COURSE: MATH 4900-010

TIME AND PLACE: 1:00-2:30 MTWRF in Burleson Hall 402

INSTRUCTOR: Dr. Robert Gardner

OFFICE: Room 402 of Burleson Hall for Summer Term 1 (traditionally, Room 308F of Gilbreath Hall)

OFFICE HOURS: 2:30-3:00 MTWRF and 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).

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 Project Gutenberg.

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.
I have also prepared notes from several other sources (in my quest to make sense out of general relativity). I have a humble amount of notes online from the following:

1. D. J. Struik, "Outline of a History of Differential Geometry: I," Isis 19(1), 92-120 (1933).
2. D. J. Struik, "Outline of a History of Differential Geometry (II)," Isis 20(1), 161-191 (1933).

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:

1. Einstein's Equation of Life and Death (from BBC, on YouTube).
2. Einstein's Big Idea (from PBS, on YouTube).
3. Inside Einstein's Mind (a PBS episode of NOVA). This aired on March 29, 2019 (5 days before the start of this class). It is available on the subscription service NetFlix.

A CENTENNIAL CELEBRATION FOR 2019: As we will see in Section 3.10, "The Bending of Light," Einstein's general relativity predicts the deflection of light as it passes near a very massive object. This prediction was famously demonstrated with a British solar eclipse expedition lead by Sir Authur Eddington. The eclipse occurred on May 29, 1919 (so the centennial was only 5 days before the first day of this class). The Eddington team took photographs of stars appearing near the sun and visible during the eclipse; the stars were in the Hyades in Taurus the Bull. The photographs were later analyzed and they revealed Einstein's predicted amount of deflection. The results were announced in the New York Times on November 10, 1919 as "Lights All Askew in the Heavens: Men of Science More or Less Agog Over Results of Eclipse" (you can read the original article here). The formal research was published in the Philosophical Transactions of the Royal Society of London as "A determination of the deflection of light by the Sun's gravitational field, from observations made at the total eclipse of May 29, 1919," Philos. Trans. Royal Soc. London 220A: 291-333 (1920), by F. W. Dyson, A. S. Eddington, and C. Davidson. This paper is in the public domain and available here. Additional information and links are given on the Wikipedia page Eddington's Experiment.

A negative image of the May 29, 1919. This negative appears as "Plate 1" in the Dyson, Eddington, Davidson paper. This image is from a European Space Agency website.

Gravitational Wave News: A prediction of general relativity which has very recently blossomed is the direct detection of gravitational waves. 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. It detected gravitational waves in September 2015. It has since made many more successful observations. These are listed on the LIGO webpage and more information is given in the online LIGO Magazine.

• The webpage for LIGO (the Laser Interferometer Gravitational-Wave Observatory) is: LIGO.
  • The formal research publication for the first detection in 2015 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.
  • 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.

• VIRGO is a 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 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.

Black Holes News:

• Related to the gravitational wave news above is the black hole wave simulation animation created by SXS, the Simulating eXtreme Spacetimes (SXS) project (http://www.black-holes.org):
  From the LIGO Caltech webpage.
• 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?" We will touch on these ideas during the last two class days.
• The Event Horizon Telescope collaboration is a global system of radio telescopes. During 2017 radio telscopes at eight sites made observations of the supermassive black hole at the center of the galaxy M87. On April 10, 2019 it was announced at six news conferences held around the world that this black hole had been imaged.
  This image is from the JPL website. A non-technical article describing the project is on the JPL website "How Scientists Captured the First Image of a Black Hole". A technical discussion is given in "First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole" published in The Astrophysical Journal Letters 875:L1 2019 April 10 (available online in PDF.

ABOUT THE INDEPENDENT STUDY: You will attend the same lectures as the graduate-level Differential Geometry (MATH 5310) class. However, you will be given a subset of the homework assignments for the undergraduate Independent Study. You will be required to do 75% of the homework problems assigned to the graduate class (you may choose which ones to do and do additional problems for bonus points). For the specific assignments and due dates, see the syllabus for the corresponding Differential Geometry (MATH 5310) class.

Return to Bob Gardner's home page
Last updated: June 2, 2019.