Problem sheets
The information "XX CW" in brackets refers to the XXth calendar week in which the corresponding sheet will be discussed.
- Sheet 1 (42): problems and solutions
- Galilean invariance of equations of motion for n gravitating point masses.
- Newtonian equations of motion in accelerated reference frames.
- The euclidean group of motions.
- The CMB background and the principle of relativity.
- Sheet 2 (43): problems and solutions
- Boost invariance of 2-body equations and non-invariance of Lagrangian.
- How does a moving cube look like?
- Rotational symmetry of simple equations for vector fields.
- Sheet 3 (44): problems and solutions
- Boost and rotations in matrix-representation.
- Polar decomposition.
- Once more: Rotational symmetry of simple equations for vector fields.
- Sheet 4 (45): problems and solutions
- An apparent paraxox concerning Lorentz contraction
- Relativitsic velocity addition explains Fresnel drag including Zeeman correction.
- General velocity addition.
- Robb's Theorem.
- Sheet 5 (46): problems and solutions
- An apparent paraxox concerning Lorentz contraction
- Relativitsic velocity addition explains Fresnel drag including Zeeman correction.
- General velocity addition.
- Robb's Theorem.
- Sheet 6 (47): problems and solutions
- Cauchy-Schwarz type inequalities in Minkoeski space.
- Surjective maps preserving non-degenerate inner products are linear.
- Sheet 7 (48): problems and solutions
- Characterisations of semi-direct group products.
- Sagnac-type arrival-time differences of light signals for rotating observers
- Sheet 8 (49): problems and solutions
- Collision between massless and massice particle (Compton-Effect).
- Lorentz-Transforations as exponentials of antisymmetric endomorphisms.
- Sheet 9 (50): problems and solutions
- Scattering A+B -> C. C cannot be massless unless A and B are.
- Energy of 2-particle system in centre-of-mass and laboratory frame.
- Decay P_0 - > P_1 + P_2. Energies and velocities of P_1 and P_2 in rest frame of P_0.
- Like 3.), now asking for modulus of relative velocity between P_1 and P_2 judged from either P_1 or P_2
- Determination of motion of a point particle in a constant electric field initially at rest.
- Sheet 10 (02): problems and solutions
- The relativistic rocket-eqtaion (i.e. the special-relativistic Tsiolkovsky equation).
- Determination of motion of a point particle in a constant electric field with initial velocity transversal to the electric field (genarlising problem 5 of previous sheet).
- Modulus of relative velocities expresses invariantly and in 3-vector form.
- Determination of most general motion with "constant acceleration" in geometric terms.
- Coordinate expressions of "constant-acceleration".
- Sheet 11 (03): problems and solutions
- Boosted Coulomb field and potential.
- Neccessary and sufficient condition for the diagonalisability of an energy-momentum tensor.
- Energy-momentum tensor of a plane wave.
- The volume 4-form ε on Minkowski vector space; all bilinear contractions of ε; Hodge duality.
- Coordinate-free characterisation of electric-magnetic-decomposition of a two-form.
- Classification of polynomial invariants built from a two-form; duality transformations and invariance of energy-momentum tensor.
- Sheet 12 (04): problems and solutions
- Distributional vector- and tensor fields for electric-current-density and energy-momentum-(current)-density of a point particle; divergncelessness in both cases.
- Action, Lagrangian and Euler-Lagrange equations for Maxwell fields with external currents. Action, Lagrangian and Euler-Lagrange equations for charged point-particle coupled to Maxwell field.
- Lorentz-Dirac equation as special-relativistic generalisation of Abraham-Lorentz equation. Solutions of Lorentz-Dirac equation equation in absence of extewrnal fields.
Supplementary notes and reading
- Mathematical background
- D. Giulini: Differentialgeometrie für Physiker (pdf)