Testing variations of Planck's constant with the GPS

Sketch of GPS satellite orbits.
Not to scale. Image source.

In a recent PRL, James Kentosh and Makan Mohageg of California State University at Northridge, report on a neat little analysis of GPS time correction data that they used to put bounds on a position dependence of Planck's constant:

Global Positioning System Test of the Local Position Invariance of Planck’s Constant
J. Kentosh and M. Mohageg
Phys. Rev. Lett. 108, 110801 (2012)

The GPS satellites orbit the Earth in about 12 hours in orbits at about 26,600 km altitude with very small eccentricity (ie the orbits are almost circular). Each satellite carries an atomic clock which is needed for the position measurements that are made, essentially, by triangulation with several satellites.

The clocks in the satellites move relative to ground-based clocks at roughly 4 km/s in a weaker gravitational field, so their proper times differ. Just how much they differ can be computed from their position data, using general relativity. Since the exact synchronization between the different clocks is crucial for the precision of position measurements, the GPS clock data that the satellites transmit is followed by a correction that synchronizes the clocks with each other and with the ground base. The correction data is updated every 15 minutes and is publicly available. It is this clock correction data that was used for the study.

Kentosh and Mohageg looked if the correction data has an altitude-dependent excess that is not explained by general relativistic corrections. For this, they selected seven of the satellites that had slightly more eccentric orbits and small standard deviations in the changes of the clock corrections. They used the clock corrections for a period somewhat longer than a year. They first extracted the relics of the corrections that are off the general relativistic predictions. The unfiltered result has several long-term oscillations, for example on annual and monthly periods, that are probably due to atmospheric effects whose exact origin is unknown. These oscillations however are not what they are looking for. In the end, they are left with average off-sets for each of the satellites. According to general relativity, one expects them to be statistically distributed around zero. The offsets they found are consistent with zero (ie consistent with general relativity) though the maximum is a little off.

So far so good. What it a little murky is the relation to a position-dependence of Planck's constant, ℏ. If Planck's constant was position-dependent and would change with altitude, then this would affect the proper time of the satellite clocks as follows. Assuming that the atomic transition energies remain constant, the number of oscillations per unit of time depends on ℏ. If ℏ was position-dependent, this would then add to the clock-correction. The problem with this argument is that ℏ itself is dimensionful and such a statement thus rests on the assumption that the units (for example for energy and time) themselves are not changing. (A point also made in this Physics Today article.)

Besides this, I can't make much sense of a position-dependent ℏ. I mean, depending on your choice of coordinate system, positions can change all the time or never. With some imagination, one could consider a dependence on a physical quantity, maybe the strength of the gravitational field. But then I'd expect there to be much stronger constraints from other astrophysical observations.

In any case, leaving aside that it's not so clear this study actually tests the constancy of ℏ, it's a great documentation for what can be done with publicly available data of technological gadgets we use every day.