A star moves directly away from Earth at v = 0.6c. Observer A treats the star as moving away from a stationary Earth; Observer B treats Earth as moving away from a stationary star. What do the two observers calculate for the redshift factor f'/f?
ADifferent values — one is treating a moving source, the other a moving observer, which give different classical results
BThe same value — special relativity depends only on relative velocity, not on who is 'really' moving
CA is correct and B is wrong — the Earth is the inertial reference frame, so the star moves
DB is correct and A is wrong — stars are more massive and thus define the true rest frame
This symmetry is the key relativistic departure from classical Doppler. In classical physics, 'moving source' and 'moving observer' give different formulas because the medium (e.g., air for sound) defines a preferred frame. Light has no medium, so there is no preferred frame — only relative velocity matters. Both observers calculate f' = f√[(1−β)/(1+β)] with the same β = 0.6, getting the same result. Options C and D both invoke the idea of a 'true' rest frame, which special relativity explicitly rejects.
Question 2 Multiple Choice
A spacecraft moves exactly perpendicular to your line of sight at a relativistic speed. What does each theory predict for the frequency of light it emits toward you?
AClassical theory: zero shift. Relativistic theory: a blueshift, because the spacecraft is approaching along the curved path
BClassical theory: zero shift. Relativistic theory: a redshift, because the spacecraft's clock runs slow due to time dilation
CClassical theory: a redshift. Relativistic theory: zero shift, since motion is transverse
DBoth theories predict the same redshift, since transverse motion affects wavefront spacing equally
The classical Doppler formula involves the component of velocity along the line of sight; purely transverse motion contributes nothing, so the classical prediction is zero shift. But relativistically, the moving spacecraft's clock is time-dilated — it ticks more slowly by a factor γ. You therefore receive fewer wave cycles per second even though the spacecraft is momentarily moving perpendicular to your line of sight. The result is a redshift f' = f/γ — purely a time-dilation effect with no classical analogue. This transverse Doppler shift was a key experimental confirmation of special relativity.
Question 3 True / False
The relativistic Doppler formula predicts the same frequency shift regardless of whether you model the source as moving toward a stationary observer, or the observer as moving toward a stationary source at the same relative speed.
TTrue
FFalse
Answer: True
This symmetry is a direct consequence of the first postulate of special relativity: the laws of physics are the same in all inertial frames. Since only the relative velocity matters, the formula f' = f√[(1+β)/(1−β)] (for approach) is the same whether you assign the motion to the source or the observer. This contrasts with the classical Doppler effect for sound, where the medium defines a rest frame and 'moving source' versus 'moving observer' give different formulas even for the same relative speed.
Question 4 True / False
Because the relativistic Doppler formula includes a time-dilation correction, it predicts noticeably different results from the classical Doppler formula even at everyday speeds like v = 100 m/s.
TTrue
FFalse
Answer: False
At speeds much less than c (β << 1), the relativistic formula reduces to approximately the classical result. The time-dilation factor γ ≈ 1 + β²/2 + ..., which deviates from 1 only at second order in β. At v = 100 m/s, β ≈ 3×10⁻⁷, so the relativistic correction is of order β² ≈ 10⁻¹³ — completely undetectable. The differences become significant only at a substantial fraction of c, which is why relativistic Doppler matters for astrophysics (distant galaxies, particle accelerators) but not for everyday acoustics.
Question 5 Short Answer
What physical effect causes the transverse Doppler shift, and why does this effect have no analogue in classical Doppler theory?
Think about your answer, then reveal below.
Model answer: The transverse Doppler shift is caused entirely by time dilation: a source moving perpendicular to your line of sight has a clock that runs slow by a factor γ relative to your frame, so it emits fewer wave cycles per second as measured by you. The result is a pure redshift f' = f/γ. Classical Doppler theory has no such effect because it assumes absolute time — clocks tick at the same rate regardless of motion. The classical formula only depends on the component of velocity along the line of sight (wavefront compression or stretching); transverse motion contributes nothing classically. Time dilation is a purely relativistic phenomenon, so the transverse shift is a distinctly relativistic prediction.
The transverse Doppler effect was one of the earliest proposed tests of special relativity that could distinguish it from the classical theory, since the classical prediction is exactly zero while the relativistic prediction is a measurable redshift. It was confirmed experimentally by Ives and Stilwell in 1938 using fast-moving atoms, and later with great precision using atomic clocks in particle accelerators. It is now routinely observed whenever relativistic particles emit radiation perpendicular to their direction of travel.