Course syllabus
CoursePM
FFM071 / FIM450 FFM071 / FIM450 Gravitation and cosmology lp3 VT21 (7.5 hp)
Course is offered by the department of Physics
Contact details
Examiner and lecturer: Riccardo Catena.

 email: catena@chalmers.se;
 telephone number: +46317726309
 webpage: https://www.chalmers.se/en/staff/pages/riccardocatena.aspx
Student representatives:

 Markus Bredberg; email: marbre@student.chalmers.se
 Christian Nilsson; email: christni@student.chalmers.se
 Ana Mari Petrova; email: petrova@student.chalmers.se
 Linus Sundberg; email: sulinus@student.chalmers.se
Course purpose
The purpose of the course is to introduce General Relativity to the student and then apply this theory to describe:
1) the dynamics of particles and electromagnetic fields in the presence of general gravitational fields;
2) gravitational waves
3) the evolution of our Universe
4) black holes
Schedule
Course literature
1) Main text book: "Gravitation and Cosmology", Steven Weinberg (Wiley 1972); available at Cremona
2) Lecture notes; available online at this webpage (see the Course design section below)
3) Parts of chapters 5 and 6 from: "Spacetime and Geometry", Sean C. Carroll (Cambridge 2019); partly available on ArXiv as hepth/9712019
4) Collected problems; available at this webpage (see the Course design section below)
Course design
The course is organised in 20 lectures (of 1h and 45 min each). Each lecture consists in a prerecorded part and in a part taught live via Zoom. The prerecorded part will be made available online at the Course design section of the Course syllabus on Canvas about one week before the foreseen start of the lecture on TimeEdit. The prerecorded part of the lectures covers all subjects I plan to teach, while the part of the lectures taught via Zoom is entirely devoted to your questions on the prerecorded videos. It is a useful exercise to watch these videos while trying to formulate questions that you could ask in our Zoom meetings. For the part of the lectures taught via Zoom we meet at the link
https://chalmers.zoom.us/j/66476832234
Password: 540246
30 minutes before the foreseen end of the lecture on TimEdit. For example, for the lecture scheduled for Monday, January the 18th, we meet at the above link at 11:15.
For each lecture, I will provide you with an online version of my notes, together with the link to the corresponding prerecorded part. I used these notes as "virtual blackboards" in the prerecorded videos. In fact, they contain what I would actually have written on a blackboard if I had not taught this course remotely. I will make an effort to keep these notes updated, and correct them if typos should be identified. Your feedback is therefore welcome.
The prerecorded parts of the lectures are organised in a short introduction, outlining the aim of the lecture, plus some "chapters", each covering a specific subject. I generated a YouTube link to the introductions and chapters of each lecture,
For each lecture, I also shared a link to a OneNote notebook with my notes. This notebook is divided into sections corresponding to introduction and chapters of the associated lecture. In addition, the last section in each notebook contains a list of questions you should be able to answer after attending the corresponding lecture.
Needless to say, only if you carefully attend the prerecorded lectures in time, we will be able to benefit from the part of the lectures taught live via Zoom. From this perspective, the lists of questions in my notes are a useful selfassessment tool.
Besides uploading the teaching material, in this section of the Course syllabus I will also upload three sets of home problems, each with a different deadline. They are part of the exam, as described below in the Examination form section. Further information on the deadlines for the three sets of problems will follow soon.
Lecture 1 (lecture notes: Link)
This lecture is primarily based on my notes and Chapters 1 and 3 (sections 3.1 and partly 3.2) from the Weinberg's book.
Introduction: Link
The Principle of Relativity: Link
The Principle of Equivalence: Link
General Relativity as a theory of gravity: Link
Lecture 2 (lecture notes: Link)
This lecture is based on Chapter 2 (sections 1 to 9) from Weinberg's book.
Introduction: Link
Lorentz transformations: Link
Vectors and Tensors: Link
Particle Mechanics: Link
Electrodynamics: Link
Lecture 3 (lecture notes: Link)
This lecture is based on my notes and Chapter 3 (sections 2 to 5) from Weinberg's book.
Introduction: Link
Locally inertial coordinate systems: Link
The geodesic equation: Link
Newtonian limit: Link
Gravitational redshift: Link
Lecture 4 (lecture notes: Link)
This lecture is based on my notes and Chapter 4 (sections 2 to 7 and section 9) from Weinberg's book.
Introduction: Link
Tensors under general coordinate transformations: Link
Derivatives of tensors: Link
Lecture 5 (lecture notes: Link)
This lecture is based on my notes and Chapters 4 (section 1) and 5 (sections 1 to 4) from Weinberg's book.
Introduction: Link
The Principle of General Covariance: Link
Particle Mechanics: Link
Electrodynamics: Link
Energymomentum tensor: Link
Hydrodynamics and Hydrostatics: Link
Lecture 6 (lecture notes: Link)
This lecture is based on my notes and Chapter 6 (sections 1 and 2 and 6 to 8) from Weinberg's book.
Introduction: Link
Riemann tensor  definition: Link
Riemann tensor  basic properties: Link
Lecture 7 (lecture notes: Link)
This lecture is based on my notes and Chapter 6 (sections 3, 4 and 10) from Weinberg's book.
Introduction: Link
The equation of geodesic deviation: Link
Riemann tensor and gravitational fields: Link
Riemann tensor and nonEuclidean geometry: Link
Riemann tensor and the Principle of General Covariance: Link
Lecture 8 (lecture notes: Link)
This lecture is based on my notes and Chapter 7 (sections 1 and 4 to 6) from Weinberg's book.
Introduction: Link
Einstein equations  derivation: Link
Einstein equations  solutions: Link
Einstein equations  nonlinearities: Link
Lecture 9 (lecture notes: Link)
This lecture is based on my notes and Chapter 8 (section 1) from Weinberg's book.
Introduction: Link
Metric for a static and isotropic gravitational field: Link
Einstein equations in a static and isotropic gravitational field: Link
Harmonic coordinates: Link
Lecture 10 (lecture notes: Link)
This lecture is based on my notes and Chapter 8 (sections 2 to 4) from Weinberg's book.
Introduction: Link
The Schwarzshild solution: Link
Eddington and Robertson corrections: Link
Geodesic equation in a static and isotropic gravitational field: Link
Lecture 11 (lecture notes: Link)
This lecture is based on my notes and Chapter 8 (sections 5 to 7) from Weinberg's book.
Introduction: Link
The deflection of light by the sun: Link
Precession of perihelia: Link
Radar echo delay: Link
Lecture 12 (lecture notes: Link)
This lecture is based on my notes and Chapter 10 (sections 1, 2 and part of section 4) from Weinberg's book.
Introduction: Link
The weakfield approximation: Link
Gravitational wave solutions: homogeneous case: Link
Gravitational wave solutions: inhomogeneous case: Link
Lecture 13 (lecture notes: Link)
This lecture is based on my notes and Chapter 10 (sections 4 and 5) from Weinberg's book.
Introduction: Link
Energy loss by gravitational radiation: Link
Nonrelativistic sources: Link
Detection of gravitational waves: Link
Lecture 14 (lecture notes: Link)
This lecture is based on my notes and Chapter 12 (sections 1 to 4) from Weinberg's book.
Introduction: Link
The principle of least action: Link
The matter action and the energymomentum tensor: Link
The gravitational action and Einstein equations: Link
Lecture 15 (lecture notes: Link)
This lecture is based on my notes and Chapter 13 (sections 1 and 2) from Weinberg's book.
Introduction: Link
Killing vectors: Link
Homogeneous, isotropic and maximally symmetric spaces: Link
Spaces of constant curvature: Link
Lecture 16 (lecture notes: Link)
This lecture is based on my notes and Chapter 13 (sections 2 to 5) from Weinberg's book.
Introduction: Link
The metric of maximally symmetric tensors: Link
Spaces with maximally symmetric subspaces: Link
Tensors in a maximally symmetric space: Link
Lecture 17 (lecture notes: Link)
This lecture is based on my notes and Chapter 14 (sections 1 to 3) from Weinberg's book.
Introduction: Link
The Cosmological Principle: Link
The RobertsonWalker metric: Link
The cosmological red shift: Link
Lecture 18 (lecture notes: Link)
This lecture is based on my notes and Chapter 15 (sections 1 to 3) from Weinberg's book.
Introduction: Link
Friedmann equation: Link
Luminosity distance in Friedmann models: Link
Lecture 19 (lecture notes: Link)
This lecture is based on my notes and Chapter 5 from Carroll's book.
Introduction: Link
Schwarzshild geodesics revisited: Link
Schwarzshild black holes: Link
Home problems
First set: problem_set_1.pdf
Deadline: February the 16th at noon
Second set: problem_set_2.pdf
Deadline: March the 3rd at 24:00
Third set: problem_set_3.pdf
Deadline: March the 10th at 24:00
Changes made since the last occasion
New lecturer and examiner. We will make use of prerecorded lectures. In the oral exam, the student will be asked to do calculations on the blackboard.
Learning objectives and syllabus
During the course the student is expected to acquire a basic understanding of the concepts and principles of General Relativity and a working knowledge of the mathematics used in this field. Specifically, the student should be able to discuss and explain the physical ideas behind the phenomena described by Einstein's theory of gravity, General Relativity, and in a skilful way use the relevant mathematical methods in problemsolving.
Learning objectives:
 The Principle of Relativity: the role of coordinate transformations in Physics
 The Principle of Equivalence and why General Relativity is a theory of gravity
 The mathematical methods used in General Relativity, including tensor analysis
 Einstein's equations: how to derive and solve them
 Why the presence of matter and energy affects the geometry of spacetime
 The basic tests of General Relativity, such as the bending of light in a gravitational field
 Gravitational waves
 The role of General Relativity in Cosmology: maximally symmetric spaces
 The Cosmological Principle and its implications
 The standard Model of Cosmology
 Black holes
Examination form
The examination is divided into two mandatory parts (with grading weights given below):
1) Home problems organised in three sets with different deadline (weight: 40%). These will be uploaded at this webpage. Each set of problems assigns 40 points. 20 points in each set are required to be admitted to the oral exam.
2) Oral exam (weight: 60%). It consists in a 10 minutelong blackboard presentation on a topic chosen by the student among the ones addressed in the course followed by 40 minutes of questions on the concepts and equations discussed in the course. The student will be asked to rederive some of these equations on the blackboard.