### RELATIVITY

**1st module**

**(one semester, about 33 hours)**

**Prof. M. Toller**

**A.A. 1998/99**

- The relativity principle and the invariance of the velocity of light
in vacuum: a brief historical introduction.
- Minkowski space-time; Lorentz and Poincaré groups; transformation of particle velocities; tensor calculus.
- The velocity of light in moving media; the aberration of light; Doppler effect; mean life of moving particles; Thomas precession.
- Electromagnetic tensor and covariant formulation of the Maxwell equations and of the Lorentz force; four-potential and gauge invariance.
- Relativistic particle dynamics; energy-momentum four-vector; conservation laws in relativistic collisions; charged particle in a given electromagnetic field; Lagrangian formulation; invariance of the action; four-current and energy-momentum tensor for a dust of charged particles; relativistic angular momentum and centre-of-mass.
- Introduction to classical field theory; action principle and field
equations; symmetries and conservation laws (first Noether theorem); energy-momentum and relativistic angular momentum tensors; Hamiltonian
formalism.
- Action principle for the electromagnetic field; second Noether theorem; singular character of the Lagrangian and constraints in the Hamiltonian formalism; energy-momentum tensor, Poynting vector and three-dimensional Maxwell tensor.
- Green functions and retarded potentials; Lienard-Wiechert potentials; radiation of accelerated charges.

#### Textbooks:

- L. D. Landau, E.M. Lifshitz, Teoria dei campi, Editori Riuniti, Roma
(1976).
- J. D. Jackson, Classical Electrodynamics, J. Wiley and Sons, New York (1962).
- C. Møller, The Theory of Relativity, Oxford University Press, London (1955).