81827 - Basics of Theory of General Relativity

Learning outcomes

The aim of the course is to provide an introduction to the principles of 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) After a brief recap of the principle of Special Relativity, 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) 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 institutional repository.

Teaching methods

Old style lectures on blackboard.

Assessment methods

Final oral examination, with

1) short written presentation of a topic of choice (for 1/3 of final grade), and

2) general questions about two of the other arguments covered in the course (for 2/3 of final grade).

Exam schedule on appointment by email. Time of examination between 30 minutes and 1 hour.

Teaching tools

The course will be presented using the blackboard.

Office hours

See the website of Roberto Casadio