Relativity 1

A) Main unit (6 credits)

1. A CRITICAL ANALYSIS OF THE GALILEIAN SPACE TIME.
Space-time as a bundle on absolute time, and as a four dimensional affine space. Light propagation, the Michelson-Morley experiment and the crisis of the Galileian model.
2. THE THEORY OF SPECIAL RELATIVITY.
The postulates of the theory. Aleksandrov's theorem and the Lorentz transformations. The effects of "length contraction" and "time dilatation" predicted by the Lorentz transformations; observation of the time dilatation in the decay of elementary particles. The relativistic composition law for velocities; light aberration; Doppler effect. The Minkowski space-time, with its affine and pseudo-Euclidean structures. Space like, time like and null vectors. World line of a particle. Proper time. The twin paradox. Four- velocity and four-acceleration of a particle. Relativistic particle dynamics: solutions of the equations of motion in the some simple cases, balance equations for energy and momentum. Maxwell's equations for the electromagnetic field: relativistically invariant formulations, by means of the esterior differential calculus and Hodge's duality. Some facts on the dynamics of perfect fluids.
Energy-momentum tensor in fluid dynamics and electromagnetism.
3. THE THEORY OF GENERAL RELATIVITY.
Physical motivations for a geometric theory of gravity. The geometry of space-time in general relativity. The general notion of frame; the problem of simultaneity with respect to the frame. The Coriolis theorem in general relativity. Particle dynamics, fluid dynamics and electromagnetism in a curved space-time; energy momentum tensor. Newton's gravity theory as a limit case of general relativity. The behaviour of clocks according to general relativity: experiments of Pound-Rebka and of Hafele-Keating. Einstein's equations for the gravitational field. Approximate solution for weak fields. The Schwarzschild solution. Motion of a test particle and of a light signal in the Schwarzschild space-time. Precession of the perihelion of a planet, and deflection of the light in the Sun's gravitational field.

B) Optional unit (3 credits)

This unit presents a deeper analysis of the basic differential-geometric notions involved in a rigorous formulation of relativity theory (differentiable manifolds, tensor fields, Lie derivative, exterior differential, distributions and Frobenius theorem, vector bundles and connections, Riemannian and pseudo-Riemannian manifolds. Some of these notions are described more abruptly in the main 6-credits unit).