Announcements:

• (5/12) note 13: added a paragraph about dark energy dominated universe.
• (5/12) note 12: added the Friedman equation for flat universe.
• (11/11) Added a few more problems (marked "extra") to note-4 and moved one problem from note-4 to note-3.
• (10/11) Added a hint to the problem of geodesics on a sphere (note 4).
• (5/11) The ip-address of the server has changed to 62.107.14.89 due to "servicearbejde" at my internet provider. Sorry for that.
• (1/11) We need a "formelsamling" (list of formulae) for the examination. I suggest that you, the students, make the list (only one list for everybody, of course) collaboratively. Here is an editable google doc, please go ahead and fill it with your favourite formulae. You have got one normal A4 page with normal fonts.
  Alternatively try this overleaf document and tell me if you like this one better.
• (31/10) Added a small subsubsection—"Detection of Gravitational Waves"—to note-8.
• (31/10) Moved keywords from notes to the webpage.
• (27/10) Changed "energy-momentum tensor" to "stress-energy-momentum tensor" to be consistent with t'Hooft.
• (24/10) An extra hint is added to the problem of the transformation rule for Christoffel symbols, note-3; A new problem ("coordinate transformation to the locally-inertial frame") is added to note-7 and a couple of others are reformulated.
• 11/10: I had made a misprint at the lecture:  g_adε^d_,b  is actually not equal  ε_a,b  but is rather equal  g_adε^d_,b=ε_a,b -g_ad,bε^d .
  However the resulting expression was correct, g_adε^d_,b + g_bdε^d_,a + g_ab,cε^c = ε_a;b + ε_b;a .
• 10/10: The prerequisite for the examination is a (copy of a) set of solutions—one free chosen exercise from each note—sent to me before the examination via en email with the subject GTR18.
• 27/9: Note 4: one problem is slightly reformulated; another one is dropped as a duplicate; and a hint is added to the problem about rays of light in polar coordinates.
• 19/9: Note 3: a hint is added to the problem about transformation rules for the Christoffel symbols.

Schedule:

Plan:

1. [pdf] Introduction. Special relativity. Relativistic kinematics. (19.08)
   Inertial frame of reference; Lorentz transformation; Special principle of relativity; Space-time interval; Metric; Euclidean and pseudo-Euclidean metric; Least action principle; Relativistic action for a massive body; Canonical momentum; Relativistic momentum and energy of a massive body.
2. [pdf] Equivalence principle. (19.08)
   Inertial forces; Equivalence principle; Flat and curved frames; Geometric theory of gravitation; General principle of relativity.
3. [pdf] Vectors in curvilinear coordinates. Covariant differentiation. (19.08)
   Covariant and contravariant vectors; Tensors; Implicit summation; Metric tensor; Covariant differential; Christoffel symbols;
4. [pdf] Motion of free particles in curved spaces. Geodesics. (19.08)
   General covariance; Geodesic; Gravitational potential; Null geodesic.
5. [pdf] Electrodynamics in curved spaces. (19.08.19)
   Principle of stationary action; Action of charges in electromagentic field; Covariant volume element; Action of electromagentic field with currents; Lorentz force equation; Maxwell equations.
6. [pdf] Action of matter in general relativity. (19.08.19)
   Action of matter in general relativity; Stress-energy-momentum tensor of matter; Stress-energy-momentum conservation law;
7. [pdf] Action of gravitation. Gravitational field equation. (19.08.19)
   Riemann tensor; Ricci tensor; Ricci scalar; Hilbert-Einstein action; Einstein equation.
8. [pdf] Newtonian limit of general relativity. Gravitational waves. (19/08/19)
   Newtonian limit; Gravitational waves.
   
9. [pdf] Schwarzschild solution. Motion in the Schwarzschild metric. (19.08)
   Schwarzschild coordinates; Schwarzschild solution.
10. [pdf] Classical tests of GTR (19.08)
    Anomalous perihelion shift; Bending of light; Gravitational red-shift.
11. [pdf] Radial fall in the Schwarzschild metric. (19.08.19)
    Lemaitre coordinates. Coordinate singularity. Gravitational singularity. Event horizons. Black holes.
12. [pdf] Geometry in a homogeneous and isotropic universe. Friedmann equation. (19.08.19)
    Space with constant curvature; Friedmann-(Lemaitre-Robertson-Walker) metric; Friedmann equation.
    
13. [pdf] Solutions to Friedman equation. Big Bang. Cosmological red-shift. Hubble constant. (19.08.19)
    Big Bang; Hubble constant; cosmological red-shift; cosmological constant; radiation-, matter-, and dark-energy-dominated universe.
14. [pdf] Proper time and length intervals in general relativity. (19.08)
    

1. Gerard 't Hooft, Introduction to General Relativity [500kB PDF file], Rinton Press, Inc., Princeton NJ, ISBN 1-58949-000-2.
2. A. Einstein, Relativity: The Special and General Theory [600kB PDF file];
3. Warren Siegel, Fields, http://arxiv.org/abs/hep-th/9912205, [1215kB PDF file].