# Foundations and Applications of Special Relativity

## Problem Sheets summer 2021

### 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

1. Galilean invariance of equations of motion for n gravitating point masses.
2. Newtonian equations of motion in accelerated reference frames.
3. The euclidean group of motions.
4. The CMB background and the principle of relativity.

• Sheet 2 (43): problems and solutions

1. Boost invariance of 2-body equations and non-invariance of Lagrangian.
2. How does a moving cube look like?
3. Rotational symmetry of simple equations for vector fields.

• Sheet 3 (44): problems and solutions

1. Boost and rotations in matrix-representation.
2. Polar decomposition.
3. Once more: Rotational symmetry of simple equations for vector fields.
• Sheet 4 (45): problems and solutions

1. An apparent paraxox concerning Lorentz contraction
2. Relativitsic velocity addition explains Fresnel drag including Zeeman correction.
4. Robb's Theorem.
• Sheet 5 (46): problems and solutions

1. An apparent paraxox concerning Lorentz contraction
2. Relativitsic velocity addition explains Fresnel drag including Zeeman correction.
4. Robb's Theorem.
• Sheet 6 (47): problems and solutions

1. Cauchy-Schwarz type inequalities in Minkoeski space.
2. Surjective maps preserving non-degenerate inner products are linear.
• Sheet 7 (48): problems and solutions

1. Characterisations of semi-direct group products.
2. Sagnac-type arrival-time differences of light signals for rotating observers
• Sheet 8 (49): problems and solutions

1. Collision between massless and massice particle (Compton-Effect).
2. Lorentz-Transforations as exponentials of antisymmetric endomorphisms.
• Sheet 9 (50): problems and solutions

1. Scattering A+B -> C. C cannot be massless unless A and B are.
2. Energy of 2-particle system in centre-of-mass and laboratory frame.
3. Decay P_0 - > P_1 + P_2. Energies and velocities of P_1 and P_2 in rest frame of P_0.
4. Like 3.), now asking for modulus of relative velocity between P_1 and P_2 judged from either P_1 or P_2
5. Determination of motion of a point particle in a constant electric field initially at rest.
• Sheet 10 (02): problems and solutions

1. The relativistic rocket-eqtaion (i.e. the special-relativistic Tsiolkovsky equation).
2. 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).
3. Modulus of relative velocities expresses invariantly and in 3-vector form.
4. Determination of most general motion with "constant acceleration" in geometric terms.
5. Coordinate expressions of "constant-acceleration".
• Sheet 11 (03): problems and solutions

1. Boosted Coulomb field and potential.
2. Neccessary and sufficient condition for the diagonalisability of an energy-momentum tensor.
3. Energy-momentum tensor of a plane wave.
4. The volume 4-form ε on Minkowski vector space; all bilinear contractions of ε; Hodge duality.
5. Coordinate-free characterisation of electric-magnetic-decomposition of a two-form.
6. Classification of polynomial invariants built from a two-form; duality transformations and invariance of energy-momentum tensor.
• Sheet 12 (04): problems and solutions

1. Distributional vector- and tensor fields for electric-current-density and energy-momentum-(current)-density of a point particle; divergncelessness in both cases.
2. 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.
3. Lorentz-Dirac equation as special-relativistic generalisation of Abraham-Lorentz equation. Solutions of Lorentz-Dirac equation equation in absence of extewrnal fields.