Announcements:

• (15/1/2021) The "spørgetime" is provisionally planned for Thursday 21/1 09:00 via zoom.
• (5/1/2021) Yet another example of examination set is put online (link above).
• (18/12/2020) Heeding the popular demand I have added a few hints and a solution to the exercise 11.1 (that is, note 11, exercise 1).
• The Formulae Collection is online on Overleaf at this address: [https://www.overleaf.com/9935614161bmxzwndwzjjj]. Go ahead and edit it. Remember that from now on you are responsible for the correctness of the formulae.
• On Wednesday 9/12 we meet virtually at Zoom (the usual time, 13:15) at this link: [https://aarhusuniversity.zoom.us/j/4106164110]. It seems that you can join the meeting using your browser if you don't wish to install the Zoom-program in your box.
• (13/10) Week 42 is "skemafri uge" (no teaching).
• (23/9) Added an exercise to note-3: calculate the covariant derivative of the Kronecker delta, Dδ^a_b.
• (23/9) You have to solve one exercise of your choice from every week-note and send your solutions to me by email (fedorov@phys.au.dk) with the subject GTR20 no later than the day before the examination.
• (31/8) First meeting: September 2nd.

Schedule:

Plan:

1. [pdf] Introduction. Special relativity. Relativistic kinematics. (14.01.21)
   Event; Distance; Space; Metric; Space-time; Inertial frame of reference; Space-time interval; Euclidean and pseudo-Euclidean metric; Lorentz transformation; Special principle of relativity; Least action principle; Relativistic action of a massive body; Canonical momentum; Relativistic momentum and energy of a massive body.
2. [pdf] Equivalence principle. (14.01.21)
   Inertial forces; Equivalence principle; Gravitation as geometry of spacetime; Curved spacetime; General principle of relativity; Principle of general covariance.
3. [pdf] Vectors in curvilinear coordinates. Covariant differentiation. (14.01.21)
   Covariant and contravariant vectors; Tensors; Implicit summation; Metric tensor; Covariant differential; Christoffel symbols;
4. [pdf] Motion of free particles in curved spaces. Geodesics. (14.01.21)
   General covariance; Geodesic; Gravitational potential; Null geodesic.
5. [pdf] Electrodynamics in curved spaces. (14.01.21)
   Principle of stationary action; Action of charges in electromagentic field; Covariant volume element; Action of electromagentic field; Lorentz force equation in gravitational fields; Maxwell equations in gravitational fields.
6. [pdf] Action of matter in general relativity. (17.01.21)
   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. (14.01.21)
   Riemann tensor; Ricci tensor; Ricci scalar; Einstein equation; Hilbert-Einstein action;
8. [pdf] Newtonian limit of general relativity. Gravitational waves. (14/01/21)
   Newtonian limit; Linearized gravity; Gravitational waves.
   
9. [pdf] Schwarzschild solution. Motion in the Schwarzschild metric. (17/01/21)
   Schwarzschild coordinates; Schwarzschild solution; Schwarzschild radius.
10. [pdf] Classical tests of GTR (17/01/21)
    Anomalous perihelion shift; Bending of light; Gravitational red-shift.
11. [pdf] Radial fall in the Schwarzschild metric. (17/01/21)
    Lemaitre coordinates. Coordinate singularity. Gravitational singularity. Event horizons. Black holes.
12. [pdf] Geometry in a homogeneous and isotropic universe. Friedmann equation. (14/01/21)
    Friedmann universe; Friedmann-(Lemaitre-Robertson-Walker) metric; Friedmann equation.
    
13. [pdf] Solutions to Friedman equation. Big Bang. Cosmological red-shift. Hubble constant. (14/01/21)
    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. (14.01)
    

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];