Master Degree


Year of erogation: 
Unit Coordinator: 
Disciplinary Sector: 
First semester
Language of instruction: 

Italian (possibly in English if all students agree)

Learning outcomes of the course unit

Knowledge and understanding: To learn the basic notion of the Physics of the gravitational force and of Einstein's theory of General Relativity.

Applying knowledge and understanding: Ability to read and understand recent scientific articles on the physics of gravitational phenomena.

Making judgments: Learning the ability to evaluate the novelties and innovations present in the possible formulations (in terms of basic variables) of Einstein equations.

Communication skills: to develop the ability to present and organize the presentation of a specialized topic of study on the themes developed.

Learning skills: The ability to deepen the knowledge of a topic from reading scientific articles.

Course contents summary

Gravitational Physics and Einstein's theory of General Relativity

Course contents

The principle of equivalence: Realization of the principle of equivalence in terms of the four-dimensional metric. Equations of geodesics and their limit for speed much lower than the speed of light. Identification of the 00 component of the metric field with the Newtonian gravitational potential.

Relativistic Kinematics: Synchronizing clocks in curved space-time measurements of distances and times. Space-time symmetries and Killing vectors. The energy-momentum tensor in special and general relativity. The Pound-Rebka experiment and the direct testing of the equivalence principle.

Part II: Einstein's equations and their consequences. Einstein's equations for the gravitational field. Variational formulation and Hilbert-Palatini action. Bianchi identity.

Exact solutions: the study of Einstein's equations in the vacuum in the presence of symmetries and their exact solutions. The spherical symmetry case and the Schwarzschild solution. Axial symmetry and the Kerr solution.

Weak field and gravitational waves: Linearization of Einstein's equations. Solutions of the linearized equations and their interpretation as gravitational waves. " Ownership of "gravitational waves" and experimental methods for their detection. Quadrupole formula to calculate the intensity of the emission of gravitational waves. Laboratory tests: The experimental tests of the classical Einstein's field equations: precession of the perihelion of Mercury; gravitational deflection of light; indirect evidence for gravitational waves from the observations on the Pulsar PSR 1913 +16.

Relativistic cosmology: The Olbers' Paradox. Homogeneous spaces and metric of Friedman-Robertson-Walker. Hubble's law. The Cosmological term of Einstein's equations. Shifting the frequency of the radiation in cosmology and the standard model of the universe. The cosmic expansion and the problem of the missing matter density.

The initial value problem: Einstein's equations in the 3+1 formalism and their Hamiltonian structure. Using the 3 +1 formalism for the numerical solution of Einstein's equation (outline)

Recommended readings

Robert M. Wald, General Relativity (University of Chicago Press 1984)

Hans C. Ohanian e Remo Ruffini, Gravitazione e spazio tempo (Zanichelli 1997)

C. W. Misner, K. S. Thorn e J.A. Wheeler, Gravitation (WH Freeman and C. 1973)

Usefull refernece avvailable on the net are:

Sean M. Carroll, Lecture Notes on General Relativity (

Matthias Blau, Lecture Notes on General Relativity (

Teaching methods

Frontal lessons on the blackboard. The lessons will be organized face to face with the possibility of adending the lessons in synchronous mode even remotely through the "Teams" platform. The information will be uploaded to the Elly page of the course.

Assessment methods and criteria

Final oral exams. The exam will consist of two parts. In the first, the student will present an in-depth analysis of his choices of the issues dealt during the course, followed by a second part in which he will answer questions on the topics discussed.