# Two “devil's advocate” questions related to LIGO measurement results interpretation

"If you would be a real seeker after truth, it is necessary that at least once in your life you doubt, as far as possible, all things."

Rene Descartes

Laymen like me typically refers to Wikipedia for basic definitions ...

"In physics, spacetime is any mathematical model that combines space and time into a single interwoven continuum."

http://en.wikipedia.org/wiki/Spacetime

"Gravitational waves are radiant energy produced in certain gravitational interactions. They are ripples in the curvature of spacetime that propagate as waves, travelling outward from the source."

http://en.wikipedia.org/wiki/Gravitational_wave#Effects_of_passing

and from the same source

"As a gravitational wave passes an observer, that observer will find spacetime distorted by the effects of strain. Distances between objects increase and decrease rhythmically as the wave passes, at a frequency corresponding to that of the wave."

Why it is assumed above that it is particular distance component (in its three dimensional space sense) of the spacetime continuum and not the time component of the spacetime continuum - which is changing under the effect of gravitational wave passage?

Let me push my question's logic further in very non-orthodox and ignorant manner and also ask, playing the role of "devil's advocate" (while risking to be accused in lack of confidence in basic concepts of General Relativity):

why one can not interprete the measurement results during LIGO detection, as a manifest of speed of light variations (versus considering the speed of light being absolutely invariant under any possible circumstances, such as the case of gravitational wave passage)?

A gravitational wave has a polarization. So when one direction transverse to the wave is expanded, the other direction transverse to the wave is contracted. Thus if you try to explain the expansion and contraction as a naïve consequence of time expansion or of time contraction then you'd have a hard time making that effect cause both an expansion and a contraction in two different directions at the same time and place.

Now I'm sure some people would prefer to have a coordinate speed of light that is different in different directions. But coordinate speeds aren't physical so that's just a model effect not a physical effect. You can do General Relativity and make your predictions without even picking any coordinates, ever. So they aren't part of physical reality, just part of how some people manipulate some mathematical models.

In reality the metric determines the lengths along 4d paths and that tells us how (where and when) clocks tick and how (where and when) tape measures have their marks. Which is what we actually see.

• Okay, that makes sense now. :) – Kyle Kanos Mar 7 '16 at 21:03
• "if you try to explain the expansion and contraction as a ... consequence of time expansion or of time contraction then you'd have a hard time making that effect cause both an expansion and a contraction in two different directions at the same time and place." - is your "hard time" based on human perception of difference between time dimension and spacial dimensions? – Alex Mar 7 '16 at 22:09
• @Alex If the wave is going in the z direction then at one moment space is contracting in the x direction while expanding in the y direction. Then at a later moment space is contracting in the y direction and expanding in the x direction. So you have to explain simultaneous contraction in one direct and expansion in another. – Timaeus Mar 8 '16 at 7:59

In addition to the explanation given by @Timaeus, when working with the practical distance scales it is much easier to detect a spatial change of a few microns by means of laser interferometry than the corresponding changes in time. The temporal dimension in seconds must be scaled by the speed of light: 1 micron divided by 300,000,000 meters per second yields a time scale of a few femtoseconds: $\delta x = c\delta t$.

One femtosecond is a millionth of a billionth of a second, and is the time scale for a single oscillation in the visible light spectrum. This accuracy would be required for two different precise clocks. Unlike the spatial measurement which can rely upon the orthogonal arms of the detector, the clocks would have to be located along a temporal line, as seen from the source of the gravitational wave.

So though the effects are equivalent between space and time, the detection in each realm requires quite different technologies. With continued improvements in time keeping, someday the temporal method will be preferred!

Your two questions actually correspond to the two different sets of coordinates that are used.

1) In the Local Lorentz coordinates Kip Thorne (in some course lectures at Caltech) derives that the space-time location of a mirror changes when the gravitational wave is present. $$\begin{bmatrix} t \\ x \\ y \\ z \end{bmatrix}_{New} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1+h_{+}/2 & h_{X}/2 & 0 \\ 0 & h_{X}/2 & 1-h_{+}/2 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} t \\ x \\ y \\ z \end{bmatrix}_{Old}$$ Notice that the matrix only strains the space coordinates x,y ….not t. Kip says the metric remains unchanged at diag(-1,1,1,1), so the coordinate speed of light remains the standard c. Two wave fronts arrive back at the splitter at different times because one arm has been made longer and one shorter and the speed of light is the same along both arms. He shows that the proper distance along the two paths have changed.

2) In the Transverse Traceless gauge coordinates Kip Thorne derives that the space-time location of a mirror remains unchanged, but the metric changes to $$g^{\mu \nu}_{New}= = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1+h_{+} & h_{X} & 0 \\ 0 & h_{X} & 1-h_{+} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ Since the metric is no longer diag(-1,1,1,1), the coordinate speed of light has changed. It is different in the x and y direction, and depends on the strains. The two wave fronts arrive back at the splitter at different times because the speed of light is different along the two arms. He shows that the proper distance along the two paths has changed, and is equal to the change for the Local Lorentz coordinates.

I have tried to reproduce here the standard argument that shows LIGO should see the effect of a gravitational wave ... as it appears to have done! None the less, there are things in the argument which seem strange to me. First, usually a transformation between frames (if that is what a gravitational wave does) would change both the metric and the coordinates together ... not just one or the other. For example, when an outside observer looks into a Schwarzschild gravitational field, he sees both strained coordinates and a strained metric. Second, in the Local Lorentz coordinates, the new t is the same as the old t. Then how can the wave front arrival times back at the splitter have changed? ...but LIGO has seen something!