Astronomical Relativity

General Book Organisation Schedule

General information

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 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:

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
Dong-Gang Wang, J.H. Oortgebouw 541


  1. 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. 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)
  2. Lecture 2: Special Relativity - September 25, 2018, 13:30-15:15, HL106/9
    Topics: Spacetime and its geometry. The lightcone. Time dilation. Lorentz transformation. Length contraction. The relativity of simultaneity. Addition of velocities. c=1. Four-vectors. 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
  3. Tutorial: September 27, 2018, 14:30-16:15, HL106/9
    Problems: pdf solutions
  4. Lecture 3: Gravity as geometry, and curved spacetime - October 2, 2018, 13:30-15:15, HL106/9
    Topics: Four-velocity. Energy-momentum four-vector. Newton's laws in terms of four-vectors. 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 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)
  5. Tutorial: October 4, 2018, 14:30-16:15, HL106/9
    Problems: pdf solutions
  6. 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
  7. Tutorial: October 11, 2018, 14:30-16:15, HL106/9
    Problems: pdf solutions
  8. 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)
  9. Tutorial: October 18, 2018, 14:30-16:15, HL106/9
    Problems: pdf solutions
  10. Lecture 6: Black holes - October 23, 2018, 13:30-15:15, HL106/9
    Topics: 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 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
  11. Tutorial: October 25, 2018, 14:30-16:15, HL106/9
    Problems: pdf solutions
  12. Lecture 7: Astrophysical black holes - October 30, 2018, 13:30-15:15, HL106/9
    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
  13. Tutorial: November 1, 2018, 14:30-16:15, HL106/9
    Problems: pdf solutions
  14. Lecture 8: Rotation - November 6, 2018, 13:30-15:15, HL106/9
    Topics: Rotational dragging of inertial reference frames. The spin 4-vector and the gyroscope equation. Geodetic precession. Metric outside a slowly rotating body. Lense-Thirring precession. Rotating black holes and their horizons.
    Material to study: Hartle, Ch. 14, and Ch. 15, only Sections 15.1 and 15.2
  15. Tutorial: November 8, 2018, 14:30-16:15, HL106/9
    Problems: pdf solutions
  16. Lecture 9: Gravitational waves - November 13, 2018, 13:30-15:15, 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
  17. Tutorial: November 15, 2018, 14:30-16:15, HL106/9
    Problems: pdf solutions
  18. 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
  19. Tutorial: November 22, 2018, 14:30-16:15, HL106/9
    Problems: pdf solutions
  20. Lecture 11: Cosmological models - November 27, 2018, 13:30-15:15, 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 18.5-18.7
  21. Tutorial: November 29, 2018, 14:30-16:15, HL106/9
    Problems: pdf solutions
  22. 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 cosmology.
    Material to study: Hartle, Ch. 19
  23. Exam: December 20, 2018, 14:00-17:00, HL204
  24. Retake exam: January 30, 2019, 14:00-17:00, HL226

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Last modified: Thu Dec 20 13:48:07 2018
Paul van der Werf