General Relativity

Description of Course Goals and Curriculum

This course provides a nice, self-contained introduction to general relativity (GR), which is the study of gravity and 4-D spacetime. General relativity is a primary branch of modern physics because it provides the currently accepted theoretical framework for how gravity works. Therefore, it is an important topic that most physics and astrophysics majors should have some familiarity with. Note that Princeton offers two courses on general relativity: this one (undergraduate-level GR) and PHY 523, which is graduate-level GR. Students looking for a formal and rigorous mathematical treatment of GR should take PHY 523, whereas students who want to understand the fundamental principles of GR and learn how to use them in everyday astrophysical applications should take this course. The course is roughly split into three components. In Weeks 1 – 3, an overview of special relativity (SR) is given. This includes topics such as Minkowski spacetime, kinematics and dynamics of SR, and relativistic electromagnetism, all of which are prerequisite knowledge for understanding GR. In Weeks 4 – 10, the fundamental principles of general relativity are taught. This includes topics such as tensor notation, geodesics, gravitational time dilation, Schwarzschild spacetime, covariant differentiation, Riemann curvature, and the Einstein field equations. Finally, in Weeks 11 – 12, applications of GR to astrophysical phenomena are presented. The main two applications that Prof. Goodman covers are gravitational waves and cosmology.

Learning From Classroom Instruction

Like most other upper-level math and science courses, there are two 80-minute lectures per week. Prof. Goodman primarily uses Powerpoint slides during lecture, with occasional derivations on the chalkboard. Prospective students should be aware that Prof. Goodman has a penchant for tangential digressions, so his delivery of the course material during lecture is often convoluted. However, his lecture slides, which he always posts to Blackboard after class, are full of important information and are a valuable resource for learning the material. The slides often contain both presentation of conceptual information as well as detailed mathematical derivations where necessary. Therefore, lectures should be viewed as a “first pass” through the material where students are first introduced to new concepts. Solid understanding of these concepts is then obtained by going back through the lecture slides, preferably on the same day after lecture. Note that there are no precepts for this course. Therefore, office hours with both Prof. Goodman and the TA are also important resources that the student should take advantage of. Office hours are particularly helpful for clearing up any confusion about concepts presented in the lecture slides. They are also a great place to ask follow-up questions that the student may have about a particular topic.

Learning For and From Assignments

Assignment and assessment-wise, there are 9 problem sets, a midterm exam, and a final exam in this course. Traditionally, Prof. Goodman uses the following grading scheme: 40% Psets (drop lowest one), 20% Midterm, and 40% Final. Problem sets are given approximately weekly. Just like other upper-level physics courses, the problem sets are quantitative in nature and are meant to familiarize the student with applying the abstract concepts learned in lecture to solving concrete problems. Most problem sets contain a mixture of derivation/proof of concepts given in lecture and quantitative application of such concepts to scenarios not seen before by the student. For maximal success on the problem sets, students should try to form or join a study group and work together. This allows students to learn how their peers approach solving problems, which often provides useful insight. Problem sessions with the TA are also very useful for getting help if you are stuck on a particular problem. Although collaboration is encouraged, it is important that each student individually understands how concepts were applied to solve each problem. The midterm exam is 80 minutes long, closed-book, and is given in-class during midterm week. It usually consists of 4 to 5 problems. The exam questions are similar to pset problems in difficulty, though with less parts per problem and simpler calculations. It is highly likely that students will encounter questions on the exam that deal with novel scenarios not covered in either lecture or homework. Don’t fear, however, because the questions can always be solved using concepts learned in class. Therefore, to succeed on this exam, it is imperative that students review their psets and understand at a fundamental level how the concepts work and are applied to solve problems. Prof. Goodman also typically gives 1-2 past exams to students for practice. The strategy that I used for studying was: a few days before the exam, pull out my old psets and rework all the problems, taking care to understand how concepts were applied. Then, handwrite a cheat sheet/study guide (usually w/o looking at my notes) to reinforce my knowledge of course material. Finally, the day before the exam, solve the past exam problems under simulated test conditions. The final exam is 3 hours long, closed-book, and is given during Final Exam period. It usually consists of 5 to 6 problems. While this exam is worth 40% of the grade, students should not be stressed because Prof. Goodman always releases a set of approx. 20 problems over Winter Break, from which he pulls the final exam problems verbatim. Students are allowed to collaborate in working on this “superset”, so for best success on the final, students should work together and utilize course resources (notes, old psets) to solve all the problems. Note, however, that these “superset” problems are typically more difficult than standard pset problems.

External Resources

Between prof. and TA office hours, your peers in the class, and the lecture slides/textbook, this course has a solid support system that usually doesn’t necessitate the use of external resources. However, there are several supplementary sources that are good to keep in mind so that you can use them if necessary. First, while the main textbook (A First Course in General Relativity by Schutz) is usually pretty good at explaining concepts, I find that the alternative textbook Gravity: An Introduction to Einstein’s General Relativity by Hartle is good for providing clarification on concepts not made clear by the prof. or Schutz. Second, the Problem Book in Relativity and Gravitation is a great resource to have if you want extra practice on solving GR problems (especially when preparing for exams). That book essentially contains ~500 problems on GR with solutions. Finally, upperclassmen in the AST or PHY depts who have previously taken the course are good resources to use for either help on a specific problem/concept or broader tutoring help.

What Students Should Know About This Course For Purposes Of Course Selection

This course is among a set of courses that are core requirements for the Astrophysics major. It is also cross-listed with the Physics department, so it can count as a departmental for Physics too. Typically, most students who take this class are upperclass Astrophysics or Physics majors looking to satisfy departmental requirements. There are also occasionally sophomores (usually prospective AST or PHY) who take the course as well. This should not deter non-AST or non-PHY majors from taking the course, however. As mentioned above, this course emphasizes conceptual understanding over formal mathematical development. Therefore, the primary prerequisites for this course are freshman-level mechanics and EM, as well as multivariable calculus. All other mathematics and physics machinery is developed as needed in the course. I encourage all students, including underclassmen, who have the appropriate preparation and are genuinely interested in understanding how gravity works at a fundamental level to take this course!