Physics 675: Introduction to relativity, gravitation and cosmology. Fall 2012


Professor: Manuel Tiglio

Lectures: Physics Building, room 4102 (notice the change of room). Tuesday and Thursday: 2pm-3:15pm.

Office: Physics Building # 4205E.

Office hours: after class or by appointment


Teaching Assistant: Rong Zhou

Office: Physics Building, room 4207A


Topics to be covered (to be expanded, depending on the pace of the class):

Differential geometry as needed to study basic General Relativity (manifolds, Lie and covariant derivatives, parallel transport, geodesics, curvature tensors, gauge invariance).

The Einstein equations

Black hole solutions and analysis of their geometry.


Linearized theory, gravitational waves.

Homework (to be added during the semester)

Required textbook:

We will be following mostly “An introduction to Spacetime and Geometry”, by Sean M. Carroll (edited by Pearson/Addison Wesley). We will not have time to cover all of it, though, and I will also take some material or examples from other books or from myself. 
Here are my lecture notes (written and rewritten during the semester, so check back for updates often). Last update: Dec 7 2011 at 11:06PM.
Below is list of other important bibliographic resources.

Other books and bibliographic resources:

  1. -“Problem book in relativity and gravitation”, by Lightman, Press, Price and Teukolsky. Edited by Princeton University Press, 1975. A very valuable list of problems with solutions. Highly recommended. If you have the time, do all of them!.

  2. -“Semi-Riemannian geometry, with applications to relativity”, by Barrett O’Neill. Mostly a book on the differential geometry aspect of GR. Very readable.

  3. - “Gravitation”, by Misner, Thorne and Wheeler. Edited by W. H. Freeman, 1973. A classic among classics. Usually referred to as “MTW”. If you are interested in GR, you should have a copy of this huge book for reference on so many topics. I would not recommend it as a text to learn GR, but certainly a must have once you have learned GR.

  4. -“Gravity, an introduction to Einstein’s General Relativity”, by James Hartle (edited by Addison Wesley). An excellent book exploring the physics of GR using little math. It is many times used as a textbook for undergraduate courses in GR.

  5. -“Gravity from the ground up”, by Bernard Schutz (edited by Cambridge). It uses very little math and instead it focuses on “high level” discussions of many aspects related to gravity. A very good complementary read.

  6. -”General Relativity”, by Robert M. Wald (edited by The University of Chicago Press). One of the best books (my favorite one), but rather terse. Carroll’s book in fact can be seen as a “more legible, with more explanations” version of around half of the material in Wald’s.

  1. -“The large scale structure of spacetime”, by Stephen Hawking and George Ellis, edited by Cambridge University Press (1975). Excellent book, with a focus on the singularity theorems, along with their proofs. Probably not the best place to start learning GR.

  1. -“Relativity”, by Albert Einstein, with an introduction by Roger Penrose and Commentary by Robert Geroch. Edited by PI Press, 2005. This is a short book written by Einstein himself for the lay public, with modern comments and and an introduction by Geroch and Penrose, respectively.

  2. -Living reviews in Relativity: a collection of online reviews written by experts in their fields and edited by the Max Planck Society. They are expected to be updated by the authors, so they present the state of the art of any topic related to GR at a high level. Excellent resource.

Topics for the final presentation: (you can get any of the references below from within campus or, if not, let me know). Some of these topics can be followed down to all the details and some are only expected to be followed at a high level or you can choose from any subtopic within any of the reviews below.

The following list and descriptions will be updated, expanded, and shortened during the semester till the time comes to pick up a project (we will hopefully be able to cover some of the following in class, in fact)

  1. - Kerr black holes, Penrose extraction if time allows. Material: Carroll's book, chapter 6.

  1. -Einstein and his initial disbelief on the “physical reality of the “Schwarzschild singularity”. Reference: “On a stationary system with spherical symmetry consisting of many gravitating masses”, Albert Einstein, Annals of Mathematics, Second Series, vol. 40, No 4, pp 922-936 (1939). Here is a a pdf copy of the article.

  1. -Stability of the Schwarzschild solution. This involves gauge invariant, odd parity, vacuum perturbations of the Schwarzschild metric, the even parity sector formalism was developed by Zerilli in the 70’s. The original reference is a paper by Regge and Wheeler from 1957. A much modern and geometrical presentation is available here and here. A complete second order covariant and gauge invariant perturbation formalism was developed here.

  1. - Precession of Mercury. Material: Carroll's book, Chapter 5, pages 212-218. 

- Experimental tests of gravity: any topic of your choice from Cliff Will's Living Review "The confrontation between General Relativity and experiment".

  1. - Critical phenomena in gravitational collapse. Living Review by Gundlach and Jose M. Martin-Garcia.

  2. - "Analogue gravity": This focuses on, in particular, black hole analogues in non-gravitational physics Living Review by Barcelo, Liberati, Visser.

  3. -"The cosmological constant", Living review by Carroll.

  4. -If anyone has a background on quantum field theory and is brave enough: Hawking effect. References: Carroll’s book has some of it (chapter 9) but I would recommend Wald’s book.

  5. -The history of gravitational waves. There is an historical account here (the title refers to a saying by Eddington, who did not believe on the existence of gravitational waves). And a pdf from the author’s webpage (i.e. I assume it is legal) is here.

- “Resolution” of the big bang singularity by loop quantum cosmology. This is a nice review by Ashtekhar.


80% of the final grade will be based on homework, and 20% on a term project paper (4-5 pages long) and presentation (20-30 mi long) of a study project (typically, studying topics that we will not have time to include in the course) that you will choose around the middle of the semester from a list that I will make final by then. Upon discussion, the term paper can be traded by a short but innovative research project. There is no final exam.


A: 90%-100%
B: 80%-89%
C: 60%-79%

D: 50%-59%.


it is not required, except for the last classes, where we will have the presentations. The exact schedule will be announced in class and posted here, once the exact number of students and schedule is known.


All announcements (new homework being posted, deadlines, etc) will be done by email, even if they have also been done during lecture time. These will be based on your email address registered with the university, so please make sure that you have the correct one registered, that you check your email often, and that my own address does not get filtered by anti-spam software.


The webpage of the course is

Check it regularly for homework, deadlines, and other.

Homework policy:

Homework assignments and their deadlines will be sent by email and posted on this webpage.

o    Please turn in the homework to Rong (your TA) by the specified deadline. You will usually have around one week to work on your homework. One week after each deadline, the graded homework will be ready to be picked up from your TA and discussed during his office hours.

o    Late homework will only be accepted under the exceptional circumstances stipulated by UMD (see below).

o    You are encouraged to discuss the homework with others. You can even work them out with others! (it is a very good way to learn, in fact). However, the work you turn in should reflect your own formulation and understanding.

Effectiveness in class participation

In order to take advantage of lecture time and the array of topics to be covered, it is strongly encouraged that you try to read the material of each lecture ahead of time, even if you only have time for a “casual” read. We will usually  proceed in an interactive manner. By the end of each class we will discuss what the next class topic will be.

Make-up exams policy

The course will adhere to the University policy for make-up exams and late homework, see for a full description of it. A summary follows:

“In order to get a make up for a missed exam you need to provide in written the reason, which has to be one of allowed by the university. Namely, due to illness (of the student or a dependent), religious observance (where the nature of the observance prevents the student from being present during the class period), participation in University activities at the request of University authorities, or compelling circumstances beyond the student's control. Students claiming excused absence must apply in writing and furnish documentary support for their assertion that absence resulted from one of these causes. “

“Students who have a concern regarding religious observances should see their instructors at the start of the semester. Although the University attempts to accommodate the religious beliefs of all of its members, it functions within a secular environment and is limited in the extent to which it can interrupt its normal operations. The President shall determine when it is appropriate for the campus community to restrict rescheduling examinations or other significant assessments on the dates of religious observance.

At this time, examinations or other significant assessments may not be scheduled on Rosh Hoshanah, Yom Kippur, Good Friday, or the first two days of Passover.”

It is the student's responsibility to inform me, by email, within the first two weeks of classes, of any intended absences or delays in handling homework, for the whole semester, due to religious observances.

Students with disabilities

Students with disabilities should contact me as soon as possible so that appropriate arrangements can be made to accommodate the student's needs.

Academic integrity

The University of Maryland, College Park has a nationally recognized Code of Academic Integrity, administered by the Student Honor Council. This Code sets standards for academic integrity at Maryland for all undergraduate and graduate students. As a student you are responsible for upholding these standards for this course. It is very important for you to be aware of the consequences of cheating, fabrication, facilitation, and plagiarism. For more information on the Code of Academic Integrity or the Student Honor Council, please visit

To further exhibit your commitment to academic integrity, remember to sign the Honor Pledge on all examinations and assignments: "I pledge on my honor that I have not given or received any unauthorized assistance on this examination (assignment)."