DiffGeom - Introduction to Differential Geometry...

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Unformatted text preview: Introduction to Differential Geometry & General Relativity 4th Printing January 2005 Lecture Notes by Stefan Waner with a Special Guest Lecture by Gregory C. Levine Departments of Mathematics and Physics, Hofstra University 2 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 ......................................................................................................3 2. Smooth Manifolds and Scalar Fields ..............................................................7 3. Tangent Vectors and the Tangent Space .......................................................14 4. Contravariant and Covariant Vector Fields ................................................24 5. Tensor Fields ....................................................................................................35 6. Riemannian Manifolds ...................................................................................40 7. Locally Minkowskian Manifolds: An Introduction to Relativity .............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 and Black Holes, by Gregory C. Levine 131 References and Further Reading ................................................................138 3 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 E n = {(y 1 , y 2 , . . . , y n ) | y i R } . Thus, E 1 is just the real line, E 2 is the Euclidean plane, and E 3 is 3- dimensional Euclidean space. The magnitude , or norm , ||y|| of y = (y 1 , y 2 , . . . , y n ) in E n is defined to be ||y|| = y 1 2 + y 2 2 + . . . + y n 2 , which we think of as its distance from the origin. Thus, the distance between two points y = (y 1 , y 2 , . . . , y n ) and z = (z 1 , z 2 , . . . , z n ) in E n is defined as the norm of z - y: Distance Formula Distance between y and z = ||z - y|| = (z 1- y 1 ) 2 + (z 2- y 2 ) 2 + . . . + (z n- y n ) 2 ....
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DiffGeom - Introduction to Differential Geometry...

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