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

PROGRAM GENERAL RELATIVITY 1

 

- Historical Motivations for General Relativity, and how it arises from Special Relativity.

 

- Review of Special Relativity: Lorentz transformations, and how to describe them covariantly in terms of vectors and tensors defined in 3+1 space-time dimensios. Discussion of the covariant formulation of Maxwell electromagnetism in terms of 3+1 gauge potential and field strength.

 

- First steps towards General Relativity. Derivatives of vectors in arbitrary coordinate systems. Concept of covariant derivative, and Christoffel symbols. The concept of the metric tensor, and its usefulness. Introduction to the physics of curved space-times.

 

- Concept of geodesics and parallel trasport. Definition of Riemann tensor, how it arises, and its expression in terms of Christoffel symbols.

 

- The Einstein-Hilbert action, and the left-hand-side of Einstein equations obtained from the extremization of this action. The concept of energy momentum tensor, and how it contributes to the right-hand side of Einstein equations.

 

- The Schwarzschild solution of Einstein equations in empty space. Study of geodesics around this solution, both for massive and massless particles. For the massive case, study of prediction for perihelion of Mercury. For the massless case, the concept of light ring.

 

- Introduction to cosmology: metric of Friedmann-Robertson-Walker. Derivation of Friedmann equation and of continuity equation. Study of several examples of cosmological space-times, depending on the universe content and its spatial curvature.

 

- Brief introduction to gravitational waves.

Readings/Bibliography

We use material and exercises from several textbooks, as indicated in the Virtuale platform. In particular,

 

Carroll: "Spacetime and Geometry: an Introduction to General Relativity"

Schutz: "A first course in General Relativity"

Blennow-Ohlsson: ``300 PROBLEMS IN SPECIAL AND GENERAL RELATIVITY''

Landau-Lifshitz: ``The classical theory of fields''

Hartle: "Gravity an introduction to Einstein's general relativity"

Kolb-Turner: "The Early Universe"

Teaching methods

Lectures on blackboard, including both theory and exercises

Assessment methods

Written exam. There will be in total six sessions of exam during the academic year.

The exam will last 1.5 hours, and might contain exercises, as well as theory questions on the material covered during the course. The exercises will be a variation of the exercises seen during the course.

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 Gianmassimo Tasinato