# Why Can't LIGO Get Its Story Straight?

What, in fact if anything, does LIGO and VIRGO measure? The commonly accepted answer is that the phase difference between the two orthogonal arms of the interferometer is measured; the phase shift being due to the jiggling of the suspended mirrors due to a passing gravitational wave.

Thorne asks:

"Does the wavelength of the light in the gravitational wave get stretched and squeezed in the same manner as the mirrors move back and forth?...The answer is "no", the spacetime curvature influences the light in a different manner than it influences the mirror separations...the influence on light is negligible, and it is only the mirrors that get moved back and forth and the light's wavelength does not get changed at all..."

In an influential paper published in the *American Journal of Physics, *entitled "If light waves are stretched by gravitational waves, how can we use light as a ruler to detect gravitational waves", Saulson makes the contrary claim that

"Light waves do indeed stretch as the gravitational wave stretches the interferometer arm..." He also affirms that "Gravitational wave or no, light travels through the arm at the speed of light, *c. *The physically observable meaning of the stretching of the space [not spacetime!] is that light in it has to cover extra distance, and so will arrive late. And, since each successive 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 arm at time *t=T* finally makes it back to the beam splitter. This phase shift is observable, and it builds up over the storage time *2L/c* of the interferometer arm [*L*]."

So the wavelength does change as light is reflected from the mirror. The frequency between the source and receiver, which are comoving, can't change for if it did so, it would cause a break in the wavetrain due to a build-up or depletion of wave crests at the detector.

Saulson recognizes this when he says that "The laser has been steadily pumping out wave crests every *1/f* seconds", where *f* is the frequency of the laser. How then can the wavelength change if the waves "travel through the arm at the speed of light". Recall that the speed of light is the product of the frequency of light and its wavelength, *c=f x l*, so that if the speed of light in the arm remains constant, and the laser constantly pumps out "wave crests" the wavelength can't change! And we are back to Thorne's assertion above. So LIGO and VIRGO measure absolutely nothing!

Now I'm not saying that the two-way (back-and-forth) speed of light is not *c, *but what I am saying is that the speed along the forward and backward tract in the interferometer arm cannot be the constant speed of light. The is evident in the original description of the Michelson-Morley interferometer where the time of the outward journey is *L/(c-v)* and the backward journey takes *L/(c+v) *seconds, where v is the speed of the ether drift. If the speed of light is constant and isotropic in all directions, as Einstein would have us believe, then the backward journey has a speed, c+v, which is greater than that of light!

Essen queried this and was subsequently forced to take early retirement at NPL. His query was that since

"Light is always propagated in empty space with a definite velocity c, which is independent of the state of motion of the emitting body", according to Einstein,

"It is not clear whether it applies to the velocity as a physical process or as a measured quantity depending on length and time; or whether it is relative to the emitting body or to an observer receiving the waves. It is usually interpreted as the velocity relative to an observer. Instead of obtaining the values of *c+v* or *c-v* for the velocity of light, where *v* is his own velocity relative to the source, an observer obtains the value *c*. Thus it appears that there should be no Doppler change in the frequency, and yet this effect is known to exist."

Essen then tells us that the way to measure the velocity of light is to send out a pulse of light "from one point to the other and back again, and the velocity is found from the time taken from the double journey. The value obtained in this way on classical theory is c(1-v^2/c^2)."

His conclusion is that "It is only the second-order term that is assumed not to be present." Yet, if we take the *arithmetic* average of the forward and backward speeds, we get

*(1/2)(c-v+c+v)=c*

which is what we would expect. Thus, even though the individual speeds are different that *c*, their arithmetic average is *c*. So much for the arithmetic average.

But, what is called for in the Michelson-Morley experiment is the *harmonic *average:

*2/{1/(c-v)+1/(c+v))}=c(1-v^2/c^2)*

which is what Essen claims! Harmonic means are always less than, or at most equal to, arithmetic means, and are usually used in averaging rates, like average travel speeds given a fixed duration because it gives equal weights to each entry while the arithmetic average tends to favor larger entries.

Let us return to the beginning of interferometer experiments, and in particular the Fizeau velocity drift experiment. Fizeau arranged to obtain the interference of two beams of light passing through columns of water moving in different directions. If *L *is the length of the tubes, and *v*, the water velocity then the two velocities through the water are *c/n+Kv* and *c/n-Kv, *where *n *is the index of refraction of water, and *K=1-1/n^2* is the Fizeau drag coefficient*. *The time difference of passage is

*dT=2L/[c/n-Kv]-2L/[c/n+Kv]=4LKv/[(c/n)^2-(Kv)^2].*

Notice that this is a first order effect because the time difference is linear in the velocity *v. I*t also depends on the fact that light is passing through another medium of index of refraction different from unity. The number of fringes is the time difference divided by the period, *T*, of the light vibration. Due to its smallness, the second term in the denominator is neglected.

A positive shift results because the wavelength changes according to Snell's law: the velocity v is proportional to the difference in wavelengths:

*1/l_1-1/l_2=4L(n^2-1)v/c l^2*

where *l* is the wavelength of the light used. Michelson and Morley repeated the experiment in 1886 and found no effect when they set the index of refraction *n=1* for air. From this they concluded that there should be no drift velocity, i.e. no relative motion between the earth and the ether, and consequently no change in wavelength, *l_1=l_2. *

Although Michelson had in mind passing light around a circuit because it didn't involve a 180 reflection at a mirror that produces standing waves, it was Sagnac who in 1913 showed that the speed of light is not constant for a rotating observer that is the crucial feature for obtaining an interference pattern. More than a half a century later, Silvertooth found an effect, that bears his name, in which there is a "detectable change in the wavelength of a laser beam in an optical apparatus fixed to the earth that is rotating with respect to the sidereal space [with respect to fixed stars] due to the earth's diurnal motion."

In Sagnac's case, the right-hand side of the above formula is proportional to the area enclosed multiplied by the frequency of rotation, *Aw*. This can be written as *Lv*, where *v=rw,*

the rotational velocity, at angular speed *w. *Silvertooth asked what would happen if we let r become increasing greater and w increasing smaller. In the limit the Sagnac interferometer should transform into a linear interferometer, and the effect interference pattern should persist.

Silvertooth seems to have measured a change in the period which measures the earth's speed "through the ether" of about 378 km/s in the direction of the constellation Leo. Surprisingly (at least to some) the cosmic microwave background radiation shows the earth to be moving in the same direction at 371 km/s. Uncannily, the dipole component lies in the ecliptic, while the quadrupole component is normal to it. Perhaps, as Marett claims, this is a consequence of more mundane "local" factors.