# General Relativity

General Relativity is an indisputable elegant edifice of pure geometry that replaces the Newtonian theory and, as its name suggests, takes into account Special Relativity. It plays a central role in the description of many astrophysical objects such as black holes or pulsars as well as in our understanding of the Universe in its entirety.

We present here a basic introduction to General Relativity prepared for a four-month, graduate-level course in the EPFL (14 weeks). The students are assumed to be familiar with Classical Mechanics and Special Relativity, without any prior knowledge of Classical Field Theory. The notes are thought to be pedagogical and physically oriented. The ubiquitous tensors will be introduced in the familiar context of classical mechanics and electromagnetism. The transition to curved spacetime will be motivated by physical concepts such as the equivalence principle or the existence of tidal forces.

Even though these notes are intended to be self-contained and comprehensible they are just a short guide to the main aspects treated in the course and do not aim to replace any the many outstanding textbooks/notes on the subject. Most of the material presented here follows the treatment of the books and notes in the bibliography, that you are “kindly” encouraged to use, the more the merrier.

#### Lecture notes

- Euclidean spacetime and newtonian physics
- Minkowski spacetime and Special Relativity
- The Equivalence Principle
- General coordinate transformations
- Tidal forces and curvature
- Einstein equations
- Gravitational waves
- The Schwarzschild-Droste solution
- General Relativity: the field theory approach
- The Cosmological Standard Model and the inflationary paradigm

**References**

The metric signature is indicated in parenthesis

**Textbooks & Lecture notes**

• **Gravitation and Cosmology by S. Weinberg.** A timeless classic. Excellent book. Great insight. Physical understanding of General Relativity is put ahead of its mathematical formulation, which is only introduced when needed. No exercises. (+ − −−).

• **Spacetime and Geometry: An Introduction to General Relativity by Sean M. Carroll.** Very readable and up-to-date book. Certainly a good complement to Weinberg’s book. Mathematical tools are presented in detail. Good discussion on black holes. Many exercises. Some lectures notes by the same author can be found in http://arxiv.org/abs/gr-qc/9712019.

Strongly recommended . (− + ++).

• **Einstein theory in a Nutshell by A. Zee.** Probably some of you know the fantastic book in QFT with a very similar title. This book about General Relativity shares the same fresh, irreverent and conversational style putting a lot on emphasis on the physical aspects. Pedagogical and ideal for someone studying General Relativity for the very first time. (− + ++)

**• Gravity, An introduction to Einstein’s General Relativity by J.B. Hartle.** An introductory course which assumes that the reader is completely new to the subject and proceeds step by step. Useful sets of downloadable Mathematica notebooks to calculate Christoffels, Riemman tensor for any metric you fill in http://web.physics.ucsb.edu/~gravitybook/

mathematica.html. (− +++).

• **A first course in general relativity by Bernard F. Schutz.** Another classic. Intuitive presentation. Mathematics are presented in an easy-to-follow way. Lots of worked-out examples throughout the text. (− +++).

• **General Relativity, An introduction for physicists by M.P. Hobson, G. Efstathiou and ****A.N. Lasenby.** Similar to Schutz but slightly less geometrical. Pros: Important ideas are presented as simply as possible. Simple notation and quite pedagogical. Cons: A bit imprecise. Some notation problems. Lots of typos. (+ −−−).

• **Gravitation: Foundations and Frontiers by T. Padmanabhan.** A modern approach based on Classical Field Theory with many topics not usually covered in other books. Around 200 (highly original) exercises. Some videos of the lectures by Prof. Padmanabhan can be found in http://gr-lectures-paddy.blogspot.ch (− +++).

**• Advanced mechanics and General Relativity by Joel Franklin.** Very didactic book based on Lagrangian mechanics. Geometric concepts are introduced early, using examples from Newtonian mechanics and Special Relativity . The structure and predictions of General Relativity are developed in analogy with familiar physical systems. Plenty of worked-out examples and well-chosen exercises. (− +++).

**• The Classical Theory of Fields by L. D. Landau and E. F. Lifshitz.** A unique book. Gradual introduction to electromagnetism and General Relativity. Good exercises. (+ − −−)

•** General Relativity by N.M.J. Woodhouse.** Excellent introductory course. Short, well organized, exceptionally clear and mathematically precise. (+ −−−).

•** Gravitation by C. W. Misner, K.S. Thorne and J. A. Wheeler.** Another classic. Called the “gravitating black brick” by many students due to its extension. Very complete, full of boxes, tables and citations. Multiple perspectives on General Relativity. Probably a bit overwhelming as a textbook for a first course in General Relativity, but recommended as a supplementary text. (− +++).

•** General Relativity by Robert. M. Wald.** More focused on the mathematical formalism than on developing the physical insight. Careful discussions of tensor formalism, the basic singularity, stability, uniqueness theorems and black hole thermodynamics. A superb treatment of some advanced topics, like ADM formulation, is presented in the appendices. Beyond the scope of this course. (− +++).

• **’t Hooft’s lecture notes.** Introduction to General Relativity (−+++).

http://www.staff.science.uu.nl/~hooft101/GtH_lectures.html

• **Jose L. F. Barbon’s lecture notes: Notes on Gravitation.** Very original and different approach. As Padmanabhan’s book, it can be good complement for those of you familiar with Classical Field Theory. (−+++) http://members.ift.uam-csic.es/jfbarbon/Teaching_files/AG12.pdf

**Problem books**

You will find many exercises in the main text of these lecture notes. They are marked with the sign and are thought to be quick exercises that you should solve while studying; just to make sure that you understood what is written in the text. If that training is not enough and you still feel the need of learning and testing your skills,you should have a look to the following books:

• Problem book in Relativity and Gravitation by A. R. Lightman, W. H. Press, R.H. Price and S. A. Teukolsky .

• Cosmology and Astrophysics through Problems by T. Padmanabhan.