Differential Geometry and Physics In Curved Spacetimes

Fall 2023    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: S4-208
  2. Class Times: Thursdays 2 pm to 4:50 pm
  3. My Office: S4-718
  4. E-mail: yizen [dot] chu [at] gmail [dot] com
  5. My Office Hours: Come find me in my office. If I'm busy, we'll set up a separate time.
  6. Jia-En Chen (Graduate Teaching Assistant):
    1. Office: S4-507
    2. E-mail: chenneymar11 [at] gmail [dot] com
    3. Office Hours: Wednesdays 10:30 am to 11:30 am
- 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 -- topics may include
    • Cosmology
    • Black Holes
    • Gravitational Waves
Because I wish to reward hard work during the semester, I will give most weight -- 75% of your total grade -- to the homework you turn in. The rest of the 25% will be based on your final presentation.

Homework (75%):     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; while GR refers to Physics in Curved Spacetimes.
  1. Due Thursday 12 October: AM 9.2, 9.4, 9.5, 9.8, 9.10, 9.11, 9.14, 9.21, 9.23; 10.1, 10.2, 10.3, 10.4, 10.6.
  2. Due Thursday 2 November: AM 9.30, 10.10, 10.12, 10.13, 11.2, 11.3, 11.4, 11.5, 11.6, 11.8, 11.9, 11.10, 11.11, 11.12. 11.15. 11.16, 11.18, 11.20; GR 5.1, 5.5
  3. Due Thursday 30 November: Vector Field Plot, AM 11.20, 11.23, 11.27, 11.31, 11.32, 11.35, 11.36, 11.38, 11.40; GR 6.1, 6.8, 6.9, 6.10, 7.1, 7.2, 7.3, 7.4.
  4. Due Thursday 14 December: AM 11.37 (Extra Credit), 11.38;  GR 7.6,.7.8, 7.10, 7.11, 7.12, 7.13. 7.14, 7.16, 7.17;  8.1, 8.2, 8.3, 8.4. 
  5. Due Thursday 4 January: AM 11.15, GR 8.7, 8.9, 8.10, 8.11, 8.12, 8.14, 8.15, 8.16, 8.18, 8.19, 8.20, 8.21.
Final Presentation (25%):     Give a 50 minute in-class presentation on a topic related to gravitational physics research. Below are a list of suggested topics; feel free to come up with your own.
  • What are Extreme-Mass-Ratio-Inspiral systems (EMRIs)? What can we learn about strong gravity from such EMRIs?
  • What can we learn about neutron stars and their interior structure from gravitational waves?
  • Discuss the recent results from the Event Horizon Telescope. Topics could include: Geodesics around Kerr, accretion disks, MHD, black hole shadows, interferometry, etc.
  • What are the "B modes" of the cosmic microwave background radiation? Why are physicists trying to look for them, and what can we learn about the early universe from their detection? Related: what sources of gravitational waves do we expect from the early universe?
  • What are Pulsar Timing Arrays, and how do they detect gravitational waves? What sources will they be sensitive to?
  • How does a gravitational wave detector like LIGO, VIRGO, KAGRA, etc. work? How is it possible to detect such a small change in displacement? How do quantum effects play a role?
  • Explore the role gravitational wave observations play in understanding our cosmic history.
  • Discuss the experimental tests of General Relativity and/or the attempts to 'modify' it.
  • Explain the basic (astro)physics behind the recent results of the NANOGrav collaboration.
Guideline for final presentation     The presentation should be in English. It 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.

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     (Chapters 8 and 9)
  • Lecture Notes for Physics in Curved Spacetime and General Relativity
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
Cosmology
Exact Solutions
Problem Book
(Mostly links to amazon.com -- out of convenience; not an endorsement of their business practices.)

Software
Acknowledgements

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

Disclaimer

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.