Slow Clock Transport

A much-discussed phenomenological scheme for clock synchronization is to define synchrony by the use of clocks transported between different locations. Arthur Eddington (1924) discusses this method of synchrony and notes that the indications of traveling clocks conform to Einstein simultaneity in the limiting case that the clocks move very slowly $(v ≪ c)$ with respect to the given inertial frame.

Synchronization by slow clock transport is problematic for foundational purposes in relativity. Slow clock transport relies on treating the high-level and somewhat ambiguous concept ‘clock’ as a primitive element, which is already unsatisfactory. In addition, the limiting case of slowly moving clocks is actually a derived result for what are called ideal clocks, i.e., devices that indicate elapsed proper time, a concept based in relativity theory.

Consider an ideal clock transported from the origin to the event $(t, x)$ at some speed $v = x / t$ ; see Fig.7.3. The amount $Δ t = t - τ$ by which the reading on the moving clock, its proper time $τ$ , differs from the coordinate time is $Δ t / t = 1 - \sqrt{1 - v^{2}}$ . The dotted red line in Fig.7.3 connects the clock at $(t, x)$ , showing time $τ$ , with the stationary clock at $(τ, 0)$ with the same reading. The slope of this line is

For small $v$ this may be approximated by $u ≃ v / 2$ .

Let us now construct a Lorentz frame ${t^{'}, x^{'}}$ moving to the right at speed $u$ . That is to say, in Minkowski space the origin of this frame moves along the $t^{'}$ - axis (red arrow in Fig.7.3,) tilted by angle $\tan α = u$ with respect to the stationary time-axis.

This construction shows that the inertial coordinates in terms of which the transported clock at $(t, x)$ and the clock at $(τ, 0)$ are simultaneous, is the inertial system moving to the right at the speed $u$ .

Hence, the stationary and transported clocks are not synchronized in terms of the stationary coordinates, but rather in terms of the coordinate system ${t^{'}, x^{'}}$ moving with about half the speed of the transported clock.

Comparing with (7.2) we conclude that slow clock time is an instance of Reichenbach time with parameter $κ = u$ .
The time needed for slow clock synchronization within a certain error $Δ t ≃ v \cdot x / 2 c^{2} < ε$ , increases as the inverse of this error and as the squared distance of the point one wants to synchronize.