OVERVIEW

The course is given in the first semester of the second year of the  “Laurea Magistrale” in Physics. It belongs to the curriculum of a theoretical physicist. Although highly characterized by mathematical tools, the physical contents are mainly stressed.

AIMS AND CONTENT

LEARNING OUTCOMES

Exposition of Einstein's theory of gravitational interactions with its main applications.

AIMS AND LEARNING OUTCOMES

This course gives an introduction to Einstein's theory of general relativity. No prior knowledge of general relativity will be assumed, and an overview of the differential geometry needed to understand the field equations and spacetime geometries will be given. Beyond this, topics covered will include black holes, gravitational waves and an introduction to cosmology.

TEACHING METHODS

lectures at blackboard (48h)

SYLLABUS/CONTENT

•    Manifolds
    ◦    Gravity as geometry
    ◦    What is a manifold?
    ◦    Vectors again
    ◦    Tensors again
    ◦    The metric
    ◦    An expanding universe
    ◦    Causality
    ◦    Tensor densities

•    Curvature
    ◦    Overview
    ◦    Covariant derivatives
    ◦    Parallel transport and geodesics
    ◦    Properties of geodesics
    ◦    The expanding universe revisited
    ◦    The Riemann curvature tensor
    ◦    Properties of the Riemann tensor
    ◦    Symmetries and Killing vectors
    ◦    Maximally symmetric spaces
    ◦    Geodesic deviation

•    Gravitation
    ◦    Physics in curved spacetime
    ◦    Einstein’s equation
    ◦    Lagrangian formulation
    ◦    Properties of Einstein’s equation
    ◦    The cosmological constant
    ◦    Energy conditions
    ◦    The Equivalence Principle revisited
    ◦    Alternative theories

•    The Schwarzschild Solution
    ◦    The Schwarzschild metric
    ◦    Birkhoff’s theorem
    ◦    Singularities
    ◦    Geodesics of Schwarzschild
    ◦    Experimental tests
    ◦    Schwarzschild black holes
    ◦    The maximally extended Schwarzschild solution
    ◦    Stars and black holes

•    More General Black Holes
    ◦    The black hole zoo
    ◦    Event Horizons
    ◦    Killing Horizons
    ◦    Mass, charge, and spin
    ◦    Charged (Reissner-Nordström) black holes
    ◦    Rotating (Kerr) black holes
    ◦    The Penrose process and black-hole thermodynamics

•    Perturbation Theory and Gravitational Radiation
    ◦    Linearized theory and gauge transformations
    ◦    Degrees of freedom
    ◦    Newtonian fields and photon trajectories
    ◦    Gravitational wave solutions
    ◦    Production of gravitational waves
    ◦    Energy loss due to gravitational radiation
    ◦    Detection of gravitational waves

•    Cosmology
    ◦    maximally symmetric universes
    ◦    Robertson-Walker metrics
    ◦    the Friedmann equations
    ◦    evolution of the scale factor
    ◦    redshifts and distances
    ◦    gravitational lensing
    ◦    our universe
    ◦    inflation

RECOMMENDED READING/BIBLIOGRAPHY

• Sean M. Carroll: Spacetime and Geometry: An Introduction to General Relativity
• James B. Hartle: Gravity: An Introduction to Einstein’s General Relativity
• Ta-Pei Cheng: Relativity, Gravitation and Cosmology: A Basic Introduction

TEACHERS AND EXAM BOARD

Exam Board

NICOLA MAGGIORE (President)

ANDREA AMORETTI

CARLA BIGGIO

CAMILLO IMBIMBO

NICODEMO MAGNOLI

SIMONE MARZANI

GIOVANNI RIDOLFI

PIERANTONIO ZANGHI'

LESSONS

LESSONS START

according to the Manifesto degli Studi

Class schedule

GENERAL RELATIVITY (6 CFU)

EXAMS

EXAM DESCRIPTION

There will be an oral examination

ASSESSMENT METHODS

The oral exam is always conducted by the responsible professor and another expert in the subject (usually a tenured professor) and has a duration that varies between about 20 and about 40 minutes. It is based on a number of questions concerning the examination program and allows the commission to judge, in addition to the preparation, the degree of achievement of communication objectives, autonomy, etc.
With these methods, given that at least one of the two teachers has many years of experience in the discipline, the committee is able to verify with high accuracy the achievement of the educational objectives of the teaching. When these are not achieved, the student is invited to deepen the study and to make use of further explanations by the professor.

Exam schedule