Introduction to General Relativity
A.Y. 2024/2025
Learning objectives
This class represents an introduction to the theory of general relativity (GR). It starts with an introduction to differential geometry, the language in which GR is written. After that, the Einstein field equations are derived heuristically, and are finally solved in certain contexts, such as spherical symmetry (leading to the Schwarzschild solution), gravity waves and cosmology
Expected learning outcomes
At the end of the course the student is expected to have the following skills:
1. Profound knowledge of differential geometry;
2. Knows the Einstein field equations and their Newtonian limit;
3. Is able to solve the Einstein equations in a context with enough symmetry;
4. Knows the physics of the Schwarzschild solution and the classical tests of GR;
5. Knowledge in modern cosmology.
1. Profound knowledge of differential geometry;
2. Knows the Einstein field equations and their Newtonian limit;
3. Is able to solve the Einstein equations in a context with enough symmetry;
4. Knows the physics of the Schwarzschild solution and the classical tests of GR;
5. Knowledge in modern cosmology.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
CORSO A
Responsible
Lesson period
First semester
Course syllabus
Introduction to differential geometry, Einstein equations, Newtonian limit, Schwarzschild solution, classical tests of GR, FLRW cosmology.
Prerequisites for admission
Basic concepts of special relativity.
Teaching methods
Blackboard lectures.
Teaching Resources
Literature will be given during the lectures.
Assessment methods and Criteria
Oral examination.
FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 6
Lessons: 48 hours
Professor:
Klemm Silke
CORSO B
Responsible
Lesson period
First semester
Course syllabus
First part: (~ 16 hrs):
- The equivalence principle;
- Elements of Special Relativity(8+2): inertial observers, Lorentz transformations, the Minkowski metric and the causal structure of the space-time, the Lorentz group, 4-vectors, the Lagrangian of the free particle, covariant formulation of E&M, scatterings and decays;
- General coordinates transformations in flat space-time: the geodetic equation and the
Rindler's metric;
- Classical tests of General Relativity;
Second part (~ 12 hrs)
An introduction to differential geometry: Differential manifolds, tangent and cotangent space, tensor analysis, differential forms, (pseudo-)Riemannian manifolds, linear connections, curvature, geodetic deviation;
Third part (~ 20 hrs) :
- Einstein's Equations: Euristic derivation, Einstein-Hilbert Action, Bianchi Identitities and
diffeomorphisms;
- Conserved quantities in GR: Killing Vectors
- Gravitational waves (GW): Einstein's equations to first order, gauge fixing, propagation of
GW in empy space. Production of GW, the quadrupole formula and the radiated power,
production in binary systems and implications for LIGO/VIRGO and pulsar timing.
- Schwarzschild's solution and black holes: Birkhoff's theorem, physics at the event horizon,
extension beyond the horizon, Kruskal diagram;
- Cosmology (briefly): the FLRW metric and some simple solutions of the cosmological
standard model.
Approximately 8-10 hours will be devoted to solving excercises in classes, either by the students or by myself.
- The equivalence principle;
- Elements of Special Relativity(8+2): inertial observers, Lorentz transformations, the Minkowski metric and the causal structure of the space-time, the Lorentz group, 4-vectors, the Lagrangian of the free particle, covariant formulation of E&M, scatterings and decays;
- General coordinates transformations in flat space-time: the geodetic equation and the
Rindler's metric;
- Classical tests of General Relativity;
Second part (~ 12 hrs)
An introduction to differential geometry: Differential manifolds, tangent and cotangent space, tensor analysis, differential forms, (pseudo-)Riemannian manifolds, linear connections, curvature, geodetic deviation;
Third part (~ 20 hrs) :
- Einstein's Equations: Euristic derivation, Einstein-Hilbert Action, Bianchi Identitities and
diffeomorphisms;
- Conserved quantities in GR: Killing Vectors
- Gravitational waves (GW): Einstein's equations to first order, gauge fixing, propagation of
GW in empy space. Production of GW, the quadrupole formula and the radiated power,
production in binary systems and implications for LIGO/VIRGO and pulsar timing.
- Schwarzschild's solution and black holes: Birkhoff's theorem, physics at the event horizon,
extension beyond the horizon, Kruskal diagram;
- Cosmology (briefly): the FLRW metric and some simple solutions of the cosmological
standard model.
Approximately 8-10 hours will be devoted to solving excercises in classes, either by the students or by myself.
Prerequisites for admission
Knowledge of special relativity and classical (Lagrangian) mechanics. Prior knowledge of differential geometry is not expected.
Teaching methods
Attendance is highly recommended.
Traditional blackboard lectures. Lecture notes will be made available.
Homeworks solved at home and in class during the semester.
Traditional blackboard lectures. Lecture notes will be made available.
Homeworks solved at home and in class during the semester.
Teaching Resources
- David Tong's lectures - http://www.damtp.cam.ac.uk/user/tong/gr.html (principale referenza)
- S. Weinberg, "Gravitation and Cosmology"
- J. Hartle, "An introduction to Einstein's general relativity"
- S. Weinberg, "Gravitation and Cosmology"
- J. Hartle, "An introduction to Einstein's general relativity"
Assessment methods and Criteria
Open book written exam and oral exam (optional). Oral exam reserved to those who scored 18/30 or more at the written exam.
FIS/02 - THEORETICAL PHYSICS, MATHEMATICAL MODELS AND METHODS - University credits: 6
Lessons: 48 hours
Professor:
Castorina Emanuele
Educational website(s)
Professor(s)