I Light speed and the LIGO experiment

[JohnnyGui]

Hm, I missed this qualifier before. If you don't correct for the distortion, yes, I think you would (erroneously) attribute an additional slowdown in motion to the object close in (if it was moving--if it was standing still relative to the massive object there would be no effect). But, as I've said before, if you don't correct for the distortion, you are making a physically meaningless calculation.

Yes, definitely keeping that in mind. Thank you for the whole clarification on this.

I got a bit off-topic regarding the physics behind LIGO and after discussing all this are my following statements regarding LIGO correct?

1. A GW passing LIGO would stretch the space of one arm and compress the other, depending on the coordinates you've chosen.

2. If someone is standing on the path of the arm itself which is stretched or compressed as a whole by a GW, he wouldn't notice any change in length of the arm compared to its length before the GW. If he runs the length of the path of the arm at a velocity v, it would take according to himself the same time duration as when there wasn't a GW.

3. Can I say that an arm of the LIGO that gets stretched by a GW is not really making light having to travel a larger distance, but is getting its wavelength stretched instead which causes the interference?

I'm sorry if I have mixed this all up (again).

 

[PeterDonis]

1. A GW passing LIGO would stretch the space of one arm and compress the other, depending on the coordinates you've chosen.

In appropriate coordinates (the ones the LIGO team uses to analyze their data), there would be a cycle--one arm stretched, the other compressed; then switch to the first arm compressed, the second stretched; then back and forth again.

2. If someone is standing on the path of the arm itself which is stretched or compressed as a whole by a GW, he wouldn't notice any change in length of the arm compared to its length before the GW.

If he's standing still, he has to adopt some system of coordinates to be able to assign a "length" to the arm. He could choose coordinates such that the length of the arm doesn't change; but he could also choose coordinates such that it does.

If he runs the length of the path of the arm at a velocity v, it would take according to himself the same time duration as when there wasn't a GW.

I'm not sure whether it would or not. I think his round-trip travel time according to the clock of a person sitting at rest at one end of the arm would change; but I'm not sure about the time according to his own clock.

3. Can I say that an arm of the LIGO that gets stretched by a GW is not really making light having to travel a larger distance, but is getting its wavelength stretched instead which causes the interference?

I think you could choose coordinates in which this would be the case, yes. (In such coordinates, the wavelength of light along each arm would alternately be stretched and compressed, in the opposite sense to the way in which the arm lengths change in the coordinates the LIGO team uses.)

 

[JohnnyGui]

If he's standing still, he has to adopt some system of coordinates to be able to assign a "length" to the arm. He could choose coordinates such that the length of the arm doesn't change; but he could also choose coordinates such that it does.

Does this mean that, in the scenario of a massive object distorting space, a faraway person who is closing in on a massive object could also use particular coordinates to notice a change in the length of his meter stick compared to when he was far from that massive object?

I think you could choose coordinates in which this would be the case, yes. (In such coordinates, the wavelength of light along each arm would alternately be stretched and compressed, in the opposite sense to the way in which the arm lengths change in the coordinates the LIGO team uses.)

The thing is, using coordinates that show that the arm lengths change (the LIGO team) instead of stretching/compressing the wavelengths makes me think that this doesn't change the wavelengths but merely makes them shift a bit further for one arm and back for the other, which then alternates as the GW passes through.
I have been plotting this as a function as well as plotting the net result that the interferometer would measure and I noticed that the "stretching/compressing" scenario would give a different shape of the net wavelengths that the interferometer would measure than when one is using the "changing arm length" scenario.

The "changing arm length" scenario (one arm getting longer and the other shorter) would give a net result of wavelengths in the interferometer that are constant in widths but only differ in amplitudes as time passes by.
In the case of the "stretching/compressing" scenario (one wavelenghts arm getting stretched and the other compressed) the interferometer would give a net result of wavelengths that differ in lenghts as well as in amplitude.

The way I did this is by drawing a sinusoidal graph like cos(x-1) and cos(x+1) for the "changing arm lengths" scenario (one arm of wavelengths get shifted back by -1 distance and the other arm of wavelengths get shifted further by +1 distance). The interferometer would then measure a net result of cos(x-1) + cos(x+1). Ofcourse, as time passes by, that shift of "+/- 1" in the function would change into a higher or lower number, but that doesn't matter for the net result: the widths of the net wavelengths stay the same and it's only the amplitude which would differ in the interferometer with time.

As for the stretching/compressing scenario I used for example cos(2x) and cos(0.5x) and for the interferometer cos(2x) + cos(0.5x). The net result of wavelengths for the interferometer would differ in widths as well as amplitudes with time. The product that the x get multiplied by ("2" and "0.5") is independent.

A great graphing website is the following: https://www.desmos.com/calculator

Which kind of net result did the LIGO researches get? In case I'm concluding this the right way that is..

 

[PeterDonis]

Does this mean that, in the scenario of a massive object distorting space, a faraway person who is closing in on a massive object could also use particular coordinates to notice a change in the length of his meter stick compared to when he was far from that massive object?

Sure, as long as he keeps in mind the fact that coordinates have no physical meaning. The length that actually has physical meaning for him--for example, the ratio of the length of the meter stick to the height of his body--doesn't change at all. Which, of course, raises the obvious question of why the person would care about some contrived system of coordinates in which the "length" of the meter stick changes. Such coordinates tell him nothing useful physically, so why would he bother using them?

using coordinates...makes me think

I probably should have emphasized this even more, but I've already said it several times: coordinates have no physical meaning. You appear to be continually confusing yourself by thinking that the various kinds of coordinates I have described in this thread are telling you something physically meaningful. They aren't. They're just telling you that coordinates have no physical meaning.

In the case of LIGO, the physically meaningful fact is the interference pattern observed at the detector. Anything you overlay on top of that as an "interpretation" is just that, an overlay you are using for "interpretation", as a conceptual crutch to help you visualize what is going on. None of it changes the physically meaningful facts at all.

Which kind of net result did the LIGO researches get?

I don't know; you will have to look at their results and compare them with your model. I don't understand your model well enough to comment.

However, at this point I would strongly suggest that, instead of trying to keep coming up with models, you take a step back and really think about the implications of what I've said: that coordinates have no physical meaning. You are trying to use coordinate-dependent concepts like "length" as if they were telling you something physically meaningful directly. They aren't. In the long run, if you are going to be considering GR problems, it really helps to train yourself out of the tendency to think in terms of coordinates, and learn to think in terms of invariants--like the interference pattern at the LIGO detector--instead. It's not easy, but in the long run it pays dividends.

 

[JohnnyGui]

snip

Ok. I thought that looking at the interference pattern would at least show which interpretation of the effect of GW's is wrong and which isn't, not necessarily coordinate-wise. I'll go dive in some literature for now.

 

[PeterDonis]

I thought that looking at the interference pattern would at least show which interpretation of the effect of GW's is wrong and which isn't, not necessarily coordinate-wise.

Since all of the interpretations depend on particular choices of coordinates, a coordinate-independent invariant like the interference pattern can't possibly show that any interpretation is "wrong"--or "right" for that matter. That's the point of "coordinates have no physical meaning".

 

[JohnnyGui]

Since all of the interpretations depend on particular choices of coordinates, a coordinate-independent invariant like the interference pattern can't possibly show that any interpretation is "wrong"--or "right" for that matter. That's the point of "coordinates have no physical meaning".

So it shouldn't matter if someone says that GW stretches space or adds additional length to it without stretching it?

 

[pervect]

So it shouldn't matter if someone says that GW stretches space or adds additional length to it without stretching it?

The coordinates that the Ligo team used can be described simply. One considers an array of freely floating force-free particles, which are uniformly spaced before the passage of the gravity wave, and one assigns coordinates to these freely floating force-free particles that remain constant as the gravity wave passes.

But the distance between the freely floating particles changes when the gravity wave passes.

What you're probably wanting to know (It's hard to be sure, but it's my best guess as to what you're trying to ask) is what happens to a "rigid object" when the gravity wave passes, such as an 1800's style meter bar prototype. The reason I'm saying this is I'm assuming that mentally, you're probably using an 1800's style meter-bar mentally as your standard of "distance" rather than a more modern definition which is based on special relativity (and which you can look up as the SI definition of the meter). For instance, if you think the speed of light is something that is measured, rather than something that is defined as a constant, you are most likely using an 1800's style meter bar as your distance standard.

Now, this physical metal meter bar isn't actually totally rigid, but it turns out that for this problem it doesn't matter, it's "rigid enough" that we can and will ignore the very the small fluctuations in its length due to the passage of the gravity wave.

The meter bar, then, doesn't change length (in our approximation) when the GW passes, because it's rigid. IT's our standard of length, and our standard of length doesn't change when the GW passes - because it's the standard of length. At least this is true to a very high degree of approximation, as I mentioned a physical meter bar isn't prefectly rigid, but as I also mentioned we can basically ignore this issue. The freely floating particles move differently than the particles attached to the bar. The distance between the freely-floating particles does change, while the distance between particles attached to the ends of the bar does notchange. In the Ligo interpretation, the freely-floating particles are force-free particles that move inertially, while the particles attached to the end of the rigid bar are not inertial, but are deflected from an inertial path by the binding forces internal to the bar that keep the bar's length constant.

So to understand the Ligo teams result, you need to know that they are NOT assigning constant coordinates to the (approximately) rigid meter bar, but ARE assigning constant coordinates to force-free particles. And you also need to know that the notion of whether "space is expanding" or "space is not expanding" is a matter of choice, it depends on which coordinates you choose. And you need to understand that the coordinates the Ligo team is using, that they are assigning constant coordinates to force-free particles.

I *could* give you an alternative and equally self consistent view of the Ligo experiment in the coordinates where the ends of the meter bar have constant coordinates,and it might even be more familiar and very useful. However, I'd rather not confuse the issue by discussing two interpretations in one post.. So I'll only explain the Ligo explanation so you know what they are saying, and leave the alternative description in a different coordinate system to another thread or post.

 

[PeterDonis]

So it shouldn't matter if someone says that GW stretches space or adds additional length to it without stretching it?

Personally, I wouldn't say either. I would say that the GW causes an interference pattern in the LIGO detector. I think it's better to stick to a limited description that accurately reflects the actual invariants of the problem, instead of giving a broader description that is heuristic and depends on things that aren't invariant, like the choice of coordinates. But I'm probably more of a purist in these matters than most.

Also, I'm not sure what "adds additional length to it without stretching it" means; that doesn't seem to match either of the interpretations we've been talking about. In one interpretation (dependent on coordinates like those the LIGO team is using), the GW alternately stretches and squeezes the arms of the LIGO detector. In another interpretation (dependent on a different choice of coordinates), the GW doesn't change the lengths of the arms, but changes the speed of the light in the detector. But, as I said above, to me both of these descriptions go beyond the actual invariant, which is the interference pattern in the detector.

 

[PeterDonis]

the coordinates the Ligo team is using, that they are assigning constant coordinates to force-free particles.

Just to clarify: in the LIGO team's coordinates, the coordinates of the mirrors at the ends of the arms are constant, but the metric coefficients change as the GW passes in such a way that the lengths of the arms change (because those lengths depend on both the coordinates of the ends and the metric coefficients). The alternate set of coordinates I was imagining, in the interpretation where the arm lengths don't change, still assigns constant coordinates to the mirrors at the ends of the arms, but is set up so that a different set of metric coefficients changes as the GW passes, the ones that govern the coordinate speed of light. But this is only a heuristic description; I have not done any computations.

 

[RockyMarciano]

Just to clarify: in the LIGO team's coordinates, the coordinates of the mirrors at the ends of the arms are constant, but the metric coefficients change as the GW passes in such a way that the lengths of the arms change (because those lengths depend on both the coordinates of the ends and the metric coefficients). The alternate set of coordinates I was imagining, in the interpretation where the arm lengths don't change, still assigns constant coordinates to the mirrors at the ends of the arms, but is set up so that a different set of metric coefficients changes as the GW passes, the ones that govern the coordinate speed of light. But this is only a heuristic description; I have not done any computations.

As discussed in a previous thread, even if the alternate set of coordinates you mention is routinely used in interferometry for instance in the maesurent of refraction variations, in the GW case only the LIGO team's class of coordinates( the family of harmonic coordinates) are mathematically compatible with the linearized EFE expressed as a wave equation with plane gravitational wave solutions(see for example Efstathiou's "General relativity" section 17.5). This is a particularity of general relativity as a mathematical model that has no bearing on the general principle which always prevails, i.e.: that the physics must be independent of the coordinates.