87957 - General Relativity 1

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) Recall of Special Relativity

1.1) Lorentz transformations in Minkowski spacetime.

1.2) four-vectors and tensors.

1.3) covariant formulation of Maxwell's electromagnetism and electromagnetic waves.

2) From Special Relativity to General Relativity: differential geometry as a language of space-time 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 Riemann tensor.

3) The general theory of relativity.

3.1) Principles of general relativity, equivalence and general covariance.

3.2) Einstein field equations and energy-momentum tensor.

3.2) The Newtonian limit.

3.3) Gravitational waves.

3.4) Classic experimental checks.

 

4) The Schwarzschild metric.

4.1) Derivation and properties.

4.2) Time-like geodesics and perihelion precession.

4.3) Light-like geodesics and gravitational lenses.

4.4) Gravitational redshift and Pound-Rebka experiment.

4.5) Physical features of Schwarzschild black holes.

4.6) Internal solution

4.7) Tolman-Oppenheimer-Volkov equation and Buchdal limit

5) Cosmological models:

5.1) Friedmann-Lemaitre-Robertson-Walker metrics.

5.2) Friedman equations.

5.3) The Hot Big Bang model and its more recent improvements.

Readings/Bibliography

Complete notes of the course are available on "Virtuale", with complete references to textbooks and research papers.

Reference textbooks:

[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

Blackboard lessons, including both theoretical content and exercises, with occasional use of a projector.

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, questions the remaining 2/3.

Final grade: sufficient knowledge 18-20; basic knowledge 21-23; fair knowledge 24-26; good knowledge 27-29; excellent knowledge 30-30L.

 

Students with temporary or permanent learning disabilities (SLD): please contact the relevant University's office promptly. The office will recommend any accommodations to affected students. These applications must be submitted to the instructor for approval 15 days in advance, who will evaluate their suitability in light of the course's educational objectives.

Teaching tools

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

Use of Mathematica (freely available for all UniBO students) or other software packages is recommended.

Office hours

See the website of Roberto Casadio