00883 - Relativity

Learning outcomes

At the end of the course, the student will acquire the fundamental knowledge in the comprehension of special and general relativity and the geometric setup of these theories. Moreover, he/she will know some experimental tests of general relativity and spherical symmetry solutions of astrophysical interest, getting in contact with phenomena like pulsar and gravitational collapse. He/she will be able to afford simple calculations of solution of the field equations in spherical symmetry or in weak field approximation.

Course contents

1. Priniciples of special relativity and Lorentz transformations
Basics of Newtonian mechanics. Michelson - Morley experiment. Invariance of light speed. Critique to the simultaneity concept Inertia principle and inertial reference frames. Relativity principle. Lorentz transformations. Time dilation and length contraction.
2. Geometrical structure of special relativity
Normed vector spaces, metric and isometries. Minkowski space. 4-vectors and tensors. Flat tensorial analysis. Group structure of Lorentz transformations. Space-time diagrams, light-cone, past, present, future. Rapidity. Twins paradox. Causality and necessity of a limit speed.
3. Relativistic dynamics
Least action principle and equations of motion. The free particle: rest energy and mass-shell relation Intercating particles, fundamental dynamical law Relativistic scattering. Centre of mass-energy. Angular momentum. Spin Stress-energy tensor Examples: light aberration, Compton effect, Doppler relativistic effect
4. Electromagnetic Field in covariant formalism
Scalar and vector potentials. 4-potential Gauge invariance. Lorentz gauge and equations for the 4-potential Electromagnetic tensor. Maxwell equations in covariant form Covariant expression for the Lorenz force Lagrangian formalism for matter coupled to electromagnetic field Stress-energy tensor of the electromagnetic field
5. Equivalence principle and physical foundations of general relativity
Weak equivalence principle. Example of the lift. Strong equivalence principle. Local inertial frames Gaussian notion of curvature. Its link with gravity
6. Curved spaces, Riemannian manifolds, geomety of space-time
Differential manifolds. Riemannian manifolds and metric. Tensor calculus of a smooth manifold. Affine connection, parallel transport and geodesics. Covariant derivative. Curvature, Riemann tensor, Bianchi identity, Ricci and Einstein tensors.
7. Dynamics in presence of gravitational field
Equations of motion of a free falling particle and geodesics. Electromagnetic fields and other physical laws in presence of gravitational field. Gravitational field equations.
8. Applications and experimental evidences of general realtivity
Gravitational time dilation. Mercury perihelion precession. Deflection of light rays. Gravitational waves.
9. Elements of black hole theory
Central field metric in absence and presence of matter. Oppenheimer-Volkoff stellar equilibrium. Gravitational collapse in co-moving metric. Eddington-Finkelstein and Kruskal metrics. Charged and spinning black holes. Area law and elements of Hawking radiation.

Readings/Bibliography

- Notes of the course (downloadable in PDF after the beginning of the course)
- Bernard F. Schutz, A first course in general relativity, Cambridge University Press, Cambridge, 1985.
- Silvio Bergia, Alessandro P. Franco, Le strutture dello spaziotempo, vol. I, CLUEB, Bologna, 2001. (in italian)
- L. D. Landau and E. M. Lifshitz, Field Theory, MIR, Moscow, 1958.
- S. Weinberg, Gravitation and Cosmology, Wiley, New York, 1972.
- R. D'inverno, Introducing Einstein' relativity, Oxford, Clarendon Press, 1992.

Teaching methods

Mainly traditional lectures at the balckboard, especially when mathematical elaboration of concepts are present.
The frontal lectures will be complemented by projections of tables, figures and animations to better illustrate visually certain concepts and results.
Exercises are proposed during the class with the active participation of students. Also, homeworks are given, that the students are warmly invited to solve to improve comprehension.

Assessment methods

Oral exam with 3 questions chosen by the examiner.

Teaching tools

Mainly balckboard lectures.
Occasionally presentation tools and projectors may be used for tables, figures, animations...

Office hours

See the website of Francesco Ravanini

See the website of Fiorenzo Bastianelli