# Franson’s speed of light correction disagrees with solar system tests

The Physics arXiv blog has an article entitled “Evidence of a Correction To the Speed of Light” which has gotten some attention (incuding mine). This article is based on a paper “Apparent correction to the speed of light in a gravitational potential”, by J.D. Franson, so I thought I would look at the original paper. Having done so, I just wanted to note here that I think this theory is not valid as it seems to be in disagreement with Solar System tests of General Relativity.

For a reference for such tests, I will use Clifford Will’s paper “The Confrontation between General Relativity and Experiment” in the Living Reviews of Relativity (specifically, the 11 June 2014 version, which I will refer to as Will, 2014).

Franson’s paper can be boiled down to his Equation 18, which predicts a change in the gravitational red shift by a factor of $\frac{9 \alpha}{64}$ for photons, where $\alpha$ is the fine structure constant ($\sim 1/137$), so the total correction is $\sim 1.08 \times 10^{-2}$. He claims this correction does not apply to neutrinos, thus providing an observable effect with Supernova 1987A. (I am not sure I agree with that conclusion but that point doesn’t matter for this note.) In his theory Equation 18 does apply to (low energy) photons, and so actually represents a change of the “the speed of light c as measured in a global reference frame,” which is his way of describing the gravitational red shift (see Equation 1). In other words, he predicts a 1% change in the gravitational red shift for low energy photons (where low energy is << 511 KeV, i.e., visible light, radio waves, etc.); this effect can be directly tested without depending on neutrino generating Supernova. In the low energy limit, the correction does not depend on the photon wavelength, and so radio observations should exhibit it too.

The gravitational red shift has indeed been tested, by a NASA experiment called Gravity Probe A, with an accuracy of a few parts in 10^-4 (see Figure 3 in WIll, 2014). This excludes the Franson correction, but it does not measure directly the apparent speed of photons. Since the Shapiro delay (the time delay of photons passing through a gravitational potential) also depends on the gravitational redshift, Franson's theory thus predicts a 1% change in that too, and that does depend on the motion of photons. Solar system measurements of the Shapiro delay provide estimates of the so called PPN $\gamma$ parameter, which is the ratio of the observed effect with that predicted by General Relativity (so that Franson is effectively predicting a $\gamma$ of ~ 0.989). This effect is much too large to be consistent with experiment (see Figure 5 in Will, 2014). Even the Viking relativity test I worked on with Irwin Shapiro in the 1970’s and 80’s (which had a relative error on PPN $\gamma$ of ~ 0.001) was good enough to rule out the Franson hypothesis.

So my conclusion is that the theory is simply wrong, and I don’t see any easy out for it. The Shapiro delay tests in the Solar System are particularly compelling, as they are based on photons moving in a gravitational potential, specifically the condition Franson’s Equation 18 is intended to address.