Relativity 1

1. A CRITICAL ANALYSIS OF GALILEIAN SPACETIME.
Spacetime 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. The Minkowski spacetime, 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: invariant formulation and viewpoint of an inertial observer,
solution of the equations of motion in the some simple cases. Conservation law of four-monentum in isolated systems. Four-momentum of a photon. Maxwell's equations for the electromagnetic field: relativistically invariant formulation, by means of the exterior differential calculus and Hodge's duality. Relativistically invariant description of the solutions of Maxwell's equations; advanced and retarded potentials. Doppler's effect. Some facts on the dynamics of perfect fluids. The energy-momentum tensor, especially in fluid dynamics and in electromagnetism.
3. THE THEORY OF GENERAL RELATIVITY.
Physical motivations for a geometric theory of gravity. The geometry of spacetime 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. The case of a freely falling particle: the principle of geodesic motion.
Basics on fluid dynamics and electromagnetism in a curved spacetime. The Newtonian law of motion for a
freely falling particle as a limit case of the principle
of geodesic motion. The behavior of clocks according to general relativity: experiments of Pound-Rebka and of Hafele-Keating. The energy momentum tensor in general relativity. Einstein's equations for the gravitational field. Approximate solution of Einstein's equations for weak fields: the Newtonian theory of gravitation as a limit case of general relativity The Schwarzschild solution of Einstein's equations. 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 light in the Sun's gravitational field.
REMARK. After attending this course, students will be given the opportunity to look into an advanced topic in the framework of general relativity, following the bibliographic indications provided of the teacher. Students interested in this opportunity will be invited to give a talk on this topic; in case of positive evaluation of the talk by the teacher, this activity will allow the acquisition of 3 extra credits of the F type.
The suggested topics for this activity are the following ones:
1) The GPS system and general relativity.
2) Cosmological models and general relativity.
3) An introduction to black holes.
4) Gravitational waves and their experimental detection.

This part of the course concerns the basic differential-geometric notions involved in a rigorous formulation of relativity theory. The subjects illustrated include: differentiable manifolds, tensor fields, Lie derivative, exterior differential, distributions and Frobenius theorem, vector bundles and connections, Riemannian and pseudo-Riemannian manifolds.