Question

A source of light moves with velocity v relative to a receiver. Demonstrate that for v << c the fractional variation of frequency of light is defined by Eq. (5.6a).

Answer 1

When v << c time dilation effect of relativity can be neglected (i.e. t’ ≈ t ) and we can use time in the reference frame fixed to the observer. Suppose the source emits short pulses with intervals T₀ . Then in the reference frame fixed to the receiver the distance between two seccessive pulses λ = c T₀- v_τ T₀ is when measured along the observation line. Here v_τ cos θ is the projection of the source velocity on the observation line. The frequency of the pulses received by the observer is

(The formula is accurate to first order only)

The frequency increases when the source is moving towards the observer.