# PHYS6810E 2018

Room 4504, Lift 25-26; Saturdays 9:30AM - 12:20PM, Spring Term, 2018

**Textbook:** Einstein Gravity in a Nutshell.

**Plan:** Note that the week counts are *very rough* estimates.

Week 1-2: I will very quickly go through Book One. They correspond to the physics that I assume that you know already, including Newtonian gravity, rotation, action principle, symmetry and conservation, electromagnetism and special relativity.

Week 3-4: Part V – Equivalence principle and curved spacetime

Week 5-6: Part VI – Einstein’s field equation

Week 7-8: Part VII – Black holes

Week 9-10: Part VIII – Introduction to our universe

Week 11-12: Part X – Gravitational waves, and other contents if we still have time.

In addition, there may be about two guest lectures on related research progress.

**Grading:** There will be of order 5 assignments and a final exam. There is no midterm exam. The grade will be determined by (assignments) x 40% + (final) x 60%.

**Lecture notes:** The lecture notes are be brief and the textbook is referred to for details.

**Assignments:** To be released.

# (Outdated) PHYS6810E 2017

Room 5506, Lift 25-26; Saturdays 9:30AM - 12:20PM, Spring Term, 2017

**Textbook:** Spacetime and Geometry: An Introduction to General Relativity.

**Lecture notes:**
Can be downloaded here

**Plan:**

- Special relativity (Chap1)
- Manifolds (Chap2)
- Curvature (Chap3)
- Gravitation (Chap4)
- Black hole (Chap5, and a little of Chap6)
- Gravitational waves (Chap7)
- FRW cosmology and thermal history
- Inflationary cosmology
- Cosmological perturbation theory

**Videos** (ITSC login required):

Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5, Lecture 6, Lecture 7, Lecture 8, Lecture 9, Lecture 10, Lecture 11, Lecture 12

**Assignments:**

Assignment 1: Consider the metric of a 3-sphere: $ds^2 = d\psi^2+\sin^2\psi \left[ d\theta^2 +\sin^2\theta d\phi^2 \right]$.

- Calculate the Christoffel connection.
- Calculate the Riemann tensor, Ricci tensor, Ricci scalar.

Assignment 2: Use computer code to reproduce Assignment 1. Send your code to the TA by email. Note:

- You may either use any existing code or write your own. Both can get full marks. Yes, the latter is harder (but recommended). But as a working scientist in GR, using existing code is perfectly fine unless tensor calculation plays a central role in your research.
- The TA has no responsibility to check the correctness of your code.

Assignment 3: Exercise 5.9.2 of Carroll (black hole in a circularly symmetric 2+1 dimensional spacetime).

Assignment 4: Exercise 7.8.3 of Carroll (Fermat’s principle with an effective refractive index $n=1-2\Phi$).

Assignment 5: For “large field inflation”, $V = \frac{1}{2} m^2 \phi^2$.

- Calculate $\epsilon_V$, $\eta_V$.
- Assuming slow roll inflation, calculate $\epsilon$, $\eta$, $w=\rho/p$.
- Given $N_*$, find $\phi_*$.
- Given $N_* \in (50,60)$, find the range of $n_s$ and $r$.