"Differential Geometry" syllabus homepage

Differential Geometry - Summer 2016

Carl Gauss (1777-1855)

Georg Riemann (1826-1866)

Albert Einstein (1879-1955)

COURSE: MATH 5310-010

TIME AND PLACE: 11:20-12:50 MTWRF in Gilbreath Hall 314

INSTRUCTOR: Dr. Robert Gardner

OFFICE: Room 308F of Gilbreath Hall

OFFICE HOURS: By appointment.

PHONE: 439-6979 (308F Gilbreath), Math Department Office 439-4349

E-MAIL: gardnerr@etsu.edu

WEBPAGE: faculty.etsu.edu/gardnerr/gardner.htm (see my webpage for a copy of this course syllabus, copies of the classnotes in PDF and Postscript formats, and updates for the course).

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. This can be found as a cheap paperback, but is also available online, for example at "Google books.''

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 and a discussion of black holes. 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"). 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 material come to the aid of gravitation theory. We will discuss gravitational redshift, precessions of orbits, the "bending of light," black holes, and the global topology of the universe.

WARNING: This is not a standard graduate-level differential geometry class! We only have 5 weeks and we will not explore tensors in any detail. Our goal is to study curvature (mostly of curves and surfaces) and use this study to inspire our exploration of general relativity.

ONLINE NOTES: We only have five weeks and will go through material at a very fast pace. The notes presented in class are online at: http://faculty.etsu.edu/gardnerr/5310/notes.htm

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. We will use a 10 point scale for letter grades with plus and minus grades given based on a 3 point subscale (so, for example, a B- corresponds to a percentage grade of 87, 88, or 89).

POWERPOINT PRESENTATION: A presentation of "Relativity and Black Holes" will be given on June 17 (tentative date). 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 presentation includes some of the observational evidence for black holes. The primary source for the presentation 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://faculty.etsu.edu/gardnerr/planetarium/relat/relatabs.htm.

VIDEOS: We will watch a video in class. "The Shape of Space" is a clever introduction to three-dimensional manifolds. A webpage by The Geometry Center accompanies the video: www.geom.uiuc.edu/video/sos/. The webpage gives additional information on the topic, as well as some hands-on projects suitable for high-school-level students. We will discuss the possible global topologies of our universe, and ways to empirically detect this structure. A PowerPoint presentation on these topics is also online.

Another video "Einstein's Universe" is available on YouTube (http://www.youtube.com/watch?v=ZZmeB8eVISU). This television show was created by the B.B.C. in 1979 to celebrate the 100 year anniversary of Einstein's birth. Though over 30 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). The companion book is Einstein's Universe by Nigel Calder (New York: Viking Press, 1979).

Some other online videos are:

Einstein's Equation of Life and Death (from BBC, on YouTube).
Einstein's Big Idea (from PBS, on YouTube).
Inside Einstein's Mind (a PBS episode of NOVA).

An interesting discussion about falling into a black hole can be found on the University of California, Riverside webpage here. Notice the section "Will you see the universe end?"

Black Hole Waves Simulation
Animation created by SXS, the Simulating eXtreme Spacetimes (SXS) project (http://www.black-holes.org).

From the LIGO Caltech webpage.

Breaking Gravitational Wave News: Gravitational waves were directly detected in September 2015. The formal research publication announcing this appeared in Physical Review Letters, 116, 061102 (2016). The paper is posted online in PDF: Observation of Gravitational Waves from a Binary Black Hole Merger, by Abbott et al.. The paper announces the detection of the merger of a 29 solar mass blackhole with a 36 solar mass black hole. Three solar masses were converted into energy and released in the form of gravitational waves. The merger occurred 1.3 billion light years away in the direction of the Magellanic Clouds. Wikipedia has a nice webpage with some pretty pictures and short videos.

A nontechnical article by the 2005 ETSU Basler Chair of Excellence for the Integration of the Arts, Rhetoric, and Science, Martin Hendry, is online at: "Gravitational waves found: the inside story". Dr. Hendry is also a coauthor on the research paper mentioned above.
LIGO, the Laser Interferometer Gravitational-Wave Observatory, consists of two interferometers separated by 1,865 miles, one in Livingston, LA and one in Richland, WA. The webpage for LIGO (the Laser Interferometer Gravitational-Wave Observatory) is: LIGO.
VIRGO is a similar gravitational wave detector located in Italy: VIRGO.
The Einstein Telescope is a proposed European gravitational wave observatory planned to start operation in the late 2020s. For more details, see ASPERA's Einstein Telescope webpage.
A space-based gravitational wave observatory would allow a tremendously larger base-line and hence increased sensitivity. NASA planned LISA (the Laser Interferometer Space Antenna), but funding was cancelled in 2011. The NASA webpage is still online: LISA. The European community is now planning a related project, called "eLISA." A launch on December 3, 2015 of a test of the eLISA concept shows that the project is alive and active. Currently, the plan is for a launch of the observatory in 2028. See eLISA.

A second detection of two merging black holes was made on December 26, 2015 and announced in June 2016. The total mass of the two black holes was 22 solar masses. A non-technical description is available from Phys.Org. If you watch the embedded video at this website, Martin Hendry is interviewed in the last minute of the video. The original paper can be accessed from Physical Review Letters

This animated gif is from Wikipedia. The blue vector is the unit tangent vector, the red vector is the principal normal vector, and the black vector is the binormal vector. This example involving a helix is explored in Example 3 on page 6 of the text.

Blackhole image from: NASA's "What is a Black Hole?" website
Wormhole image from: http://casa.colorado.edu/~ajsh/schww.html#worm
All websites on this webpage were last accessed February 20, 2016.

Tentative Schedule

DAY	DATE	TOPIC
1	MON 6/6	1.1=Curves: arclength, tangent vector, curvature
2	TUE 6/7	1.1 (cont.): binormal vector, torsion 1.2=Gauss Curvature: normal section, principal curvature
3	WED 6/8	1.3=Surfaces in E³: surfaces of revolution, parallels 1.4=First Fundamental Form: metric form, intrinsic property
4	THR 6/9	1.5=Second Fundamental Form: Frenet Frame, normal curvature
5	FRI 6/10	1.6=Gauss Curvature in Detail: principal curvature
6	MON 6/13	1.6 (cont.), 1.7=Geodesics: Christoffel symbols
7	TUE 6/14	1.7 (cont.): "straight lines," more geodesics
8	WED 6/15	1.8=Curvature Tensor: Theorema Egregium
9	THR 6/16	1.9=Manifolds: coordinates
10	FRI 6/17	1.9(cont.): smooth manifold, vectors as operators, inner products Chapter 2 of Wald's General Relativity
11	MON 6/20	Video: Shape of Space The Shape of Space
12	TUE 6/21	PowerPoint: Relativity and Black Holes
13	WED 6/22	2.1=Inertial Frames, 2.2=Michelson-Morley Experiment: stellar aberration Einstein: Preface, 1.1-1.6
14	THR 6/23	2.3=Postulates of Relativity, 2.4=Simultaneity, 2.5=Coordinates Einstein: 1.7-1.12
15	FRI 6/24	2.6=Invariance of the Interval 2.7=Lorentz Transformation: invariance of the interval
16	MON 6/27	2.7 (cont.), Einstein: Appendix I, 1.13-1.17
17	TUE 6/28	2.8=Spacetime Diagrams
18	WED 6/29	2.9=Lorentz Geometry, 2.10=Twin Paradox: Doppler effect, 2.11=Causality
19	THR 6/30	3.1=Principle of Equivalence, 3.2=Gravity as Spacetime Curvature
20	FRI 7/1	3.3=Consequences of General Relativity 3.6=Geodesics: timelike, lightlike, spacelike 3.7=Field Equations: Ricci tensor, Einstein: 2.18-2.11, Appendix III
-	MON 7/4	Independence Day Holiday, No Class!
21	TUE 7/5	3.8=Schwarzschild solution, Einstein: 2.23-2.29, Appendix IV
22	WED 7/6	3.9=Orbits in General Relativity: precessions
23	THR 7/7	3.10=Bending of Light, Einstein: 3.30-3.32, Appendix V Black Holes: Schwarzschild radius
24	FRI 7/8	Black Holes (cont.): Eddington-Finkelstein coordinates, gravitational redshift

"Einstein" refers to readings from the supplemental text.

Section	Problems	Due Date	Points
1.1	6a, 6b, 7a, 7b, 9, 10, 11c	Friday 6/10	5 + 5 + 5 + 5 + 5 + 5 + 5 = 35
1.2 1.3	3 2(d), 7(a,b), 7(c)	Wednesday 6/15	5 + 5 + 5 + 5 = 20
1.4 1.5	3(a), 5, 13 3(a)	Friday 6/17	5 + 5 + 5 + 5 = 20
1.6 1.7	1(a), 2(a), 4(a) 1	Wednesday 6/22	5 + 5 + 5 + 5 = 20
1.8 1.9	1(a), 4 6(a), 6(b)	Friday 6/24	5 + 5 + 5 + 5 = 20
2.2 2.4 2.5	3(a) 1, 4 2	Wednesday 6/29	5 + 5 + 5 + 5 = 20
2.6 2.7	3, 4, 5 3, Theorem	Tuesday 7/5	5 + 5 + 5 + 5 +5 = 25
3.8 3.9 3.10 Black Holes	2, 4 3 2 Black Hole	Friday 7/8	5 + 5 + 5 + 5 + (5) = 20 + (5)
-	-	Total	180 + (5)

Bonus points are in parentheses.

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Last updated: July 3, 2016.