00883 - Relativity

Learning outcomes

The aim of the course is to provide an introduction to the principles of (special and general) relativity and some of their main observational consequences (relativistic kinematics, cosmology, black holes).

Course contents

The course is divided into three main parts:

1) Principles and main applications of Special Relativity.
Starting from the incompatibility between the Newtonina mechanics and Maxwell's electrodynamics, unsuccessful attempts to detect the aether are reviewed briefly, and Einstein;s formulation of the principles of relativity introduced. Relativity of simultaneity is then discussed, and Lorentz transformations derived, from which the main kinematical consequences are deduced (length contraction, time delation, velocity addition, Doppler effect, light aberration, etc). Four-momentum is then defined and the dynamics of collisions studied (energy-momentum conservation). Finally, the covariant formalism is introduced (Minkowski space-time, Lorentz tensors) in order to write the laws of electrodynamics in a simple form. This part ends with a brief analysis of the Lorentz group and its representations (including spinors).

2) Elements of differential geometry.

 The student is introduced with the necessary notions and tools to describe geometric spaces independently of the reference frame. Differential manifolds are defined as well as general tensors and tensorial operations. In particular, the Lie and covariant derivatives are introduced. The role of the metric tensor is studied in details, given its key role in general relativity.

3) Introduction to General Relativity.

 The principles of general relativity, of equivalence and of general covariance are introduced. We show how geodesics determine the motion of test particles on a given space-time, and how Einstein equations determine the latter from the energy-momentum tensor of a source. The three classical tests re reviewed: Mercury's perihelion precession, light deflection and gravitational redshift. The general formalism is applied to the two most relevant cases:
a) the space outside a compact spherical source, described by the Schwarzschild metric. Radial geodesics are studied and the nature of the Schwarzschild horizon uncovered, thus introducing the notion of black hole.

b) gravitational waves in the weak field approximation;

c) the evolution of the universe is investigated from the cosmological principle of homogeneity and isotropy, leading to simple Friedman-Robertson-Walker models. The course ends with the Hubble law.

Readings/Bibliography

Lecture notes available from Alma DL

Teaching methods

Old style lectures on blackboard

Assessment methods

Final oral examination, with a short presentation of a topic of student's choice, and general questions regarding the arguments of the course.

Teaching tools

The course will be presented using the blackboard.

Office hours

See the website of Roberto Casadio