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

Table of contents
- Solution iii

Solution

Put

Δ r = r_{2} - r_{1}

. Expanding the quotient

S / T

consistently to first order gives

v_{eff} = 1 + \frac{M}{r _{2}} - \frac{M}{Δ r} lo g \frac{r _{2}}{r _{1}} .

Thus

v_{eff} = 1 + \frac{M}{r _{2}} - \frac{M}{r _{2} - r _{1}} lo g \frac{r _{2}}{r _{1}} .

Because

lo g (r_{2} / r_{1}) / (r_{2} - r_{1}) \to 1/ r_{2}

r_{1} \to r_{2}

,

r_{1} \to r_{2} lim v_{eff} = 1 .

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.

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