Fisica Matematica. Stefano Ansoldi. Dipartimento di Matematica e Informatica. Università degli Studi di Udine. Corso di Laurea in Matematica

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1 Fisica Matematica Stefano Ansoldi Dipartimento di Matematica e Informatica Università degli Studi di Udine Corso di Laurea in Matematica Anno Accademico 2003/2004 c 2004 Copyright by Stefano Ansoldi and the University of Udine, All Italian and international copyrights reserved for all original material presented in this course and lecture notes through any medium, including lecture or print. Individuals are prohibited from being paid for taking, selling, or otherwise transferring for value, this material or part of it without the express written permission of Stefano Ansoldi. Individuals are prohibited from distributing by any means this material or part of it without the express written permission of Stefano Ansoldi. Individuals are granted the right of using this material for personal, educational purposes only without the express written permission of Stefano Ansoldi.

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3 Contents Contents List of Figures List of Tables List of Definitions List of Propositions List of Notations i v vii ix xi xiii II Acknowledgments xv 1 Lecture Introduction A glimpse at Mathematical Physics Operative definition of physical concepts Axiomatic definition of Mathematical Concepts Some contact points Our program in Mathematical Physics Lecture From discrete to continuous systems Some naming conventions The equations of motion Field equations Other dynamical quantities in field theory Synopsis Lecture Euler-Lagrange equations in field theory Preliminaries Functional Derivatives Estremals and field equations Stationary action principle Synopsis i

4 ii Contents. 4 Lecture Reflections on space and time Classical mechanics and its framework A fundamental law of electrodynamics The consistency problem Einstein solution: a reflection about time Simultaneity Lorentz transformations The algebraic derivation Synopsis Lecture The group of Lorentz transformations dimensional case A comment about the 4-dimensional case Synopsis Lecture Tensors Tensor product Properties of tensor product Lecture Tensors Additional properties of tensor product Isomorphism with multilinear transformations Tensors and components Synopsis Lecture Some reminders of topology Some reminders of differential geometry Vector bundles and sections Partition of unity Tensors Tensors (Tensor Fields) on Manifolds Lecture Some algebraic preliminaries Scalar products on a vector space Riemannian (Lorentzian) geometry Riemannian and Lorentzian manifolds Connections on manifolds Connections and symmetric connections Lecture Connections on manifolds Characterization of symmetric connections Smooth curves and covariant derivative along a curve.. 63 c 2004 by Stefano Ansoldi Please, read statement on cover page

5 Contents. iii 11 Lecture Tensors A few more concepts about tensors Connections on manifolds Parallel vector fields and parallel translation Extension of covariant derivative to tensors Lecture Connections on manifolds Component expression of the covariant derivative Curvature Definition of the Riemann tensor and a basic property Lecture Curvature Components of the Riemann tensor, symmetries and Ricci tensor Lecture More about curves on manifolds Autoparallel curves and the exponential map Lecture Riemannian (Lorentzian) geometry Interplay between connection and metric Curvature Curvature on Riemannian (Lorentzian) Manifolds Lecture Notation, Greek indices, General Relativity Problems of the special theory Einstein s elevator thought experiment Synopsis and a word of caution Lecture More about curves on manifolds Geodesics Mathematical formulation of General Relativity The classical limit of the geodesic equation Lecture Einstein Equations Action for the gravitational field Variational principle The Lagrangian density Derivations of Einstein equations in vacuo c 2004 by Stefano Ansoldi Please, read statement on cover page

6 iv Contents. 19 Lecture Properties of Einstein s field equation Physical meaning of the metric fields Time measurements Space measurements Clock synchronization More about the classical limit Synopsis Lecture Conservation of energy in classical mechanics Conservation laws in a special relativistic field theory Conservation laws and general covariance Einstein equations Problems 123 A Parallelism and Autoparallelism 125 A.1 Autoparallel curves c 2004 by Stefano Ansoldi Please, read statement on cover page

7 List of Figures 8.1 Typical example of a non-hausdorff topological space Vector bundle Section of a vector bundle Timelike, spacelike and null vectors Exponential of a vector v

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9 List of Tables vii

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11 List of Definitions 3.1 Field Functional of fields Field fluctuation (or variation) Finite variation of a functional Extremal of a functional Euler-Lagrange equations Tensor product Universal factorization property Topology and open sets Topological space Neighborhood Cover Subcover Refinement Open cover Locally finite open cover Compact topological space Paracompact topological space Hausdorff topological space Vector bundle Section of a vector bundle Differentiable partition of unity Tensor bundle of the (r, s) type Smooth tensor field Tangent and Cotangent bundle Vector fields and 1-form fields Line element field Lie Brackets Scalar product Signature and Lorentzian metric Timelike, spacelike and null vectors Riemannian metric Lorentzian metric Isometry between manifolds Connection at m M Connection on a manifold Symmetric connection Connection in coordinates ix

12 x Contents Smooth curve on a manifold Tensor algebra Contractions of a tensor Parallel vector field along a curve Covariant derivative of vector fields Extension of covariant derivative Riemann curvature tensor Ricci tensor Autoparallel Compatibility condition Ricci scalar Einstein tensor Timelike, spacelike, null vectors on a manifold Local character of a curve at a point Global character of a curve Length of a curve Stress Energy Tensor c 2004 by Stefano Ansoldi Please, read statement on cover page

13 List of Propositions 3.1 Conditions for an extremal Galilean law of composition of velocities Tensor product: universal factorization Isomorphism of V W into W V Isomorphism of F U onto U Isomorphism of (U V ) W onto U (V W ) Tensor product of functions Distributive properties of with respect to Basis of tensor product Tensor product and linear applications Tensor product and duals Tensor product and linear mappings Existence of partition of unity Characterization of smooth tensor fields Characterization of smooth vector fields Properties of the Lie Brackets Existence of Riemannian metric Existence of Lorentzian metric Characterization of symmetric connections Covariant derivative along a curve Characterization of parallel vector field Existence of parallel vector fields Parallel translation is an isomorphism Riemann tensor and covariant derivatives Riemann tensor and coordinate basis Properties of the Riemann tensor Autoparallelism equation I characterization of compatible connections II characterization of compatible connections ! symmetric compatible connection More symmetries of the Riemann tensor Symmetries of the Ricci tensor Differential identities of curvature tensors Character of geodesics Local conservation laws xi

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15 List of Notations Particular cases of bundles and spaces of fields Covariant derivative components Compatible Symmetric Covariant Derivative xiii

16 xiv Contents. c 2004 by Stefano Ansoldi Please, read statement on cover page