Geometría Diferencial

Introduction to Differential Geometry & General Relativity
4th Printing January 2005

Lecture Notes Stefan Waner with a Special Guest Lecture by Gregory C. Levine
by

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 ofMathematics, Hofstra University These notes are dedicated to the memory of Hanno Rund. TABLE OF CONTENTS 1. Preliminaries......................................................................................................3 2. Smooth Manifolds and Scalar Fields..............................................................7 3. Tangent Vectors and the TangentSpace.......................................................14 4. Contravariant and Covariant Vector Fields ................................................24 5. Tensor Fields....................................................................................................35 6. Riemannian Manifolds...................................................................................40 7. Locally Minkowskian Manifolds: An Introduction toRelativity.............50 8. Covariant Differentiation...............................................................................61 9. Geodesics and Local Inertial Frames.............................................................69 10. The Riemann Curvature Tensor..................................................................82 11. A Little More Relativity: Comoving Frames and Proper Time...............94 12. The Stress Tensor and the Relativistic Stress-Energy Tensor................100 13. Two Basic Premises of General Relativity................................................109 14. The Einstein Field Equations and Derivation of Newton's Law...........114 15. The Schwarzschild Metric and Event Horizons ......................................124 16. White Dwarfs, Neutron Stars andBlack Holes, by Gregory C. Levine131 References and Further Reading................................................................138

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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 = {(y1, y2, . . . , yn) | yi é R}. Thus, E1 is just the real line, E2 is the Euclideanplane, and E3 is 3-dimensional Euclidean space. The magnitude, or norm, ||y|| of y = (y1, y2, . . . , yn) in En is defined to be ||y|| = y12 + 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 = ||z - y|| =(z1 - y1)2 + (z2 - y2)2 + . . . + (zn - yn)2 .

Proposition 1.1 (Properties of the norm) The norm satisfies the following: (a) ||y|| ≥ 0, and ||y|| = 0 iff y = 0 (positive definite) (b) ||¬y|| = |¬|||y|| for every ¬ é R and y é En. (c) ||y + z|| ≤ ||y|| + ||z|| for every y, z é En (triangle inequality 1) (d) ||y - z|| ≤ ||y - w|| + ||w - z|| for every y, z, w é En (triangle inequality 2) Theproof of Proposition 1.1 is an exercise which may require reference to a linear algebra text (see “inner products”). Definition 1.2 A Subset U of En is called open if, for every y in U, all points of En within some positive distance r of y are also in U. (The size of r may depend on the point y chosen. Illustration in class). Intuitively, an open set is a solid region minus its boundary. If weinclude the boundary, we get a closed set, which formally is defined as the complement of an open set. Examples 1.3 (a) If a é En, then the open ball with center a and radius r is the subset B(a, r) = {x é En | ||x-a|| < r}.

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Open balls are open sets: If x é B(a, r), then, with s = r - ||x-a||, one has B(x, s) ¯ B(a, r). (b) E n is open. (c) Ø is open. (d) Unions of open sets are open. (e)...