Differential Geometry (and Relativity) - Summer 2002

COURSE: MATH 5310-010, CALL #21394

TIME: 11:20 - 12:50 MTWThF, PLACE: 302 Sam Wilson Hall

INSTRUCTOR: Dr. Robert Gardner, OFFICE: Gilbreath 308G

OFFICE HOURS: TThF 9:30--10:30, MW by appointment, PHONE: 439-6977 (Math Office 439-4349)

E-MAIL ADDRESS: gardnerr@etsu.edu

WEBPAGE: www.etsu.edu/math/gardner/gardner.htm.

CLASS WEBPAGE: A copy of the syllabus and class notes are available on the internet at: www.etsu.edu/math/gardner/5310/sil5310_2.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 TEXTS: "Relativity: The Special and the General Theory" by Albert Einstein, available from Random House. "Cosmic Time Travel by Barry Parker, Perseus Publishing.

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. We will discuss gravitational redshift, precessions of orbits, the ``bending of light,'' black holes, 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. For a menu of the notes (which are available in PDF, PostScript, and DVI formats) see: http://www.etsu.edu/math/gardner/5310/notes.htm. In addition, there is a "class" homepage which is linked to sites of interest to us: http://www.etsu.edu/math/gardner/5310/5310.htm. I will encourage student input on this particular project.

THE PLANETARIUM SHOW: On the second day of class (Tuesday June 4), we will meet in the planetarium. A presentation of ``Relativity and Black Holes'' will be given. This show includes a survey of the results we will see this semester. It also includes extensive historical references to the individuals responsible for these results (Lorentz, Einstein, Minkowski, and Schwarzschild). Since this is a math class, we will not spend any time on observational astronomy, but the planetarium show includes some of the observational evidence for black holes. The primary source for the planetarium show is Kip Thorne's excellent Black Holes and Time Warps: Einstein's Outrageous Legacy (1994, W. W. Norton Publishing). A web-based version of the show is available at http://www.etsu.edu/physics/plntrm/relat/relatabs.htm.

VIDEOS: We will watch two videos in class. "The Shape of Space" is a clever introduction to three-dimensional manifolds. We will discuss the possible global topologies of our universe, and ways to empirically detect this structure. A webpage by The Geometry Center accompanies the video: http://www.geom.umn.edu/video/sos/. The webpage gives additional information on the topic, as well as some hands-on projects suitable for high-school-level students. "Time Travel" is an episode of the P.B.S. series Nova. It includes interviews with Carl Sagan and Kip Thorne, and discusses the use of wormholes and exotic matter in the use of time travel. P.B.S. has a website which accompanies this video at http://www.pbs.org/wgbh/nova/time/. A third video "Einstein's Universe" will be available for students to borrow. This television show was created by the B.B.C. in 1979 to celebrate the 100 year anniversary of Einstein's birth. Though over 20 years old, the video still contains excellent explanations of time dilation, length contraction, and the effects of a strong gravitational field (such as that experienced by someone orbiting a black hole). A companion book Einstein's Universe by Nigel Calder (New York: Viking Press, 1979) is also available.

Important Dates

Blackhole Albert Einstein Wormhole

Tentative Schedule
DAY
DATE
TOPIC
1
MON 6/3
1.1=Curves: arclength, tangent vector, curvature
2
TUE 6/4
1.1 (cont.): binormal vector, torsion, Planetarium Show
3
WED 6/5
1.2=Gauss Curvature: normal section, principal curvature
1.3=Surfaces in E3: surfaces of revolution, parallels
4
THR 6/6
1.3 (cont.), 1.4=First Fundamental Form: metric form, intrinsic property
5
FRI 6/7
1.5=Second Fundamental Form: Frenet Frame, normal curvature,
Parker: Chapter 3
6
MON 6/10
1.6=Gauss Curvature in Detail: principal curvature
7
TUE 6/11
1.7=Geodesics: Christoffel symbols
8
WED 6/12
1.7 (cont.): "straight lines," more geodesics,
1.8=Curvature Tensor: Theorema Egregium
9
THR 6/13
1.8 (cont.)
10
FRI 6/14
1.9=Manifolds: coordinates
11
MON 6/15
1.9(cont.): smooth manifold, vectors as operators, inner products
Einstein: Preface, 1.1-1.6, video: Shape of Space
12
TUE 6/18
2.1=Inertial Frames, 2.2=Michelson-Morley Experiment: stellar aberration
2.3=Postulates of Relativity, Einstein: 1.7-1.12, Parker: Chapters 1, 2
13
WED 6/19
2.4=Simultaneity, 2.5=Coordinates
14
THR 6/20
2.6=Invariance of the Interval
Einstein: Appendix I, 1.13-1.17
15
FRI 6/21
2.7=Lorentz Transformation: invariance of the interval
16
MON 6/24
2.8=Spacetime Diagrams
17
TUE 6/25
2.9=Lorentz Geometry, 2.10=Twin Paradox: Doppler effect, 2.11=Causality
18
WED 6/26
3.1=Principle of Equivalence, 3.2=Gravity as Spacetime Curvature
3.3=Consequences of General Relativity
19
THR 6/27
3.6=Geodesics: timelike, lightlike, spacelike
3.7=Field Equations: Ricci tensor, Einstein: 2.18-2.11, Appendix III
20
FRI 6/28
3.8=Schwarzschild solution, Einstein: 2.23-2.29, Appendix IV
21
MON 7/1
3.9=Orbits in General Relativity: precessions
22
TUE 7/2
3.10=Bending of Light, Einstein: 3.30-3.32, Appendix V
Parker: Chapter 6
23
WED 7/3
Black Holes: Schwarzschild radius, Eddington-Finkelstein coordinates,
gravitational redshift
-
THR 7/4
Independence Day Holiday
24
FRI 7/5
Parker: Chapters 7, 8, 10
25
MON 7/9
video: Time Travel, Parker: Chapter 11
"Einstein" and "Parker" refer to readings from the supplemental texts.

SOME REFERENCES: The following is a list of books on relativity, geometry, and cosmology which I find particularly interesting. They range from easy-to-read popular books, to extremely difficult technical textbooks. Since we only have 5 weeks for our class, we will only lightly introduce most of the topics. If you are interested in studying these topics in more detail, then these references are a good place to start. The books marked with a star * are my favorites!

  1. M. Bartusiak, Einstein's unfinished Symphony: Listening to the Sounds of Space-Time N. Calder, Einstein's Universe (1979) NY: Viking Press. This is a popular book which is the companion to the BBC video by the same name.
  2. J. Callahan, The Geometry of Spacetime: An Introduction to Special and General Relativity, Undergraduate Texts in Mathematics (2000) NY: Springer-Verlag. A nice senior-level book that would make a good alternative to the Faber text.
  3. A. Das, The Special Theory of Relativity: A Mathematical Exposition* (1993) Universitext, NY: Springer-Verlag. This is my favorite book on special relativity! It is a math book and hence reads much like a geometry or linear algebra text. Minkowski space is axiomatically developed.
  4. C. Dodson and T. Poston, Tensor Geometry: The Geometric Viewpoint and Its Uses, 2nd Edition, Graduate Texts in Mathematics Series #130 (1991) NY: Springer-Verlag. This is a technical math book with applications to relativity.
  5. A. Einstein, Relativity: The Special and the General Theory (1961) NY: Random House. This is a popular book by the master himself!
  6. G Ellis and R. Williams, Flat and Curved Space-Time (1988) Oxford: Oxford University Press. More technical than a "popular" book, this text is a readable "semi-technical" work.
  7. L. Epstein, Relativity Visualized (1985) San Francisco: Insight Press. This is a popular book sort of in the "for Dummies" style.
  8. R. Faber, Differential Geometry and Relativity Theory, An Introduction, Pure and Applied Mathematics, A Program of Monographs, Textbooks, and Lecture Notes #76 (1983) NY: Marcel Dekker. This is a very nice introductory text. The level of mathematical rigor isn't bad. The motivation for Einstein's field equations is a bit weak, though, but this helps make the book a good deal more readable (than, say, a text with lots of tensor analysis in it).
  9. R. Gott, Time Travel in Einstein's Universe: The Physical Possibilities of Travel Through Time (2000) Boston: Houghton-Mifflin. This is a popular-level book on time travel.
  10. S. Hawking, A Brief History of Time: From the Big Bang to Black Holes 10th Anniversary Edition (1998) NY: Bantam Books. This is a (surprisingly) popular book on the origin and evolution of the universe.
  11. S. Hawking, Black Holes and Baby Universes, and Other Essays (1993) NY: Bantam Books. A popular book which addresses some (sometimes) speculative topics.
  12. S. Hawking, The Universe in a Nutshell (2001) NY: Bantam Books. This is a colorful presentation of cosmology, relativity, and hyperspace at the popular level.
  13. S. Hawking and G. Ellis, The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics (1973) Cambridge: Cambridge University Press. This is a classic text with a nice development of the differential geometry relevant to relativity theory.
  14. S. Hawking and R. Penrose, The Nature of Space and Time Issac Newton Institute Series of Lectures (1996) Princeton: Princeton University Press. This is a descriptive book which contains a debate between Hawking and Penrose (both top figures in gravitation theory).
  15. M. Kaku, Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension (1994) Oxford: Oxford University Press. A popular book, sometimes rather speculative, on higher dimensional manifolds and how they can be embedded in larger dimensional spaces.
  16. M. Lachieze-Rey, Cosmology: A First Course (1995) Cambridge: Cambridge University Press. This was first published in French as Initiotion a la Cosmologie. This is a nice undergraduate text on cosmology which requires some calculus background.
  17. H. Lorentz, A. Einstein, H. Minkowski, and H. Weyl, The Principle of Relativity: A collection of Original Memoirs on the Special and General Theory of Relativity (1923, 1952) NY: Dover Publications. This contains translations of original work of Lorentz and Minkowski. It also includes translations of Einstein's two papers which established special and general relativity.
  18. W. Misner, K. Thorne, and J. Wheeler, Gravitation (1973) NY: W. H. Freeman. This is the classic text from the early `70's. It is readable, but very long (over 1,000 pages).
  19. B. O'Neill, Semi-Riemannian Geometry: With Applications to Relativity, Pure and Applied Mathematics, A Series of Monographs and Textbooks #103 (1983) NY: Academic Press. This is a rigorous (and hard) math text on semi-Riemannian manifolds, with a chapter on special relativity and a chapter on general relativity (and cosmology).
  20. B. Parker, Cosmic Time Travel: A Scientific Odyssey (1991) Cambridge: Perseus Publishing. This is a popular level book on relativity and time travel. It gives a nice description of what it "looks like" to fall into a black hole.
  21. P. Peebles, Principles of Physical Cosmology (1993) Princeton: Princeton University Press. This is a very long (over 700 pages) and technical book that is a "modern-day classic."
  22. R. Sachs and H. Wu, General Relativity for Mathematicians, Graduate Texts in mathematics #48 (1977) NY: Springer-Verlag. This is not for the faint of heart! It is mathematically rigorous - maybe not the perfect text for a physicist, though.
  23. W. Sullivan, Black Holes: The Edge of Space, the End of Time (1979) NY: Warner Books. This the classic popular book on black holes. However, the information on observational evidence for black holes is very dated.
  24. K. Thorne, Black Holes and Time Warps: Einstein's Outrageous Legacy* (1994) NY: W. W. Norton. This is my favorite popular book on relativity! Thorne is a leading character in modern-day gravitational theory and, hence, he has the "inside story." This book includes a detailed history of the development of our understanding of relativity and black holes. My planetarium show "Relativity and Black Holes" is primarily based on this book.
  25. W. Thurston (ed. S. Levy), Three-Dimensional Geometry and Topology, Volume 1* (1997) Princeton: Princeton University Press. This book gives a technical discussion of the topology of three-manifolds and is the best technical book available on the "shape of space."
  26. R. Wald, General Relativity* (1984) Chicago: University of Chicago Press. This is my favorite book on general relativity! It is mathematically clean and rigorous.
  27. J. Weeks, The Shape of Space*, 2nd Edition, Pure and Applied Mathematics: A Program of Monographs, Textbooks, and Lecture Notes (2002) NY: Marcel Dekker. This is a very nice book on the global topology of the universe. It only requires a high school-level knowledge of math.
  28. S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (1972) NY: Wiley. This is a very technical text which includes a derivation of the Robertson-Walker metric (which results from an application of general relativity to cosmology).

Return to Bob Gardner's home page