Derive the expression for the Doppler shift, ω' = γω(1 - (v/c)cosθ), by applying the Lorentz transformation to the momentum 4-vector of a photon.
Let reference frame K' move with velocity v with respect to reference frame K. In K a sinusoidal electromagnetic plane wave has an angular frequency ω and a wave vector k. Find ω' and k' in reference frame K'.
Reference frame K' moves with velocity vi with respect to reference frame K. An electromagnetic plane wave is observed in K propagating in a direction -i + j with frequency ν. Find the frequency and direction of propagation of the plane wave when it is observed in K'.
tanθ' = ky'/kx' = -1/[γ(β√2 + 1)] is the angle the wave vector makes with the x-axis in K'. θ' defines the direction of propagation of the plane wave in K'.
A source emits electromagnetic waves with frequency f into a 4π solid angle. What is the frequency f' of the waves observed by an observer moving with speed v in a circular orbit around the source?
A star is moving towards the earth at a speed of 3 * 106
m/s. This speed was determined by observing that the wavelength of a
particular spectral line was shifted by 1 nm.
(a) What is the wavelength of the spectral line that must have been used for this measurement?
(b) Was the shift towards shorter or longer wavelengths?
(c) When observing a different star from earth, the frequency for that particular line is observed to have increased by 80%. How fast is that star moving relative to earth?
(d) Is it moving towards or away from Earth?
Light from Sirius A shows a shift in wavelength due to the influence of a
companion star, Sirius B, with a period of 50 years.
(a) If the Balmer α line of hydrogen (λrest = 656 nanometers) exhibits a maximum Doppler shift of 0.025 nm, what is the orbital velocity of Sirius A?
(b) Given this orbital velocity, what is the radius of Sirius's orbit, if one assumes a circular orbit?
(c) What is the combined mass of Sirius A and Sirius B?
A, located on earth, signals with a laser pulse every six minutes. B is on a space station that is stationary with respect to earth. C is in a rocket traveling from A to B with a velocity v = 0.6c relative to A.
(a) At what intervals does B receive signals from A?
(b) At what intervals does C receive signals from A?
(c) If C reflects light pulses back to A, at what intervals does A receive these pulses?
The relativistic Doppler effect is the change in frequency f of light, caused
by relative motion of the source and the observer. Assume that the source
and the observer are moving away from
each other with a relative velocity v. Consider the problem in the
reference frame of the source. Let fs be the frequency of the
wave the source emits. Suppose one wave front arrives at the observer.
(a) What is the distance of the next wave front away from him?
(b) What is the time t between crest (of the wave front) arrivals at the observer?
(c) Due to relativistic effect, what will the observer measure this time t0 to be?
(d) What is the corresponding observed frequency f0?
An excited nucleus of 57Fe formed by the radioactive decay of
57Co emits a gamma ray of 1.44 *104 eV. In the process,
there is conservation of energy and m0c2 = γma0c2
+ hf, where m0c2 is the initial mass of the nucleus
and ma0c2 is its mass after the emission of the gamma ray.
There is also conservation of momentum, hf/c = γma0u, where u is the
recoil velocity of the iron nucleus. The energy released by the reaction
is Er = (m0 - ma0)c2.
(a) Show that hf = Er(m0 + ma0)/(2m0) = (1 - Er/(2m0c2))Er.
Thus hf < Er: part of Er goes to the photon, and the other part supplies kinetic energy to the recoiling nucleus.
(b) Set m0 = 57*1.7*10-27 kg, and show that Er/(2m0c2)) ~ 1.3*10-7.
Thus the fraction of the available energy Er that appears as recoil is small.
(c) Moessbauer discovered in 1958 that, with solid iron, a significant fraction of the atoms recoil as if they were locked rigidly to the rest of the solid. This is the Moessbauer effect. If the sample has a mass of 1 gram, by what fraction is the gamma ray energy shifted in the recoil process?
(d) A sample of normal 57Fe absorbs gamma rays of 14.4 keV by the inverse recoilless process much more strongly than it absorbs gamma rays of any nearby energy. The excited nuclei thus formed reemit 14.4 keV radiation in random directions some time later. This is resonant scattering. If a sample of activated 57Fe moves in the direction of a sample of normal 57Fe, what must be the value of the velocity v that will shift the frequency of the gamma rays, as seen by the normal nuclei, by 3 parts in 1013? This is one line width.
(e) A Doppler shift in the gamma ray results in a much lower absorption by a nucleus if the shift is of the order of one line width or more. What happens to the counting rate of a gamma-ray detector placed behind the sample of normal 57Fe when the source of activated Fe moves
(i) toward the normal 57Fe,
(ii) away from it?
(f) If a 14.4 keV gamma ray travels 22.5 meters vertically upward, by what fraction will its energy decrease?
[Gravitation redshift, a thought experiment: Suppose a particle of rest mass m is dropped from the top of a tower and falls freely with acceleration g. It reaches the ground with a velocity v = (2gh)1/2, so its total energy E, as measured by an observer at the foot of the tower is E = mc2 + ½mv2 + O(v4) = mc2 + mgh + O(v4).
Suppose an observer has some magical method of converting all this energy into a photon of the same energy. Upon its arrival at the top of the tower with energy E the photon is again magically changed into a particle of rest mass m' = E'/c. Energy conservation requires that m' = m.
Therefore E'/E = mc2/(mc2 + mgh + O(v4)) = (1 + gh/c2 + O(v4))-1 = 1 - gh/c2 + O(v4).]
(g) A normal 57Fe absorber located at this height must move in what direction and at what speed in order for resonant scattering to occur?