TAILIEUCHUNG - Introduction to Differential Geometry & General Relativity Third Printing January 2002

Introduction to Differential Geometry and General Relativity Lecture Notes by Stefan Waner, with a Special Guest Lecture by Gregory C. Levine Department of Mathematics, Hofstra University These notes are dedicated to the memory of Hanno Rund. TABLE OF CONTENTS 1. Preliminaries: Distance, Open Sets, Parametric Surfaces and Smooth Functions 2. Smooth Manifolds and Scalar Fields 3. Tangent Vectors and the Tangent Space 4. Contravariant and Covariant Vector Fields 5. Tensor Fields 6. Riemannian Manifolds 7. Locally Minkowskian Manifolds: An Introduction to Relativity 8. Covariant Differentiation 9. Geodesics and Local Inertial Frames 10. The Riemann Curvature Tensor 11. A Little More. | Introduction to Differential Geometry General Relativity Third Printing January 2002 Lecture Notes by Stefan Waner with a SpeciaC Guest Lecture by Gregory C. Levine Departments of Mathematics and Physics Hofstra University Introduction to Differential Geometry and General Relativity Lecture Notes by Stefan Waner with a Special Guest Lecture by Gregory C. Levine Department of Mathematics Hofstra University These notes are dedicated to the memory of Hanno Rund. TABLE OF CONTENTS 1. Preliminaries Distance Open Sets Parametric Surfaces and Smooth Functions 2. Smooth Manifolds and Scalar Fields 3. Tangent Vectors and the Tangent Space 4. Contravariant and Covariant Vector Fields 5. Tensor Fields 6. Riemannian Manifolds 7. Locally Minkowskian Manifolds An Introduction to Relativity 8. Covariant Differentiation 9. Geodesics and Local Inertial Frames 10. The Riemann Curvature Tensor 11. A Little More Relativity Comoving Frames and Proper Time 12. The Stress Tensor and the Relativistic Stress-Energy Tensor 13. Two Basic Premises of General Relativity 14. The Einstein Field Equations and Derivation of Newton s Law 15. The Schwarzschild Metric and Event Horizons 16. White Dwarfs Neutron Stars and Black Holes by Gregory C. Levine 2 1. Preliminaries Distance and Open Sets Here we do just enough topology so as to be able to talk about smooth manifolds. We begin with n-dimensional Euclidean space En yp y2 yj 1 yi é . Thus E1 is just the real line E2 is the Euclidean plane and E3 is 3-dimensional Euclidean space. The magnitude or norm llyII of y y1 y2 yn in En is defined to be llyll Vyi2 y22 yn2 which we think of as its distance from the origin. Thus the distance between two points y y1 y2 yn and z z1 z2 zn in En is defined as the norm of z - y Distance Formula Distance between y and z llz - yII V z1 - y1 2 z2 - y2 2 zn - yn 2 Proposition Properties of the norm The norm satisfies the following a llyll 0 and llyll 0 iff y 0 positive definite b ll yll l lllyll for every é JR and

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