文档介绍：Introduction
to
Differential Geometry
&
General Relativity
Thiiird Priiintiiing January 2002
Lecture Notes
by
Stefan Waner
with a Specialll 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
R
En = {(y1, y2, . . . , yn) | 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, ||yy|| of y = (y1, y2, . . . , yn) in En is defined to be
2 2 2
||yy|| = y1 +y2 +...+yn ,
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
yy 2 2 2
Distance between and z