Astronomical Relativity
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
selfstudy after the lectures, in order to absorb all
details. Furthermore, while the book is selfcontained, skipping the
lectures or tutorials is one of the best recipes for failing the
course.
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:

Gravity  An introduction to Einstein's General Relativity
James B. Hartle
Addison Wesley (2003 or more recent edition)
ISBN 9780805386622
All students
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.
Teacher 
Prof. P. (Paul) van der Werf

Assistants 
Nastasha Wijers, J.H. Oortgebouw 440
DongGang Wang, J.H. Oortgebouw 541


Lecture 1: Introduction: Gravity, geometry, Newtonian physics and introduction to Special Relativity
 September 18, 2018, 13:3015:15, HL106/9
Topics: Class organisation and introduction.
The position of gravity in astronomy and physics.
Gravity as geometry.
Specifying geometries.
Coordinates, line element and invariance.
Newtonian mechanics.
Inertial frames of reference.
The principle of relativity.
Newtonian gravity.
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, 13:3015:15, HL106/9
Topics:
Spacetime and its geometry.
The lightcone.
Time dilation.
Lorentz transformation.
Length contraction.
The relativity of simultaneity.
Addition of velocities.
c=1.
Fourvectors.
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, 14:3016:15, HL106/9
Problems: pdf
solutions

Lecture 3: Gravity as geometry, and curved spacetime  October 2,
2018, 13:3015:15, HL106/9
Topics:
Fourvelocity.
Energymomentum fourvector.
Newton's laws in terms of fourvectors.
Variational principle for free particle motion.
Light rays.
Observers and observations.
The equivalence
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 fourvolume in curved spacetime. Embedding diagrams.
Material to study: Hartle, Ch. 5, Section 5.25.6 (except Box 5.1), Ch. 6 (except Section 6.4 and Box 6.1)
and Ch. 7, Sections 7.17.7 (except Box 7.1)

Tutorial: October 4, 2018, 14:3016:15, HL106/9
Problems: pdf
solutions

Lecture 4: Motion in curved spacetime
 October 9, 2018, 13:3015:15, HL106/9
Topics: Vectors in curved spacetime. Orthonormal and coordinate
bases. 3dimensional surfaces in 4dimensional 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, 14:3016:15, HL106/9
Problems: pdf
solutions

Lecture 5: The geometry outside a spherical star  October 16,
2018, 13:3015:15, 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 9.19.3, but skip the part of
Section 9.3 below Eq. (9.48)

Tutorial: October 18, 2018, 14:3016:15, HL106/9
Problems: pdf
solutions

Lecture 6: Black holes  October 23, 2018, 13:3015:15, HL106/9
Topics:
Orbits of light rays in Schwarzschild geometry. Deflection of light far from the
Schwarzschild radius. The Schwarzschild black
hole. EddingtonFinkelstein coordinates. Lightcones of the
Schwarzschild geometry. Horizon and singularity. Gravitational
collapse.
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, 14:3016:15, HL106/9
Problems: pdf
solutions

Lecture 7: Astrophysical black holes  October 30, 2018,
13:3015:15, HL106/9
Topics: Black holes in Xray 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, 14:3016:15, HL106/9
Problems: pdf
solutions

Lecture 8: Rotation  November 6, 2018, 13:3015:15, HL106/9
Topics:
Rotational dragging of inertial reference frames.
The spin 4vector and the gyroscope equation.
Geodetic precession.
Metric outside a slowly rotating body.
LenseThirring precession.
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, 14:3016:15, HL106/9
Problems: pdf
solutions

Lecture 9: Gravitational waves  November 13, 2018, 13:3015:15, HL106/9
Topics:
Gravitational waves.
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, 14:3016:15, HL106/9
Problems: pdf
solutions

Lecture 10: Cosmology  November 20, 2018, 13:3015:15, HL106/9
Topics: Homogeneous, isotropic spacetimes. The
RobertsonWalker 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 18.118.4

Tutorial: November 22, 2018, 14:3016:15, HL106/9
Problems: pdf
solutions

Lecture 11: Cosmological models  November 27, 2018,
13:3015:15, HL106/9
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 18.518.7

Tutorial: November 29, 2018, 14:3016:15, HL106/9
Problems: pdf
solutions

Lecture 12: Presentday cosmology  December 4, 2018, 13:3015:15, HL106/9
Topics: The values of the cosmological parameters. BigBang
nucleosynthesis. Standard candels and standard rulers. Causality and
particle horizon sizes. Inflation. The limits of presentday
cosmology.
Material to study: Hartle, Ch. 19

Exam: December 20, 2018, 14:0017:00, HL204

Retake exam: January 30, 2019, 14:0017:00, HL226
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Last modified: Thu Dec 20 13:48:07 2018
Paul van der Werf