇μ γα = 0, ds

the identities (17.7.20)3−4 are proved.

(17.7.24) 

 = 0, rigid Definition 17.11 We say that a frame of reference R is irrotational if   if K = 0, geodesic if the curves of  are geodesics, or equivalently, if Cα = 0. Remark 17.8 The differential form (17.6.3) is integrable if and only if αβ ≡ ∂α γβ − ∂β γα = ∇α γβ − ∇α γβ = 0.

(17.7.25)

In this case, there exists a function F(x α ) such that γα = ∂α F. Consequently, the 4field γ is orthogonal to the surfaces F(x α ) = const, and the curves of the congruence , which define the frame of reference R, are orthogonal to the level surfaces F(x α ) = const. Furthermore, c dT = d F, and there exists a global standard time T whose simultaneity surfaces coincide with the surfaces F(x α ) = const. From (17.7.25) and  = C = 0, i.e., if (17.7.17), it follows that the integrability condition is satisfied if  the frame R is irrotational and geodesic.

17.8 Relative Momentum Equation In view of what we said in Sect. 7.2, the observer R at x adopts the space and time measures carried out by the observer in the Lorentz frame Ix ≡ (x, ai , γ(x)).

468

17 Relative Formulation of Physical Laws

Consequently, the space distance dσ and the time interval dT between two events x and x + dx are given by (17.6.2) and (17.6.3), while the distance ds in the spacetime between the events x and x + dx is given by (17.6.4). Let P be a material point moving relative to the frame R defined by the time congruence . We denote by x(τ ) the equation of the world line σ of P, where τ is the proper time. Applying the decomposition (17.2.8) to the 4-velocity U of P and taking into account (17.6.2), (17.6.3), and (17.6.4), we obtain U=

dT dx dT dx = = v + cγ) , ( dτ dτ dT dτ

where the space vector

  v = P

dx dT

 =

dx dT

(17.8.1)

(17.8.2)

is called the standard velocity relative to R. In view of (17.8.2), we have v α vα = γαβ v α v β = (gαβ + γα γβ ) v α v β = gαβ v α vβ .  v2 = 

(17.8.3)

The line element ds 2 in the spacetime V4 relative to the events x(τ ) and x(τ + dτ ) is given by (17.8.4) ds 2 = dσ 2 − c2 dT 2 = −c2 dτ 2 , so that

1 dT = . dτ  v2 1− 2 c

(17.8.5)

Let us consider the 4-momentum of P, P α = m0

dxα = m 0U α , dτ

(17.8.6)

where m 0 is the proper mass of P. At a point x(τ ), in view of (17.2.10) and (17.2.11), we obtain the space and time projections of P α , Pα =  pα +

E α γ ,  pα = γαβ P β , c

so that  pα = m 0 γαβ

E = −γα P α , c

d x β dT = m vα , dT dτ

where m=

m0  v2 1− 2 c

(17.8.7)

(17.8.8)

(17.8.9)

17.8 Relative Momentum Equation

469

is the standard mass. Finally, from (17.8.5), (17.8.6), and (17.8.7), we have E = −c m 0 γα

d x α dT = mc2 . dT dτ

(17.8.10)

The space vector  p is called the standard momentum, and the scalar E is called the standard energy. All the above quantities  v,  p, m, E, . . . are evaluated in the proper frame according to Assumption 17.1, that is, in the absence of gravitation. We underline that the vectors  p, F,  v and the scalar E are invariant with respect to coordinate changes internal to the frame R. The equation x(τ ) of the world-line of a moving particle P is a solution of the equation dU dp = m0 = , (17.8.11) dτ dτ where the 4-force  acting on P satisfies the condition  · U = 0.

(17.8.12)

We now propose to derive from (17.8.11) the equations describing the dynamics of P relative to an arbitrary frame of reference R, defined by a timelike congruence . First, we recall that in view of (17.7.4), we can use the standard time as a parameter on x(λ). Consequently, the equation of this curve can also be written in the form x(T ). As usual, we denote by Tx(T ) the tangent space to the spacetime V4 at the different points of x(T ), and we decompose Tx(T ) into the direct sum of the subspaces x(T ) and x(T ) . In x(T ) , we choose a triad (ai (T )) of space vectors that undergoes a Fermi–Walker transport along x(T ), and we denote by I (T ) the family of geodesic frames along x(T ) with Lorentz frames (ai (T ), γ(T )). If pi (T ) are the components of the standard momentum  p(T ) in (ai (T )), we have that  p = pi (T )ai (T ).

(17.8.13)

In view of Assumptions 17.1–17.3, we state that the momentum and energy equations of P relative to R have the following form:  p  d F  −  ≡ G, dT   dE   · −  · v =G v, dT 

where

 = dτ P (),  dT

(17.8.14) (17.8.15)

(17.8.16)

470

17 Relative Formulation of Physical Laws

and

p  = d F − P G dT



dp dT

 (17.8.17)

is the gravitational force relative to the observer R. It is worthwhile to notice that in special relativity, (17.8.14) and (17.8.15) supply the momentum equation and the energy equation relative to an inertial frame. In fact, in this case, the space and time projections are equivalent to the first three components of (17.8.11) and its fourth component, respectively, the standard time coincides with the coordinate time in an inertial frame, and the F–W derivation becomes the ordinary derivation with respect to time. From the decomposition (17.8.7) of the 4-momentum p, we get   d p d E dp = + γ dT dT dT c   E dγ d E d p + +γ = dT c dT dT c dτ dτ P () + P (). = dT dT

(17.8.18)

The space projection of the above relation on x(T ) gives us  P

dp dT





   d p dγ E = P + P dT c dT   dγ E p d F + P = dT c dT dτ  P () ≡ . = dT

(17.8.19)

From this relation, taking into account (17.8.14)–(17.8.17), we obtain p  d F −  = −mc P dT



dγ dT



 ≡ G.

(17.8.20)

On the other hand, owing to (17.7.17) and (17.8.1), we have also dxμ dγ α = ∇μ γα dT dT μα +  μα − γμ Cα )+ = ( v μ + cγ μ )( K μα +  μα ) + c Cα .  vμ( K

(17.8.21)

In conclusion, we have that dγ = P dT



dγ dT



  + ), = c C + v · (K

(17.8.22)

17.8 Relative Momentum Equation

471

and (17.8.20) becomes   d F p  +  + )  + . ≡G  = −m c2 C + c v · (K dT

(17.8.23)

Remark 17.9 The F–W derivative on the left-hand side of (17.8.23) allows us to regard the right-hand side as the total force. The choice of this derivation operator leads to (17.8.23), which differs from the equation proposed by Cattaneo in [25, 26], for the term mc v · K. Remark 17.10 The momentum equation relative to the frame R contains only quantities independent of the coordinates adapted to R. The total force acting on the material point P is given by adding the projection on x(T ) of the 4-force  and the  If the 4-force vanishes, i.e., if the point P is freely gravitating, then its world term G. line is a geodesic. In conclusion, for the observer R there are forces that have the same structure of the fictitious forces of classical mechanics, except for the presence of the term related to a nonrigid frame. In general relativity, these forces, which are due to the existence of the gravitational field produced by the choice of the frame R, are absent if and only if R is an inertial frame.

17.9 Relative Energy Equation Now we want to evaluate the energy equation relative to R. By the scalar product of (17.8.11) with γ, i.e., by projecting (17.8.11) on x(T ) , we get dp d p E dγ 1 dE = −γ · +γ· + γ· . c dT dT dT c dT

(17.9.1)

But we have dτ dp P () · γ = −γ · , dT dT dγ 1 d γ· = (γ · γ) = 0, dT 2 dT so that (17.9.1) becomes dE dτ d p =c P () · γ + cγ · . dT dT dT

(17.9.2)

The last term in (17.9.2) can be written as cγ ·

d p d( p · γ) dγ =c − c p· . dT dT dT

(17.9.3)

472

17 Relative Formulation of Physical Laws

The first term on the right-hand side of the above equation vanishes, since γ and  p are orthogonal. Then we have the relation cγ ·

dγ d p = −c p· . dT dT

(17.9.4)

The above computations allow us to write (17.9.2) in the form dE dγ dτ p· =c P () · γ − c . dT dT dT

(17.9.5)

Taking into account (17.7.17) and (17.8.23), we prove that the second term on the right-hand side of (17.9.5) is equal to  v · G. In fact, to transform the first term, we use (17.8.12) after decomposing  and U into space and time vectors    − dτ γ · γ · ( ·U=  v − cγ) = 0, dT

(17.9.6)

1 dτ γ · γ =  · v, dT c

(17.9.7)

dE  ·  · = v+G v. dT

(17.9.8)

so that

and (17.9.5) becomes

17.10 Continuity Equation Let v be an arbitrary vector field of V4 such that ∇ · v = 0.

(17.10.1)

Applying to v the standard decomposition v = v + aγ, a = −γ · v,

(17.10.2)

Equation (17.10.1) can be written as follows: ∇ · v = ∇ · v + a∇ · γ + γ · ∇a = 0.

(17.10.3)

In view of (17.7.17), we have also αα − γ · C = K αα , αα + K ∇ ·γ =

(17.10.4)

17.10 Continuity Equation

473

 is skew-symmetric and C is a space vector. Furthermore, we have since  1 ∂a da ≡ , ds c ∂T

γ · ∇a =

(17.10.5)

where T is the standard time. Taking into account (17.10.4) and (17.10.5), Eq. (17.10.3) can also be written as follows: ∇ · v = ∇ · v+

1 ∂a αα = 0. + aK c ∂T

(17.10.6)

Applying to ∇ v the standard decomposition, we have v) + (∇ v) + (∇ v) , ∇ v = (∇ v) + (∇

(17.10.7)

where vμ , ((∇ v) )αβ = γβλ γαμ ∇λ ((∇ v) )αβ = ((∇ v) )αβ = ((∇ v) )αβ =

−γαλ γβ γ μ ∇λ vμ λ μ −γα γ γβ ∇λ vμ λ μ γα γ γβ γ ∇λ vμ .

(17.10.8) (17.10.9) (17.10.10) (17.10.11)

Since γ αβ is a space tensor and γ α is a timelike vector, the divergence of (17.10.9) and (17.10.10) vanishes. On the other hand, the divergence of (17.10.11), recalling vμ = 0 and taking into account (17.7.17), gives that γ μ vμ = γ λ v μ ∇λ γμ =  v · C. − γ μ γ λ ∇λ

(17.10.12)

Finally, from (17.4.11) we derive F

(∇ v)αα = ((∇ v) )αα , so that

F

∇ · v = ∇ · v + v · C.

(17.10.13)

The results obtained allow us to write (17.10.6) in the form F

∇ · v+

1 ∂a αα +  + aK v · C = 0. c ∂T

(17.10.14)

If v is identified with the 4-current j,  v with the space current density  j = P (j) relative to the observer R, and a with c  = −γ · j, where  is the charge density relative to R, then (17.10.14) becomes

474

17 Relative Formulation of Physical Laws F

∇ · j+

∂ αα + j · C = 0, + K ∂T

(17.10.15)

and we can say that the charge conservation relative to the observer R does not retain its classical form, since the frame R is not rigid and the current density interacts with the gravitational field.

17.11 Divergence of a Skew-Symmetric Tensor Let  v be a space vector and denote by x(λ) a curve of the congruence  defining the frame R. If we choose the standard time as a parameter along this curve, then the F–W derivative of  v along the curve x(λ), in view of (17.3.4), (17.7.17), and (17.7.18), can be written as follows: d F d v v v ∂ F ≡ = − c v · Cγ, ∂T dT dT since γ=

(17.11.1)

dx 1 dx = . ds c dT

In this section we determine the relative form with respect to a frame of reference R of the following absolute equation: ∇β F αβ = J α ,

(17.11.2)

where F is a skew-symmetric 2-tensor. From the skew-symmetry of F there follows the following decomposition into the subspaces  and : αβ + F α γ β − γ α F β , F αβ = F

αβ = − F βα . F

(17.11.3)

Introducing (17.11.3) into (17.11.2), we obtain the equation  αβ   +F α γ β − γ α F β = J α . ∇β F

(17.11.4)

Now we analyze the single terms on the right-hand side of (17.11.4). First, in view of (17.11.1), (17.7.18), and (17.3.4), we have that α γ β ) = γ β ∇β F α ∇β γ β α + F ∇β ( F α 1 dF α K β +F = β c dT α 1 ∂F F β Cβ γ α + F α K β . +F = β c ∂T

(17.11.5)

17.11 Divergence of a Skew-Symmetric Tensor

475

Moreover, from (17.10.13) we derive the relation β ∇β γ α β ) = −γ α ∇β F β − F −∇β (γ α F   F  α  α   β  βα . β + K = −γ ∇ · F + F · C − F

(17.11.6)

αβ is skew-symmetric, we have that μβ = 0 and F Recalling that γμ F αβ = P (∇β F αβ ) − γ α γμ ∇β F μβ ∇β F αβ ) + γ α F μβ ∇β γμ = P (∇β F βμ − γβ Cμ ) αβ ) + γ α F μβ ( βμ + K = P (∇β F αβ ) − γ α F βμ  βμ . = P (∇β F In order to evaluate the term

(17.11.7)

αβ ), P (∇β F

αβ over the spaces we begin by considering all the projections of the 2-tensor ∇λ F obtained by the tensor product ( + ) ⊗ ( + ) ⊗ ( + ) and then evaluating their divergence. On  ⊗  ⊗ , we have μ β νδ ⇒ γ μ γνα ∇μ F νδ . γλ γνα γδ ∇μ F δ

(17.11.8)

On  ⊗  ⊗ , we have μ νδ ⇒ −γ μ γνα γ β γδ ∇μ F νδ = 0. − γλ γνα γ β γδ ∇μ F β

On  ⊗  ⊗ , we have μ β νδ ⇒ −γ μ γ α γν ∇μ F νδ . −γλ γ α γν γδ ∇μ F δ

νδ γν = 0, we obtain On the other hand, since F μ νδ = γ μ γ α F νδ ∇μ γν r −γδ γ α γν ∇μ F δ μ μν − γμ Cν ) νδ ( μν + K = γδ γ α F δν − γδ Cν ) νδ ( δν + K = γα F

νδ  δν , = γα F

(17.11.9)

476

17 Relative Formulation of Physical Laws

μ μν is skew-symmetric and K μν is symmetric. Finally, the diversince γδ γμ = 0, F gence of the projection on  ⊗  ⊗  is β

νδ = −γ α F μν  μν . − γ α γν γδ ∇μ F

(17.11.10)

On  ⊗  ⊗ , we have μ

μ

νδ ⇒ γ γ α γν γ β γδ ∇μ F νδ = 0. γλ γ α γν γ β γδ ∇μ F β

(17.11.11)

On  ⊗  ⊗ , we have β

β

νδ ⇒ −γβ γ μ γνα γ ∇μ F νδ = 0. − γ μ γλ γνα γδ ∇μ F δ

(17.11.12)

On  ⊗  ⊗ , we have νδ ⇒ γβ γ μ γνα γ β γδ ∇μ F νδ . γ μ γλ γνα γ β γδ ∇μ F Moreover, νδ = −γ μ γνα γδ ∇μ F νδ γβ γ μ γνα γ β γδ ∇μ F μδ − γμ Cδ ). νδ ∇μ γδ = γ μ F αδ ( μδ + K = γ μ γνα F νδ , there follows Therefore, from the space character of F β αδ Cδ . νδ = F γβ γ μ γνα γδ ∇μ F

(17.11.13)

On  ⊗  ⊗ , we have β νδ ⇒ γβ γ μ γ α γν γ β ∇μ F νδ = 0. γ μ γλ γ α γν γδ ∇μ F δ

(17.11.14)

On  ⊗  ⊗ , we have νδ ⇒ γ μ γν γ α γδ ∇μ F νδ . −γ μ γλ γ α γν γ β γδ ∇μ F On the other hand, we have νδ = −γ μ γν γ α F νδ ∇μ γδ = 0. γ μ γν γ α γδ ∇μ F

(17.11.15)

Collecting the results (17.11.8)–(17.11.15), we obtain the relation μν Cν − γ α F αβ = γ μ γνα ∇μ F νδ + F μν  μν , ∇β F δ

(17.11.16)

17.11 Divergence of a Skew-Symmetric Tensor

477

which in view of (17.4.15) gives F

μν Cν − γ α F αβ = ∇ β F αβ + F μν  μν . ∇β F

(17.11.17)

Finally, (17.11.5)–(17.11.8) and (17.11.17) supply α 1 ∂F F + c ∂T α K β ( βα )− β − F αβ Cβ + F αβ + K F β F

αβ + ∇β F αβ =∇ β F

(17.11.18)

F

μν  β + F μν ). γ α (∇ β F Applying to J α the standard decomposition j α + aγ α , a = −γβ J β Jα = 

(17.11.19)

and taking into account (17.11.18), we see that the absolute Eq. (17.11.2) is equivalent to the following relative equations: α F α K αβ Cβ + F β ( βα ) +  αβ + 1 ∂ F F = − F β − F αβ + K ∇β F j α, β c ∂T

(17.11.20)

F

μν  β = − F μν + a. ∇β F

(17.11.21)

To give (17.11.20) and (17.11.21) a more expressive equivalent form, we begin by proving the following formula: √ ∂F γ 1 = √ K αα , ∂T γ

(17.11.22)

where γ = det(γαβ ) and ∂ F /∂T denotes the F–W derivative along a curve x(T ) of the congruence  (see (17.11.2)). First, we note that the F–W derivative of a scalar quantity is equal to the ordinary derivative and recall the derivation rule of a determinant. Then along x(T ), we have that √ √ ∂ γ ∂F γ ∂ √ = = γμ μ γ ∂T ∂T ∂x γ μ ∂γ ∂γαβ γμ = √ = √ γγ αβ . μ 2 γ ∂x 2 γ ∂x μ Since in [27] it is shown that γ μ ∂γαβ /∂x μ = 2K αβ , formula (17.11.22) is proved.

478

17 Relative Formulation of Physical Laws

Taking into account this result, (17.11.20) and (17.11.21) become F 1 ∂ F √ α αβ Cβ + F β ( βα ) +  αβ + √ αβ + K ∇β F j α, ( γ F ) = −F c γ ∂T

(17.11.23)

F

μν  β = − F μν + a. ∇β F

(17.11.24)

17.12 Relative Maxwell’s Equations It is well known that the first two Maxwell’s equations can be written in the following absolute form: (17.12.1) ∇β F αβ = J α , where F αβ is the skew-symmetric electromagnetic 4-tensor and J α the 4-current. This equation has the form (17.11.2), which we have analyzed in the above section. Then the corresponding relative form is just given by (17.11.23) and (17.11.24). Referring to the decomposition (17.11.3), we decompose the electromagnetic tensor as follows: μ + cD α γ β − cD β γ α ,

αβμ H (17.12.2) F αβ =  μ is the adjoint space 4-vector of the skewwhere 

αβμ is defined in (17.2.13), H αβ  α = γ α γμ F βμ . It is a simple exercise to  symmetric space tensor F , and cDα = F β verify that Eqs. (17.11.23) and (17.11.24) assume the form  F × C + D × ω  · K + j,  + √1 ∂ F D = −H +D ∇ ×H γ ∂T F × ω  = −1H  + , ∇ ·D c where

1  ωα = 

αβμ βμ ,  = − γβ J β . c

(17.12.3) (17.12.4)

(17.12.5)

Moreover, the second group of Maxwell’s equations can be written in the following absolute form: ∇β F ∗αβ = 0, (17.12.6) where F ∗αβ is a skew-symmetric electromagnetic 4-tensor, which in vacuum is equal to the adjoint tensor of F αβ . Also, this equation has the form (17.11.2). Then the corresponding relative form is again given by (17.11.23) and (17.11.24). Referring to the decomposition (17.11.3), we decompose the electromagnetic tensor F ∗αβ as follows:

αβμ Eμ + c Cα γ β − c Cβ γ α , (17.12.7) F ∗αβ = 

17.12 Relative Maxwell’s Equations

479

∗αβ , and where Eμ is the adjoint space 4-vector of the skew-symmetric space tensor F α α ∗α ∗βμ  = γ γμ F . We can easily verify that Eqs. (17.11.23) and (17.11.24) cC = F β assume the form  F × ω  · K,  + √1 ∂ F B = −E × C + B +B ∇ ×B γ ∂T F  = − 1 E × ω . ∇ ·B c

(17.12.8) (17.12.9)

17.13 Divergence of a Symmetric Tensor If T is a symmetric 2-tensor, instead of the decomposition (17.11.3), we have the following alternative decomposition: αβ + T α γ β + γ α T β + hγ α γ β , T αβ = T

(17.13.1)

αβ is symmetric. where T We propose to apply the procedure of the above section to the absolute equation ∇β T αβ = 0.

(17.13.2)

In view of (17.13.1), Eq. (17.13.2) can be written as  αβ   +T α γ β + γ α T β + hγ α γ β = 0. ∇β T

(17.13.3)

Note that (17.13.3) differs from (17.11.4) in the sign of the third term in the parenαβ , as well as in the term ∇β (hγ α γ β ). This last one can theses, in the symmetry of T also be written in the form ∇β (hγ α γ β ) =γ α γ β ∇β h + hγ β ∇β γ α + hγ α ∇β γ β 1 ∂h βα − γβ C α )+ αβ + K + hγ β ( = γα c ∂T β − γβ C β ). ββ + K hγ α ( β Moreover, since γ α is timelike and C α is spacelike, we have that α β

α

∇β (hγ γ ) = hC + γ

α



 1 ∂h β  + h Kβ . c ∂T

(17.13.4)

α γ β ) in (17.12.4) can be written in the form (17.11.9); the term The term ∇β (T α β ∇β (γ T ) can be substituted by the opposite of (17.11.10). Furthermore, owing to μβ , the condition (17.11.20) is substituted by the equation the symmetry of T

480

17 Relative Formulation of Physical Laws μ

μν Cν + γ α T αβ = γ γνα ∇μ T νδ + T μν K μν , ∇β T δ

(17.13.5)

so that (17.11.21) becomes T

αν Cν . αβ ) = ∇ β T αβ + T P (∇β T

(17.13.6)

Finally, (17.13.5) can be put in the form α 1 ∂T + hC α + c ∂T αβ Cβ + T α K β ( βα )+ β − T αβ + K T β   1 ∂h β + γα + hK β c ∂T F

αβ + ∇β T αβ =∇ β T

(17.13.7)

F

β Cβ + T μν K β + T μν ), γ α (∇ β T whose spatial projection supplies the momentum balance α F αβ Cβ + T α K β ( βα ), αβ + 1 ∂ T = −hC α − T β + T αβ + K ∇β T β c ∂T

(17.13.8)

whereas the time projection gives the energy balance equation F 1 ∂h β − T β Cβ − T μν K β = −h K μν . + ∇β T β c ∂T

(17.13.9)

17.14 Momentum–Energy Tensor of Dust Matter The momentum–energy tensor of dust matter is T αβ = 0 U α U β ,

(17.14.1)

where (U α ) is the 4-velocity and 0 the proper mass density. Applying to (U α ) the decomposition U α = (v α + cγ α ),

(17.14.2)

we obtain 0 U α U β = v α v β + c(v α γ β + v β γ α + c2 γ α γ β ),

(17.14.3)

17.14 Momentum–Energy Tensor of Dust Matter

481

where  = 0  2 . Consequently, we have that

αβ = v α v β , T α = cv α T

(17.14.4)

h = c . 2

These relations allow us to write the relative momentum balance (17.12.7) and the relative balance of energy as follows: F

∇ β (v α v β ) +

∂v α = − c2 C α − v α v β Cβ ∂T βα ), β cv β ( αβ + K + cv α K β

F 1 ∂c2 β − cv β Cβ − v μ v ν K μν . + ∇ β (v β ) = − c2 K β c ∂T

(17.14.5)

(17.14.6)

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Index

Symbols (0, 2)-tensor, 6 degenerate, 26 nondegenerate, 26 (1, 1)-tensor, 9 (2, 0)-tensor, 8 contravariant, 8 2-tensor components contravariant, 8 covariant, 7 covariant, 6 mixed, 9 4-current, 248 4-force, 247 4-momentum, 247 4-vector, 240 lightlike, 240 null, 240 spacelike, 240 timelike, 240 4-velocity, 247 C h curve, 63 C k function, 60, 63 p-form, 69 r -chain, 97 r -cube, 95 faces of, 96 oriented, 96

A Acceleration Coriolis, 137 drag, 137 relative, 137 Ampère electromagnetic fields, 302

Anomalies periodic, 314 secular, 314 Antiderivation, 90 Atlas, 61 maximal, 61

B Bases controvariant, 59 global, 59 local, 59 reciprocal, 59 Basis controvariant, 59 covariant, 59 holonomic, 44, 67 natural, 67 reciprocal, 59 Bianchi identity, 116 theorem, 120 Big bang, 443 Big crunch, 443 Bivector, 122 Black hole, 418, 420 Black hole information loss paradox, 421 Body rigid, 133 Body of reference, 133 Bracket, 69

C Canonical basis, 25

© Springer Nature Switzerland AG 2019 A. Romano and M. Mango Furnari, The Physical and Mathematical Foundations of the Theory of Relativity, https://doi.org/10.1007/978-3-030-27237-1

491

492 form, 24, 25 Cauchy–Schwarz inequality, 28 Cauchy’s problem, 362, 367 Central inversion, 35 Chandrasekhar limit, 410 Characteristic equation, 34 polynomial, 34 Charge conservation, 221 Chart, 61 compatible, 61 Christoffel symbols, 47 Clausius–Duhem inequality, 154 Clock hypothesis, 230 Clocks, 133 Cochain, 97 Codazzi–Mainardi equations, 58 Codifferential, 71 Cohomology class, 93 relation, 93 space, 93 Comoving frame of reference, 229 Complete vector field, 85 Components controvariant, 60 covariant, 60 Configurations actual, 150 initial, 150 Connection affine, 106 coefficients, 106 Levi-Civita, 120 Constitutive equations, 155 Continuity equation, 158 Contracted multiplication, 12 Contraction, 11 Contravariance law, 5 Convective term, 152 Coordinates adapted, 329, 377, 432, 453, 463 bases, 44 comoving, 432 curves, 59, 64 curvilinear, 46, 59 Gaussian, 375 geodesic, 453 Lagrangian, 149 material, 150, 278 mathematical, 55 natural, 74 orthogonal, 45

Index physical, 55 Quasi-Minkowskian, 344 spatial, 150 surface, 59 transformation, 61 Cotangent vector space, 67 Covariance law, 5 Covariant derivative, 51, 52 differential tensor field, 110 Covector, 3 Critical mass density, 440 Curvature scalar of, 325 Curve autoparallel, 109 binormal vector, 43 canonical parameters, 110 closed, 42 curvature, 43 curvature of, 41, 81 curvature vector, 42 ending point, 42 geodesic, 109 image of a curve, 42 length, 77 natural parameter of, 42 osculating plane, 42 parameter of, 42 principal normal, 43 radius of curvature, 43 regular, 42 simple, 42 starting point, 42 tangent unit vector, 42 torsion, 43 torsion of, 41

D Deformation gradient, 150 homogeneous, 151 rate of, 152 Description Eulerian, 151 Lagrangian, 151 material, 151 Diffeomorphism, 60 Differential, 67, 70 form closed, 92

Index exact, 92 operator curl, 91 curl tensor, 124 divergence, 92, 124 gradient, 91, 124 Laplace, 125 Direction of the future, 241 Dissipation inequality reduced, 154, 285 Domain, 61 E Eigenbivector, 122 Eigenvalue, 33 algebraic multiplicity, 34 equation, 33 geometric multiplicity, 33 problem, 33 simple, 33 spectrum, 33 Eigenvector, 33 Einstein–de Sitter model, 443 Einstein’s disk, 56 Einstein’s gravitational equations, 325 Einstein tensor, 325 Electromagnetic Ampère’s formulation, 301 Boffi’s formulation, 306 Chu’s formulation, 308 Minkowski’s formulation, 296 moving body, 295 Electromagnetic momentum–energy tensor, 252 Electromagnetic tensor, 249 Elliptic geometry, 40 Ergosphere, 429 Ether, 156 Euler–Lagrange equations, 78 Event, 133, 240, 319 Event horizon, 410, 413 Exterior algebra, 20 product, 13, 17, 19 F Fermi derivative, 263 Fermi–Walker connection, 461 covariant derivative, 452, 458 derivative, 457 gradient, 459

493 symbols, 459 transport, 262, 450, 456 Fermi–Walker derivative, 457 Fiber, 74 Focal points, 79 Force, 139 Coriolis, 141 drag, 141 fictitious, 141 inertial, 141 law, 139 of Lorentz, 222 Four-vector, 240 Frame geodesic, 324 local inertial, 324 Lorentz, 324 of reference, 133, 463 geodesic, 467 irrotational, 467 rigid, 467 optical isotropic, 201 proper, 221 Frame of reference, 133 Frenet trihedron, 43 Friedmann’s equations, 438 Future, 241

G Gamma identity, 210 Gauss equations, 58 Gaussian curvature, 49 Gauss–Weingarten equations, 47 Generalized orthonormal bases, 29 Geodesic, 51, 78 Geodesic deviation equation, 118 Geodesic rest frame, 463 Gram–Schmidt orthonormalization procedure, 31 Gravitational collapse, 418 Gravitational potential energy, 162 Gravitational tensor, 354 Gravitation tidal force, 170 Gravitoelectric vector, 358 Gravitomagnetic vector, 358 Group infinitesimal generators, 84 of rotations, 85 of translations, 84 orbit of, 84

494 H Harmonic function, 125, 368 Hausdorff space, 61 Heat-conducting perfect fluid, 276 Homeomorphism, 60 Hubble constant, 440 Hyperbolic geometry, 40

I Incoherent matter, 268, 275 Inequalities periodic, 189 secular, 189 Inertial frame, 138 Integral of a form, 97 Integral of an r -form, 95 Interior product, 89 Intrinsic Cauchy–Green tensor, 282 Intrinsic deformation gradient, 279 Isolated point, 138 Isothermal surfaces, 368

K Killing vector field, 379

L Lane–Emden equation, 182 Laplace’s equation, 164 Length proper, 214 rest, 214 Leray PDE systems, 368 Leray’s theorem, 368 Lie algebra, 70 derivative, 86, 87 Light cone, 240, 447 Linear form, 3 Local inertial frame, 169, 193 time, 133 Lorentz 4-force, 250 boost transformation, 246 condition, 251 frame, 240, 453 gauge conditions, 347 general transformation, 207, 245 homogeneous transformation, 245

Index improper transformation, 245 infinitesimal transformation, 259 inhomogenous transformation, 245 proper transformation, 245 special transformation, 206, 246 transformation, 240 transformation without rotation, 208, 245

M Manifold, 61 coordinates, 61 map, 61 differential, 61 Riemannian, 77 with spherical symmetry, 379 Mass, 140 Matching conditions, 376 Material simple, 155 Maxwell stress tensor, 253 Measurement process, 132 Mechanical determinism, 141 Metric coefficients, 45 Minkowski’s inequality, 28 Model of universe, 376 Momentum–energy tensor, 269, 274

N Newton’s first law, 138 No-hair conjecture, 421

O One-parameter global group of diffeomorphisms, 83 group Invariant, 89 local group of diffeomorphisms, 85 Oppenheimer–Volkoff limit, 420 Orthogonal group, 35 matrix, 35

P Paracompact space, 61 Parallel transport, 51, 108 along a curve, 108 Parametric curve, 42 Parametric equations, 64 Past, 241

Index Penrose process, 429 Perfect fluid, 276 Permutation, 16 even, 17 inversion, 16 odd, 17 Petrov types, 123 Poincaré group of, 245 transformation, 245 Point elliptic, 49 hyperbolic, 49 parabolic, 49 Poisson’s equation, 164 Polytropic transformation, 179 Potential scalar, 251, 358 vector, 250, 358 Poynting vector, 253, 304 Principal normal vector, 80 Principle cosmological, 191 of action–reaction, 139 of equivalence, 313 of frame-indifference, 142 of general relativity, 146, 147, 313 of inertia, 138 of Mach, 313 Process thermodynamic, 285 thermokinetic, 285 Projection map, 74 on space, 256 on time, 256 Proper density of matter, 283 frame, 246 internal energy, 283 rotations, 35 specific entropy, 283 time, 247

Q Quadratic form, 27 Quadrupole potential energy, 168

R Regular surface, 44 Relativistic

495 energy, 225 kinetic energy, 225 mass, 223, 247 rest energy, 225 thermoelastic materials, 285 Rest frame, 246 Ricci decomposition, 121 Ricci tensor, 117, 325 Riemann manifold, 76 tensor, 58 Robertson–Walker metric, 435 Rotations, 35

S Scalar curvature, 117 Scalar product, 28 Schwarzschild black hole, 416 exterior solution, 383 interior solution, 387 radius, 387 universe model, 392 Self-gravitating bodies, 172 Set smoothly contractible, 93 star-shaped, 93 Space curly tensor, 466 pseudotensor, 455 stretching tensor, 466 vectors, 455 Spacelike hypersurface, 366 Spacetime, 240 algebraically Special, 123 quasi-Minkowskian, 343 static, 377 spatial spherical symmetry, 383 stationary, 378 Specific free energy, 284 Squared distance, 77 Standard time interval, 329 Stress tensor of Maxwell, 156 Strictly hyperbolic Leray system, 368 operator, 366 Surface first fundamental form, 45 locally oriented, 46 mean curvature, 49 normal curvature, 47 principal curvatures, 49

496 principal directions, 49 second fundamental form, 47

T Tangent fiber bundle, 74 vector, 65 vector space, 66 Tensor algebra, 11 alternating, 13, 17 contracted product, 11 contraction, 11 curvature, 112 electric field–magnetic induction, 297 field, 69 index, 27 metric, 76 signature, 77 positive definite, 27 semi-definite, 27 product, 6, 8–10 signature, 27 skew-symmetric, 13, 17 strict components, 14 symmetric, 22 torsion, 111 Tetrad fields, 448 Thermokinetic process history of, 154 Thomas precession, 266 Thomas–Wigner rotation, 212 Time dynamic, 138 proper, 214 rest, 214 universal, 133 vectors, 455 Timelike congruence, 453 path, 366 Tolman–Oppenheimer–Volkoff equation, 389, 418

Index limit, 410 Transformations Eddington–Finkelstein, 411 Galilean, 139 isometric, 142 of gauge, 344 orthochronous, 243 orthogonal, 35, 142 Transverse derivative, 460 TT gauge, 355

U Uniform vector field, 108 Universe tube, 278, 341, 375

V Vector angle, 30 covariant components, 30 field, 68 length, 30 modulus, 30 orthogonal, 29 unit, 29 Vector space Euclidean, 30 oriented, 21 orthogonal complement, 32 pseudo-Euclidean, 28 Velocity, 152 absolute, 136 drag, 137 relative, 136

W Weak-field approximation, 343 WeylTensor, 121 World acceleration, 247 line, 246 trajectory, 246 velocity, 247