### Teachings

Select Academic Year:     2017/2018 2018/2019 2019/2020 2020/2021 2021/2022 2022/2023
Professor
FRANCESCO DEMONTIS (Tit.)
Period
First Semester
Teaching style
Convenzionale
Lingua Insegnamento
ITALIANO

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/65]  MATHEMATICS [65/00 - Ord. 2012]  PERCORSO COMUNE 9 72

### Objectives

1. Acquiring knowledge and understanding.
The course is devoted to students of the second year of the Master’s degree in Mathematics. It aims to provide a good knowledge of the main concepts of the Special and General Relativity. More specifically, the objectives of this course are:
a) to investigte the basis of the Special and General Relativity;
b) to be able to obtain the Newtonian theory of gravitation starting from the equations of General Relativity (as a first approximation);
c) to introduce, in a rigorous way, the the tensor calculus as an instrument to develop the theory of the General Relativity.

2. Applying knowledge and understanding.
Possible applications of the methods treated during the course will be discussed whether for the solution of purely mathematical problems or for their applications in Physics.

3. Making informed judgements and choices.
This course allows assiduous students to achieve knowledge for applying the learned techniques to the solution of problems which can be encountered in Relativity.
4. Communicating knowledge and understanding.
The evaluation of the oral test keeps into account the ability of the student to give a methodical and consistent exposition of the topics analyzed in the course.

5. Abilities to continue learning.
This course allows assiduous students to acquire a basic expertise which is sufficient to understand advanced mathematical texts for widening autonomously their knowledge in Relativity (Special and General).

### Prerequisites

The course requires a good knowledge of the basic concepts introduced in the following courses of the Bachelor Degree: Calculus 1, 2 and 3, Geometry 1, 2, 3 and 4, Mechanics 1 and 2, and Physics 1 and 2.

### Contents

1. An overview on the Newton gravitation theory. Kepler laws and their dynamical interpretation. Newton’s law of gravitation. Inertial and gravitational mass. Property of the gravitational field. Newtonian potential. Laplace and Poisson theorems.

2. Special Relativity. Axiomatic formulation of the Special Relativity. Lorentz transformations and their consequences (time dilation, lenght contaction, relativistic composition of velocities). Some experimental verification of the special relativity. Minkowski space and four-vectors (four-velocity and four-acceleration). The concept of mass in relativity. Four- momentum. The equivalence of mass and energy.

3. Tensor Calculus and Riemannian Geometry. Controvariant and covariant vectors. Tensors: definition. Criteria to distinguish if an “object” represent a tensor. Riemannian manifolds. Geodesics. Geodesics coordinates. Covariant and absolute differentiation. The Riemann curvature tensor, Ricci's tensor, Einstein's tensor.

4. General Relativity. Axiomatic formulation of the General Relativity: Mach principle, Equivalence principle, Covariant principle and principle of correspondence. Einstein's gravitational equations. Energy-momentum tensor for fluid and not aggregate matter. The cosmological constant. The newtonian limit of the Einstein's equations. External Schwarzschild' s solution. Gravitational red-shift.

5. Introduction to the Cosmology. Difficulty in the newtonian cosmology: Olbers's paradox. Cosmological principle. Weyl’s axiom. The Robertson-Walker. Friedmann’s equations with (and without) the cosmological constant. Classification of Friedmann’s models.

### Teaching Methods

The course consists of 72 lecture hours. Lectures will be given by using either chalk and blackboard or slides. In order to make the teaching as much efficient as possible, the theoretical topics are immediately supported by exercises. The teacher offers constant assistance to the students during the whole year both by personal interviews and by means of e-mail messages. The main tools to support teaching are the teacher's personal web site http://people.unica.it/francescodemontis/
It provides information updated in real time, including: a diary reporting the topics treated in each lecture, information on teaching activities and additional documents to support learning.

### Verification of learning

The course is assessed by means of an oral exam which, usually, lasts 40 minutes. The student has to choose two different topics: one of them has to be an argument of Special Relativity and the second one about the General Relativity. During the exposition of these two arguments, the student should show the global understanding of the entire program.

To pass the exam, the student should show to have acquired a sufficient knowledge of all of the topics of the course. To obtain the maximal grade (30-30 with “cum laude”), the student should instead show to have acquired an excellent knowledge of all of the topics of the course.

### Texts

Lecture notes will be released at the end of each lecture.

An excellent text book (where all the topics of the course can be found) is:

W. Rindler, Relativity (Special, General and Cosmological), Second Edition, Oxford Academic Press, 2006.