Modeling the effects of GW and the "Earth Frame"

[PeterDonis]

the answer to (a) must be "no", all the test masses must be motionless.

I don't think this is correct. I haven't been following all the details of your exchange with pervect, but a couple of items should be noted:

(1) The "transverse-traceless" approach only applies in a small patch of spacetime in which the gravitational wave can be idealized as a purely transverse plane wave. It certainly can't be applied in a global coordinate chart that includes an entire sphere at some distance from the source.

(2) If you are trying to visualize the whole gravitational wave in a global coordinate chart that includes an entire sphere at some distance from the source, you cannot simply assume that the entire wave front is just a sphere (or annulus) to which the transverse plane wave in a small patch, described in #1 above, is a local approximation. In other words, you cannot assume that the wave amplitude at a given radius from the source, at a given time in a chart in which the source is at rest, is the same at all angular coordinates around a sphere at that radius. (I'm not positive that you can even assume this everywhere on a circle of a given radius in the source's orbital plane.)

 

[GeorgeDishman]

I don't think this is correct. I haven't been following all the details of your exchange with pervect, but a couple of items should be noted:

(1) The "transverse-traceless" approach only applies in a small patch of spacetime in which the gravitational wave can be idealized as a purely transverse plane wave. It certainly can't be applied in a global coordinate chart that includes an entire sphere at some distance from the source.

I think the local conditions can be iterated around the equator and perhaps over the whole surface again creating the global view by overlapping small regions but it is certainly a point where I have an ongoing concern. However, any problems are significantly smaller than in the alternative.

(2) If you are trying to visualize the whole gravitational wave in a global coordinate chart that includes an entire sphere at some distance from the source, you cannot simply assume that the entire wave front is just a sphere (or annulus) to which the transverse plane wave in a small patch, described in #1 above, is a local approximation. In other words, you cannot assume that the wave amplitude at a given radius from the source, at a given time in a chart in which the source is at rest, is the same at all angular coordinates around a sphere at that radius. (I'm not positive that you can even assume this everywhere on a circle of a given radius in the source's orbital plane.)

I totally agree, as you go round the circle, the phase of the GW changes and remember the GW has half the period of the binary orbit. I've attached the diagram I included some time ago which shows how two cycles fit round the "equator".

Based on what has been said, what I am suggesting is that we should not think of test particles "moving together then apart" but instead staying static with an equivalent variation of the speed of light causing the variation in the interferometer output.

The thought experiment that helped me on this is to imagine a variant of the RingWorld concept.

Usually that is shown with living space on the inner surface, which presupposes rotation to create artificial gravity through "centrifugal force". Instead, think of a non-rotating RingWorld with suspended test masses hanging towards the star by suspensions that leave them free to move in the direction along the ring, then make the star a hard binary. How do the masses move relative to the material of RingWorld if that is nearly rigid (as close as GR allows)?

Sorry, I just drew that image by hand so it's a bit rough but I'm sure you'll get the idea.

To avoid the complexity of the material behaviour, think of three sets of test masses half a GW wavelength apart in radius and ignore the RingWorld material. Do the middle set of masses move relative to the top and bottom sets (which must act in unison). The Ringworld material is shown at the top this time in cross section with three test masses, one from each set.

A little thought should show that either the middle ring moves in opposition to the outer two with equal magnitude or none move at all.

Do you have a metric that can describe this setup?

 

[PeterDonis]

I think the local conditions can be iterated around the equator and perhaps over the whole surface again creating the global view by overlapping small regions but it is certainly a point where I have an ongoing concern.

I have a concern about this too, as I've expressed before. Unfortunately I'm not familiar enough with the field to know what, if any, math has been developed to deal with this.

 

[GeorgeDishman]

Let me put it another way. Suppose that, relative to one of the mass suspension points (mass 0), the mass beneath it has some motion x₀=f(t) in the direction of the ring. Because the effects come from nothing more than the rotation of the binary, another mass n farther round the ring by θ must have the same motion but delayed relative to the first mass, x_n=f(t(1-2θ)) The separation of two adjacent masses is what LIGO measures and we can think of a LIGO between between every adjacent mass. I think this argument is the same as the definition of distance in the Hubble Law, as explained in Ned Wright's tutorial:

http://www.astro.ucla.edu/~wright/cosmo_02.htm#MD

D_now = D(us to Z) = D(us to A) + D(A to B) + ... D(X to Y) + D(Y to Z)​

If θ is small then the beam length L is small in comparison to the circumference and the strain dL/L in the limit must be the derivative of the displacement. Now I'm not sure that the usual symmetry holds in GR but naively I have been assuming that if the measured strain is the derivative of the displacement, then I can calculate the displacement by integrating the strain.

Incidentally, if I use Wright's diagram locally to illustrate the "expanding space" version of the light in the LIGO beam tube, the fact that "the lightcones must tip over" as he puts it means the speed of the light isn't quite c for the whole length relative to an end, but that's a second order effect so not significant, just a curiosity.

 

[pervect]

Do you have a metric that can describe this setup?

Currently, no :(

I've been doing some reading, but no answer yet - I may start a technical thread on the issue.

 

[GeorgeDishman]

Currently, no :(

I've been doing some reading, but no answer yet - I may start a technical thread on the issue.

That would certainly be interesting but probably beyond my present mathematical level. I do think I've got enough of an understanding to put the rambling conversation together as a more coherent write-up now, if only at a qualitative level, but I'll have to find the time between some DIY tasks to make an attempt at that.

I am very grateful for the effort you've put in already in helping me get past my mental logjam, thank you.

 

[pervect]

I'm curious myself about some of the issues that were raised. For quite a while I've thought of a metric (or line element) as defining the space-time geometry. In this case, though, I realized I don't have one and so far I haven't found one. So it's an interesting question in and of itself.

I have a feeling we'd differ about how well the metric / line element can be represented in 3d - but I think I've already explained my concerns about that as well as I can.