LEARNING OUTCOMES

Exposition of Einstein’s theory of gravitational interactions with its main consequences and applications: black holes (by Schwarzschild, Reisner-Nordström and Kerr), gravitational waves and cosmology.

AIMS AND LEARNING OUTCOMES

The course provides an introduction to Einstein’s theory of General Relativity. No preliminary mathematical course is required, but only familiarity with covariant formalism and the principles of field theory (action, equations of motion as functional derivatives of action versus fields). The minimum differential geometry tools needed to construct the Einstein-Hilbert action and deduce Einstein equations as field equations will be introduced. The main purpose of the course is to transmit to the student the rich physical content of Einstein’s equations: black holes, gravitational waves and an introduction to cosmology

PREREQUISITES

No preliminary mathematical course is required, but only familiarity with covariant formalism and the principles of field theory (action, equations of motion as functional derivatives of action with respect to fields)

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

•    Gravitation
    ◦    Physics in curved spacetime
    ◦    Einstein’s equation
    ◦    Lagrangian formulation
    ◦    Properties of Einstein’s equation
    ◦    The cosmological constant

•    The Schwarzschild Solution
    ◦    The Schwarzschild metric
    ◦    Birkhoff’s theorem
    ◦    Singularities
    ◦    Geodesics of Schwarzschild
    ◦    Experimental tests
    ◦    Schwarzschild black holes
    ◦    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

•    Perturbation Theory and Gravitational Radiation
    ◦    Linearized theory and gauge transformations
    ◦    Degrees of freedom
    ◦    Newtonian fields and photon trajectories
    ◦    Gravitational wave solutions

•    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