Research report | A Study of Relativistic Bounds on Clock Synchronisation on Earth
We investigate the extent to which a high-precision clock might be able to synchronise another to its own time, when they are part of a network that requires time-stamping events to a very high precision. Synchronisation at an ultra-fine level is strongly subject to the rules of relativity: clocks that are at rest in a single inertial frame can (in principle) synchronise each other to any accuracy required, whereas clocks at rest on Earth or on satellites are not inertial, and hence cannot necessarily synchronise each other to an arbitrary level of accuracy. Aside from the standard result that clocks over a wide area of Earth cannot agree on the timing of events on Earth to better than some tens of nanoseconds (a result which does not contradict the success of satellite-positioning technology), we discuss simultaneity in detail, and prove a related and new result for the extent to which two clocks at rest on Earth at the same height might agree on the meaning of "now". Although the answer requires no change in current technology, it must be understood in context. This report describes that context in detail.
The question of the extent and meaning of clock synchronisation is becoming pertinent as clocks become ever more accurate, and are used in networks that place increasing demands on the accuracy of their time-stamps. At such high levels of accuracy, relativity has an important role in timing analyses. Because both gravity and Earth’s rotation affect a clock’s tick rate relativistically, the extent to which one clock on Earth can synchronise with another, within the bounds imposed by relativity, must be addressed. The necessary analysis will not be straightforward and will require some caveats, because synchronisation can be problematic in relativity. This scenario differs from the well-understood synchronisation of clocks by, say, a GPS satellite, where the satellite is acting as a master clock. In this report, we are assuming that no GPS satellite is available to perform the synchronisation. We must be very careful to specify the frame in which the synchronisation is being performed.
Simultaneity is an old subject in relativity, but that does not mean that all questions pertaining to it have long ago been answered. It is well understood for inertial frames, but it is less straightforward for the non-inertial frames that are relevant to a rotating Earth. All analysis in this report uses orthodox relativity: that simultaneity is always defined in an inertial frame, and if no such frame is available, then the next best thing is used if possible: a series of “momentarily comoving inertial frames”.
In this report, the use of momentarily comoving inertial frames allows us to place a bound of about 10−19 seconds on the extent to which two clocks at rest on Earth can agree (even in principle) on the meaning of a “shared now”. This time interval is far smaller than current accuracies preservable within communications networks—meaning, things are okay for now: current technology has not advanced to the level where the demands of relativity become essential to it. Nonetheless, we must not infer that simultaneity can be defined over the whole of Earth to this level. In particular, clocks that are stationed over a wide area of Earth cannot agree on the timing of events on Earth to better than some tens of nanoseconds. This report discusses how these numbers are calculated, what they mean, and why they do not clash with the success of satellite-positioning technology.
This report contains several appendices that cover well-known concepts in relativistic timing that I think are not fully explained in the literature, and are more or less absent from relativity textbooks, since these books tend to avoid discussions of precise timing in a technological context.