General Relativity

# PHGN 418 General Relativity Spring 2021

### The Facts:

Lecture: T,TR 5-6:15pm in Brown Building West 280
Suggested Text: "Spacetime and Geometry: An Introduction to General Relativity" by Sean Carroll
My notes as typed up by a non-student, but nonetheless incredibly useful: (Notes)
Alex's Office Hours: Wednesdays 3:30-7pm Madison's Office Hours: Thursdays 3-4:50pm
Grading: Your grade will be based on lecture participation (10%), homework (50%), an in-class midterm exam (20%) and a take-home final exam (20%).

Lecture participation will be assessed by your response to questions posed during lecture. I will draw names randomly to answer questions and if you are not present or fail to respond as if paying attention, you will lose 1% of the 10% for lecture participation.

### Description:

This course will develop and apply Einstein's General Theory of Relativity. One of the most interesting aspects of this subject is that it brings the student to our modern understanding of the earliest recognized of the fundamental forces of nature, i.e. gravitation. The student sees this first as a source of constant acceleration near the surface of the Earth (projectile motion), and then matures their understanding, with Newton's Universal Law of Gravitation, to a notion of gravitation as a central force acting between any two bodies with nonzero mass. While the Newtonian view works in many applications, it fails when contemplating systems that are ginormously massive. The (more) correct theory, General Relativity, is not a simple "extension" of Newtonian gravity, but rather a complete revision of our understanding of gravity. In some cases, e.g planetary orbits, the predictions of General Relativity provide small corrections to the "for the most part correct" results of the Newtonian theory. However in other circumstances, e.g. black holes and cosmology, General Relativity gives us results that quite honestly would have sent Newton running scared and have left many other physicists scratching their heads in disbelief!

This subject is often tauted as too mathematically sophisticated for undergraduates, which I find to be a shame since the concepts underlying the theory can be expressed concisely and clearly given a reasonable effort on the part of the instructor. While the subject does involve mathematics that extend beyond what has already been encountered in most other physics courses, the truth of the matter is that it really boils down to taking what you already know and being as general as possible with it. When you leave this course, you will know far better what a "vector" is than many professional physicists. There is a wealth of material that we can study and the extent of what we do will rely in part on your diligence to the course.

### Objectives:

• We will aim to cover as a minimum chapters 1,2,3,4,5 and 8 of Carroll. Chapters 1-4 constitute the formal development of the theory while chapters 5 and 8 discuss two of its most important applications, i.e. black holes and cosmology respectively. You will find that the flavor of he course changes quite a bit once we have completed the formal development and moved into applications.
• Additional topics can be discussed if time permits.

### Tutorials:

These tutorials were developed by my senior design student Amanda Casner a couple of years ago. They are meant to be a context independent introduction to tensors. For those who have taken my Particle Phsyics course, a lot of the material may already be familiar. For new students, these will help ease you into the rather steep learning curve that this course begins with. I would recommend giving them a view.
(Lesson 1) 4 group requirements, examples of groups & sets that are not groups, finite groups, different representations of different groups, multiplication tables.
(Lesson 2) Rotation groups in 2D and 3D, matrix representations of rotation groups, rotations acting on a vector, visualizing consecutive rotations & net rotations
(Lesson 3) Conditions for SO(3)
(Lesson 4) Index notation, representing matrices using index notation, transformations between primed and unprimed systems, transformations of multiple indices, why index notation is better than matrix representation (showing how order doesn't matter)
(Lesson 5) Obtaining invariants using matrix representation and index notation, dual vectors, the metric, transformation rule for the metric, isometries, obtaining the metric given isometries and vice-versa, the dot product
(Lesson 6) Defining tensors, classifying (p,q) tensors, vector & dual vector transformation rules, tensor transformation rules, representing 2 single-index tensors as a symmetric 2-index tensor, expanding to coordinate dependence and connection to GR

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