Past exam of the philosophy of space-time and philosophy of quantum mechanics paper of Parts A and B of the physics course of the University of Oxford / 2019 / 4 / Solution

The ideal slow clock transport synchronization procedure reproduces Einstein synchronisation in an inertial frame, as a limiting result about proper time. It does not determine distant simultaneity without assumptions about clocks and their dynamics. Begin with identical ideal clocks together at $x=0$, with matching rates and readings. Leave one there and transport the other to a laboratory position $L$ at small constant speed $u$, neglecting gravity. Use Einstein-synchronised laboratory coordinates to calculate the result, rather than assuming that the transported clock already displays coordinate time.

The journey takes laboratory time $T=L/u$. The transported clock's proper time is $\tau =T\sqrt{1-u^2/c^2}$, giving the lagConsequentlyThe small time dilation accumulates over a long journey, but its quadratic dependence on $u$ dominates the journey's inverse dependence on $u$. Saying that the journey is arbitrarily long is therefore not an objection to this limit. For a nonuniform journey with $|v|\le u$, $\delta =\int (1-\sqrt{1-v^2/c^2})dt$; a simple monotonic path of fixed length has the same vanishing bound. Acceleration stages can be arranged with vanishing displacement and proper time deficit, assuming the clock hypothesis.^[ref]

This procedure establishes synchronization operationally for stationary laboratory clocks, once ideal-clock behaviour has empirical support. It does not show that every actual device is insensitive to transport: thermal shifts, acceleration-dependent mechanisms or differing gravitational potentials can spoil the idealization. Nor does the same construction define global synchronization for rotating laboratories, where path dependence is significant.

The conventionality question is separate. Resynchronize the same laboratory with $t_{\kappa }=t+\kappa x/c$, where $|\kappa |<1$. At the arrival event the transported clock approaches the reading $t$, whereas a stationary clock displaying $t_{\kappa }$ reads $t+\kappa L/c$. Slow transport has not eliminated that difference. The transformed metric tensor has an off-diagonal term, and the proper time calculation still gives the same physical clock reading: changing labels cannot change its ticks. This is the invariance of reunion ages under clock resynchronization, applied to transport comparisons.

Slow clock transport synchronization is thus a useful physical realization of a preferred conventional coordinate choice given Lorentz-invariant dynamics and ideal clocks. It does not independently prove that the one-way speed of light must be isotropic in every admissible synchronization. It supplies a reproducible convention with a strong physical rationale, rather than a convention-free observation of an absolute present.