D7    Problems and Suggestions


  1. A fighter jet flies with 1000 m/s and shoots a projectile off in its flight direction with a muzzle velocity of likewise 1000 m/s. Add these velocities ‘classically’ and ‘relativistically’.
  2. Derive the Lorentz transformations for t' and x' algebraically from those for t and x, which we deduced first in D2 and then a second time in D3.
    Why is that actually unnecessary?
  3. Show algebraically that the Lorentz transformations from the non-prime to the prime system and vice-versa mutually cancel each other.
  4. Derive the Lorentz transformations for t' and x' from an Epstein diagram!
  5. Examine our formula from D4 for relativistic velocity addition. Are v = 0.5 • c, w = 0.8 • c, u = -0.5 • c and c all parallel velocities. Form
    a) v ⊕ v     b) v ⊕ w     c) v ⊕ c    d) c ⊕ w    e) c ⊕ c     f) c ⊕ u     g) u ⊕ -c     h) w ⊕ w
  6. How quickly is a star approaching us, given that the Hα line for an excited hydrogen atom is not found to be 656 nm as in a laboratory on earth, but rather at 649 nm? (The emission line is thus a little ‘blue’ shifted).
  7. How fast does one have to approach a traffic light, so that one sees the red light (wavelength of 620 nm) as green (wavelength of 520 nm)?
  8. A laser produces light at 632 nm wavelength. What wavelength do we measure, if this laser is at the tail of a UFO, which is moving away from us
    at 0.5 • c?
  9. Why does the rotation of a star show up as a widening of its spectral lines?
  10. Why do spectral lines widen, if an emitting gas exhibits a high temperature and high pressure? (The effects in problems 9 and 10 express themselves quantitatively differently and can be partly computationally separated, if they arise superimposed.)
  11. Derive the optical Doppler formula from the acoustic Doppler formulas for the wavelengths, by additionally considering length contraction!
  12. Read pages 140-142 as well as 146-149 from Einstein's original publication in [09]. 
  13. In addition to our 2 coordinate systems (black and red with the points A and B) there is a ‘middle’ system C, in which A moves equally fast to the left as B to the right. Ascertain, in general, the velocity of this middle system C for both A and B. Without STR the answer would naturally be  v/2 and -v/2…
    The existence of this middle system C, by the way, provides a beautiful argument for the fact that the relative velocities of B for A and of A for B must be quantitatively equal: From the point of view of C the situation is perfectly symmetrical!
  14. How do the radar speed measurements of the traffic police function? Consider the whole from the frame of the reflecting car.
  15. What velocity w is 'half of the speed v', so that  w ⊕ w = v  ? Read the short paper
  16. Do add velocities and Epstein-angles by simple ruler-and-compass constructions (since september 2012 !). Read the following short paper:


Albert Einstein’s Swiss Military "Service Booklet". He was declared exempt from service when found to have flat and sweaty feet - certainly to his great satisfaction.


Einstein fled from Munich not only because he found the prevailing spirit at the Luitpold Gymnasium unbearable. He also feared being conscripted by the military - an idea that certainly filled him with horror and persuaded him to abandon German citizenship. He deeply hated everything militaristic, and repeatedly supported conscientious objectors and lobbied his entire life for disarmament and the strengthening of supranational institutions.

“This topic brings me to that worst outcrop of the herd nature, the military system, which I abhor. That a man can take pleasure in marching in formation to the strains of a band is enough to make me despise him. He has only been given his big brain by mistake; a backbone was all he needed. This plague-spot of civilization ought to be abolished with all possible speed.”       [20-6]

Yet Einstein was no naive pacifist. In view of what was brewing in the early thirties in Germany on his Berlin doorstep, he forsook his previous strictly pacifist line and wrote to a Belgian military objector:

“Organized power can be opposed only by organized power. Much as I regret this, there is no other way.”      [17-168]