ID:
SCC1297
Duration (hours):
64
CFU:
8
SSD:
Astrofisica, cosmologia e scienza dello spazio
Year:
2026
Overview
Date/time interval
Primo Semestre (21/09/2026 - 15/01/2027)
Syllabus
Course Objectives
Educational objectives
The course aims to provide students with an introduction to Einstein's theory of General Relativity and to its main applications in modern cosmology.
The first part of the course develops the geometrical description of gravitation, introduces Einstein’s field equations, and discusses their main physical consequences, including classical tests of General Relativity, black holes, and gravitational waves. The second part applies the relativistic framework to cosmology, with particular emphasis on the expansion history of the Universe, its thermal evolution, cosmological perturbations, and the observational evidence for dark matter and dark energy.
Expected learning outcomes
At the end of the course, the student will be able to:
• formulate the basic principles of General Relativity;
• calculate the geometrical quantities associated with simple spacetime metrics, including Christoffel symbols, curvature tensors, and the Einstein tensor;
• derive the geodesic equations for test particles and light rays in simple spacetimes;
• write Einstein’s field equations for a specified metric and matter content;
• apply the weak-field approximation to gravitational phenomena;
• calculate gravitational redshift and the perihelion precession of Mercury;
• characterize the Schwarzschild spacetime and its physical interpretation as the exterior field of a spherical gravitating body or a non-rotating black hole;
• describe the origin, propagation, and observational relevance of gravitational waves;
• derive the Friedmann equations from the Friedmann-Lemaître-Robertson-Walker metric;
• use cosmological redshift and distance measures to describe the expansion history of the Universe;
• explain the observational evidence for dark matter, dark energy, and the accelerated expansion of the Universe;
• calculate the main physical processes involved in the thermal history of the Universe;
• describe the origin and statistical properties of cosmological perturbations;
• analyse the linear evolution of cosmological perturbations and its connection with large-scale structure and cosmic microwave background anisotropies.
The course aims to provide students with an introduction to Einstein's theory of General Relativity and to its main applications in modern cosmology.
The first part of the course develops the geometrical description of gravitation, introduces Einstein’s field equations, and discusses their main physical consequences, including classical tests of General Relativity, black holes, and gravitational waves. The second part applies the relativistic framework to cosmology, with particular emphasis on the expansion history of the Universe, its thermal evolution, cosmological perturbations, and the observational evidence for dark matter and dark energy.
Expected learning outcomes
At the end of the course, the student will be able to:
• formulate the basic principles of General Relativity;
• calculate the geometrical quantities associated with simple spacetime metrics, including Christoffel symbols, curvature tensors, and the Einstein tensor;
• derive the geodesic equations for test particles and light rays in simple spacetimes;
• write Einstein’s field equations for a specified metric and matter content;
• apply the weak-field approximation to gravitational phenomena;
• calculate gravitational redshift and the perihelion precession of Mercury;
• characterize the Schwarzschild spacetime and its physical interpretation as the exterior field of a spherical gravitating body or a non-rotating black hole;
• describe the origin, propagation, and observational relevance of gravitational waves;
• derive the Friedmann equations from the Friedmann-Lemaître-Robertson-Walker metric;
• use cosmological redshift and distance measures to describe the expansion history of the Universe;
• explain the observational evidence for dark matter, dark energy, and the accelerated expansion of the Universe;
• calculate the main physical processes involved in the thermal history of the Universe;
• describe the origin and statistical properties of cosmological perturbations;
• analyse the linear evolution of cosmological perturbations and its connection with large-scale structure and cosmic microwave background anisotropies.
Course Prerequisites
No specific prerequisites are required beyond those expected for admission to the Master’s Degree Programme in Physics. Familiarity with classical mechanics, electromagnetism, special relativity, multivariable calculus, and basic linear algebra is assumed. Previous knowledge of General Relativity is not required.
Teaching Methods
The course is delivered through in-person lectures. The theoretical framework is developed through detailed derivations, physical discussion, and worked examples.
During the first part of the course, the mathematical and physical foundations of General Relativity are introduced and applied to selected gravitational phenomena. During the second part, the relativistic framework is applied to homogeneous, isotropic, and perturbed cosmological models.
Calculations, examples, and short problem-solving activities are integrated into the lectures. Students are encouraged to take part in the discussion and to work through selected derivations during the course. Lecture notes and supplementary material are made available progressively during the semester.
During the first part of the course, the mathematical and physical foundations of General Relativity are introduced and applied to selected gravitational phenomena. During the second part, the relativistic framework is applied to homogeneous, isotropic, and perturbed cosmological models.
Calculations, examples, and short problem-solving activities are integrated into the lectures. Students are encouraged to take part in the discussion and to work through selected derivations during the course. Lecture notes and supplementary material are made available progressively during the semester.
Assessment Methods
The final assessment consists of an oral examination structured in three parts of equal weight.
1. Presentation of a scientific article.
During the course, the instructor proposes a list of scientific articles relevant to General Relativity, gravitational physics, and cosmology. The student selects one article in agreement with the instructor at least one week before the examination and presents its scientific context, main results, and physical relevance.
2. Presentation of a course topic.
The student presents a topic selected from those covered during the course, illustrating its physical motivation, theoretical framework, and main mathematical steps.
3. Discussion of the whole programme.
The student answers questions covering the entire programme, including both General Relativity and cosmology.
The examination assesses knowledge and understanding of the course contents, ability to carry out and discuss basic calculations, rigour in the use of scientific language, clarity of exposition, ability to establish connections between different topics, and critical understanding of the selected scientific article.
Honours may be awarded to students who demonstrate excellent and comprehensive mastery of the course contents, rigour in the discussion of calculations, and a particularly mature and autonomous presentation of the selected article.
1. Presentation of a scientific article.
During the course, the instructor proposes a list of scientific articles relevant to General Relativity, gravitational physics, and cosmology. The student selects one article in agreement with the instructor at least one week before the examination and presents its scientific context, main results, and physical relevance.
2. Presentation of a course topic.
The student presents a topic selected from those covered during the course, illustrating its physical motivation, theoretical framework, and main mathematical steps.
3. Discussion of the whole programme.
The student answers questions covering the entire programme, including both General Relativity and cosmology.
The examination assesses knowledge and understanding of the course contents, ability to carry out and discuss basic calculations, rigour in the use of scientific language, clarity of exposition, ability to establish connections between different topics, and critical understanding of the selected scientific article.
Honours may be awarded to students who demonstrate excellent and comprehensive mastery of the course contents, rigour in the discussion of calculations, and a particularly mature and autonomous presentation of the selected article.
Contents
The course is divided into two closely connected parts.
Part I - General Relativity and gravitational phenomena (approximately 32 hours)
• The equivalence principle and the geometrical interpretation of gravitation
• Spacetime intervals, metrics, coordinates, and tensor notation
• Covariant derivatives, Christoffel symbols, and geodesic equations
• Curvature tensors, the Ricci tensor, the Ricci scalar, and the Einstein tensor
• Einstein’s field equations and the relation with Newtonian gravity
• The weak-field approximation
• Gravitational redshift and classical tests of General Relativity
• The perihelion precession of Mercury
• The Schwarzschild metric and the exterior gravitational field of spherical bodies
• Black holes, event horizons, and basic physical properties of Schwarzschild black holes
• Linearized General Relativity and gravitational waves
• Gravitational-wave sources, detection, and their role in contemporary astrophysics
Part II - Relativistic cosmology (approximately 32 hours)
• The cosmological principle and the Friedmann-Lemaître-Robertson-Walker metric
• Friedmann equations and the dynamics of the expanding Universe
• Cosmological redshift and distance measures
• The content of the Universe: radiation, baryonic matter, dark matter, and dark energy
• Thermal equilibrium and the Boltzmann equation
• Primordial nucleosynthesis
• Recombination, decoupling, and thermal relics
• Cosmological perturbations and their initial conditions
• Power spectra, Gaussian random fields, and non-Gaussianity
• Inflation and the generation of primordial perturbations
• Linear evolution of cosmological perturbations
• Cosmic microwave background anisotropies and large-scale structure
Part I - General Relativity and gravitational phenomena (approximately 32 hours)
• The equivalence principle and the geometrical interpretation of gravitation
• Spacetime intervals, metrics, coordinates, and tensor notation
• Covariant derivatives, Christoffel symbols, and geodesic equations
• Curvature tensors, the Ricci tensor, the Ricci scalar, and the Einstein tensor
• Einstein’s field equations and the relation with Newtonian gravity
• The weak-field approximation
• Gravitational redshift and classical tests of General Relativity
• The perihelion precession of Mercury
• The Schwarzschild metric and the exterior gravitational field of spherical bodies
• Black holes, event horizons, and basic physical properties of Schwarzschild black holes
• Linearized General Relativity and gravitational waves
• Gravitational-wave sources, detection, and their role in contemporary astrophysics
Part II - Relativistic cosmology (approximately 32 hours)
• The cosmological principle and the Friedmann-Lemaître-Robertson-Walker metric
• Friedmann equations and the dynamics of the expanding Universe
• Cosmological redshift and distance measures
• The content of the Universe: radiation, baryonic matter, dark matter, and dark energy
• Thermal equilibrium and the Boltzmann equation
• Primordial nucleosynthesis
• Recombination, decoupling, and thermal relics
• Cosmological perturbations and their initial conditions
• Power spectra, Gaussian random fields, and non-Gaussianity
• Inflation and the generation of primordial perturbations
• Linear evolution of cosmological perturbations
• Cosmic microwave background anisotropies and large-scale structure
Course Language
English
More information
The instructor can be contacted by e-mail at: of.piattella@uninsubria.it. Meetings with students can be arranged by e-mail.
Degrees
Degrees
PHYSICS
Master’s Degree
2 years
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Docenti di ruolo di IIa fascia
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