Problem Session Log

Log of old sessions:

2007:

Session 1 (18/1): a) We discussed Lorentz covariance of the wave equation, concept of a 4-vector in Minkowski space, invariance of the space-time interval under Lorentz transformations, definition of space-like, time-like and light-like 4-vectors, natural units, c=1.
The following problems were solved: 1.7, 1.8 in Mickelsson and 21 in Schutz.

Session 2 (22/1): Relativistic Doppler effect from covariance of the wave equation.
The following problems were solved: 1.1, 1.2, 1.10, 1.16, 1.20, 1.21

Session 3 (24/1): We discussed the Minkowski space and how Lorentz transformations are used to relate measurements of different observers.
The following problems were solved: 1.3

Session 4 (26/1): We discussed point particle mechanics and how to solve scattering problems in general.
The following problems were solved: 1.5, 1.26

Session 5 (30/1): We discussed Lorentz transformations in different directions and between several observers, using matrix notation. We solved scattering and decay problems.
The following problems were solved: 1.22, 1.27, 1.35, 1.36, 1.40

Session 6 (1/2): We computed Lorentz invariant quantities from the electromagnetic field tensor, with emphasis on index calculus. We performed Lorentz transformation of the electromagnetic field tensor and briefly discussed the twin paradox.
The following problems were solved: 1.54, 1.60, 1.61(a)

Session 7 (5/2): We discussed non-linear coordinate transformations in special relativity and computed how the metric changes when going from an inertial frame to a constantly accelerated system. We discussed variational calculus.

Session 8 (7/2): We derived the geodesic equation and applied it in the case of a free particle in Minkowski space.

Session 9 (12/2): We computed the shortest path on a sphere and discussed Minkowski space in non-linear coordinate systems.

Session 10 (15/2): We computed the metric, Christoffel symbols and curvature tensors on the 2-sphere in a systematic way.

Session 11 (19/2): We did a similar computation as last time, but for the paraboloid. We also pointed out the crucial difference between extrinsic and intrinsic curvature.

Session 12 (21/2): We discussed the Laplace operator in curvilinear coordinates and contracted indices of Christoffel symbols.
The following exercises were solved: 2.19, 2.20

Session 13 (26/2): We derived the geodesic equation, Christoffel symbols and Riemann curvature tensor for the Schwarzschild metric. The latter was computed using matrix notation.
The following exercises were solved: 3.3, 3.4

Session 14 (28/2): We looked at solutions of the Schwarzschild geodesic equations and computed the perihelium shift of a planet's orbit around a star.

Session 15 (2/3): We solved problems from old exams.


2006:

Session I (20/1):
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.
The following problems were solved: 1.1, 1.7, 1.8, 1.21, 1.12

Session II (24/1):
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 following problems were solved: 1.2, 1.10, 1.15, 1.16, 1.22, 1.20

Session III (27/1):
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).
The following problems were solved: 1.24, 1.35, 1.26, 1.27, 1.36

Session IV (3/2):
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 ≥ √(x2y2)], which is valid for all non-space-like vectors x and y. The part on electrodynamics mainly included the Lorentz invariants FμνFμν and εμνλωFμνFλω and free electromagnetic plane waves.
The following problems were solved: 1.40, 1.60, 1.66, 1.61, 1.37

Session V (6/2):
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.
The following problems were solved: 1.57, 1.58, 1.68

Session VI (17/2):
We started the differential geometry part of the course by explicitly constructing atlases for the circle S1 and the sphere S2. We moved on by computing the Christoffel symbols for the sphere in spherical coordinates. In order to do this, we used variational calculus to extremize the curve length and then compared the result with the geodesic equations. Finally, we solved some problems in the compendium regarding parallel transport on the sphere and the computation of a commutator of vector fields.
The following problems were solved: 2.2, 2.3, 2.7a

Session VII (20/2):
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 following problems were solved: 2.7b, T2.12, 2.12, 2.16

Session VIII (22/2):
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.
The following problems were solved: 2.15, 2.17, 3.3, 3.17

Session IX (24/2):
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 3rS/2, where rS is the Schwarzschild radius. We also computed the wavelength shift for a space-ship travelling towards a black hole. In addition, we also studied the wavelength shift when a signal is sent from outside of the event horizon to some observer inside the event horizon in Kruskal-Szekeres coordinates.
The following problems were solved: 3.12, 3.18

Session X (28/2):
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 r-coordinate for a light-like geodesic near a black hole and one problem regarding the electromagnetic field tensor in general relativity.
The following problems were solved: 3.27, 3.30, 3.23, 3.5

Extra session (3/3):
An extra session was inserted where possible solutions to the homework problems were presented.
The following problems were solved: HW1.1, HW1.2, HW1.3, HW1.4, HW2.1, HW2.2, HW2.3, HW2.4, HW3.1, HW3.2, HW3.3, HW4.1, HW4.2, HW4.3

2005:
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 et.al. and problem numbers starting with "HW" refer to the homework problems.

Session 1 (20/1): The geometry of Minkowski space and transformations between different inerial systems were studied. We discussed the breakdown of absolute simultaneity. An example of how Lorentz invariance can be used was also given.
The following problems were solved: 1.1, 1.5, 1.8, 1.11

Session 2 (25/1): We solved a number of problems concerning the relativistic addition of velocities, length contraction and time dilation. In the end, we also discussed and resolved the so-called "twin paradox" using the concept of proper time.
The following problems were solved: 1.16, 1.2, 1.10, 1.15, 1.22

Session 3 (27/1): The main part of the session was used for doing calculations using the conservation of 4-momenta and the Lorentz invariance of the Minkowski product. Some problems involving the proper time, including the twin paradox, were also studied.
The following problems were solved: 1.24, 1.26, 1.32, 1.35, 1.37

Session 4 (4/2): An additional problem using the conservation of 4-momenta as well as a problem using the spinorial representation of Lorentz transformations were solved. The rest of the session was used to do some fundamental problems on electrodynamics in special relativity as well as to go through the solutions to some of the exercises given in Homework 1.
The following problems were solved: 1.41, 1.50, 1.60, 1.64, 1.65, HW 1.1, HW 1.2, HW 1.4

Session 5 (9/2): The SR part of the course was concluded. A number of problems involving electromagnetic fields were solved and we derived how the electric and magnetic fields transform under Lorentz transformations. Finally, some of the Homework problems were solved.
The following problems were solved: 1.57, 1.58, 1.68, HW 1.3, HW 2.4d

Session 6 (18/2): Start of the differential geometry part. We constructed an explicit atlas for the torus. We derived the Christoffel symbols on the sphere induced by the embedding into R3. Finally, we performed a parallell transport on the sphere and saw how the commutator of two vector fields is computed.
The following problems were solved: 2.2, 2.7a

Session 7 (21/2): The meaning of parallell transport and the interpretation of Christoffel symbols as a measure of the change in the basis vectors were discussed. In addition, some problems concerning manifolds embedded in higher dimensional manifolds as well as some general differential geometry problems were solved.
The following problems were solved: 2.7b, T2.12, 2.16, 2.14

Session 8 (23/2): The final session on differential geometry, we discussed problems involving non-coordinate bases, where there is a basis of vectorfields which are not just the partial derivatives, but rather, vectorfields that do not commute. We also did one problem from the general relativity part of the course.
The following problems were solved: 2.12, 2.17, 3.7

Session 9 (25/2): We did some problems concerning the Schwarzschild metric and the Schwarzschild black hold (inlcuding the Kruskal-Szekeres coordinates). The problems included the derivation of geodesic equations and wavelength shift in general relativity. Apart from the problems below, the wavelength shift for a light signal sent from an observer outside the event horizon of a black hole to an observer inside the event horizon was discussed.
The following problems were solved: 3.12, 3.18, 3.11

Session 10 (28/2): The final problem session of the course. We studied two problems on Robertson-Walker space-times and one problem where we should derive an effective differential equation for the r-coordinate as a function of the proper time s in the Schwarzschild metric.
The following problems were solved: 3.27, 3.30, 3.23


2004:
The problem numbers refer to the 2003 edition of the course compendium.
Session 1 (23/1): 1.1, 1.2, 1.3, 1.6, 1.9, 1.15, 1.16, 1.18
Session 2 (30/1): 1.12, 1.22/1.23, 1.25, Twin "paradox", Addition of rapidities, 1.20, 1.21
Session 3 (4/2): 1.27, 1.28, 1.32, 1.33, 1.37, 1.35
Session 4 (6/2): 1.44, 1.46, 1.52, 1.61, 1.64
Session 5 (11/2): 1.57, 1.54, 1.65, 1.59, Homework problem 1.3, SR questionary
Session 6 (13/2): The torus (S1 x S1), 2.7a, Exer 2.1, Coordinate change in R2, Exer 2.7, Exer 2.2, Exer 2.5
Session 7 (18/2): Christoffel symbols on the sphere S2 induced from R3, 2.3, 2.13a, 2.9, Parallel transport coordinate independence
Session 8 (20/2): 2.7b, 2.14, The metric tensor, Metric after coordinate change in R2, 2.17
Session 9 (25/2): Homework problem 3.4, 3.10/3.11, 3.17, 3.27
Session 10 (27/2): Homework problem 3.3, Exer 3.5, Kruskal-Szekeres coordinates and Wavelength shift in a black hole, 3.9, Homework problem 3.1