First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SCM0014353, A.A. 2018/19

Information concerning the students who enrolled in A.Y. 2016/17

Information on the course unit
Degree course First cycle degree in
SC1160, Degree course structure A.Y. 2008/09, A.Y. 2018/19
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Number of ECTS credits allocated 10.0
Type of assessment Mark
Course unit English denomination FUNDAMENTS OF RELATIVITY
Website of the academic structure
Department of reference Department of Physics and Astronomy
E-Learning website
Mandatory attendance
Language of instruction Italian
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Teacher in charge LUCA MARTUCCI FIS/02

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses FIS/05 Astronomy and Astrophysics 4.0
Core courses FIS/02 Theoretical Physics, Mathematical Models and Methods 6.0

Course unit organization
Period First semester
Year 3rd Year
Teaching method frontal

Type of hours Credits Teaching
Hours of
Individual study
Lecture 10.0 80 170.0 No turn

Start of activities 01/10/2018
End of activities 18/01/2019
Show course schedule 2019/20 Reg.2008 course timetable

Examination board
Board From To Members of the board
6 Commissione Istituzioni di relatività 2018-2019 01/10/2018 30/11/2019 MARTUCCI LUCA (Presidente)

Prerequisites: Calculus 1, 2, and 3. Physics 1 and 2. Geometry. Analytical Mechanics.
Target skills and knowledge: Relativity: Understanding the foundations of Special Relativity. Ability to solve elementary problems in relativistic mechanics. Ability to use tensor calculus.
Mathematical methods: Basic knowledge of distributions, Fourier transforms, tensor calculus.
Examination methods: Relativity: Written exam with exercises and oral exam with questions about the course topics.
Mathematical methods: Mandatory written exam with exercises and optional oral exam with questions about the course topics.
Assessment criteria: Knowledge and understanding of the course topics, ability in the solution of elementary problems related to the course topics.
Course unit contents: Relativity:
1) Symmetries and invariances. Intertial frames, physical laws, Galilean group. Simultaneity and Newtonian measurements. Speed of light, the aether. Experimental evidences of the finiteness and constance of the speed of light.
2) The new mechanics: the principle of relativity. Postulates of special relativity. Minkowski diagrams. Worldline. Simultaneity. Lorentz transformations.
3) Relativistic kinematics - basics: The interval, the Minkowski metric. Matrix representation of Lorentz transformations. Rapidity. Composition of velocities. Time dilation. Lengths contraction. Penrose-Terrell rotation. The twin paradox.
4) Covariant Relativistic kinematics: Poincaré and Lorentz symmetry groups. Covariant formalism. Tensors. Relativistic momentum. Kinetic energy, rest energy and energy-mass equivalence.
5) Relativistic kinematics - optics: Angle transformations; Stellar aberration. Relativistic Doppler effect.
6) Scattering: Momentum conservation. Particle dacays. Threshold energy. Invariant mass.
7) Electromagnetism: 4-potential and covariant formulation of Maxwell's equations. Gauge invariance. Lorentz transformations of the electromagnetic field. Relativistic invariants. Maxwell equations in vacuo. Solution of the wave equations and its properties. Continuity law for electric currents. Motion of charges in constant electric and magnetic fields.
8) Introduction to General Relativity: The equivalence principle. The metric. Inertial and gravitational forces. Accelerated observers. Rindler spacetime. Time dilation.

Mathematical Methods:
1) Tensors and special relativity: Rotations in euclidean space. Minkowski space, Lorentz and Poincaré transformations. Proper ortochronous transformations. Covariant and contravariant basis. Scalar fields, vector and tensor fields. Divergence and continuity equation in covariant form. Symmetry and Antisymmetry of indices. Differential forms. Exterior derivative. Invariant (Pseudo)tensors: Kronecker delta, metric, Levi-Civita pseudo-tensor. Tensor products and contractions.
2) Distributions: General motivations. Test functions and S(R) space. Regular and singular tempered distributions. Examples: Dirac delta and principal value. Weak convergence. Approximation of singular distributions by means of regular ones. Operations on distributions: derivative, complex conjugation, rescaling and shift of a variable. Dirac delta of a function. Fourier transform and Fourier theorem in S(R). Convolution of functions and their properties. Convolution theorem. Symmetry and reality. Parseval formula. Fourier transform of distributions and Fourier theorem for distributions. Applications: Fourier transform of the Dirac delta function, of a constant function, of the sign function, of the stepfunction. Convolutions of distributions and their properties.
Planned learning activities and teaching methods: Lectures and individual and group activities. Lectures are given in Italian.
Additional notes about suggested reading: Suggested textbooks. Exercises on the course topics are available through the website of the course on the e-learning platform of the Department of Physics and Astronomy "G. Galilei" (
Textbooks (and optional supplementary readings)
  • Gasperini, Maurizio, Manuale di relatività ristretta per la laurea triennale in fisica. Milano: Springer, 2010. Cerca nel catalogo
  • Barone, Vincenzo, Relatività, principi e applicazioni. Torino: Bollati Boringhieri, 2004. Cerca nel catalogo
  • Rindler, Wolfgang, Relativity: Special, general, and cosmological. Oxford: Oxford University Press, 2016. Cerca nel catalogo

Innovative teaching methods: Teaching and learning strategies
  • Case study
  • Working in group
  • Loading of files and pages (web pages, Moodle, ...)

Innovative teaching methods: Software or applications used
  • Moodle (files, quizzes, workshops, ...)