 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-73/1/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Take

v > 0

for recession. By the definition of kinematic redshift,

1 + z_{kin} = ν_{e} / ν_{0}

. Inverting the supplied relativistic Doppler effect and expanding with

β = v / c

gives

1 + z_{kin} = \frac{1 + β}{1 - β} = (1 + \frac{1}{2} β + O (β^{2})) (1 + \frac{1}{2} β + O (β^{2})) = 1 + β + O (β^{2}) .

Thus

z_{kin} ≃ v / c for ∣ v ∣ ≪ c .

A negative radial velocity gives a blueshift at this order.

 New to topics? Read the docs here!