We solved a number of problems involving the geometry of Minkowski space, how to use Lorentz invariant quantities in calculations, and how to apply Lorentz transformations in different setups.

We discussed the basic physical implications of the special theory of relativity such as length contraction, relativistic addition of velocities and relativistic doppler shift. The emphasis of the session was put to when the different formulas derived from the theory can be applied. Some emphasis was also put on the necessity of using Taylor expansions to numerically compute different quantities in the non-relativistic limit.

The first problem concerned how to use the formula for the transformation of accelerations and what acceleration an accelerating observer experiences for himself. The other problems concerned relativistic kinematics and its applications. We derived some formulae which can be used to simplify some expressions, such as γ = E/m, v = p/E, etc. We saw examples of when problems can be easily solved by squaring a vector to obtain Lorentz invariants and one example when this was not enough (so that we also had to use the conservation of the 0-component).

We concluded the part on particle kinematics and began the discussion on electrodynamics within the theory of special relativity. In the particle kinematics part, we studied how to compute threshold energies and how to simplify calculations using Lorentz invariants. We also derived the Schwarz inequality in Minkowski space [x⋅y ≥ √(x

The part on special relativity was conclueded. We studied how electromagnetic fields transform from one inertial framt to another by explicitly performing a Lorentz boost in the x-direction. The result was used to solve some problems involving the generation of electromagnetic fields in different inertial frames as well as the movement of charged particles in electromagnetic fields. In addition, we also derived the Lorentz force law from the least action principle.

We started the differential geometry part of the course by explicitly constructing atlases for the circle S

The Riemann curvature tensor was computed for some different setups. In particular, we computed the curvature tensor on the sphere in spherical coordinates. We also studied the action of the covariant derivative on some special tensors and proved that the parallell transport of a vector around some loop on the sphere rotates the vector by an angle equal to the solid angle enclosed by the loop. Finally, we studied a Lorentzian metric induced by an embedding into a pseudo-Riemannian space and derived some constants of motion for it.

The differential geometry part of the course was concluded with the calculation of the metric and the length of a geodesic curve on a hyperboloid embedded into Minkowski space and the treatment of how to compute Christoffel symbols and the Riemann curvature tensor given a general set of basis vectors. We then moved on to the third chapter of the course compendium where we showed that any metric on a two-dimensional manifold satisfies Einstein's equation in vacuum. Finally, an example of this was examined, the hyperboloid from the first problem of this session.

We considered some issues related to the Schwarzschild metric, which describes the space-time around a black hole. We computed the global and proper periods for circular orbits around a black hole and saw that there are no such non-spacelike orbits for any radius which is less than 3r

We solved some problems regarding cosmology and the Robertson-Walker metric. The geodesic equations for the flat RW-metric were derived and we explicitly constructed the set of points in causal contact with a given point. In the case of a curved RW-metric, we again derived the geodesic equations and then calculated the cosmological redshift. Furthermore, we did one problem regarding the differential equation for the

An extra session was inserted where possible solutions to the homework problems were presented.

The problem numbers refer to the 2005 edition of the course compendium. Problem numbers starting with "T" refer to problems in the compendium "Tensoranalys" by A. Ramgard

The problem numbers refer to the 2003 edition of the course compendium.