| Differential Geometry (and Relativity) - Summer 2000 |
COURSE: MATH 5310-010, CALL #20383
TIME: 11:20 - 12:50 MTWThF, PLACE: 313 Gilbreath
INSTRUCTOR: Dr. Robert Gardner, OFFICE: Gilbreath 308G and Brown 201
OFFICE HOURS: TRF 9:30--10:30, MW by appointment, PHONE: 439-6977 (Math Office 439-4349)
E-MAIL ADDRESS: gardnerr@etsu.edu
HOMEPAGE: www.etsu.edu/math/gardner/gardner.htm.
CLASS HOMEPAGE: A copy of the syllabus and class notes are available on the internet at: www.etsu.edu/math/gardner/5310/5310.htm and www.etsu.edu/math/gardner/5310/notes.htm.
TEXT: "Differential Geometry and Relativity Theory, An Introduction" by Richard L. Faber, Monographs and Textbooks in Pure and Applied Mathematics, Volume 75, copyright 1983 by Marcel Dekker, Inc. (ISBN 0-8247-1749-X).
SUPPLEMENTARY TEXT: "Relativity: The Special and the General Theory" by Albert Einstein, available from Random House.
PREREQUISITES: Multivariable calculus and linear algebra (the more, the better!).
ABOUT THE CLASS: This course will be roughly broken into three parts: (1) differential geometry (with an emphasis on curvature), (2) special relativity, and (3) general relativity. We will spend about half of our time on differential geometry. We will then take a "break" and address special relativity. The class will finish (and climax) with general relativity. We will deal at length with the (differential geometry) topics of curvature, intrinsic and extrinsic properties of a surface and manifold. We will briefly survey special relativity (giving coverage that a physicist would consider fairly thorough, but which a geometer would consider a "shallow survey"). We will cover general relativity as time permits. In particular, we will "outline" (as the text puts it) Einstein's field equations and derive the Schwarzschild solution (which involves a nonrotating, spherical mass). We will see the differential geometry concepts come to the aid of gravitation theory and the study ob black holes. Time permitting, we will discuss the global topology of the universe, and philosophical implications of relativity.
MY TEACHING STYLE FOR THIS CLASS: We will go at a maddening rate. My lectures will follow from the overheads which I present in class. You will be given copies of these overheads before we cover them in class. You may also find copies of the notes on the internet in PDF format.
GRADING: Your grade will be determined based on your performance on assigned homework problems. Very roughly, you will be assigned 3 or 4 problems per section we cover. Click here for the list of homework problems. Class discussion may also be a factor in determining your grade. In particular, discussion of the reading assignments (from the Einstein book) is strongly encouraged.
A NOTE ABOUT THE INTERNET: I have put the overheads I use on the web. In addition, there is a "class" homepage which is linked to sites of interest to us. I will encourage student input on this particular project.
| DATE | TOPIC |
| MON. 6/5 | 1.1=Curves: arclength, tangent vector, curvature |
| TUE. 6/6 | 1.1(cont.): binormal vector, torsion, Planetarium Show |
| WED. 6/7 | 1.2=Gauss Curvature: normal section, principal curvature, 1.3=Surfaces in E3: surfaces of revolution, parallels |
| THR. 6/8 | 1.4=First Fundamental Form: metric form, intrinsic property |
| FRI. 6/9 | 1.5=Second Fundamental Form: Frenet Frame, normal curvature |
| MON. 6/12 | 1.6=Gauss Curvature in Detail: principal curvature |
| TUE. 6/13 | 1.6(cont.), 1.7=Geodesics: Christoffel symbols |
| WED. 6/14 | 1.7(cont.): "straight lines," more geodesics |
| THR. 6/15 | 1.7(cont.), 1.8=Curvature Tensor: Theorema Egregium, Einstein: Preface, 1.1-1.6 |
| FRI. 6/16 | 1.8(cont.), 1.9=Manifolds: coordinates |
| MON. 6/19 | 1.9(cont.): smooth manifold, vectors as operators, inner products |
| TUE. 6/20 | 1.9(cont.), 2.1=Inertial Frames, video: Shape of Space |
| WED. 6/21 | 2.2=Michelson-Morley Experiment: stellar aberration, Einstein: 1.7-1.12 |
| THR. 6/22 | 2.3=Postulates of Relativity, 2.4=Simultaneity |
| FRI. 6/23 | 2.5=Coordinates, 2.6=Invariance of the Interval |
| MON. 6/26 | 2.7=Lorentz Transformation: invariance of the interval Einstein: Appendix I |
| TUE. 6/27 | 2.8=Spacetime Diagrams, 2.9=Lorentz Geometry |
| WED. 6/28 | 2.10=Twin Paradox: Doppler effect, 2.11=Causality Einstein: 1.13-1.17 |
| THR. 6/29 | 3.1=Principle of Equivalence, 3.2=Gravity as Spacetime Curvature |
| FRI. 6/30 | 3.3=Consequences of General Relativity, 3.6=Geodesics: timelike, lightlike, spacelike |
| MON. 7/3 | 3.7=Field Equations: Ricci tensor, Einstein: 2.18-2.11, Appendix III |
| TUE. 7/4 | Independence Day Holiday |
| WED. 7/5 | 3.8=Schwarzschild solution |
| THR. 7/6 | 3.9=Orbits in General Relativity: precessions Einstein: 2.23-2.29, Appendix IV |
| FRI. 7/7 | 3.9(cont.), 3.10=Bending of Light, Einstein: 3.30-3.32, Appendix V |
| MON. 7/10 | Black Holes: Schwarzschild radius, Eddington-Finkelstein coordinates, gravitational redshift |
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