Differential Geometry and Physics In Curved Spacetimes

Fall-Winter 2017    Welcome! This course will introduce you to the mathematics and physics of curved spacetimes. This includes Einstein's theory of General Relativity, humanity's current understanding of gravitation. Depending on your background, this may not be an easy class, but if you do exert the mental effort I believe you'll find the course intellectually rewarding!
  1. Class Venue: S4-623
  2. Class Times: Mondays 3 - 4:50 pm and Thursdays 2 - 2:50 pm (Subject to change)
  3. My Office: S4-718
  4. E-mail: yizen [dot] chu [at] gmail [dot] com
  5. My Office Hours: Wednesday 2:15-3:15 pm in my office. You should e-mail me if you wish to set up a time to meet outside of these times.
  6. Jason Payne (Graduate Teaching Assistant):
    1. Office: S4-507
    2. E-mail: jasonpayne16 [at] gmail [dot] com
    3. Office Hours: Tuesdays 11 am - 12:45 pm; and Wednesdays, 12 pm - 1 pm.
- Yi-Zen
  • Disability     If you have a disability that you think I should know about, and if you need special accomodations, please feel free to speak to me after class or e-mail me to set up a meeting.    
  • Academic Integerity     You are encouraged to discuss with your classmates the material covered in class, and even work together on your assignments. However, the work you turn in must be the result of your own effort. If I find that you copied your work from some place else, you will immediately receive zero credit for that particular piece of work. If you plagarized your classmate, your classmate will also receive zero credit for her/his/their work, unless (s)he/they can prove to my satisfaction (s)he/they were unwilling participant(s) of your dishonesty.
Syllabus and Grading Scheme

We will be covering aspects of:
  • Differential Geometry
  • Lorentz Symmetry and Physics in Flat Spacetime
  • Physics in Curved Spacetime
  • General Relativity, including
    • Cosmology
    • Black Holes
    • Gravitational Waves
Because I wish to reward hard work during the semester, I will give most weight -- 60% of your total grade -- to the homework you turn in. The rest of the 40% will be split evenly between the midterm and final.

Homework (60%):     I will assign problem sets from the lecture notes posted here. I recommend starting your homework as soon as possible -- do not wait until the day before it is due to do it!

Note: I will not accept late homework -- just turn in whatever you have done at the time/day it is due. Below, AM refers to Analytical Methods Chapter 7; while GR refers to Physics in Curved Spacetimes.
  1. Due Thursday 28 September, 4 pm (i.e., right after my office hours): AM Problems 7.1 through 7.18.
  2. Due Thursday 26 October, 4 pm (i.e., right after my office hours): AM 7.21 through 7.27, 7.29 through 7.31, 7.33 through 7.39; 7.41 and 7.42.
  3. Due Monday 13 November, 5 pm (i.e., right after class): GR 1.1, 2.1-2.4, 2.6-2.15, 2.17-2.19
  4. Due Monday 4 December, 5 pm (i.e., right after class): GR 3.1-3.7; 4.1-4.2; 5.1-5.9
  5. Due Thursday 28 December, 4 pm (i.e., right after class): GR 6.2-6.9, 7.3; either 7.4 or 7.5; 7.6-7.7, 7.9-7.11; either 7.14 or 7.15; 7.16, 7.19-7.21.
  6. Due Monday 8 January, 12 noon: GR 7.22, 7.26, 8.1-8.9.
Midterm (20%) (Due Thursday 16 November 2017, 11:59 pm):     Gravitational waves are expected to arise from a wide range of frequencies and astrophysical/cosmological sources -- quantum gravity fluctuations during a (hypothetical) inflationary epoch; potential phase transitions in the early universe; gravitational wave bursts from (hypothetical) cosmic strings; astrophysical binary systems; vibrations of neutron stars; core-collapse supernovae; Extreme-Mass-Ratio-Inspiral systems; etc. Write a 10-20 page paper overviewing the range and types of observations that are taking place and would be undertaken over the next 2 decades or so. What sources of gravitational waves are expected in each frequency bandwidth? Address the physics and astrophysics that we hope to learn from each type of gravitational wave observation.

As inspiration/guide, you may wish to begin by checking out Moore, Cole and Berry.

Final (20%) (Due Monday 8 January 2017, 11:59 pm):     Write a 10-20 page paper discussing the basic physics behind the gravitational waves generated by the binary black hole inspirals and mergers aLIGO has heard to date: (from earliest to latest) GW150914, GW151226, GW170104 and GW170814. Because this is a graduate-level class, the paper needs to contain detailed equations describing -- at the very minimum -- the Keplerian motion of the binary systems; their loss of energy due to gravitational radiation; as well as the backreaction of this energy loss on the frequency of the Keplerian motion and the gravitational waves emitted. Explain the bottom panel of Fig. 1 of arXiv: 1602.03837. Can you perform an order-of-magnitude estimate to argue these events indeed involved black holes? Etc.

As a starting point, you may wish to read the pedagogical articles of arXiv: 1608.01940 and 1609.09349. Make sure you do not just lift material from these papers!

Writing guidelines for midterm and final papers     The papers should be written in English, and the font size should be 12 points. Your writing will be judged firstly by the accuracy, breadth and depth of the content; but also by the clarity of the exposition. Make sure you cite your sources carefully and provide proper credit whenever appropriate. Turn in your papers by e-mailing them to both Jason Payne and I. If you write your paper on MS Word, please convert it to Open Office format before sending it to me. Extra credit will be given if you write your paper in LaTeX; if you do, just e-mail me your TeX file.

Lecture Notes & Problems

I will be teaching from my lecture notes below. The main shortcoming of my lecture notes is that there are no figures -- this is why you need to come to class, where I will supply them whenever necessary...
  • Lecture Notes for Differential Geometry     (Chapter 7)
  • Lecture Notes for Physics in Curved Spacetime and General Relativity     (More to come!)
I will continue to update/edit these notes throughout the semester, so check back regularly. Do let me know if you find any errors, typos, etc.

Differential Geometry General Relativity
Exact Solutions
Problem Book
(Mostly links to amazon.com -- out of convenience; not an endorsement of their business practices.)


While developing this course, I have taken inspiration from several of the textbooks listed above.


The views and opinions expressed in this page are strictly those of mine (Yi-Zen Chu). The contents of this page have not been reviewed or approved by the National Central University.