Geometry, Symmetries, and Classical Physics

Geometry, Symmetries, and Classical Physics Geometry, Symmetries, and Classical Physics A Mosaic Manousos Markoutsakis First edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2022 Manousos Markoutsakis Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. 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ISBN: 978-0-367-53523-0 (hbk) ISBN: 978-0-367-54141-5 (pbk) ISBN: 978-1-003-08774-8 (ebk) DOI: 10.1201/9781003087748 Publisher’s note: This book has been prepared from camera-ready copy provided by the authors. To my parents Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I: 1 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7 11 16 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 31 32 39 43 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Galileian Spacetime . . . . . . . . . . . . . . . . . . Newton’s Laws of Mechanics . . . . . . . . . . . . . Systems of Particles and Conserved Quantities . . . Gravitation and the Shell Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transformations and the Lie Derivative . Symmetry Transformations of Manifolds . Isometric and Conformal Killing Vectors . Euclidean and Scale Transformations . . . Part II: . . . . . 23 53 53 58 60 61 67 67 70 74 78 . . . . . . . . . . . . . . . . . . . . 85 Applying the Principle of Stationary Action . . . . . . . . . . . . . . . . Noether’s Theorem in Mechanics . . . . . . . . . . . . . . . . . . . . . . Galilei Symmetry and Conservation . . . . . . . . . . . . . . . . . . . . 85 91 94 Relativistic Mechanics 6.1 6.2 6.3 6.4 3 Mechanics and Symmetry Lagrangian Methods and Symmetry 5.1 5.2 5.3 6 . . . . . Newtonian Mechanics 4.1 4.2 4.3 4.4 5 . . . . . Geometry and Metric . . . . . . . . . . . . Isometry and Conformality . . . . . . . . . Examples of Geometries . . . . . . . . . . Differential Forms and the Exterior Derivative Integrals of Differential Forms . . . . . . . Theorem of Stokes . . . . . . . . . . . . . Symmetries of Manifolds 3.1 3.2 3.3 3.4 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Differentiation in Several Dimensions . . Differentiable Manifolds . . . . . . . . . Tangent Structure, Vectors and Covectors Vector Fields and the Commutator . . . Tensor Fields on Manifolds . . . . . . . Geometry and Integration on Manifolds 2.1 2.2 2.3 2.4 2.5 2.6 3 Geometric Manifolds Manifolds and Tensors 1.1 1.2 1.3 1.4 1.5 xi . . . . . . Lorentz Transformations . . . . Minkowski Spacetime . . . . . . Relativistic Particle Mechanics . Lagrangian Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 99 102 112 116 vii viii Contents 6.5 Part III: 7 9 11 12 13 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 123 126 128 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 149 151 152 155 155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 162 164 166 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 172 173 173 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 177 183 187 188 194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 197 200 202 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 212 216 218 Rotation Group . . . . . . . . . . . . . . . Rotation Algebra . . . . . . . . . . . . . . Translations and the Euclidean Group . . . Euclidean Algebra . . . . . . . . . . . . . . . . . . . . . . . Group of Boosts . . . . . . . . . . . Group of Boosts and Rotations . . . Galilei Group . . . . . . . . . . . . Galilei Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . Lorentz Group . . . . . . . . . . . . . . . . Spinor Representation of the Lorentz Group Lorentz Algebra . . . . . . . . . . . . . . . Representation on Scalars, Vectors and Tensors Representation on Weyl and Dirac Spinors . Representation on Fields . . . . . . . . . . Meaning of Poincaré Transformations . . . . . . Poincaré Group . . . . . . . . . . . . . . . . . . Poincaré Algebra and Field Representations . . . Correspondence of Spacetime Symmetries . . . . Conformal Symmetry 14.1 14.2 14.3 14.4 . . . . . . . . . . . . . Representations of Groups and Algebras . Adjoint Representations . . . . . . . . . . Tensor and Function Representations . . Symmetry Transformations of Tensor Fields Induced Representations . . . . . . . . . Lie Algebra of Killing Vector Fields . . . Poincaré Symmetry 13.1 13.2 13.3 13.4 . . . . 135 Lorentz Symmetry 12.1 12.2 12.3 12.4 12.5 12.6 . . . . 135 139 141 Boosts and Galilei Symmetry 11.1 11.2 11.3 11.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotations and Euclidean Symmetry 10.1 10.2 10.3 10.4 . . . . Matrix Exponential and the BCH Formula . . . . . . . . . . . . . . . . . Lie Algebra of a Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . Abstract Lie Algebras and Matrix Algebras . . . . . . . . . . . . . . . . Representations 9.1 9.2 9.3 9.4 9.5 9.6 10 . . . . . . . . . . . . . . . . Notion of a Group . . . . . . . . . . . Notion of a Group Representation . . Lie Groups and Matrix Groups . . . . Lie Algebras 8.1 8.2 8.3 118 Symmetry Groups and Algebras Lie Groups 7.1 7.2 7.3 8 . . . . . . . . . . . . . . . . . . Relativistic Symmetry and Conservation . . . . . . . . . . . . . . . . . Conformal Group . . . . . . . . . . . . . Conformal Algebra . . . . . . . . . . . . Field Transformations . . . . . . . . . . . Linearization of the Conformal Group . . . . . . . . . . . . . . . . . . . . . . 147 159 169 175 197 205 ix Contents Part IV: 15 Lagrangians and Noether’s Theorem 15.1 15.2 15.3 15.4 15.5 15.6 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 228 231 233 235 242 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 251 259 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 268 271 272 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Manifestation of Curvature . . . . . . . The Riemann Curvature Tensor . . . . Algebraic Symmetries . . . . . . . . . . Bianchi Identity and the Einstein Tensor Ricci Decomposition and the Weyl Tensor Symmetric Spaces . . . . . . . . . Weyl Rescalings . . . . . . . . . . The Weyl-Schouten Theorem . . . Group of Diffeomorphisms . . . . Part VI: . . . . . . . 225 247 267 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 282 287 292 279 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 301 305 306 308 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 319 321 325 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 337 342 346 299 315 General Relativity and Symmetry Einstein’s Gravitation 21.1 21.2 21.3 21.4 . . . . . . . Connection and the Covariant Derivative . Formulae for the Covariant Derivative . . . The Levi-Civita Connection . . . . . . . . Parallel Transport and Geodesic Curves . . Symmetries of Riemannian Manifolds 20.1 20.2 20.3 20.4 . . . . . . . Riemannian Geometry Riemannian Curvature 19.1 19.2 19.3 19.4 19.5 21 . . . . . . . Connection and Geodesics 18.1 18.2 18.3 18.4 20 . . . . . . . . . . . . Internal Symmetries and Charge Conservation . . Interactions and the Gauge Principle . . . . . . . Scalar Electrodynamics . . . . . . . . . . . . . . Spinor Electrodynamics . . . . . . . . . . . . . . Part V: 19 . . . . . . . . . . . . Spacetime Symmetries and Currents . . . . . . Versions of the Energy-Momentum Tensor . . . Conserved Integrals . . . . . . . . . . . . . . . Conditions for Conformal Symmetry . . . . . . Gauge Symmetry 17.1 17.2 17.3 17.4 18 . . . . . . . Introducing Fields . . . . . . . . Action Principle for Fields . . . Scalar Fields . . . . . . . . . . . Spinor Fields . . . . . . . . . . Maxwell Vector Field . . . . . . Noether’s Theorem in Field Theory Spacetime Symmetries of Fields 16.1 16.2 16.3 16.4 17 Classical Fields . . . . . . . . . . . . Physics in Curved Spacetimes . . The Einstein Equations . . . . . . Schwarzschild Metric . . . . . . . Asymptotically Flat Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 x 22 Contents Lagrangian Formulation 22.1 22.2 22.3 22.4 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 358 362 368 Locally and Globally Conserved Quantities . On the Energy of Spacetime . . . . . . . . . Komar Integrals . . . . . . . . . . . . . . . . Weyl Rescaling Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 381 383 387 Conservation Laws and Further Symmetries 23.1 23.2 23.3 23.4 Part VII: A B 355 377 Appendices Notation and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . 399 A.1 Physical Units and Dimensions . . . . . . . . . . . . . . . . . . . . . . . A.2 Mathematical Conventions . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 401 403 Mathematical Tools B.1 B.2 B.3 B.4 B.5 B.6 B.7 B.8 C D . . . . . . . . . . . . . . . Action Principle in Curved Spacetimes . . The Action for Matter Fields . . . . . . . The Action for the Gravitational Field . . Diffeomorphisms and Noether Currents . . . . . . . . . . . . . . . . Tensor Algebra . . . . . . . . . Matrix Exponential . . . . . . . Pauli and Dirac Matrices . . . . Dirac Delta Distribution . . . . Poisson and Wave Equation . . Variational Calculus . . . . . . . Volume Element and Hyperspheres Hypersurface Elements . . . . . Weyl Rescaling Formulae . . . . Spaces and Symmetry Groups . Bibliography . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 418 419 422 425 430 434 440 . . . . . . . . . . . . . . . . . . . . . . 445 . . . . . . . . . . . . . . . . . . . . . . 451 . . . . . . . . . . . . . . . . . . . . . . 453 . . . . . . . . . . . . . . . . . . . . . . 457 405 Preface Classical theoretical physics is a remarkably coherent and beautiful subject. The particular viewpoint which we adopt in this book is that symmetry principles play a decisive role in the foundations of the theory. Indeed, the core of classical theoretical physics can be derived from the following three symmetry principles: • The underlying spacetime symmetry along with its representations defines the basic elements (i.e. the fields) of the theory. • The dynamical evolution of the theory is encoded in the action principle. • The fundamental interactions of the theory are determined by a gauge symmetry principle. This book provides a systematic discussion of classical spacetime and gauge symmetries and the associated physical invariances. The book covers geometric manifolds, the foundational continuous symmetry groups of classical mechanics and classical fields, as well as symmetries in geometry and general relativity. The beauty of theoretical physics derives in large part from the beauty of mathematics. Therefore, the book develops the central notions of differentiable manifolds, Lie groups and algebras, and Riemannian geometry from the outset and in the necessary conceptual depth. In addition, several nontrivial and exciting topics are covered, such as the discussion of conformal symmetry, Weyl symmetry, and the discussion of conserved quantities in general relativity, to name a few. The treatment of all topics is technically complete and this kind of presentation is necessary in order to develop a true understanding. It is indeed one of the primary goals of this text to provide a high degree of transparency. The content of the book is divided into six parts and an appendix. In Part I, we begin with the mathematical foundations of differentiable manifolds, tangent spaces, and tensor fields. We introduce the metric and discuss geometric manifolds with a selection of relevant examples. We develop the machinery of differential forms and derive Stokes’ theorem. We then introduce the notion of the Lie derivative and describe how it can be used to formulate symmetry on geometric manifolds. Part II deals with classical dynamics and summarizes the Newtonian, Lagrangian and relativistic formulations of particle mechanics. We invoke the action principle and establish the Euler-Lagrange equations of motion. We then derive and apply Noether’s theorem for nonrelativistic and for relativistic mechanics. The relation between symmetry and conservation represents the recurring theme. Part III covers the algebraic aspects of symmetry. First, we provide the mathematical background on Lie groups, Lie algebras, and representations as needed for subsequent purposes. We develop in detail the rotation group, the Euclidean group, the Galilei group, the Lorentz group, the Poincaré group, and the conformal group. We disclose how relativistic symmetry leads to the existence of Weyl and Dirac spinors. For all spacetime symmetries, we systematically derive the generators of the Lie group in their field representation and the associated commutator relations. Part IV is about classical field theory. The action principle is again the starting point, this time for field Lagrangians. We provide the examples of real and complex scalar fields, spinor fields and the Maxwell field. Noether’s theorem for fields is derived and we deduce the conserved quantities. An important notion introduced here is that of the energy-momentum tensor in the canonical version and the symmetric version based on Belinfante’s prescription. We derive the concrete conserved quantities for each of the spacetime transformations xii Preface contained in the full conformal group. Then we discuss the notion of a conformally invariant field theory and give some examples. Finally, we examine the interaction of fields, with the gauge symmetry as the guiding principle. Part V covers Riemannian geometry and related symmetry aspects. We first introduce the covariant derivative and the notion of connection on a differentiable manifold. After introducing Riemannian curvature, we discuss symmetry properties as well as the Ricci decomposition of the Riemann tensor. We then revisit symmetry on geometric manifolds and identify the properties of a manifold at maximal symmetry. Then Weyl rescalings and the associated Weyl-Schouten theorem are studied. Finally, we consider differentiable transformations from an algebraic point of view and introduce the corresponding infinitedimensional group and algebra. In Part VI, first the basic conceptions of general relativity as the most important classical theory of gravity are covered. In addition to Einstein’s field equations and the Schwarzschild solution, the concept of an asymptotically flat spacetime is discussed. The subsequent complete treatment of the Lagrangian formalism in general relativity includes matter fields as well as the metric field and the case of manifolds with a boundary. Within the Lagrangian framework, the metric energy-momentum tensor is introduced in a natural way. We provide a thorough discussion of internal diffeomorphisms and the associated identically conserved Noether currents. In the last chapter, we discuss locally and globally conserved quantities in general relativity. In particular, the Komar integral quantities are introduced and discussed. The last section addresses the question of how Weyl rescaling symmetry can be achieved. Among other things, we discuss here the conformally coupled scalar field. In the appendix we summarize the conventions used and some relevant mathematical results. This includes a detailed exposition of tensor algebra, matrix groups, Dirac delta distribution, Poisson and wave equations, calculus of variations, spheres in arbitrary dimensions, and hypersurfaces. We also provide a fairly complete collection of formulae for Weyl rescalings of the tensor fields that are relevant to us. An overview of all major spacetime symmetries concludes this part. This book has a special focus on conformal symmetry and Weyl rescaling symmetry in d ≥ 3 dimensions. Conformal symmetry in d = 2 is not treated in detail, since it is a topic of its own. The discussion of conformal symmetry begins with rigorous mathematical definitions, but is then translated into a more practical form. Topics discussed include conformality between manifolds, the Lie group and Lie algebra of conformal transformations, and the notion of conformal symmetry for Lagrangian field theories with concrete examples. Moreover, we cover Weyl rescalings of geometries, the Weyl-Schouten theorem describing the conditions for achieving conformal flatness, the implications of Weyl rescalings in general relativity, and finally the conformally coupled scalar field. The book is designed to be self-contained, but it is assumed that the reader has a solid knowledge of linear algebra, analysis in several dimensions, classical mechanics, electromagnetism, and special relativity. In addition, a basic knowledge of general relativity is an advantage. Mathematical notation is kept consistent throughout the book. Central mathematical notions are first introduced rigorously, and then the emphasis is on their application. We make extensive use of the tensor indices notation, as this is the most economical way to formulate complex tensor equations. The Lie derivative and the covariant derivative are defined in an algebraic way, using symmetry principles as a guide. In writing this text, I have avoided using too involved mathematical constructions, such as the pullback of maps or the Hodge duality. Instead, we use the local transformations and the explicit expressions that employ the epsilon tensor. Thematically, we do not treat discrete symmetries such as space or time inversion, since these symmetries acquire their proper relevance within quantum theory. Furthermore, the Hamiltonian formalism unfortunately had to be omitted entirely in order to keep the size of the book manageable. Preface xiii The bibliography at the end of each chapter and at the end of the book provides the curious reader with references to explore topics in greater depth. We list three types of references. First, there are original articles that address specific topics. Then there are textbooks that provide a wealth of further developments and applications that could not be covered in the limited space available. Finally, there is a selection of classic reference texts that have stood the test of time and continue to provide valuable insights today. Scattered throughout the text, the reader will encounter the sign (exercise). At these points, the reader is encouraged to tackle a straightforward exercise to solidify understanding. I would like to take this opportunity to express my gratitude to the editorial team at CRC Press for making this book possible. My special thanks go to Editorial Assistant, Dr. Kirsten Barr and to Acquiring Editor, Rebecca Davies for supporting the book concept and for guiding me through the final stages of writing. My thanks also go to Shashi Kumar for helping me with the many intricacies within the TeX system. Finally, I would like to thank my family for their patience and steady support during the writing process. I Geometric Manifolds 1 Manifolds and Tensors 1.1 1.2 1.3 1.4 1.5 Differentiation in Several Dimensions . . . . . . . . . . . . . . Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tangent Structure, Vectors and Covectors . . . . . . . . . Vector Fields and the Commutator . . . . . . . . . . . . . . . . . Tensor Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7 11 16 20 In this chapter we introduce the foundational notions of differentiable manifolds, vectors at a point, vector fields, and tensor fields. We begin with a summary of basic facts about differentiation in RD , since we aim to transfer the known calculational concepts to the case of differentiable manifolds. We provide the general definition of a differentiable manifold, introduce coordinates, and discuss diffeomorphisms. The directional derivative leads us to the algebraic definition of a vector as an element of the tangent space at a point. In the next step, we move from vectors at a point to vector fields defined over the entire manifold. Finally, we generalize to the multilinear structure and introduce general tensor fields on manifolds. 1.1 Differentiation in Several Dimensions Euclidean Space ED One of the cornerstones of classical physics is the use of the continuum of real numbers R, or, for higher dimensions, the vector space RD with integer dimensionality D = 1, 2, 3, . . .. The elements of RD are represented as column vectors  1  x  ..  x =  . , (1.1) xD or, in abbreviated form, by the component notation xk . The canonical basis vectors are given by ek = (0, . . . , 1, . . . , 0)T , where the entry 1 is at the kth row. The vector space RD represents also a raw model for space in classical physics, provided we endow it with an additional structure. This structure is a metric, or equivalently, a scalar product, which for any two vectors x, y of RD is defined by hx, yi ≡ D X xk y k . (1.2) k=1 The scalar product, in turn, introduces a norm (or length) of a vector, q p 2 2 |x| ≡ hx, xi = (x1 ) + · · · + (xD ) . DOI: 10.1201/9781003087748-1 (1.3) 3 4 Geometry, Symmetries, and Classical Physics – A Mosaic If the origin of the vector space can be freely shifted, we have an affine space at hand and we can identify vectors with points. In this way, we can introduce the notion of distance d(x, y) between any two points x, y of RD as d(x, y) ≡ |x − y|. (1.4) The above structure defines the D-dimensional Euclidean space, which we denote by ED . In general, we will not make a distinction between the linear RD and the affine-linear ED . Functions, Maps and Curves Let us consider the linear spaces RD and RN , with integer dimensions D, N = 1, 2, 3, . . ., each space being equipped with its scalar product. Let us also consider open subsets U , V , etc. of these linear spaces on which we will define our maps. Open sets U are those for which every point x ∈ U has a neighborhood that is completely contained in U . A function f is a map from an open subset U ⊂ RD to the real numbers, i.e. f : U → R, x 7→ f (x). More generally, a map F assigns elements of U ⊂ RD to elements of RN , i.e. F : U → RN , x 7→ F (x), and the image is represented as a column vector,   1 F (x)   .. (1.5) F (x) =  , . F N (x) or as F k (x) in the index notation. The special case N = D corresponds to the case of a vector field on U . For N = 1, we recover the case of functions again. Another special case is for D = 1, where the elements of an open interval U of R are mapped to N -dimensional vectors. Then we speak about curves; i.e. a curve γ is a map γ : U → RN , t 7→ γ(t). Differentiability Given a function f : U → R, x 7→ f (x), the partial derivative of f (x) with respect to the variable xk (for a certain index value k) at the point x is the limit f (x + tek ) − f (x) ∂f (x) ≡ lim , k t→0 ∂x t (1.6) where ek is the basis vector of RD in the kth direction. The collection of all partial derivatives of a function constitutes the gradient of this function and is of great importance. The gradient gradf (x) of a function f (x) is written as the row vector   ∂f ∂f (1.7) , . . . , gradf ≡ ∂x1 ∂xD and belongs to the dual space of RD . If the gradient or, equivalently, all partial derivatives of a function exist, the function is said to be differentiable. Let us now approach the property of differentiability from a more conceptual point of view. We consider a map F : U (⊂ RD ) → RN , x 7→ F (x). In principle, differentiability means that locally a linear approximation is possible. A map F is called (totally) differentiable at the point x ∈ U if there is a linear map DF (x) : RD → RN , so that for ξ ∈ RD , |ξ| ≪ 1, it is F (x + ξ) = F (x) + DF (x) ξ + o(|ξ|), with lim ξ→0 o(|ξ|) = 0. |ξ| (1.8) (1.9) 5 Manifolds and Tensors This means that the remainder function o(|ξ|) as a power series must be higher than first order in |ξ|. The linear map DF (x) is called the differential , or derivative, of F at the point x. For functions f (x), the symbol df (x) is used, while for curves γ(t), the symbol γ̇(t) is common. Obviously, the differential is represented by an N ×D-matrix. This matrix is easily seen to be the Jacobian matrix of F (Carl Gustav Jacob Jacobi ) at the point x, DF =  k ∂F ∂xl    ≡ ∂F 1 ∂x1 .. . ∂F N ∂x1 ··· .. . ··· ∂F 1 ∂xD .. . ∂F N ∂xD   , (1.10) i.e. the differential is given by the ordered collection of all partial derivatives. The differential for functions f (x) is simply the gradient gradf (x), while for curves γ(t) it is the velocity vector γ̇(t). A map that is n-fold differentiable with continuous derivatives is called C n . A map that can be differentiated arbitrarily often with continuous derivatives is called smooth and said to be C ∞ . The set of all C ∞ functions on a open set U ⊂ RD is denoted by C ∞ (U ) and similarly C ∞ (RD ) for the case the entire RD is considered. We recall here three central results regarding differentiation in RD without proofs. The first one is the chain rule. If we have a nested map F ◦ G(x) = F (G(x)) with G : U (⊂ RD ) → RN and F : V (⊂ RN ) → RM , where G is differentiable in x and F is differentiable in G(x), then the composed map F ◦ G is differentiable in x and its differential is given by D(F ◦ G)(x) = DF (G(x)) · DG(x). (1.11) In other words, the Jacobi matrix of the composed map is equal to the product of the Jacobi matrices of the single maps. The second result deals with the question under which conditions a map can be locally inverted. The answer is given by the inverse function theorem, which in essence asserts that a map with an invertible Jacobi matrix can be inverted if we restrict the domain of definition in a suitable way. More precisely, let us consider a map F : U (⊂ RD ) → RD which is C 1 and the points x and y = F (x) of RD . If the Jacobi matrix DF (x) is invertible, then there is an open subset V ⊂ U containing x and an open subset V ′ ⊂ RD containing y, so that F maps V one-to-one and onto V ′ and the inverse map F −1 : V ′ → V is also C 1 . The Jacobi matrix of F −1 is given by D(F −1 )(y) = (DF (x))−1 , (1.12) i.e. by the inverse of the original Jacobi matrix. The Jacobi matrix DF of a map F from RD to RD is of square type and one can take the determinant of it, which we denote by J(x),  k ∂F ∂(F 1 , . . . , F D ) ∂F , (1.13) ≡ ≡ J ≡ det l 1 D ∂x ∂(x , . . . , x ) ∂x and call the Jacobian determinant, or simply the Jacobian of F . As stated above, for an invertible map F it is, in coordinate notation, D X ∂F k ∂xl k = δm , ∂xl ∂F m (1.14) l=1 and thus we obtain the formula ∂F ∂x = ∂x ∂F −1 . (1.15) 6 Geometry, Symmetries, and Classical Physics – A Mosaic The Jacobian determinant J(x) is a crucial quantity in volume integrals, tensor densities, and in conformal transformations. The third result is about how we can approximate a function by a Taylor expansion (Brook Taylor ). Suppose we have a C 2 function f : U (⊂ RD ) → R and a vector ξ ∈ RD , with |ξ| ≪ 1. Then the function can be approximated by the Taylor expansion in the form f (x + ξ) = f (x) + D D X 1 X ∂2f ∂f 2 k (x) ξ + (x) ξ k ξ l + o(|ξ| ), ∂xk 2 ∂xk ∂xl (1.16) k,l=1 k=1 2 with the remainder function o(|ξ| ) fulfilling 2 lim ξ→0 o(|ξ| ) |ξ| 2 = 0. (1.17) The Taylor expansion for a C n function is similar and employs the respective higher order partial derivatives. Directional Derivative We consider a function f (x) defined on RD and a vector v of RD with unit length, |v| = 1. We also consider the straight line given by the expression x + tv, with a real parameter t. The directional derivative, denoted vx f (x), of the function f (x) at the point x in the direction of v is the real number defined by vx f ≡ d f (x + tv) dt = lim t=0 t→0 f (x + tv) − f (x) . t (1.18) The definition is almost exactly the one for the partial derivative, with the only difference that the direction is not along one of the prime axes of RD , but along the vector v. By using the chain rule, we can see that the directional derivative can be expressed as vx f = v k ∂f (x) = hv, gradf (x)i. ∂xk (1.19) Note that we have used the summation convention above. In modern differential geometry, actually one takes a new point of view and considers the directional derivative along v to be a differential operator of the form ∂ (1.20) vk k , ∂x which acts on functions f (x). This is a central idea when considering vectors on differentiable manifolds. A tangent vector at the point x is simply an expression of the form v = vk ∂ , ∂xk (1.21)  with the local coordinate basis ∂/∂xk and the specific vector components v k . The directional derivative, defined as an operator acting on functions of the space C ∞ (RD ), obeys the algebraic properties of linearity vx (af + bg) = a(vx f ) + b(vx g), (1.22) vx (f g) = (vx f )g(x) + f (x)(vx g), (1.23) and Leibniz rule Manifolds and Tensors 7 for any real numbers a, b and any functions f (x) and g(x) (Gottfried Wilhelm von Leibniz ). The two above algebraic properties will actually be our starting point for the definition of tangent vectors. As a technical remark, note that in the definition 1.18 of the directional derivative we employed a straight line γ(t) = x + tv, for which γ̇(0) = v holds. If we were to take any other curve γ(t) with γ̇(0) = v, the result of the directional derivative would be exactly the same. In other words, the directional derivative encompasses an entire equivalence class of curves within its definition. Two curves are considered equivalent, in this context, if their velocities γ̇(0) at the point t = 0 coincide. 1.2 Differentiable Manifolds Notion of a Manifold We think of a differentiable manifold , denoted by a calligraphic M, as a point set which locally looks like RD but globally may have a completely different form. In addition, all points of a differentiable manifold should be describable by D-tuples of real numbers, the coordinates of the points. The raw model of a differentiable manifold is a two-dimensional surface embedded in Euclidean space. The properties arising from this simple description are sufficient in many situations. Nonetheless, this description is conceptually too special, since it assumes that we have a means to measure distances between points, as it is possible in RD . In general, however, a differentiable manifold has no distance measure defined. The metric structure allowing the measurement of distances is an additional structure, which a manifold may or may not have. Another conceptual trap arises from the notion that the manifold is embedded in a higher-dimensional space. In fact, we want to have a definition of the manifold that is intrinsic and need not to refer to a higher dimensional space. Moreover, we require that a differentiable manifold includes closeness and differentiability as basic properties from the outset. Before we proceed to the general definition, some elementary notions from topology are needed. Given an arbitrary point set M , a topology on this set is a collection of subsets of M , called open sets, with the property that unions of open sets are open and finite intersections of open sets are also open. We also require that M itself and the empty set ∅ are considered open sets. A topological space is a set M with a topology. Given a point p of M , a neighborhood of p is an open set containing p. A topological space is called a Hausdorff space (Felix Hausdorff ) if for every pair of points p and q one finds neighborhoods of them that are disjoint. A topology essentially defines a notion of closeness between the points of the set M . A map F : M → N between two topological spaces M , N is called continuous if the inverse image of any open set of N is an open set of M . A map F : M → N between two topological spaces is a homeomorphism if it is continuous and has a continuous inverse. Two topological spaces with a homeomorphism between them are considered topologically equivalent. A differentiable (or smooth) manifold M is a topological space with a differentiable (or smooth) structure, which means: • M has a family of pairs {(Ui , ψi )}, with open subsets Ui of M and maps ψi : U i → RD . S • The union of the open subsets covers the entire set M, i.e. i Ui = M. The maps ψi : Ui → RD are homeomorphisms from Ui to open subsets of RD . • For any pair of subsets Ui , Uj with non-vanishing intersection, Ui ∩ Uj 6= Ø, the map ψj ◦ ψi−1 from the subset ψi (Ui ∩ Uj ) of RD to the subset ψj (Ui ∩ Uj ) of RD is C ∞ , in the usual sense of analysis on RD . Each pair (Ui , ψi ) is called a coordinate chart. The open subset Ui is called a coordinate 8 Geometry, Symmetries, and Classical Physics – A Mosaic patch and the map ψi is called a coordinate map. Within a coordinate chart (Ui , ψi ), every point manifold is mapped to a D-tuple of numbers ψi (p) of RD . This D-tuple  1 p of the  D ψi (p), . . . , ψi (p) comprises the coordinates of the point p and can be considered as a representation of the point p of M in the linear space RD . In this sense, a manifold is locally, within a patch, homeomorphic to a subset of Euclidean space. Globally, however, the manifold has in general a different structure than Euclidean space. The dimension of the manifold M is the integer number D of coordinates needed to describe a point. The definition above also specifies how two different sets of coordinates relate to each other. For a subset Ui ∩ Uj = Ø of M, we can use the coordinate representation of ψi (Ui ∩ Uj ) or the one of ψj (Ui ∩ Uj ). If we change the coordinates with the map ψj ◦ ψi−1 , then this must be done in a smooth way. This ensures that when we move within the manifold, the assignment of coordinates happens smoothly throughout. By taking a holistic view, we can consider the collection of all chosen coordinate charts {(Ui , ψi )}, which is called an atlas covering the manifold. An atlas is not unique to a differentiable manifold. Given two different atlases, they are consistent if their union is also an atlas according to the definition of a manifold. The atlas of a manifold which is the union of all possible ones, is called the maximal atlas. It should be noted that in physics it is more common to use the term coordinate system or reference system instead of coordinate chart. p Ui Uj M ψj ψi ψj ◦ ψi−1 ψi (p) ψj (p) ψj (Uj ) ψi (Ui ) RD RD Figure 1.1: Manifold and coordinates Let us provide a few examples of differentiable manifolds. The vector space RD , the Euclidean space ED , and any open subspace of these spaces is a differentiable manifold. One specialty of RD and ED is that it is possible to cover these by a single, global coordinate system, which of course is the Cartesian coordinate system. The group GL(n, R) of all 2 invertible real n × n-matrices M , as a subset of Rn , GL(n, R) ≡ {M ∈ M at(n, R) | det M = 0}, (1.24) is a differentiable manifold. Further, the 1-dimensional unit circle S 1 , the 2-dimensional unit sphere S 2 S 2 ≡ {x ∈ R3 | |x| = 1}, (1.25) and, more generally, the n-dimensional unit sphere S n are all differentiable manifolds. In order to cover these manifolds by coordinates, however, we need more than one coordinate 9 Manifolds and Tensors system. The 2-dimensional torus S 1 × S 1 , as a Cartesian set product, is also a manifold. Generally, given any two differentiable manifolds M and N with respective dimensions D and N , their Cartesian product M × N is also a differentiable manifold with dimension D + N and a coordinate representation by (D + N )-tuples. Pictorially, we can grasp such a product manifold by imagining that to every point of M a copy of the entire manifold N is attached. A special type of manifolds M are those with a boundary. These manifolds consist of their boundary ∂M and their interior Int(M) ≡ M \∂M, which are both manifolds in their own right, as defined above. In order to define correctly the notion of a manifold with boundary, we consider the closed upper half-space in D dimensions  H̄ D ≡ (x1 , . . . , xD ) ∈ RD | xD ≥ 0 . (1.26) The boundary ∂ H̄ D of the closed upper half-space H̄ D is the point set  ∂ H̄ D ≡ (x1 , . . . , xD−1 , 0) ∈ RD . (1.27) A D-dimensional differentiable manifold with boundary M is defined as before, with the exception that it contains an atlas with two types of coordinate charts. A coordinate chart either maps to RD or a coordinate chart maps to H̄ D . All other requirements for manifolds remain as stated before. The boundary ∂M of the manifold consists of all boundary points p of M which are mapped to the tuples of the form (x1 , . . . , xD−1 , 0). The interior Int(M) of the manifold consists of the so-called regular points which are mapped to general tuples (x1 , . . . , xD ). We can see that the boundary is a (D − 1)-dimensional manifold, while the interior is a D-dimensional manifold. The closed real interval [a, b], for example, is a manifold with a boundary consisting of the two points a, b (indeed a discrete set has manifold dimension zero), while the open interval (a, b) represents the interior. An arbitrary manifold M can be considered to be a manifold with boundary ∂M = Ø and Int(M) ≡ M. Manifolds with a boundary will be relevant when we discuss the Stokes integral formula. Now that we know what a differentiable manifold is, let us emphasize again that a manifold has no measure of distance by itself. A differentiable manifold encodes only the notions of closeness (through its topology) and differentiability (through its coverage by an atlas). The points of the manifold can be described by D-tuples, which represent the assigned coordinates and are to be thought of as suitable labels of the points. Coordinates are not unique and can be changed. Coordinates provide a description of the points, so that we can use standard methods of analysis of RD , such as differentiation and integration. Coordinates Let us look more closely to the notion of coordinates. For the space RD , we simply identify each of its points with its coordinates and consequently we write out a D-tuple for both. For a general manifold M, we need to make the distinction between a point and its coordinates clearer. To this end, coordinates are introduced to be functions. In the case RD , a coordinate function xµ is the map that assigns to a vector a ∈ RD its µth coordinate aµ , i.e. the coordinate function xµ : RD → R is the projection defined as xµ (a) = aµ . (1.28) In the case of a general manifold M, we first make a choice of a coordinate chart (U, ψ). Suppose the point p of M has the coordinates ψ(p) = a. Then the coordinate function is considered to act locally xµ ◦ ψ : U → R as xµ (ψ(p)) = aµ . (1.29) 10 Geometry, Symmetries, and Classical Physics – A Mosaic In abuse of notation, the coordinate functions are thought to be functions of the points of the manifold itself and we write xµ (p) = aµ then. This type of equation makes sense if we remember that it is valid within the choice of a coordinate chart. We speak about local coordinates xµ then. We must point out that the local coordinates xµ , µ = 1, . . . , D, are not vectors. The local coordinates xµ are real D-tuples describing points of a manifold. There is no vector space structure defined for general coordinates. When we consider a point p of a manifold and we change from one coordinate chart (U, ψ) to another one (U ′ , ψ ′ ), the coordinate values change from ψ(p) = x = (x1 , . . . , xD ) to ψ ′ (p) = x′ = (x′1 , . . . , x′D ). These coordinate values are related to each other by (x′1 , . . . , x′D ) = (x′1 (x), . . . , x′D (x)), (1.30) (x1 , . . . , xD ) = (x1 (x′ ), . . . , xD (x′ )). (1.31) and inversely by According to the third condition in the definition of a manifold, these functional relationships are smooth. Hence, we can apply standard methods of analysis in RD to study this local coordinate transformation. The Jacobian J = det (∂x′ /∂x), for instance, is non-vanishing within the overlap of the two chosen coordinate patches. Coordinate transformations will be a central and recurring topic in this book. The choice of local coordinates allows us to define an orientation on a manifold. A manifold M is called orientable if there exists an atlas {(Ui , ψi )} so that for every point p ∈ M the coordinate transformations functions ψj ◦ ψi−1 for all possible charts for p have a positive Jacobian determinant, det (∂x′ /∂x) > 0. Here we use the symbols x = ψi (p) and x′ = ψj (p). An orientable manifold has two opposite global orientations, the “positive” and the “negative” one, where the name assignment is based on convention. This is the generalization of the notion of positive (right hand) orientation and negative (left hand) orientation in R3 . Submanifolds The idea of a submanifold should be intuitively clear, although the formal definition is somewhat technical if done in a coordinate-free fashion. Here we will restrict ourselves to a practical coordinate-based definition. The idea is the following: a manifold is a set which essentially has D degrees of freedom. If we want to define a subset with n ≤ D degrees of freedom, we need to have D − n conditions that implement the restriction in a smooth way. Let us formulate this. Consider a D-dimensional manifold M and a subset S of M. The set S is called an n-dimensional submanifold of M if it can be described in local coordinates by D − n equations of the form f 1 (x1 , . . . , xD ) f D−n 1 D (x , . . . , x ) = .. . =  0      0  . (1.32) We require for the D − n differentiable real functions f 1 , . . . , f D−n that the corresponding Jacobian matrix (∂f α /∂xµ ) has in all points the maximal rank, equal D − n. In fact, the submanifold S itself is an n-dimensional differentiable manifold. We call a one-dimensional submanifold a curve and a two-dimensional submanifold a surface. A submanifold with dimensionality D − 1, defined by one scalar equation f (x1 , . . . , xD ) = 0, is called a hypersurface. 11 Manifolds and Tensors Maps between Manifolds We have seen that we can deal with the points of a manifold by using a coordinate representation of them. The idea of using coordinates applies also when we study maps between manifolds. Let us start with a function f : M → R defined on a manifold. Instead of looking at the points of the manifold, we use a coordinate representation ψ and view the function f ◦ ψ −1 that maps points of RD to R. It is standard practice to treat the function f as if it were a function of the coordinates and write f (x) or f (x1 , . . . , xD ) for its values. If we have a map F : M → N between two differentiable manifolds M and N , we can define a local representative map ϕ ◦ F ◦ ψ −1 by choosing a coordinate representation ψ on M and a representation ϕ on N . Once again, the original map is identified with its local representation and we use the simplified notation F (x) = F (x1 , . . . , xD ) for its values. Within coordinates, we say that a map F is smooth if its local coordinate representation is smooth in the sense of RD . A smooth map F : M → N which is such that its inverse F −1 exists and is also smooth is called a diffeomorphism. Two manifolds which can be related to each other by a diffeomorphism are considered equivalent from the differentiability point of view. Every diffeomorphism can be seen from two different points of view. Either the diffeomorphism defines a coordinate transformation and leaves the manifold untouched, which corresponds to the passive view. Or the diffeomorphism is viewed as a deformation of the manifold itself, which corresponds to the active view. It is important that we keep a clear understanding on how we interpret and use diffeomorphisms in applications. The smooth functions f : M → R on a given manifold M constitute a real vector space denoted by C ∞ (M). For any two functions f and g of C ∞ (M) and any real number a, the vector space operations are defined pointwise by (f + g)(p) = f (p) + g(p) and (af )(p) = a f (p). Of course, we can restrict functions on any open subset U of a given manifold and consider only the vector space of smooth functions on U , which is then denoted by C ∞ (U ). 1.3 Tangent Structure, Vectors and Covectors Tangent Space We have seen that within RD each vector defines a directional derivative, which in turn has certain algebraic properties. We now take these algebraic properties as the starting point to define vectors on differentiable manifolds. Given a manifold M and a point p of the manifold, we consider the space of functions C ∞ (M).∗ A tangent vector at the point p, denoted Xp , is a map Xp : C ∞ (M) → R, f 7→ Xp (f ), which satisfies linearity and the Leibniz rule. I.e. for any two functions f (x), g(x) of C ∞ (M) and any real numbers a, b, it is Xp (af + bg) = a Xp (f ) + b Xp (g), (1.33) and Xp (f g) = Xp (f ) g(p) + f (p) Xp (g). (1.34) Now the set of all tangent vectors at the point p constitutes a real vector space, denoted as Tp M, and called the tangent space of M at the point p. The vector space operations are naturally defined by (Xp + Yp )(f ) = Xp f + Yp f for any two vectors Xp , Yp , and (aXp )(f ) = a Xp f for any real number a. It is common to use the simplified notation Xp f without brackets. ∗ We may consider alternatively the function space C ∞ (U ) for an open set U containing p. 12 Geometry, Symmetries, and Classical Physics – A Mosaic We would like to emphasize that tangent vectors are not elements of the manifold, as in the case of RD . In the figure 1.2 we give an illustration of a manifold M and its tangent space Tp M. The vectors Xp and Yp are elements of the tangent space. We like to imagine Xp p Yp Tp M M Figure 1.2: Manifold and tangent space that the point p of the manifold defines the origin of the linear space Tp M. This pictorial representation stems from planes tangent to surfaces, but our tangent space definition is completely independent of any notion of an embedding in a higher-dimensional space. Vectors in Local Coordinates We can write any vector Xp of the tangent space Tp M as a linear combination of basis vectors. A natural basis is defined if we choose local coordinates xµ around the point p. We can assume that the coordinates of p have the particular value xµ (p) = 0 at the point p, i.e. p is mapped to the origin of the tangent space Tp M. We consider an arbitrary function f ∈ C ∞ (M) that depends on the coordinates x and takes the values f (x) = f (x1 , . . . , xD ). We Taylor-expand this function around the origin xµ (p) = 0 and obtain, to first order  ∂f  f (x) = f (0) + xµ . (1.35) ∂xµ x(p) Note that we use the summation convention here and henceforth. Now we let Xp operate on the function f itself. By using the Leibniz rule, we have     ∂f  ∂f  µ µ . (1.36) + x (p) Xp Xp f = Xp (f (0)) + Xp (x ) ∂xµ p ∂xµ p It is Xp (c) = 0 for any constant c. We note further that xµ (p) = 0. Thus, it is  ∂  µ Xp = Xp (x ) . ∂xµ p (1.37) This is the basis expansion we are looking for. The D vectors  ∂  ∂xµ p (1.39) The real numbers X µ ≡ Xp (xµ ) are the vector components of the vector Xp in the local coordinates xµ . Hence, we can write   µ ∂  . (1.38) Xp = X ∂xµ p 13 Manifolds and Tensors constitute a basis of the tangent space Tp M, the so-called local coordinate basis (or natural basis). It is reassuring that we have derived the same natural basis of differential operators as in the case of RD . Moreover, again in analogy to RD , the real number Xp f given by ∂f ∂xµ Xp f = X µ (1.40) p is the directional derivative of f at the point p along Xp . We can always choose another set of local coordinates around the point p, let us call them x′µ . Then the tangent vector Xp can be written as ∂ Xp = X ′µ , (1.41) ∂x′µ p where the new components are given by X ′µ = Xp (x′µ ). Because of the chain rule, it is ∂xν ∂ = ∂x′µ ∂x′µ p ∂ , ∂xν (1.42) which here represents the transformation law for basis vectors. Thus, the components of the vector Xp transform as ∂x′µ Xν. (1.43) X ′µ = ∂xν p This is the well-known contravariant transformation law for vector components, now expressed for the general case of manifolds. Differential of a Map The notions of tangent space and vectors at a point give us the means to linearize maps between manifolds. Consider a D-dimensional manifold M, an N -dimensional manifold N , and a smooth map F : M → N between them. Then we can define at each point p of M a linear map DFp : Tp M → TF (p) N , Xp 7→ DFp (Xp ) between the corresponding tangent spaces in the following way. To each vector Xp of Tp M, the vector DFp (Xp ) ≡  ∂F α ∂xν X p ν  ∂ ∂y α (1.44) F (p) of TF (p) N is assigned. Here F α is the αth component of the representative function of F between RD and RN , α = 1, . . . , N . The local coordinates {xµ } and {y α } in the respective spaces define the basis vectors. The linear map DFp in coordinates is defined simply by the Jacobian matrix of the map F . The map DF generalizes the concept of the differential, known from calculus in RD to the case of differentiable manifolds. The naming for DF varies and we will call it simply the differential of the map F . Mathematicians prefer the notation F ∗ and speak about the pushforward , but we will not use this terminology. Let us remark that one can define the differential in a coordinate-free manner, see for example [12]. Vectors Tangent to Curves We have introduced the tangent space by using the algebraic properties of a differential operator. There is an alternative way to introduce the tangent space, which is based on the idea of vectors being tangent to curves. On our manifold M we consider smooth 1-parameter curves. These are maps γ : I → M, t 7→ γ(t) assigning to a real parameter t of the interval 14 Geometry, Symmetries, and Classical Physics – A Mosaic I ⊂ R a point p of the manifold M. Two curves γ1 and γ2 crossing at a point p of M for the parameter value t = 0 as γ1 (0) = γ2 (0) = p are said to be tangent at the point p iff dxµ dxµ (γ1 (0)) = (γ2 (0)), dt dt (1.45) for all µ = 1, . . . , D in local coordinates. In fact, this definition is independent of the coordinate choice. Now two curves crossing the point p are defined to be equivalent if they are tangent to each other. This leads to the notion of a vector as the equivalence class of all curves being tangent at the point p. This definition, despite being geometric and visual in its character, can be used also in the case of infinite-dimensional manifolds. In figure 1.3 we illustrate the concept of the equivalence class, denoted [γ], of tangent curves γ at a point γ(0). The tangent vector [γ] belongs to the tangent space Tp M. The [γ] {γ(t)} γ(0) = p M Figure 1.3: Tangent vector on curve action of a tangent vector Xp = [γ] on functions f defined on the manifold M is once again given by the directional derivative, which takes now the form Xp f = d f (γ(t)) dt . (1.46) t=0 We have encountered this expression already in the case of RD . Let us view the function f and the curve γ in local coordinates. By using the chain rule, we obtain for the directional derivative the expression dxµ ∂f (γ(t)) Xp f = . (1.47) µ dt ∂x p t=0 In other words, the vector Xp = [γ] is given as a differential operator with its components X µ being dxµ (γ(0)). (1.48) Xµ = dt We call dxµ (γ(0))/dt the components of the velocity and they uniquely define the tangent vector Xp = [γ], given here as an equivalence class of curves. It is useful to look at an example. Let us consider the unit helix curve γ(t) embedded in R3 and given in components by   cos t γ(t) =  sin t , (1.49) t with t being a positive real parameter. By using the Cartesian coordinates (x, y, z), the tangent vector γ̇(t) at the curve point γ(t) is given as γ̇(t) = − sin t ∂ ∂ ∂ + cos t + . ∂x ∂y ∂z The basis {∂/∂x, ∂/∂y, ∂/∂z} is, in fact, the Cartesian basis. (1.50) 15 Manifolds and Tensors Cotangent Space Now that we have introduced tangent vectors, the definition of covectors is straightforward. The cotangent space at the point p ∈ M is simply the algebraic dual∗ of the tangent space, ∼ Tep M ≡ (Tp M) . It is comprised of the linear maps αp : Tp M → R, Xp 7→ αp (Xp ), called covectors, which assign to each vector Xp a real number αp (Xp ). The cotangent space is a linear space. It is common to use the notation hαp , Xp i ≡ αp (Xp ) for the pairing of αp and Xp , even if no scalar product is present. Given any smooth function f defined around a point p, we can define uniquely a new covector dfp through the relation hdfp , Xp i = Xp f. (1.51) When we choose the function f to be the coordinate function xµ , the above formula says dxµp , Xp = X µ , (1.52) i.e. the covector dxµp picks the µth component of the vector Xp . By specializing to the case where the vector is a basis vector, we obtain + * ∂ = δνµ . (1.53) dxµp , ∂xν p  This means that the set dxµp of D covectors at the point p constitutes a basis of the cotangent space Tep M that is dual to the coordinate basis of Tp M. Every element αp of the cotangent space can be written as linear combination as αp = αµ dxµp , with the covector components αµ in local coordinates given by * + ∂ . αµ = αp , ∂xµ p (1.54) (1.55) The pairing hαp , Xp i in local coordinates is calculated as the contraction hαp , Xp i = αµ X µ . (1.56) Considering a covector given as dfp , its coordinate basis expansion is easily seen to be (exercise 1.1 ) ∂f dxµ . (1.57) dfp = ∂xµ p p This expression is consistent with the notion of the differential of a function, as known from analysis in RD . In particular, the expression dxµp has two meanings. On one hand, it represents a basis vector of the cotangent space Tep M. On the other hand, it is truly the differential of the coordinate function xµ evaluated at the point p, in the sense of analysis in RD . If we have an equation of the form f (x1 , . . . , xD ) = 0 involving the coordinate functions xµ and differentiate, we obtain an equation of the form h(x1 , . . . , xD , dx1 , . . . , dxD ) = 0. The expressions dxµ can then be interpreted in both of their meanings. We will make use ∗ Some authors use an asterisk for denoting the algebraic dual. We use the tilde so that we can reserve the asterisk for the complex conjugation. 16 Geometry, Symmetries, and Classical Physics – A Mosaic of this fact when we study metrics on manifolds. We can view a covector αp in two different local coordinate systems, αp = αµ dxµp = αµ′ dx′µ (1.58) p . For relating the two component representations αµ and αµ′ to each other, we note that dx′µ p = ∂x′µ ∂xν dxνp , (1.59) p which is the expression of the transformation law for basis covectors. This relation is equivalent to the transformation law of covector components αµ′ = ∂xν ∂x′µ αν . (1.60) p This formula is the known covariant transformation law for covector components, expressed here for differentiable manifolds. 1.4 Vector Fields and the Commutator Vector Fields In the previous section, we have introduced vectors and covectors defined at each point of the manifold. The next natural step is to view fields of vectors (or covectors) as maps assigning to each point of the manifold a vector (or covector) in a smooth way. Now we make this notion more concrete and start with the vector case. Given a manifold M, let us consider the union of all tangent vector spaces Tp M for all points p of the manifold as [ Tp M (1.61) TM ≡ p∈M and call this set the tangent bundle of M. Now a vector field is a map X : M → T M assigning to each point p of the manifold a vector Xp in a smooth way. There are different ways to express the smoothness and one of these can be to demand that the vector field components X µ (x) as functions of local coordinate variables xµ depend smoothly upon them. From now on, we will use the simplified notation ∂µ = ∂ ∂xµ (1.62) for the local basis vectors, varying from point to point in the manifold. There is no reference to the point p anymore. In terms of such a local basis {∂µ }, a vector field X is written as X = X µ (x) ∂µ (1.63) The transformation rule for the locally varying components X µ of the vector field is written as ∂x′µ ν X ′µ (x′ ) = (1.64) X (x) ∂xν Pictorially, we can imagine a vector field as in the sketch 1.4 below. We attach each local vector at its corresponding point of the manifold. However, we should remember that each one of these vectors actually belongs to a different tangent space. 17 Manifolds and Tensors X M Figure 1.4: Vector field To give a concrete example, consider the 2-dimensional manifold R2 \ {0} and the vector field ∂ ∂ y x + (1.65) X= 3/2 3/2 (x2 + y 2 ) ∂x (x2 + y 2 ) ∂y in Cartesian coordinates (x, y). This is the vector field generated by a positive point charge at the coordinate origin. We can transform to the coordinate system employing polar coordinates (r, θ), which are defined by  x = r cos θ , (1.66) y = r sin θ or, inversely, by r θ = =   x2 + y 2 . arctan (y/x) (1.67) The new basis vectors are found to be ∂ ∂x ∂ ∂y ∂ ∂ ∂ = + = cos θ + sin θ , ∂r ∂r ∂x ∂r ∂y ∂x ∂y (1.68) and ∂ ∂x ∂ ∂y ∂ ∂ ∂ = + = −r sin θ + r cos θ . ∂θ ∂θ ∂x ∂θ ∂y ∂x ∂y The vector field X in the new basis reads X= 1 ∂ , r2 ∂r (1.69) (1.70) as expected. (exercise 1.2 ) In figure 1.5 we provide a pictorial illustration of this r−2 -law vector field. The locally varying basis vector ∂r is pointing away from the origin. Covector Fields Conceptually, the case of covector fields is very similar to vector fields. First, we consider the cotangent bundle TM of the manifold M as the union of all cotangent spaces,  (1.71) Tp M. TM ≡ p∈M A covector field is a map α : M → TM which assigns to each point p of the manifold a covector αp in a smooth way. Smoothness is attained if the covector field components αµ (x) depend smoothly on the local coordinates x. In terms of a local dual basis {dxµ }, a covector field α is written as α = αµ (x) dxµ (1.72) 18 Geometry, Symmetries, and Classical Physics – A Mosaic y R2 X x Figure 1.5: Two-dimensional vector field X = r−2 ∂r The transformation law for the components αµ (x) reads αµ′ (x′ ) = ∂xν αν (x) ∂x′µ (1.73) For the directional derivative of a function f (x) on the manifold M along the vector field X, the basic relation Xf = df (X) = hdf, Xi (1.74) holds. Considering a smooth curve in the manifold γ : I → M, t 7→ γ(t), for which all its tangent vectors coincide with a vector field X on the points along the curve, the directional derivative of a function f (x) along the curve is given as df (X) = d f (γ(t)). dt (1.75) Finally, the duality between vector and covector basis elements is expressed as hdxµ , ∂ν i = δνµ (1.76) Active and Passive Transformations We define the notions of active and passive transformations by considering vector fields here, although the ideas apply equally well to covector fields. Consider a manifold M and its points p, which are mapped to coordinates x = ψ(p) under a certain coordinate system ψ. If we change the coordinate system to another one ψ ′ , then we obtain new coordinates x′ = ψ ′ (p) for each point of the manifold. This corresponds to a passive transformation. Any vector field X defined over the manifold remains unchanged. The vector field can be written in components in either of the two coordinate systems as X = X µ (x)∂µ = X ′µ (x′ )∂µ′ . (1.77) This means that the basis changes, ∂µ 7→ ∂µ′ , and the components change, X µ (x′ ) 7→ X ′µ (x), but the vector field remains as it is. The new coordinates are given by X ′µ (x′ ) = ∂x′µ ν X (x). ∂xν (1.78) 19 Manifolds and Tensors In the above constellation, one talks about the passive view of transformations. In contrast, an active transformation changes the points of the manifold itself, while the coordinate system used remains unchanged. An active transformation is a diffeomorphism of the manifold M that assigns to each point p a new point p′ . For both manifold points we use the same coordinate system ψ, so that we have the coordinates x = ψ(p) for the original point and x′ = ψ(p′ ) for the new point. The change in the points induces a change in any vector field X defined over the manifold. The active transformation maps the vector field X = X µ (x)∂µ to the new vector field X ′ , which is given as X ′ = X ′µ (x′ )∂ µ , (1.79) with the components ∂x′µ ν X (x). (1.80) ∂xν This is the very same formula as before, but now within a different context. Here we talk about the active view of transformations. In the following, we will always describe how a considered transformation is meant to act. X ′µ (x′ ) = Algebra of Vector Fields Technically, a vector field X defines a map that assigns to each smooth function f on the manifold a new smooth function Xf . Thus, we can view the vector field as a map X : C ∞ (M) → C ∞ (M), f 7→ Xf defined by (Xf )(p) ≡ Xp f for all points p on the manifold. In this way, we can let multiple vector fields X, Y, . . . act on smooth functions f . The composition of two vector fields X and Y denoted XY is naturally given as (XY )f ≡ X(Y f ). Interestingly, the composition XY is linear in f , but it does not fulfill the Leibniz rule and consequently it is not a vector field. However, the combination XY − Y X is linear and satisfies the Leibniz rule. (exercise 1.3 ) The notation [X, Y ] ≡ XY − Y X (1.81) is used here and is called the commutator of the vector fields X and Y , or, the Lie bracket (Marius Sophus Lie). The Leibniz rule reads then [X, Y ] (f g) = ([X, Y ] f ) g + f ([X, Y ] g). (1.82) By using a coordinate representation, the commutator [X, Y ] of two vector fields X, Y can be expressed as (exercise 1.4 ) µ [X, Y ] = X ν ∂ν Y µ − Y ν ∂ν X µ . (1.83) From an algebraic point of view, the commutator itself [·, ·], besides being bilinear in its arguments X, Y , is also antisymmetric, [X, Y ] = − [Y, X], (1.84) [[X, Y ] , Z] + [[Y, Z] , X] + [[Z, X] , Y ] = 0, (1.85) and obeys the Jacobi identity, for any three vector fields X, Y , Z. A real vector space equipped with a product that is bilinear, antisymmetric and satisfies the Jacobi identity is called a real Lie algebra, see also the definitions in appendix B.1. This means that the space of all vector fields X on a manifold M, in combination with the commutator, constitutes a real Lie algebra, denoted X M. The dual space XeM contains all covector fields α on the manifold. 20 1.5 Geometry, Symmetries, and Classical Physics – A Mosaic Tensor Fields on Manifolds Tensors at a Point We proceed now to the definition of tensors on manifolds. The reader should recall the algebraic concepts described in appendix B.1. For any single point p of the manifold M, we can build (m, n)-type tensors Tp at the point p, which are elements of the Dm+n -dimensional (m,n) tensor product space Tp M, where the latter is given as Tp(m,n) M ≡ O m  Tp M ⊗ O n  e Tp M . (1.86) The tensor product space at the point p is constructed from multiple copies of the tangent space Tp M and the cotangent space Tep M. Every (m, n)-type tensor Tp of the space (m,n) Tp M is a multilinear map Tp : Tep M × · · · × Tep M × Tp M × · · · × Tp M → R | {z } | {z } (1.87) ∂µ1 |p ⊗ · · · ⊗ ∂µm |p ⊗ dxνp1 ⊗ · · · ⊗ dxνpn , (1.88) m-fold n-fold that assigns to each (m + n)-tuple (α1 , . . . , αm , X1 , . . . , Xn ) of m covectors and n vectors (m,n) at the point p the real number Tp (α1 , . . . , αm , X1 , . . . , Xn ). A tensor of the space Tp M is called m-fold contravariant and n-fold covariant. Whenever we choose a local coordinate (m,n) system xµ around the point p, we induce a local basis in the tensor space Tp M consisting of the Dm+n elements which are constructed as tensor products of m basis vectors ∂µ |p and n basis covectors dxνp . Hence, any tensor Tp at the point p is written as a linear combination Tp = T µ1 ...µm ν1 ...νn · ∂µ1 |p ⊗ · · · ⊗ ∂µm |p ⊗ dxνp1 ⊗ · · · ⊗ dxνpn . (1.89) The local components of the tensor Tp are given as
T µ1 ...µm ν1 ...νn = Tp dxµp 1 , . . . , dxµp m , ∂ν1 |p , . . . , ∂νn |p . (1.90) When we say “local” we mean that the tensor Tp is defined as an element of a vector space at the point p. The components of Tp are defined by a choice of basis at p. However, we can move out of the point p within the manifold and ask if we can define a tensorial object, which has Tp or its components T µ1 ...µm ν1 ...νn as local values at the point p. This chain of thoughts leads to the notion of a tensor field, to which we now turn. Tensor Fields An (m, n)-type tensor field φ on the manifold M is a smooth assignment of an (m, n)type tensor φp at each point p of the manifold. Once again smoothness can be imposed by requiring that the components of the tensor field depend smoothly on the local coordinates xµ . Formally, we write [ Tp(m,n) M, (1.91) φ : M → T (m,n) M ≡ p∈M using the (m, n)-tensor bundle T (m,n) M as the range. Tensor fields φ provide a most important mathematical notion in order to formulate classical physical laws. A tensor field 21 Manifolds and Tensors equation F = 0, with a certain tensor field F , is the same in all coordinate systems, exactly as we would require from a universally valid (physical) law. In terms of notation, in our applications we will occasionally write φ(x) to display the local dependency of the tensor field φ. This is, strictly speaking, an abuse of notation, since the tensor field φ is independent on any coordinate choice. Still, this notation is useful to make the distinction between rigid tensors (at a point) and tensor fields (defined for each point of a manifold) clear. The dependence on local coordinates x will also be displayed for the components of tensor fields. As soon as local coordinates x are chosen, the corresponding locally varying (m, n)-tensor basis ∂µ1 ⊗ · · · ⊗ ∂µm ⊗ dxν1 ⊗ · · · ⊗ dxνn (1.92) is introduced. In terms of this basis, the (m, n)-tensor field φ is expanded as φ = φµ1 ...µm ν1 ...νn (x) ∂µ1 ⊗ · · · ⊗ ∂µm ⊗ dxν1 ⊗ · · · ⊗ dxνn (1.93) where the tensor field components φµ1 ...µm ν1 ...νn (x) are given as φµ1 ...µm ν1 ...νn (x) = φ (dxµ1 , . . . , dxµm , ∂ν1 , . . . , ∂νn ). (1.94) Now let us examine what happens when we change the coordinates of a particular point in the manifold. An active (or passive) transformation from xµ to x′µ , given by x′µ = ∂x′µ ν x, ∂xν (1.95) leads to the following general transformation of the tensor field components: φ′µ1 ...µm ν1 ...νn (x′ ) = ∂x′µm ∂xσ1 ∂xσn ρ1 ...ρm ∂x′µ1 · · · · · · φ σ1 ...σn (x) ∂xρ1 ∂xρm ∂x′ν1 ∂x′νn (1.96) This is the tensor field transformation law , as employed in classical tensor analysis for the basic definition of a tensor field. Let us remark that, similar to tensors at a point, tensor fields over a manifold can be also regarded as multilinear maps. As explained before, a tensor at a point assigns to a tuple of vectors and covectors at that point a real number. In analogy, a tensor field over a manifold assigns to a tuple of vector fields and covector fields a smooth function. The straightest way to see this, is to use the component representation of a tensor field φ. The components φµ1 ...µm ν1 ...νn (x), defined over the entire manifold, are used to assign to a tuple (α1 , . . . , αm , X1 , . . . , Xn ) of m covector fields and n vector fields the smooth function φµ1 ...µm ν1 ...νn (x) · α1µ1 (x) · · · αmµm (x) · X1ν1 (x) · · · Xnνn (x). (1.97) This means that the tensor field φ is a multilinear map of the form φ : XeM × · · · × XeM × X M × · · · × X M → C ∞ (M). {z } | {z } | m-fold (1.98) n-fold Multilinearity here means especially that φ is homogeneous in any scalar smooth function f (x) appearing in the arguments, so that it is φ(. . . , f X, . . .) = f φ(. . . , X, . . .). 22 Geometry, Symmetries, and Classical Physics – A Mosaic Tensor Densities Extending the purely algebraic constructions of tensor densities, as explained in appendix B.1, to the case of tensor fields on manifolds is straightforward. Per definition, a tensor density field T with components T µ1 ...µm ν1 ...νn (x) transforms as T ′µ1 ...µm ν1 ...νn (x′ ) = J(x)w ∂x′µ1 ∂x′µm ∂xσ1 ∂xσn ρ1 ...ρm · · · · · · T σ1 ...σn (x), ∂xρ1 ∂xρm ∂x′ν1 ∂x′νn (1.99) with the Jacobian determinant J(x) ≡ det   ∂x′µ (x) . ∂xν (1.100) and the weight w. In addition to tensor density fields, the so-called numerical tensors are important for conceptual aspects and for practical applications. The numerical tensors are ...µn the generalized Kronecker delta δνµ11...ν and the Levi-Civita epsilon symbol εµ1 ...µD . They n extend the algebraic definitions from appendix B.1 to the case of manifolds. The numerical tensors not only retain their components invariant under coordinate transformations, but they keep their component values constant throughout all the points of the manifold. This means that under arbitrary diffeomorphisms of the manifold, it is always for the Kronecker delta ...µn ...µn , (1.101) = δνµ11...ν δ ′µν11...ν n n and, similarly, for the epsilon symbol ε′µ1 ...µD = εµ1 ...µD . (1.102) We should notice that the epsilon symbol has as many components as the dimensionality D of the manifold. In addition to the epsilon symbol εµ1 ...µD , which is a (0, D)-tensor density field of weight w = 1, one defines the Levi-Civita epsilon tensor ǫµ1 ...µD by p (1.103) ǫµ1 ...µD ≡ |g| εµ1 ...µD , where g ≡ det(gµν ) is the determinant of a symmetric (0, 2)-tensor field gµν (x), called the metric, whose primary task is to define a geometry on the manifold, as we will see in the next chapter. With the above definition, the epsilon tensor ǫµ1 ...µD represents an absolute (0, D)-tensor field. In the following, we will use both, the epsilon symbol and the epsilon tensor, depending on which of the two is more appropriate for the specific application. Further Reading Our discussion of differentiable manifolds focused primarily on the needs in classical physics. The reader interested to encounter a mathematically rigorous treatment of manifolds and tangent spaces can turn, for example, to Boothby [12]. 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