87957 - General Relativity 1

Academic Year 2023/2024

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Physics (cod. 9245)

Learning outcomes

At the end of the course the student will learn elements of General Relativity. In particular he/she will acquire a basic knowledge of Einstein field equations and their Schwarzschild and gravitational wave solutions. He/she will be able to describe classical tests of General Relativity and master simple applications of it.

Course contents

1) Review of Special Relativity: Lorentz transformations in the Minkowski spacetime, 4-vectors and tensors. Covariant formulation of Maxwell electromagnetism and electromagnetic waves.


2) From Special Relativity to General Relativity: differential geometry as the language of spacetime manifolds. 

2.1) Manifolds, coordinates, curves, vectors and tensors.

2.2) Lengths and angles: the metric tensor.

2.3) Lie derivatives and symmetries.

2.4) Area and volume: differential forms.

2.5) Curvature: parallel transport, covariant derivatives, geodesics and the Riemann tensor. 


3) The general theory of relativity.

3.1) Principles of General Relativity, Equivalence and General covariance.

3.2) Einstein field equations and the energy-momentum tensor.

3.2) The Newtonian limit.

3.3) Gravitational waves.

3.4) The classical tests.


4) The Schwarzschild metric.

4.1) Derivation and properties.

4.2) Timelike geodesics and the precession of perihelia.

4.3) Lightlike geodesics and gravitational lensing.

4.4) Gravitational redshift and Pound-Rebka experiment.

4.5) The Schwarzschild black hole.


5) Cosmological models:

5.1) Friedmann-Lemaitre-Robertson-Walker metrics.

5.2) Friedman equations.

5.3) The hot Big Bang model.


Notes of the course will be made available on "Virtuale".

Some reference books:

[1] B. Schutz, Geometrical methods of mathematical physics, Cambridge Univ. Press (1980)

[2] B. Schutz, A first course in general relativity, Cambridge Univ. Press (2009).


Teaching methods

Lectures on blackboard, including both theory and exercises

Assessment methods

There will be no scheduled exams, but the tests will be held by appointment with the teacher via email.

Oral exam with discussion of a short essay to be sent in advance and questions both on the theoretical part and carrying out exercises from the course notes. Essay will weigh approximately 1/3 of the final grade.

Teaching tools

I advise the students to consult the list of topics covered and information on exercises on the Virtuale platform.

Office hours

See the website of Roberto Casadio


Quality education

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.