Antonio Romano Mario Mango Furnari The Physical and Mathematical Foundations of the Theory of Relativity A Critical Analysis Antonio Romano Mario Mango Furnari (cid:129) The Physical and Mathematical Foundations of the Theory of Relativity A Critical Analysis AntonioRomano Mario MangoFurnari Dipartimento di Matematica e Istituto di Cibernetica Applicazioni “RenatoCaccioppoli” Naples, Italy Universitàdegli Studi diNapoli Federico II Naples, Italy ISBN978-3-030-27236-4 ISBN978-3-030-27237-1 (eBook) https://doi.org/10.1007/978-3-030-27237-1 MathematicsSubjectClassification(2010): 83AXX,83BXX,83CXX,83FXX ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Writingabookonrelativitytodayisequivalenttoaddingagrainofsandtoavery 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 prin- ciplesofrelativity,whereasinbooksofthesecondcategory,itispossibletodeepen 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 text- book. First, it is our rooted opinion that the reader should understand which clas- sicalconceptsheorsheisleavingbehindinfollowingthepaththatwillleadtothe 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,thisbookbeginswithtwochaptersinwhichtheclassicalideasthatwillbe abandonedorsaved arerecalled indetail. Therefore,inthese chapterstheclassical procedure to measure space and time, the basic ideas of classical dynamics, the Galilean relativity principle, and some fundamental results of Newtonian gravita- tion are recalled. Afterintroducingspecialrelativityfollowingthephysicalapproachproposedby Einstein (Chap. 7), the Minkowski mathematical model is discussed (Chap. 8). In thisregard,werecallthatamodelcanbeconsideredamathematicaltranscriptionof 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 correspondenceforMinkowski’smodel,allthecharacteristicsandparticularitiesof this correspondence are widely analyzed with the aim of putting in evidence the deepdifferencesexistingbetweenthecorrespondencethatinspecialrelativityallow 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 vi Preface Chapters 9 and 10 are devoted to topics that are usually presented hastily; the relativisticthermodynamicsofcontinuaandtheelectromagneticfieldsinmatter.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 experi- mentallyobserved.Thiscircumstancejustifiesthemanyproposalsthatcanbefound 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 beenproposedaswellastheequivalenceofallthemodelsproposedtodescribethe interactionbetweenmatterandelectromagneticfields.Moreprecisely,weshowthat 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 generalrelativitywiththeawarenessthatthereisnoagreementontheirformulation. Oneofthemostcontroversialprinciplesisthatofgeneralrelativity.Itshouldbethe generalization ofthe special principle of relativity,since it statesthat 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”hastobeunderstoodasthecollectionoftheobservers’ 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, accordingtowhichtheeffectsofanygravitationalfieldonphysicalphenomenacan 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 assumptionthatintheseregionsspecialrelativityholds,allowstheintroductionofa first partial correspondence between physical reality and geometric objects of the spacetimemanifoldV .Infact,thegeodesiccoordinatesinanarbitrarypointofV 4 4 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 vii describethegravitationalfieldwiththemetricofaRiemannianmanifoldthatinturn isrelatedtothematterandenergyoccupyingaregionofspacetime.Thenthemain 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(cid:2)vTlm, where the tensor Glm involves only the spacetime geometry, Tlm is the momentum–energy tensor satisfying the conservation laws rmTlm ¼0, and v is an unknown constant. Inordertodeterminetheconstantv,Einsteinresortedtoalinearapproximation 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 v¼8ph=c4. It should be noted that the identification of Einstein’s equations and Poisson’s equation is purely formal, since the interpreta- tion of the gravitational potential in the two equations is completely different. Furthermore, in the linear nonstatic case, every metric coefficient satisfies d’Alemebert’sequation,sothattheexistenceofgravitationalwavesisforeseenand 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 equationsandtheconservationlaws.Inotherwords,beforeknowingthemetric,we cannotspeakaboutspace,time,geodesics,etc.Inparticular,thedefinitionsoflocal 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 V . This means that 4 coordinates ðxaÞ must be defined on an open set U of V to which it is possible to 4 associate a frame of reference for an observer O. In particular, experimental pro- cedures must be defined that allow O to evaluate the coordinate ðxaÞ of an event belongingtoasetE.Afterdefiningthisone-to-onecorrespondencebetweenpoints of U and physical events of E, we can also adopt in U arbitrary coordinates ðx0aÞ, provided that the coordinate transformation x0a ¼x0aðxbÞ is known. viii Preface This is the approach we follow in Chaps. 14–16. Specifically, in Chap. 14, the modelofspacetimeproposedbySchwarzschild ispresented.Thismodel describes the relativistic gravitational field produced by a spherically symmetric mass dis- tribution S inside and outside S. In the coordinates introduced to solve separately the interior and exterior Einstein equations, the matching conditions on the gravi- tational 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 V , except for the origin r ¼0 of radial coordinates, in 4 Chap. 15 this metric is interpreted as representing the gravitational field of a mass concentratedatr ¼0.Theeventhorizonr ¼r isdefined,andthecomplexphysics s 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 cosmo- logicalmodelsdescribedbythoseequations.Inparticular,weshowtheexistenceof global coordinates inthespacetimetowhichaphysical meaning canbeattributed. 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 Fab isknown, what is the relation between the components ofFab andtheelectricandmagneticfieldsastheyaremeasuredbyanobserver?Is it possible to formulate the tensor laws in V in terms of quantities and operators 4 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: differentialcalculus,exterioralgebra,differentialmanifolds,Riemannianmanifolds, exterior derivation and integration, and transformation groups. Naples, Italy Antonio Romano Mario Mango Furnari Contents Part I Elements of Differential Geometry 1 Tensor Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Linear Forms and Dual Vector Spaces . . . . . . . . . . . . . . . . . . 3 1.2 Biduality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Covariant 2-Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 ðr;sÞ-Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Contraction and Contracted Multiplication . . . . . . . . . . . . . . . 11 1.6 Skew-Symmetric (0, 2)-Tensors . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Skew-Symmetric ð0;rÞ-Tensors . . . . . . . . . . . . . . . . . . . . . . . 16 1.8 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.9 Oriented Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.10 Representation Theorems for Symmetric and Skew-Symmetric ð0;2Þ-Tensors. . . . . . . . . . . . . . . . . . . . 22 1.11 Degenerate and Nondegenerate ð0:2Þ-Tensors . . . . . . . . . . . . . 25 1.12 Pseudo-Euclidean Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 28 1.13 Euclidean Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.14 Eigenvectors of Euclidean 2-Tensors . . . . . . . . . . . . . . . . . . . 33 1.15 Orthogonal Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2 Introduction to Differentiable Manifolds. . . . . . . . . . . . . . . . . . . . . 39 2.1 Historical Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2 Elements of the Geometry of Curves . . . . . . . . . . . . . . . . . . . 41 2.3 Elements of Geometry of Surfaces. . . . . . . . . . . . . . . . . . . . . 44 2.4 The Second Fundamental Form . . . . . . . . . . . . . . . . . . . . . . . 46 2.5 Parallel Transport and Geodesics . . . . . . . . . . . . . . . . . . . . . . 50 2.6 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.7 Riemann’s Tensor and the Theorema Egregium . . . . . . . . . . . 57 2.8 Curvilinear Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ix x Contents 2.9 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.10 Differentiable Functions and Curves on Manifolds . . . . . . . . . 63 2.11 Tangent Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.12 Cotangent Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.13 Differential and Codifferential of a Map . . . . . . . . . . . . . . . . . 70 2.14 Tangent and Cotangent Fiber Bundles . . . . . . . . . . . . . . . . . . 73 2.15 Riemannian Manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.16 Geodesics over Riemannian Manifolds. . . . . . . . . . . . . . . . . . 77 2.17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3 Transformation Groups, Exterior Differentiation and Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.1 Global and Local One-Parameter Groups . . . . . . . . . . . . . . . . 83 3.2 Lie Derivative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.3 Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.4 Closed and Exact Differential Forms . . . . . . . . . . . . . . . . . . . 92 3.5 Properties of the Exterior Derivative . . . . . . . . . . . . . . . . . . . 94 3.6 An Introduction to the Integration of r-Forms. . . . . . . . . . . . . 95 3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4 Absolute Differential Calculus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.1 Preliminary Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2 Affine Connection on Manifolds . . . . . . . . . . . . . . . . . . . . . . 106 4.3 Parallel Transport and Autoparallel Curves. . . . . . . . . . . . . . . 108 4.4 Covariant Differential of Tensor Fields. . . . . . . . . . . . . . . . . . 110 4.5 Torsion Tensor and Curvature Tensor . . . . . . . . . . . . . . . . . . 111 4.6 Properties of the Riemann Tensor . . . . . . . . . . . . . . . . . . . . . 115 4.7 Geodesic Deviation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.8 Levi-Civita Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.9 Ricci Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.10 Differential Operators on a Riemannian Manifold . . . . . . . . . . 124 4.11 Riemann’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Part II Newtonian Dynamics, Gravitation, and Cosmology 5 Review of Classical Mechanics and Electrodynamics . . . . . . . . . . . 131 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.2 Foundations of Classical Kinematics . . . . . . . . . . . . . . . . . . . 132 5.2.1 Change of the Frame of Reference . . . . . . . . . . . . . . 133 5.2.2 Absolute and Relative Velocity and Acceleration. . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3 Laws of Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . 137 5.3.1 Force Laws and the Action–Reaction Principle. . . . . . 139 5.3.2 Newton’s Second Law . . . . . . . . . . . . . . . . . . . . . . . 140