Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 72 / 2 / i / Solution

Follow successive photons from the same comoving source. Their comoving radial distance is fixed, soIt follows that $d{t}_e/d{t}_0=a(t_e)/a(t_0)=1/(1+z)$. Taking the logarithmic derivative of $1+z=a(t_0)/a(t_e)$, with both endpoints allowed to vary, givesTherefore the observer-time redshift drift isHolding the emission time fixed would omit the second term and would not compare successive observations of the same source.