Level and credits - BSc Astronomy, level 300, 6EC
Course description - This course provides a
first introduction to the theory of General Relativity at BSc level,
and highlights its principal applications in astronomy. A full
understanding of General Relativity requires a significant amount of
mathematics, including differential geometry and tensor calculus, and
a typical General Relativity course at MSc level first develops this
mathematical background. The present course deliberately sidesteps
several of these mathematical foundations and focuses on physical
concepts. The mathematics is introduced as needed. The textbook for
the course uses the same approach. The metric is used as the central
concept, from which the properties of spacetime and the astrophysical
applications are derived. Among these applications, black holes,
cosmology and gravitational waves are discussed in some detail.
Language - The course will be taught completely
in English. The exam also has to be done in English.
Prerequisites - Required background is a
knowledge of calculus and linear algebra at BSc level, of special
relativity (although that subject is reviewed at the beginning of the
present course), and of classical mechanics, including its Lagrangian
formulation. In terms of the Leiden curriculum, the student must have
obtained the Propedeuse, and in addition must have succesfully
completed the courses Klassieke Mechanica B and Lineaire Algebra 2.
Without this full set of prerequisites, enrolment will not be allowed.
Lectures - The course will have 12 weekly
lectures of 2 hours each. During the lectures, the material will be
discussed at a conceptual level, but lengthy mathematical derivations
will be skipped. Also, several of the examples presented in the book
will be skipped over in the lectures. On the other hand, more
astronomical context will be added during the lectures. It is
crucially important that you spend significant time (several hours) on
self-study after the lectures, in order to absorb all
details. Furthermore, while the book is self-contained, skipping the
lectures or tutorials is one of the best recipes for failing the
Tutorials - There will be weekly tutorials of 2
hours each in which problems can be practiced. It is essential to
participate in the tutorials; without this, you have no chance to pass
the exam. The tutorials will be supervised by the Teaching
Assistants. The problems to covered at the tutorials (time permitting)
will be made available in advance at this website.
The following problems from
Hartle's book form good practice material for the exam:
Ch. 5: 1, 3, 8, 20
Ch. 6: 4, 6
Ch. 7: 2, 10, 14, 15, 17, 18, 25
Ch. 8: 1, 2, 5, 6, 9, 11
Ch. 9: 1, 6, 7, 8, 16
Ch. 12: 6
Ch. 13: 5
Ch. 14: 3
Ch. 15: 3, 13
Ch. 16: 6, 7
Ch. 18: 2, 5, 13, 16, 24
Note that some
of these are more easy and some more difficult than what you should
expect at the exam. For the more diificult ones, if I were to give you
such a problem at the exam, I would also give you a hint to set you
off in the right direction, as in the problem sets.
Blackboard - This course does not use
Blackboard. All information exchange will be through this website.
Exam - The course will end with a written exam
in December 2018. The exam will mostly consist of problems that are
quite similar (but not identical) to the problems done during the
tutorials. You may not bring the book or your notes to the exam, but a
formula sheet will be provided, containing all formulae that you
require. A sample exam is available here, to give
you an idea of what to expect. A retake exam will also be scheduled in
January 2019. There will be no further opportunities to take the exam.
Time investment - The typical time
investment for this course (corresponding to 6EC) will be about 8
hours per week (since 30EC corresponds to 40 hours per week).
This is a significant time investment, so make
sure that you have this time available.
We will closely follow the following textbook:
must have a copy of this book. There is a hardcover version and a
paperback version; either is fine. Note that the author maintains a
list of errata at his website.
Gravity - An introduction to Einstein's General Relativity
James B. Hartle
Addison Wesley (2003 or more recent edition)
||Prof. P. (Paul) van der Werf|
Nastasha Wijers, J.H. Oortgebouw 440
Dong-Gang Wang, J.H. Oortgebouw 541
Lecture 1: Introduction: Gravity, geometry, Newtonian physics and introduction to Special Relativity
- September 18, 2018, 13:30-15:15, HL106/9
Topics: Class organisation and introduction.
The position of gravity in astronomy and physics.
Gravity as geometry.
Coordinates, line element and invariance.
Inertial frames of reference.
The principle of relativity.
Gravitational and inertial mass.
Variational principle for Newtonian mechanics.
Material to study: Hartle, Ch. 1, Ch. 2 (except Box 2.3) and Ch. 3 (except Box 3.1)
Lecture 2: Special Relativity
- September 25, 2018, 13:30-15:15, HL106/9
Spacetime and its geometry.
The relativity of simultaneity.
Addition of velocities.
The metric and line element of flat spacetime.
Material to study: Hartle, Ch. 4, except Boxes 4.3 & 4.4, and Ch. 5, Section 5.1
Tutorial: September 27, 2018, 14:30-16:15, HL106/9
Lecture 3: Gravity as geometry, and curved spacetime - October 2,
2018, 13:30-15:15, HL106/9
Newton's laws in terms of four-vectors.
Variational principle for free particle motion.
Observers and observations.
principle. Clocks in a gravitational field. Gravitational
redshift. Spacetime curvature. Static weak field metric. Newtonian
motion in spacetime terms. Coordinates, line element and metric in
curved spacetime. The summation convention. Local inertial
frames. Light cones and world lines in curved spacetime. Length, area,
volume and four-volume in curved spacetime. Embedding diagrams.
Material to study: Hartle, Ch. 5, Section 5.2-5.6 (except Box 5.1), Ch. 6 (except Section 6.4 and Box 6.1)
and Ch. 7, Sections 7.1-7.7 (except Box 7.1)
Tutorial: October 4, 2018, 14:30-16:15, HL106/9
Lecture 4: Motion in curved spacetime
- October 9, 2018, 13:30-15:15, HL106/9
Topics: Vectors in curved spacetime. Orthonormal and coordinate
bases. 3-dimensional surfaces in 4-dimensional spacetime. Variational
principle for free test particle motion. The geodesic equation.
Christoffel symbols. Killing vectors. Geodesic equation for light rays.
Material to study: Hartle, Ch. 7, Sections 7.8 and 7.9, and Ch. 8
Tutorial: October 11, 2018, 14:30-16:15, HL106/9
Lecture 5: The geometry outside a spherical star - October 16,
2018, 13:30-15:15, HL106/9
Topics: The Schwarzschild geometry. Geometrized
units. Gravitational redshift in Schwarzschild geometry. Orbits of
particles and light rays in Schwarzschild geometry. Escape
velocity. Stable circular orbits.
Material to study: Hartle, Ch. 9, Sections 9.1-9.3, but skip the part of
Section 9.3 below Eq. (9.48)
Tutorial: October 18, 2018, 14:30-16:15, HL106/9
Lecture 6: Black holes - October 23, 2018, 13:30-15:15, HL106/9
Orbits of light rays in Schwarzschild geometry. Deflection of light far from the
Schwarzschild radius. The Schwarzschild black
hole. Eddington-Finkelstein coordinates. Lightcones of the
Schwarzschild geometry. Horizon and singularity. Gravitational
Material to study: Hartle, Ch. 9, Section 9.4, but
skip the part on "The Time Delay of Light", which starts on
page 212; Hartle, Ch. 12 except Section 12.3
Tutorial: October 25, 2018, 14:30-16:15, HL106/9
Lecture 7: Astrophysical black holes - October 30, 2018,
Topics: Black holes in X-ray binaries. Accretion disks around
compact objects. Black holes in galactic nuclei. Quantum evaporation
of black holes. Hawking radiation. Black hole thermodynamics
Material to study: Hartle, Ch. 11, only Section 11.2; and Ch. 13 except Fig. 13.6
Tutorial: November 1, 2018, 14:30-16:15, HL106/9
Lecture 8: Rotation - November 6, 2018, 13:30-15:15, HL106/9
Rotational dragging of inertial reference frames.
The spin 4-vector and the gyroscope equation.
Metric outside a slowly rotating body.
Rotating black holes and their horizons.
Material to study: Hartle, Ch. 14, and Ch. 15,
only Sections 15.1 and 15.2
Tutorial: November 8, 2018, 14:30-16:15, HL106/9
Lecture 9: Gravitational waves - November 13, 2018, 13:30-15:15, HL106/9
Gravitational wave detection and polarization.
Energy in gravitational waves.
The LIGO result and its implications
Material to study: Hartle Ch. 16
Tutorial: November 15, 2018, 14:30-16:15, HL106/9
Lecture 10: Cosmology - November 20, 2018, 13:30-15:15, HL106/9
Topics: Homogeneous, isotropic spacetimes. The
Robertson-Walker metric. Scale factor and comoving
coordinates. Cosmological redshift and expansion. The first law of
thermodynamics in cosmology. The Friedman equation in a spatially
flat universe. Evolution of flat FRW models.
Material to study: Hartle, Ch. 18, Sections 18.1-18.4
Tutorial: November 22, 2018, 14:30-16:15, HL106/9
Lecture 11: Cosmological models - November 27, 2018,
Topics: The Big Bang. Spatially curved
FRW models. Open, closed and flat universes and their spatial
geometry. The general FRW metric and Friedman equation. General
solutions. Bouncing universes.
Material to study: Hartle, Ch. 18, Sections 18.5-18.7
Tutorial: November 29, 2018, 14:30-16:15, HL106/9
Lecture 12: Present-day cosmology - December 4, 2018, 13:30-15:15, HL106/9
Topics: The values of the cosmological parameters. Big-Bang
nucleosynthesis. Standard candels and standard rulers. Causality and
particle horizon sizes. Inflation. The limits of present-day
Material to study: Hartle, Ch. 19
Exam: December 20, 2018, 14:00-17:00, HL204
Retake exam: January 30, 2019, 14:00-17:00, HL226
[ to Paul van der Werf's homepage |
to Leiden Observatory ]
Last modified: Thu Dec 20 13:48:07 2018
Paul van der Werf