This is also http://arxiv.org/pdf/1209.1876v1.pdf and see

http://arxiv.org/pdf/1203.5592v1.pdf and

http://arxiv.org/pdf/1209.1876v1.pdf

I personally found Berengut and. Flambaum's comment to be more cogent.

Measuring how Plank constant change is fundamentally meaningless. It is mathematically impossible to have a consistent theory of quantum mechanics if the Plank constant changes http://arxiv.org/abs/quant-ph/0301044 .

If the experiment results can prove Planck constant is changed with the gravitational potential, then it implies theory of quantum mechanics and special relativity are only valid in a reference frame with a constant background gravitational potential, such as the earth ground reference frame.

The original paper is http://arxiv.org/abs/1203.0102 Sorry for copying the wrong one before.

Here is my trouble with this. GPS clocks (including measurement errors) are only good to ~ 10^-13 (fractional frequency stability). The "signal" here comes from the eccentricity of the orbit. Since the clock depends on ~ GM / R, the change is GM del_R / R^2. The GPS eccentricity amounts to +- ~300 km at 23,600 km. This yields a geometrical sensitivity of ~ 4 x 10^-7 * GM. Compare with clocks on Earth, where delta R could be 5 km, R is ~ 6370 km, and the geometrical sensitivity is thus ~ 10^-7 * GM, or a factor of 4 worse. However, ground based clocks are better than 10^-15, or at least a factor of 100 better than the GPS clocks, so the ground based clocks should be a factor of 25 times more sensitive to any such effects, over all. The case for velocities is worse - the difference in velocity for the GPS satellites is ~ 100 m/sec, and the Earth's rotational velocity (proportional to cos(latitude) can be several 100 m/sec.

Either way, if the GPS satellites are sensitive to any radial changes in the fine structure constant (or if there is something else going on that mimics such an effect), I would expect to see clear, highly significant, differences between highly accurate clocks here on Earth. In the absence of reports of such clock anomalies, I do not have a lot of confidence in the Kentosh & Mohageg results being due to some sort of new physics.

Usually the clocks on the ground will not report any anomalies because the earth ground reference frame has a close to constant background gravitational potential. But if the experiment is designed to compare the clocks in the lowest spot and the highest spot of the earth, then it might be able to detect the anomalies.

Well, you need to be a little careful there. Take Boulder, Colorado versus Paris, France (both have good clocks). Boulder is 1.6 km above sea level, Paris is basically at sea level. However, Boulder is at a lower latitude, so it is further away from the Earth's center of Mass thanks to the equatorial bulge by ~ 2.3 km, and it is moving faster than Paris by ~ 50 m/sec. In General Relativity, these largely cancel and the important is the altitude above the (geopotential) sea level - see

http://www.scirp.org/journal/PaperDownload.aspx?paperID=4864 for the theory while

http://tf.boulder.nist.gov/general/pdf/1846.pdf discusses the height of the clocks in Boulder above the geoid. Using the latter, the signal in Boulder (for Beta = 0.007) is 1.3 x 10^-15, which the new Boulder Cesium fountain (at 3 x 10^-16 accuracy) should be able to see.

Another effect is the clock error due to variation in the Earth's potential in the Sun, which varies due to the eccentricity in the Earth's orbit by about 1.6 x 10^-10 over the course of a year. If there was a clock error of 0.007 times that, or 1.15 x 10^-12, that would be potentially visible in Earth-Mars ranging (although, unless it was properly parameterized, it might be absorbed into errors in the orbital elements of the Earth and Mars).

I agree that 2 to 3km difference in distance to the center of the earth should yield effects if the clocks are so accurate. When we talk about the gravitational potential, we have to consider the total gravitational potential on the earth ground, which is the cosmic background gravitational potential plus the local gravitational potential from the mass of the earth. A bigger Planck constant in a gravitational field can provide a microscopic explanation to gravitational redshift and time dilation.

Another effect occurred to me - a 1.5 x 10^-12 clock error would be ~ 50 microseconds integrated over a year, and that would be easily visible in the pulsar timing arrays looking for gravitational radiation. Given that the pulsar time scale is good to a few microseconds - see Figure 1 of http://arxiv.org/pdf/1210.2774v1.pdf - that says that pulsars can estimate this effect at least an order of magnitude better than the reported GPS signal, and it doesn't see it.

How do you know the gravitational potential on pulsars?

The potential on pulsars doesn't really matter. it is the Earth that is swinging in and out of the Sun's gravitational potential on a seasonal basis. All that is required of the pulsars is that they don't _all_ have the _same_ seasonal error, and in just such a form as to cancel the Earth's potential effect, which seems not just reasonable but unescapable to me.

Only the rest mass inside a sphere has contribution to the gravitational potential on the surface of the sphere, so it is important to define the maximum size of a gravitational field by comparing its gravitational field strength with the nearby gravitational field strength. For example the earth has a maximum sphere radius of about 2.6X10^8m where the gravitational strength is equals the one of the sun. The seasonal effects from the sun's gravitational field will change the size the maximum sphere a little bit, but don't have direct effect on the cosmic background gravitational potential on the earth within the maximum sphere. If you want to detect any effect within the maximum sphere, you can only count on the difference from the earth's gravitational potential. But if you can use a satellite has an orbital radius bigger than 2.6X10^8m, then you should be able to detect the effect from the sun's gravitational potential.

I am sorry, but I do not agree on with that at all. If you live near the sea coast, you might notice the tides. They (at least their solar part) are an effect of the varying solar potential, as are the solar perturbations on Earth orbiting satellites and the Moon.

The cosmic background gravitational potential is defined by the rest mass inside a cosmic sphere, the mass of the sun has contribution to it, but the mass of other stars have even more contribution to it, so the seasonal change of the sun's gravitational potential can be ignored. While talking about the gravitational force, the sun or the moon for sure cannot be ignored, but the gravitational force from other stars can be ignored in this case.

FYI - see http://physicsworld.com/cws/article/news/2012/mar/28/can-gps-find-variations-in-plancks-constant

Also see http://arxiv.org/abs/1203.5592 - its abstract states:

"In their Letter, Kentosh and Mohageg [Phys. Rev. Lett. 108, 110801 (2012)] seek to use data from clocks aboard global positioning system (GPS) satellites to place limits on local position invariance (LPI) violations of Planck's constant, h. It is the purpose of this comment to show that discussing limits on variation of dimensional constants (such as h) is not meaningful; that even within a correct framework it is not possible to extract limits on variation of fundamental constants from a single type of clock aboard GPS satellites; and to correct an important misconception in the authors' interpretation of previous Earth-based LPI experiments."

Also see http://arxiv-web3.library.cornell.edu/abs/1209.1876 - its abstract:

"We reply to a Comment [Berengut and Flambaum, Phys. Rev. Lett. 109, 068901 (2012), arXiv:1203.5592] on our recent Letter [Kentosh and Mohageg, Phys. Rev. Lett. 108, 110801 (2012), arXiv:1203.0102]. The Comment objects to our work partly because h has dimensions and therefore measuring any variation of it is meaningless. We reply that the relevance of dimensions in the study of fundamental constants has been the subject of some debate. We also note that our model differs from the one normally used to compare hydrogen masers and cryogenic optical resonators. Ours is based on a different treatment of proper length and energy to conform to general relativity. We also enclose in an Appendix our transmittal to the editors of Physical Review Letters."

I don't find where this dispute was ever resolved, but the discussion seems to have ended with "Reply to Comment..."

I actually disagree with this part of the comment by Berengut and Flambaum : "limits on variation of dimensional constants (such as h) is not meaningful"

With modern technology, to estimate variations in the fundamental constants is ultimately to compare clocks (or frequency standards, including spectral lines) and can, if you want, be expressed in terms of variation in some dimensionless number (such as the fine structure constant).

The study of the variation of G (AKA G dot determinations) has a long history, and was originally driven by the Dirac large numbers hypothesis (which predicted a change in G inversely proportional to the age of the universe). That hypothesis has actually been disproven experimentally, at least in its simplest form - see http://arxiv.org/abs/1108.0246 . What is really being compared here is the time rate of change of orbital period as measured against an atomic clock (in other words, do the planets keep the same time as an atomic clock). This can be expressed in terms of a dimensionless constant, but it is not actually necessary to do that to do the experimental tests.

There are also a long running set of efforts to find variations in the fine structure constant, which are also expressible in terms of variations of clocks or spectral lines, and I place the Kentosh and Mohageg work in that category. I think they are wrong, but observationally, not because their test is not meaningful.

Experimental results can give final judgment to any physics theory, no matter how beautiful it was. So I think the Kentosh and Mohageg work is worth to upgrade to an even more accurate level.