Problem Session Log
Log of old sessions:
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.
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
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
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
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
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
The following problems were solved: 1.57, 1.58, 1.68, HW 1.3,
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
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
The following problems were solved: 3.27, 3.30, 3.23
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
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