Spacetime curvature and the universe
Nabil Iqbal
Fall 2022  Spring 2023
The Cosmic Microwave Background, a snapshot of the universe as seen a fraction of a second after the Big Bang.
The force that we conventionally call "gravity" is actually better understood as the curvature of space and time. In this project, we will seek to understand this idea. We begin by developing some aspects of Einstein's theory of General Relativity, learning the geometric concepts that are needed to describe the curvature of spacetime and how the presence of matter (galaxies, planets, etc.) affects this curvature.
We will then apply these ideas to the dynamics of our own universe at very long scales. There are several possible directions for further development, depending on the tastes of the student. They will follow roughly one of the following directions:

Cosmology: the universe began roughly 14 billion years ago in a cataclysmic event called the Big Bang; we will follow the development of our own universe as it expanded outwards, and study the possible fates that could befall universes that are slightly different from our own (e.g. with different proportions of matter, extra dimensions, and so forth). This may require some numerical work solving differential equations.

Gravitational Waves: Einstein's equations predict that any disturbance of matter creates gravitational waves, disturbances in the fabric of space and time that ripple outwards. These ripples were recently detected in an exquisitely precise and Nobelprize winning experiment. We will study the propagation, detection, and (possibly) the production of gravitational waves.

Black holes: I feel these really need no further justification as being interesting objects to study. We will learn how Einstein's equations inevitably lead to the formation of black holes, which turn out to be simultaneously the most and least complicated objects in the universe.
Prerequisites:

MATH2071 Mathematical Physics II.

Some exposure to Special Relativity (i.e. through MATH2657 or physics equivalent) is strongly recommended  if you don't have this and still want to take the project please contact me.
Resources:
The Wikipedia articles on cosmology and gravitational waves aren't bad places to start. Some nice textbooks on general relativity are:

Spacetime and Geometry, S. Carroll (A free online version of the lecture notes on which this book are based are here).

Gravity in a nutshell, A. Zee (An informal yet surprisingly weighty introduction to GR)

A First Course in General Relativity, B. Schutz (A simpler textbook with a nice treatment of gravitational waves).
More detailed and specialized readings will be provided in due course.