Course syllabus

Course-PM

FFM071 / FIM450 FFM071 / FIM450 Gravitation and cosmology lp3 VT23 (7.5 hp)

Course is offered by the department of Physics

 

Contact details

Examiner: Riccardo Catena. e-mail: catena@chalmers.se;

Student representatives:

Rickard Dahlgren Blumenau;   rickardd4@icloud.com       
Hampus Hansen;   hampus26@hotmail.se     
Lars Frederik Marten;   lars.marten@rwth-aachen.de   
Ludvig Rodung;    rodung@student.chalmers.se     
Anton Julius Skoglund;   anton.julius.skoglund@gmail.com      

 

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 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

 

Course design

The course is organised in 19 parts for which I provide links to my own notes below.

The material contained in these 19 parts is divided into 22 in-person lectures of 1h and 45 min each (including an about 15-minute long break). In-person lectures will be held in the lecture rooms indicated on Canvas.

 

Part 1 (pdf)

This part is primarily based on my notes and Chapters 1 and 3 (sections 3.1 and partly 3.2) from Weinberg's book. It focuses on:

The Principle of Relativity;

The Principle of Equivalence;

General Relativity as a theory of gravity.

 

Part 2 (pdf)

This part is based on Chapter 2 (sections 1 to 9) from Weinberg's book. It focuses on:

Lorentz transformations;

Lorentz Vectors and Tensors;

Particle Mechanics in Special Relativity

Electrodynamics in Special Relativity 

 

Part 3 (pdf)

This part is based on my notes and Chapter 3 (sections 2 to 5) from Weinberg's book. It focuses on:

Locally inertial coordinate systems;

The geodesic equation;

Newtonian limit;

Gravitational redshift.

 

Part 4 (pdf)

This part is based on my notes and Chapter 4 (sections 2 to 7 and section 9) from Weinberg's book. It focuses on:

Tensors under general coordinate transformations;

Derivatives of tensors.

Questions not covered during lecture: 3, second part (definition of tensor density)

 

Part 5 (pdf)

This part is based on my notes and Chapters 4 (section 1) and 5 (sections 1 to 4) from Weinberg's book.

The Principle of General Covariance;

Particle Mechanics in General Relativity;

Electrodynamics in General Relativity;

Energy-momentum tensor;

Hydrodynamics and Hydrostatics in General Relativity.

Questions not covered during lecture: 7 

 

Part 6 (pdf)

This part is based on my notes and Chapter 6 (sections 1 and 2 and 6 to 8) from Weinberg's book. It focuses on:

Riemann tensor - definition;

Riemann tensor - basic properties;

Questions not covered during lecture: 3

 

Part 7 (pdf)

This part is based on my notes and Chapter 6 (sections 3, 4 and 10) from Weinberg's book. It focuses on:

The equation of geodesic deviation;

Riemann tensor and gravitational fields;

Riemann tensor and non-Euclidean geometry;

Riemann tensor and the Principle of General Covariance.

Questions not covered during lecture: 3

 

Part 8 (pdf)

This part is based on my notes and Chapter 7 (sections 1 and 4 to 6) from Weinberg's book. It focuses on:

Einstein equations - derivation;

Einstein equations - solutions;

Einstein equations - nonlinearities.

 

Part 9 (pdf)

This part is based on my notes and Chapter 8 (section 1) from Weinberg's book. It focuses on:

Metric for a static and isotropic gravitational field;

Einstein equations in a static and isotropic gravitational field;

Harmonic coordinates.

Questions not covered during lecture: 4

 

Part 10 (pdf)

This part is based on my notes and Chapter 8 (sections 2 to 4) from Weinberg's book. It focuses on:

The Schwarzshild solution;

Eddington and Robertson corrections;

Geodesic equation in a static and isotropic gravitational field.

 

Part 11 (pdf)

This part is based on my notes and Chapter 8 (sections 5 to 7) from Weinberg's book. It focuses on:

The deflection of light by the sun;

Precession of perihelia;

Radar echo delay.

Questions not covered during lecture: 7, 8 and 9

 

Part 12 (pdf)

This part is based on my notes and Chapter 13 (sections 1 and 2) from Weinberg's book. It focuses on:

Killing vectors;

Homogeneous, isotropic and maximally symmetric spaces;

Spaces of constant curvature.

Questions not covered during lecture: 4 first and second part (we did define a maximally symmetric space) and 5 

 

Part 13 (pdf)

This part is based on my notes and Chapter 13 (sections 2 to 5) from Weinberg's book. It focuses on:

The metric of maximally symmetric tensors;

Spaces with maximally symmetric subspaces;

Tensors in a maximally symmetric space.

Questions not covered during lecture: 2, 3 and 6 

 

Part 14 (pdf)

This part is based on my notes and Chapter 14 (sections 1 to 3) from Weinberg's book. It focuses on:

The Cosmological Principle;

The Robertson-Walker metric;

The cosmological red shift.

Questions not covered during lecture: 2

 

Part 15 (pdf)

This part is based on my notes and Chapter 15 (sections 1 to 3) from Weinberg's book. It focuses on:

Friedmann equation;

Luminosity distance in Friedmann models.

Questions not covered during lecture: 4, 5 and 6

 

Part 16 (pdf)

This part is based on my notes and Chapter 5 from Carroll's book. It focuses on:

Schwarzshild geodesics revisited;

Schwarzshild black holes.

 

Part 17 (pdf)

This part is based on my notes and Chapter 10 (sections 1, 2 and part of section 4) from Weinberg's book. It focuses on:

The weak-field approximation;

Gravitational wave solutions - homogeneous case;

Gravitational wave solutions - inhomogeneous case.

Questions not covered during lecture: 7 first part (we did not perform a complete decomposition, but only focused on the physically relevant degrees of freedom)

 

Part 18 (pdf)

This part is based on my notes and Chapter 10 (sections 4 and 5) from Weinberg's book. It focuses on:

Energy loss by gravitational radiation;

Non-relativistic sources;

Detection of gravitational waves.

Questions not covered during lecture: 1 (I only quickly discussed how to relate T_0i to the purely spatial components of the energy momentum tensor) and 4

 

Part 19 (pdf)

This part is based on my notes and Chapter 12 (sections 1 to 4) from Weinberg's book. It focuses on:

The principle of least action;

The matter action and the energy-momentum tensor;

The gravitational action and Einstein equations.

Questions not covered during lecture: 1,2 and 3. We did not address this subject.

 

 

Changes made since the last occasion

None.

 

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 (weight: 40%) to be published on Canvas. 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 20 minutes of questions on the concepts and equations discussed in the course. In this second part of the oral exam, students will be asked to re-derive some basic equations on the blackboard. Examples of questions that you can receive during the oral exam are listed at the end of each lecture note.