The Doppler shift

Problem:

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.

Solution:

Problem:

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'.

Solution:

Problem:

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'.

Solution:

Problem:

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?

Solution:

Problem:

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?

Solution:

Problem:

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?

Solution:

Problem:

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.

image

(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?

Solution:

Problem:

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.image
(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?

Solution:

More Doppler effect problems