Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 37E / b / iii / Solution

Put $\Delta r=r_2-r_1$. Expanding the quotient $S/T$ consistently to first order givesThusBecause $\log (r_2/r_1)/(r_2-r_1)\to 1/r_2$ as $r_1\to r_2$,This local limit is the expected speed of a massless particle measured in a freely falling local inertial frame; the finite-interval effective velocity combines nonlocal length and clock comparisons.