I Light speed and the LIGO experiment

[JohnnyGui]

He would also have to allow for the effect of spacetime curvature on the paths of light rays. If he corrected for that, he would find that B's stick is the same length as his own.

So if he didn't correct for that, he would measure a different length of the meter stick?

 

[PeterDonis]

So if he didn't correct for that, he would measure a different length of the meter stick?

No; if he didn't correct for that, he would not be making a "measurement", he would be making a physically meaningless calculation.

 

[JohnnyGui]

No; if he didn't correct for that, he would not be making a "measurement", he would be making a physically meaningless calculation.

Ok, so not correcting for a distance that lies in a curved space would be meaningless. Now, when it's about a gravitational wave "stretching/curving" the space around one of the LIGO arms making that arm increase in length, and you correct for that "stretch/curving", shouldn't the length of that LIGO arm be the same as its initial length (i.e. without the presence of a gravitational wave)?

 

[PeterDonis]

not correcting for a distance that lies in a curved space would be meaningless

You're stating this wrong. You observe light rays from each end of the object coming into your detector separated by a certain angle. You calculate from that angle and the known distance of the object the "size" of the object. But in doing that calculation, you have to use the correct paths of the light rays, in the actual curved spacetime they are traveling through. The usual formula for "size" in terms of distance and angle subtended in your field of view does not do that; it assumes that spacetime is flat. So you have to "correct" that formula to take into account the actual paths of the light rays in the actual curved spacetime. That's not "correcting for a distance"; it's using the correct formula instead of an incorrect one.

when it's about a gravitational wave "stretching/curving" the space around one of the LIGO arms making that arm increase in length, and you correct for that "stretch/curving",

No, you don't "correct for the stretch/curving". You calculate the correct path of the light rays, in the curved spacetime. Whether the geometry of that curved spacetime can be interpreted as "stretching/curving space" depends on your choice of coordinates. But the paths of light rays in the curved spacetime geometry do not; they are what they are regardless of what coordinates you choose. And in LIGO, the paths of the light rays in the curved spacetime geometry are what determine the interference pattern at the detector. You don't have to "correct" for anything.

 

[JohnnyGui]

No, you don't "correct for the stretch/curving". You calculate the correct path of the light rays, in the curved spacetime. Whether the geometry of that curved spacetime can be interpreted as "stretching/curving space" depends on your choice of coordinates. But the paths of light rays in the curved spacetime geometry do not; they are what they are regardless of what coordinates you choose. And in LIGO, the paths of the light rays in the curved spacetime geometry are what determine the interference pattern at the detector. You don't have to "correct" for anything.

I understand that it depends on the choice of coordinates and that you don't need to correct for anything to measure the interference pattern. I'm only focusing on physical length of the LIGO arm as well as when you choose such a particular set of coordinates in that length that can be interpreted as stretching/curving the space around that LIGO arm. In that case, if you correct the formula for calculating the length of the arm and take into account the actual paths of the light rays in the curved space caused by the gravitational wave, would you calculate the same length as the initial length of the LIGO arm in the absence of a gravitaitonal wave? Just like you'd calculate the same length of the meter stick previously mentioned if you correct for the effect of spacetime curvature on the paths of the lightrays?

 

[PeterDonis]

I'm only focusing on physical length of the LIGO arm

And how is "physical length" defined? You need to pick a simultaneity convention, so you can identify which events on the worldlines of the different parts of the arm happen at the same time. That means you have to pick coordinates. There is no "length" independent of coordinates. In spacetime, independent of coordinates, the LIGO arm occupies a "world tube"; a world tube doesn't have a "length".

Note that such a choice of coordinates was necessary in the meter stick case too; I said I was assuming standard Schwarzschild coordinates. The meter stick does not have a "length" independent of coordinates; it only has a spacetime world tube.

if you correct the formula for calculating the length of the arm and take into account the actual paths of the light rays in the curved space caused by the gravitational wave, would you calculate the same length as the initial length of the LIGO arm in the absence of a gravitaitonal wave?

This doesn't make sense. LIGO is not observing the length of the arm from a distance, using light rays emitted from each end and looking at the angle between them. It's observing the interference pattern caused by light rays bouncing back and forth along two perpendicular arms.

 

[JohnnyGui]

And how is "physical length" defined? You need to pick a simultaneity convention, so you can identify which events on the worldlines of the different parts of the arm happen at the same time. That means you have to pick coordinates. There is no "length" independent of coordinates. In spacetime, independent of coordinates, the LIGO arm occupies a "world tube"; a world tube doesn't have a "length".

Just as physical length was defined for the meter stick, when it is said that the meter stick, using Schwarzschild coordinates and correcting for the paths of the lightrays in a curved space, will have the same length as a meter stick that's not in a curved space (person A's meter stick). Such that we compare the length of a LIGO arm in the absence of a gravitational wave to the length of the LIGO arm if a gravitational wave stretches/curves (again, using that particular set of coordinates that can be interpreted as stretching) the whole arm. And then correcting for the effect of that stretch/curve on the path of lightrays.

This doesn't make sense. LIGO is not observing the length of the arm from a distance, using light rays emitted from each end and looking at the angle between them. It's observing the interference pattern caused by light rays bouncing back and forth along two perpendicular arms.

I'm not talking about the purpose of the LIGO experiment at all or how the interference pattern is observed, I understand that you don't have to observe the length or the angle for that. I'm merely looking at the effect of a gravitational field on the physical length of the LIGO arm, and that if correcting for the effect of the stretching on the light rays, caused by the gravitational wave, would yield a same length as the length in the absence of a gravitational wave.

 

[PeterDonis]

Just as physical length was defined for the meter stick, when it is said that the meter stick, using Schwarzschild coordinates and correcting for the paths of the lightrays in a curved space, will have the same length as a meter stick that's not in a curved space (person A's meter stick).

"The same length" was defined in terms of a specific measurement method--looking at the angle between light rays from the meter stick close to the massive object, and using that plus the known distance plus the paths of the light rays to calculate a length of the meter stick. That calculated number is what is the same as the length of the meter stick far away from the massive object. Note that we are also assuming some well-defined measurement method for the meter stick far away from the massive object--the simplest assumption would be to use the same method as we use for the first meter stick. But without specified measurement methods, there is no way to compare the "lengths" because "length" is not well-defined in the absence of such a specified method.

Such that we compare the length of a LIGO arm in the absence of a gravitational wave to the length of the LIGO arm if a gravitational wave stretches/curves (again, using that particular set of coordinates that can be interpreted as stretching) the whole arm.

And how do you measure the lengths of these two arms? (We'll assume you have a second LIGO apparatus far away from the first one, such that the gravitational wave is only passing through one of them.)

And then correcting for the effect of that stretch/curve on the path of lightrays.

What light rays? If you don't have a specified measurement method, you don't know what light rays to compute the paths for.

I'm merely looking at the effect of a gravitational field on the physical length of the LIGO arm

And I'm saying that the "physical length" of the arm depends on how you measure it. Specify a measurement method, and we can then do the comparison between its results for a LIGO apparatus with a GW passing through, and a LIGO apparatus with no GW passing through. But in the absence of a specific measurement method, there's no way to make the comparison.

 

[JohnnyGui]

And I'm saying that the "physical length" of the arm depends on how you measure it. Specify a measurement method, and we can then do the comparison between its results for a LIGO apparatus with a GW passing through, and a LIGO apparatus with no GW passing through. But in the absence of a specific measurement method, there's no way to make the comparison.

Ok. What I was assuming is the same measurement method as the meter sticks, but here it goes: just like you said, having a LIGO arm A far away from another copy LIGO arm B. A gravitational wave passes by LIGO arm B curving the whole arm. You're standing at LIGO arm A and you know the distance to LIGO arm B which lays perpendicular to your viewing angle. Furthermore, you measure the angle of both endpoints of LIGO arm B and you correct for the effect of the curved space at LIGO arm B, which is caused by the gravitational wave there, on the course of the lightrays. What length would one, standing at LIGO arm A, measure?

 

[PeterDonis]

you know the distance to LIGO arm B

How do you know the distance? You knew the distance before the GW passed by, but how do you know it stayed the same?

In the case of the meter sticks, we knew the distance wouldn't change because the spacetime is static. (We also have to assume that both meter sticks are following static worldlines, i.e., worldlines that remain at the same radial coordinate in standard Schwarzschild coordinates. But we were assuming that anyway.) But a region of spacetime where GWs are present, the spacetime is not static. So you can't assume the distance when the GW passes is the same as it was before the GW passed.

Note, btw, that we are also assuming that we have a well-defined measurement method for the distance. Different methods could yield different results, so we really should specify what method we are using.

What length would one, standing at LIGO arm A, measure?

Let's take a step back and ask: would the angle subtended by LIGO arm B, as observed at A, change when the GW passed? I think the answer to that question is yes (assuming sufficiently precise measurements).

If you want to interpret that observation as meaning that the length of arm B changed, that's a permissible interpretation. But it's still an interpretation. You could also interpret it this way: the passing GW distorts the paths of light rays so that the angle observed at A changes even though the length of arm B didn't change. Or you could interpret it as the passing GW changing the distance from arm B to A, so the arm subtends a different angle even though its length hasn't changed.

Let's go back and look at the meter stick scenario again, for comparison. A question we didn't ask before is this: suppose we place an observer so that he is at the same distance from both meter sticks ("distance" measured in the same way in both cases--we won't specify what that way is). Will the angle subtended by both meter sticks, as observed by that observer, be the same? I believe the answer to that question is no.

So can we also interpret this result as telling us that the length of the meter stick close to the massive object is different? Strictly speaking, we could. But now let's imagine other experiments we could run to test this. Suppose we place both meter sticks so they are oriented vertically. We take a very long rod and place it radially so that it lies alongside the meter sticks as it passes them. We make marks on the rod that are 1 meter stick length apart in both places, and verify that each meter stick lies exactly between two adjacent marks (i.e., each end of the stick coincides with one of two adjacent marks).

Now we very slowly and carefully move the rod vertically so that it shifts, relative to the meter stick far away from the object, by exactly one mark. For concreteness, say we move the rod downwards, so at the end of the movement, the mark that was at the top of the faraway meter stick is now at the bottom, and the mark that was next above the top mark is now at the top. Then we ask: did the rod also shift by exactly one mark relative to the meter stick close to the massive object? The answer to that will be yes.

In fact, we could go even further. We could very slowly and carefully lower the rod so that the two marks that initially coincided with the faraway meter stick now coincide with the close-in meter stick. And the faraway meter stick would now lie exactly between an adjacent pair of marks that were initially much further up the rod.

The bottom line is that the "length" of an object in a general curved spacetime is not a single well-defined property of the object. Different experiments involving "lengths" can give different results, even experiments that you would expect to give the same results in flat spacetime. And the relationship between various different experimental results can be different for different curved spacetime geometries.

 

[JohnnyGui]

You have practically hit the nail on my question with your great reply and I thank you for that.

Let's take a step back and ask: would the angle subtended by LIGO arm B, as observed at A, change when the GW passed? I think the answer to that question is yes (assuming sufficiently precise measurements).

If you want to interpret that observation as meaning that the length of arm B changed, that's a permissible interpretation. But it's still an interpretation. You could also interpret it this way: the passing GW distorts the paths of light rays so that the angle observed at A changes even though the length of arm B didn't change. Or you could interpret it as the passing GW changing the distance from arm B to A, so the arm subtends a different angle even though its length hasn't changed.

This is indeed what I was expecting. Also, that's what I actually meant in the beginning of the thread, when I (poorly worded) said that curved space of a massive object and curved space of a GW is practically "the same thing". What I was trying to say is that a curved space around a massive object and a curved space by a GW would both give a distortion effect on the length of something (although the amount and way of distortion would differ ofcourse and that it's not static in the case of a GW)

Let's go back and look at the meter stick scenario again, for comparison. A question we didn't ask before is this: suppose we place an observer so that he is at the same distance from both meter sticks ("distance" measured in the same way in both cases--we won't specify what that way is). Will the angle subtended by both meter sticks, as observed by that observer, be the same? I believe the answer to that question is no.

I was thinking about the same thing and I agree that the answer is no.Regarding your example of the rod; there's something that helps me although it could be a wrong way to interpret it. I like to think of the path along the rod, passing a flat space into a curved space, as a partially distorted mirror that has a normal half and a distorted half. If you lay down the rod that has 1 meter marks parallel to the mirror, you'd see that the reflection of the distance between the 1 meter marks in the distorted part of the mirror looks longer than the reflection of the 1 meter marks on the normal part of the mirror. Now, if you push the rod exactly 1 meter alongside the mirror (1 meter according to the normal reflection) in the parallel direction of the distorted part of the mirror, that "1 meter push" would look longer in the distorted reflection compared to the normal reflection but it's still 1 meter according to the distorted part of the mirror and thus anyone sitting "in" that distorted part, would consider that 1 meter of the rod has got pushed into his side. Just like, according to anyone sitting "in" the normal part of the mirror, would say that 1 meter of the rod has been pushed away from him.

Come to think of it, if one sitting in a flat space is looking at the part of the rod getting pushed into the distorted space, wouldn't he say that the speed of the rod's push into the distorted part is different compared to the push of the rod in his own flat space, even if it's all just one rod? He'd see the push of the rod in the distorted part is covering a longer length in the same time as the push of the rod in his own flat part is covering 1 meter.

 

[PeterDonis]

I like to think of the path along the rod, passing a flat space into a curved space, as a partially distorted mirror that has a normal half and a distorted half.

The curvature doesn't have sharp boundaries; it changes gradually. It is almost flat far away from the massive object, more curved closer in, and changing continuously in between. Strictly speaking, there is no place where it is perfectly flat.

If you lay down the rod that has 1 meter marks parallel to the mirror, you'd see that the reflection of the distance between the 1 meter marks in the distorted part of the mirror looks longer than the reflection of the 1 meter marks on the normal part of the mirror.

No, you wouldn't. You would see it as shorter. That is, the angle between light rays from two adjacent marks close to the massive object, when they reached your eye or telescope, would be less than the angle between light rays from two adjacent marks far away from the massive object, assuming the same distance traveled for both.

Now, if you push the rod exactly 1 meter alongside the mirror (1 meter according to the normal reflection) in the parallel direction of the distorted part of the mirror, that "1 meter push" would look longer in the distorted reflection compared to the normal reflection

No, it would look shorter.

anyone sitting "in" that distorted part, would consider that 1 meter of the rod has got pushed into his side. Just like, according to anyone sitting "in" the normal part of the mirror, would say that 1 meter of the rod has been pushed away from him.

This part is correct.

if one sitting in a flat space is looking at the part of the rod getting pushed into the distorted space, wouldn't he say that the speed of the rod's push into the distorted part is different compared to the push of the rod in his own flat space, even if it's all just one rod?

If he treats the change in apparent angle of the light rays as "speed", yes--the speed will appear to be slower for the part of the rod closer to the massive object. But the speed measured locally by an observer watching the rod pass him will be the same everywhere.

He'd see the push of the rod in the distorted part is covering a longer length in the same time as the push of the rod in his own flat part is covering 1 meter.

Shorter, not longer. See above.

 

[JohnnyGui]

The curvature doesn't have sharp boundaries; it changes gradually. It is almost flat far away from the massive object, more curved closer in, and changing continuously in between. Strictly speaking, there is no place where it is perfectly flat.

Yes. I forgot to say the mirror having an increasing distortion as you walk parallel to it. Nevertheless, it wouldn't be as continuous as space curvature ofcourse.

No, you wouldn't. You would see it as shorter. That is, the angle between light rays from two adjacent marks close to the massive object, when they reached your eye or telescope, would be less than the angle between light rays from two adjacent marks far away from the massive object, assuming the same distance traveled for both.

Ah, ofcourse. Didn't think this through.

If he treats the change in apparent angle of the light rays as "speed", yes--the speed will appear to be slower for the part of the rod closer to the massive object. But the speed measured locally by an observer watching the rod pass him will be the same everywhere.

Couldn't this "change in angle of the light rays" be used as an analogous interpretation of why time dilates for things closer to a massive object? Since this distortion also shows that the speed will apear to be slower closer to the massive object with respect to someone far away from it?
However, thinking about it I think it can't because this way of interpreting time dilation only explains time dilation in space displacement but not other phenomenons such as a twin aging slower compared to his sibling. Unless, as vague as it may sound, change in anything can all be dumbed down to a change in space but I'm not sure about this.

 

[PeterDonis]

Couldn't this "change in angle of the light rays" be used as an analogous interpretation of why time dilates for things closer to a massive object?

No. Time dilation is shown by the change in the time interval between successive light pulses when they arrive at the faraway observer (according to his clock), compared to the time interval between when they were emitted by the observer close to the massive object (according to his clock). It has nothing to do with the angle between light rays that arrive at the faraway observer at the same time.

Since this distortion also shows that the speed will apear to be slower closer to the massive object with respect to someone far away from it?

The coordinate speed of light slows down as you get closer to the massive object; but coordinates have no physical meaning. An observer close to the massive object that measures the speed of light locally (for example, by measuring the speed of the light rays emitted from each end of his meter stick, as they are on their way out to the faraway observer) will measure it to be $c$.

Also, the observation of time dilation--different time intervals between successive light pulses according to the two clocks (faraway vs. close-in)--can't be due to any change in the speed of the light anyway, because the spacetime is static, so if the first of a pair of light pulses changes it speed on the way from the close-in observer to the faraway observer, the second light pulse of the pair will change its speed in exactly the same way. So the total "travel time" of the two must be the same. The difference in time intervals according to the two clocks therefore can't have anything to do with the behavior of the light; it has to be something to do with the behavior of the clocks themselves.

Unless, as vague as it may sound, change in anything can all be dumbed down to a change in space

No, it can't.

 

[JohnnyGui]

Also, the observation of time dilation--different time intervals between successive light pulses according to the two clocks (faraway vs. close-in)--can't be due to any change in the speed of the light anyway, because the spacetime is static, so if the first of a pair of light pulses changes it speed on the way from the close-in observer to the faraway observer, the second light pulse of the pair will change its speed in exactly the same way. So the total "travel time" of the two must be the same. The difference in time intervals according to the two clocks therefore can't have anything to do with the behavior of the light; it has to be something to do with the behavior of the clocks themselves.

This is exactly what I concluded as well but after posting my statement and what made me realize my theory is false. I should've thought twice.

It seems that I was focused on time dilation as being a disagreement on space displacement while it's actually a disagreement on time duration/interval.

However, wouldn't the distortion of the space be an additional effect next to time dilation that makes someone seem to move even more slower than by time dilation alone, if one wouldn't correct for the effect of that distortion? Notice that I'm not talking about time intervals of light signals that a faraway person is receiving but that he's merely looking at someone moving at a massive object.

 

[PeterDonis]

wouldn't the distortion of the space be an additional effect next to time dilation that makes someone seem to move even more slower than by time dilation alone, if one wouldn't correct for the effect of that distortion?

I don't think so, because, as I said, the spacetime is static, so the distortion affects all light rays the same; it doesn't change from one light ray to the next, so it doesn't affect the time interval between successive light rays.

I'm not talking about time intervals of light signals

You are if you are talking about time dilation, because that's what time dilation is: a difference in the time interval between the same pair of light signals according to two different clocks, the emitter's and the receiver's.

 

[JohnnyGui]

You are if you are talking about time dilation, because that's what time dilation is: a difference in the time interval between the same pair of light signals according to two different clocks, the emitter's and the receiver's.

I'm not only talking about time dilation. What I mean is, that there are 2 mechanisms involved here in making someone at a massive object seem to move slower with respect to someone far away. First, there is the distortion which, as you said, makes a path or meter stick look shorter at the massive object so that someone faraway would say that someone at the massive object moves slower (your post #42), space displacement-wise (if there's no correction on the distortion effect). Then, on top of that, time dilation is also present which makes someone moving in that already "shortened" path move even more slowly with respect to someone faraway but time interval-wise.
So, the faraway person would say that someone at the massive object is moving a smaller distance in a Δt (because of the distortion) as well as say that he'd move that same smaller distance in a longer Δt than what the moving person would say about himself (because of time dilation).

 

[PeterDonis]

(if there's no correction on the distortion effect).

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.

The time dilation effect, OTOH, is there even after you've applied any corrections you need to apply for the effects of curvature on the paths of light rays.

 

[Dale]

You are if you are talking about time dilation, because that's what time dilation is: a difference in the time interval between the same pair of light signals according to two different clocks, the emitter's and the receiver's.

I think I would call that gravitational redshift. I don't know if my distinction is standard terminology.

 

[PeterDonis]

I think I would call that gravitational redshift. I don't know if my distinction is standard terminology.

Terminology in this area is, um, mixed. I have seen the observation I described referred to both ways.

 

[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.