How to Use A Michelson Interferometer to Measure the Newtonian Force

LIGO chose to measure the strength of gravitational waves by using a Michelson type interferometer while recognizing that the "ripples in the fabric of spacetime cause the frequency of the laser light to fluctuate "ever so slightly" as well as admitting to the fact that "a gravitational wave does stretch and squeeze the wavelengths of light in the arms." However, LIGO adds that "it turns out that it doesn't matter. What matters is how long the laser beam spends traveling in each arm. When a gravitational wave passes, it changes the length of the arms,which changes how far each laser beam needs to travel before being reunited with its partner beam."

How then can you get interference, either constructive or destructive if the wavelength of the laser beam changes, and the lengths of the arms also change? Are we back to Weber's dumbbells which does not use the principle of the interferometer? LIGO also insists that the speed of light will always be $c$---which is impossible if the wavelength changes. The frequency of the beam must always be the same in any Michelson interferometer. So what is going on? Why haven't we seen any interference patterns and fringes? Obviously, you can't if everything is affected by the passage of a gravitational wave.

I believe it is possible to measure the strength of the gravitational (or electrical) force using a Michelson interferometer. Gravity will affect the passage of a ray of light like a medium with an index of refraction, $n>1$. A Michelson interferometer is know to measure the index of refraction, say of air, by comparing one arm with nothing in it and the other arm filled with air. Instead of air, two masses can be placed in one arm, and this will change the index of refraction. If $n$ changes by an amount $\Delta n$, the path length, which was originally $2nL$ changes to $2\Delta n L$, $L$ is the length of the arm and the $2$ means that light passes twice because it is reflected from the far mirror.

Moving the masses together, the pattern will shift by an amount $\Delta n=\lambda/2L$, where $\lambda$ is the wavelength of the laser light that is used: it must be constant for fringes to occur, contrary to what LIGO claims. A shift of $m$ fringes will occur when $\Delta n=m\lambda/2L$.

Consider a refractive index of the form

$$n=1+\frac{GM}{r},$$

compatible with Kepler's laws. When the space between the two masses changes, the index of refraction will be changed to 

$$|\Delta n|=F_N|\Delta r|,$$

where $F_N=GM/r^2$, Newton's inverse square law. This can be related to the number of fringes shifted $m$ by

$$F_N|\Delta r|=m\lambda/2L.$$

The phase shift will be $\Delta\phi=c\Delta t/\lambda$ so that the strength of the force can be measured by the phase shift according to

$$F_N=\frac{m}{2L\Delta\phi}.$$

This is completely static as it must be, and shows that the gravitational force changes the index of refraction of the medium. Light cannot propagate at speed $c$ when $n\neq 1$.

A similar expression can be found in Saulson's article "If light waves are stretched by gravitational waves, how can be use light as a ruler to detect gravitational waves" [AJP 65 (1997) 501]. After a lengthy discussion about how gravitational waves do stretch and squeeze both the laser beam and interferometer arm, he merely claims that "the phase shift is related to the strength of the gravitational wave by the relation, $\Delta\phi=2\pi L h_0/\lambda$", without specifying what $\lambda$ is, and where $h_0$ is the amplitude of the wave.

This is in contradiction with his earlier statement that "there is no direct sense in which we observe the wavelength of light in the arms. Our observations are instead of the phase of light, that is, the arrival time of the wave crests." But,"[t]he distance between successive wave crests (i.e., the wavelength) is $\lambda=c\nu^{-1}$ throughout the arm." So on the one hand the distance between successive wave crests is the wavelength, but because gravitational waves squeeze and stretch the wavelength, it is not!

It is clear from Saulson's discussion that the speed of light cannot be constant, traveling at $1\pm h(t)$ times $c$ in each of the orthogonal arms. It is therefore difficult to see how "we can use the travel time of light as a reliable ruler under most conditions, in spite of the stretching of light that goes on when space expands." 

Other unsubstantiated statements are: "the gravitational wave causes no phase shift at the beam splitter immediately after its arrival." As we have seen above, the Michelson interferometer is completely stationary. It cannot distinguish at time $\tau$ when "[a]ll the other wave crests suddenly arrive at $t=\tau$ become farther from the beam splitter than they were before. Gravitational wave or no, light travels through the arm at the speed of light, $c$. The physically observable meaning of the stretching of space is that the light in it has to cover extra distance, and so will arrive late. And, since each successive wave crest has to cover a larger extra distance to make it back to the beam splitter, the total time delay (or phase shift) builds up steadily until all of the light that was in the interferometer are at $t=\tau$ finally makes it back to the beam splitter. This phase shift is observable, and it builds up over the storage time $\tau_s=2L/c$ of the interferometer arm."

Whatever type of interferometer accomplishes this, it is certainly not  of the Michelson type!