Transverse Relativistic Doppler Effect

A qualitative difference between classical mechanics and relativity is the existence of the transverse Doppler effect in relativity (when light propagates perpendicular to its source in the observer’s frame). Calculate the frequency of the photon ω' in the observer’s frame in terms of its frequency ω in the rest frame.

SOLUTION

Intuitively it is clear that the effect is due to a time dilation t' = γt so ω' = ω/γ (ω α 1/t). More formally, we can use the energy and momentum transformation, but it is more convenient to introduce a 4-vector k^μ (k is a wave vector)

and consider its transformation from the rest frame K to the observer’s frame K'

(1)

where k' ⁰ = ω'/c, k'¹ = k' cos α = 0 (α = π/2). Substituting this into (1), we find

so

Note that the transverse Doppler effect gives only a second order correction to the frequency

where β << 1, whereas the longitudinal Doppler effect yields a first order correction.