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Syllabus for

Academic year
FAS060 - Black holes
Owner: TTFYA
5,0 Credits (ECTS 7,5)
Grading: TH - Five, Four, Three, Not passed
Level: C

Teaching language: English

Course module   Credit distribution   Examination dates
Sp1 Sp2 Sp3 Sp4 No Sp
0102 Examination 5,0 c Grading: TH   5,0 c   Contact examiner,  Contact examiner

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Professor  Marek Abramowicz


For single subject courses within Chalmers programmes the same eligibility requirements apply, as to the programme(s) that the course is part of.

Course specific prerequisites

The course is open to 4th/5th-year students (or students at an equivalent
level) and graduate students with background in physics and mathematics.

The course will be quite demanding, but not too difficult for those who are
determined to learn about the subject and are prepared for a serious effort.
Some knowledge of general physics, in particular special relativity, is
assumed. Knowledge of differential geometry and general relativity is not
required as the subject will be introduced during the course.


Black holes are fascinating objects. The course will cover all aspects of
black hole physics and astrophysics. In particular, important results
obtained by Stephen Hawking, Roger Penrose and others will be explainded in
details, including the most important ones concerning singularities and black
hole entropy. String theory explanation of black hole entropy will be

It will be described how high-tech X-ray satellites, the Hubble Space
Telescope, and large radio telescopes, helped astronomers find several black
holes in the sky. Supermassive black holes found in centers of many
galaxies, including our own Galaxy, have masses millions to billions times
greater than that of the Sun. They provide power for quasars, the greatest
energy sources in the Universe. In a relatively small volume, comparable to
the size of a planetary system, a quasar generates energy at a rate thousands
of times exceeding the combined output of several billion stars that exist in
a galaxy. A scaled-down version of quasars, the stellar mass black holes,
were found in some binary systems in our Galaxy. They are only about ten
times more massive than the Sun.


(1) Geometry on curved surfaces. The metric. Geodesic lines. (2) Covariant
derivative. Geodesic deviation. Intrinsic and extrinsic curvatures and their
various interpretations. (3) Motion on curved surfaces. Inertial forces:
centrifugal, Coriolis, Euler. Gravitational force. Tidal forces. (4)
Einstein's special relativity reminder: Minkowski spacetime. Lorentz
transformations. Mass, energy, momentum. Relativistic dynamics. (5)
Einstein's general relativity: Principle of equivalence. Curved spacetimes.
Riemann and Ricci tensors. Field equations. Geodesic motion. Newtonian limit.
(6) Stationary and static spacetimes. Killing vectors. Kinematic invariants.
Preferred observers. ZAMO. (7) Acceleration formula. Conformal
transformations. Optical reference geometry. (8) Schwarzschild geometry.
Horizon. Effective potential. Motion of particles, photons and gyroscopes.
Inside-outsite reversal. Eddington-Finkelstein and Kruskal coordinates. (9)
Penrose diagram. Rindler's metric. Event horizons. Singularities. Hawking and
Penrose singularity theorems. Cosmic Censor. White holes and wormholes.
Topological issues. (10) Kerr geometry. Horizon and ergosphere. Motion of
particles, photons and gyroscopes. Penrose process. (11) Black hole
electrodynamics. Superradiance. (12) Reversible and irreversible processes.
Area of the horizon increase theorem. Black hole thermodynamics. Quantum
processes. Hawking radiation. (13) Black hole entropy. String theory
interpretation of black hole entropy. (14) The endpoint of stellar
evolution. Chandrasekhar mass. (15) Astrophysical black holes: accretion
disks. Quasars.




Wiley, New York, 1983.


Two possible ways. (1) An oral essay presentation in the classroom (during
the last week of the course). The subject must be agreed with examiners. Up
to two students may work on one presentation. (2) Written exam. Four
questions from a list of twenty will be asked at the exam. The list will be
announced two weeks before the exam.

Page manager Published: Thu 03 Nov 2022.