Course syllabus

Course-PM

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.

Student representatives:

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

TimeEdit

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 hep-th/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 pre-recorded part and in a part taught live via Zoom. The pre-recorded 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 pre-recorded 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 pre-recorded 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 pre-recorded part. I used these notes as "virtual blackboards" in the pre-recorded 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 pre-recorded 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 pre-recorded 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 self-assessment 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 

Energy-momentum 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 non-Euclidean 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 weak-field 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

Non-relativistic 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 energy-momentum 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 Robertson-Walker 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 pre-recorded 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 problem-solving.

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 minute-long 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 re-derive some of these equations on the blackboard.