Appendix: Exercises... 419
Separation of variables yieldsa 3 −^2 (t)−a− 32 (t 0 )= 4 (t−t 0 )and, witht 0 =0,
a 3 (t)=a 3 ( 0 )
√
1 + 4 a 3 ( 0 )^2 t.
Still, the distortion relaxes to zero, but it is a non-exponential decay witha 3 (t)∼
t−^1 /^2 for long times.
Exercises Chapter 18
18.1 Doppler Effect(p.378)
Letω 0 be the circular frequency of the electromagnetic radiation in a system which
moves with velocityv=vexwith respect to the observer, who records the frequency
ω.Determine the Doppler-shiftδω=ω 0 −ωfor the two cases where the wave vector
of the radiation is parallel (longitudinal effect) and perpendicular (transverse effect)
to the velocity, respectively.
The Lorentz transformation rule (18.16) implies
(k′)x=γ(kx−βω), (k′)y=ky,(k′)z=kz,ω′=γ(ω−vkx). (A.16)Whenkis parallel or anti-parallel to the velocity, one haskx=±ω/cand conse-
quently
ω=ω 0√
1 −β^2 /( 1 ∓β).The longitudinal relative Doppler shift is
(ω−ω 0 )/ω 0 =√
1 −β^2 /( 1 ∓β)− 1 ≈±β+..., (A.17)where...stands for terms of second and higher order inβ=v/c.The±signs
indicate: the frequency is enhanced when the light source is approaching the observer,
when it is moving away, the frequency is lowered.
When the velocity is perpendicular to the direction of observation, one haskx=0.
Then the transverse relative Doppler shift is
(ω−ω 0 )/ω 0 =√
1 −β^2 ≈1
2
β^2 +..., (A.18)where now...stands for terms of fourth and higher order inβ.