LIGO light changes frequency not wavelength

[RockyMarciano]

No, it isn't. A pseudotensor is something that doesn't transform properly under a change of coordinates. The expansion tensor, like all genuine tensors, transforms properly regardless of which metric signature convention you are using.

A pseudotensor transforms like a tensor under proper transformations but changes sign under an orientation reversing coordinate transformation, which is what a transformation that changes signature convention does, therefore you must additionally impose the physically reasonable condition that only time and space orientation preserving transformations are allowed.

 

[PeterDonis]

How do you figure the measure of "intensity" is performed if not with respect to a standard measurement gauge which is an object with a certain length(that you claim is not invariant).

Intensity is wave amplitude (actually amplitude squared). The amplitude of a wave at a given event in spacetime is an invariant.

A pseudotensor transforms like a tensor under proper transformations but changes sign under an orientation reversing coordinate transformation

Ah, I see, we are using different definitions of the word "pseudotensor". You are using the first definition on the Wiki page linked below, and I am using the second (which, as the article notes, is the usual definition in GR). With the first definition, yes, I agree that you have to restrict to orientation-preserving transformations.

https://en.wikipedia.org/wiki/Pseudotensor

 

[PeterDonis]

Could you please extend some courtesy and explain your remark that what I wrote was inconsistent?

Oops, I see I had incorrectly thought the remark about round-trip travel times and the SI definition of the meter was yours. It was pervect's. I'll correct my previous posts accordingly; I apologize for the confusion on my part.

I agree with your remark that there is no coordinate-independent definition of "length" (more precisely, of "proper length") in a non-stationary spacetime.

 

[MeJennifer]

Oops, I see I had incorrectly thought the remark about round-trip travel times and the SI definition of the meter was yours. It was pervect's. I'll correct my previous posts accordingly; I apologize for the confusion on my part.

I agree with your remark that there is no coordinate-independent definition of "length" (more precisely, of "proper length") in a non-stationary spacetime.

I am glad we sorted that out!

 

[RockyMarciano]

Intensity is wave amplitude (actually amplitude squared). The amplitude of a wave at a given event in spacetime is an invariant.

Right. But I'm trying to decipher how you can consider an amplitude measurement which is obviously measured as a displacement of the fringes, a measure of distance, i.e. a length in the end, as invariant in this case and at the same time say that length is coordinate dependent by definition.For ilustrative purposes this is how interference patterns are measured explained in a basic and clear fashion. The difference in the LIGO case is that the problem instead of computing the wavelength from the fringes count and the displacement of the mirrors consists of converting the measure of the fringes to a displacement of mirrors, with a fixed laser wavelength. And the displacement is so small that instead of counting fringes, the pattern must be highly magnified and displacement with respect to the initial pattern must be measured in multiple points of the pattern.

 

[PeterDonis]

an amplitude measurement which is obviously measured as a displacement of the fringes

No, it isn't.

First, did you read my previous description? In the "default case" of no interference, there are no fringes. The fringes only appear at all where there is interference. So whatever is being measured, it can't be a displacement of the fringes from the "no interference" case to the "interference" case, since there are no fringes at all in the former case.

Second, what I have called an amplitude measurement is really an intensity measurement (amplitude squared). In LIGO, which is typical, this is measured by a photodetector, which produces an electrical signal proportional to the intensity of the light. There is no distance measurement involved at all.

Finally, while it is possible to measure the separation between the interference fringes in a sufficiently sophisticated detector, that measurement does not tell you the amplitude (or intensity) of the light. It gives information about the wavelength of the light (more precisely, the wavelength of the "beats" between the two different laser signals that are not precisely in phase). And this wavelength is measured in the rest frame of the detector; a detector in a different state of motion relative to the interferometer arms would measure a different wavelength (different separation between the fringes)--and, indeed, if we used a different inertial frame, even without a detector in such a state of motion as to be at rest in that frame, we would assign a different value to the "separation between the fringes" in that frame. So such a measurement does "depend on the coordinates", whereas the light intensity measurement does not.

 

[pervect]

This is only true if the spacetime is stationary, at least during the travel time of the light, and if the objects between which the proper length is being measured are at rest relative to each other during the travel time of the light. If these requirements are not met, the concept of "proper length" is not well-defined.

[Edited to delete mistaken further comments.]

We all agree, I hope, that the round-trip time is observer independent? And we are debating whether half the round-trip corresponds to a meaningful notion of length or not?

In order to define the length corresponding to half the round-trip-time, we need to define a frame, with an associated notion of simultaneity. There is a logical frame to use - this is the frame of the Earth, or rather the Frame attached to the Earth in the vicinity of Ligo, henceforth the "Ligo Frame". We routinely measure distances on the Earth, so it is meaningful to leverage this pre-existing notion of distance that we use everyday in our lives for this notion of distance in the Ligo frame.

We note in passing that the Ligo frame is not inertial. Hopefully we don't need to discuss that in depth. Basically, the point is that we can (and do) measure distances on the surface of the Earth, in spite of the fact that it's not an inertial frame (due to the presence of gravity, and even more potentially confusing, due to the fact that it's rotating).

Concerns were raised about the presence of gravity waves upsetting the usual notion of distance on the Earth. As I mentioned eariler, the appropriate mathematical notion we need to address these concerns is the notion of measuring the distances in the tangent space. Once we realize that we can measure the distances in the tangent space, we don't have to worry about whether the manifold is stationary or not - it's totally irrelevant once we've made this approximation.So we have a tangent four-space to our manifold, and we use the usual process of projection operators to create a notion of 3-space in which we can measure the distance. There are tricky aspects here, due to the rotation of the Earth, but those tricky aspects aren't unique to gravity waves, they're the usual confusion with respect to relativistic rotating frames. And they're not particularly relevant to Ligo, we could avoid them entirely if we analyzed a Ligo-alike that was floating out in space and not rotating and had zero proper acceleration.

There is one other approximation we need to make. The test masses on the Ligo interferometer are not quite at rest with respect to the Ligo frame. If they were at rest, they would maintain a constant distance from each other, as the Ligo frame, as we've defined it, is rigid.. The test masses DO move relative to each other, and hence they move relative to the Ligo frame. Because the test masses are moving relative to our frame, one may be concerned with the Lorentz contraction induced by their motion. SInce the velocity of the test masses with respect to the Ligo frame is less than a nanometer per second (it's probably much less, I haven't calculated it in detail), the amount of Lorentz contraction is negligible.

 

[MeJennifer]

We all agree, I hope, that the round-trip time is observer independent? And we are debating whether half the round-trip corresponds to a meaningful notion of length or not?

In order to define the length corresponding to half the round-trip-time, we need to define a frame, with an associated notion of simultaneity. There is a logical frame to use - this is the frame of the Earth, or rather the Frame attached to the Earth in the vicinity of Ligo, henceforth the "Ligo Frame". We routinely measure distances on the Earth, so it is meaningful to leverage this pre-existing notion of distance that we use everyday in our lives for this notion of distance in the Ligo frame.

We note in passing that the Ligo frame is not inertial. Hopefully we don't need to discuss that in depth. Basically, the point is that we can (and do) measure distances on the surface of the Earth, in spite of the fact that it's not an inertial frame (due to the presence of gravity, and even more potentially confusing, due to the fact that it's rotating).

Concerns were raised about the presence of gravity waves upsetting the usual notion of distance on the Earth. As I mentioned eariler, the appropriate mathematical notion we need to address these concerns is the notion of measuring the distances in the tangent space. Once we realize that we can measure the distances in the tangent space, we don't have to worry about whether the manifold is stationary or not - it's totally irrelevant once we've made this approximation.So we have a tangent four-space to our manifold, and we use the usual process of projection operators to create a notion of 3-space in which we can measure the distance. There are tricky aspects here, due to the rotation of the Earth, but those tricky aspects aren't unique to gravity waves, they're the usual confusion with respect to relativistic rotating frames. And they're not particularly relevant to Ligo, we could avoid them entirely if we analyzed a Ligo-alike that was floating out in space and not rotating and had zero proper acceleration.

There is one other approximation we need to make. The test masses on the Ligo interferometer are not quite at rest with respect to the Ligo frame. If they were at rest, they would maintain a constant distance from each other, as the Ligo frame, as we've defined it, is rigid.. The test masses DO move relative to each other, and hence they move relative to the Ligo frame. Because the test masses are moving relative to our frame, one may be concerned with the Lorentz contraction induced by their motion. SInce the velocity of the test masses with respect to the Ligo frame is less than a nanometer per second (it's probably much less, I haven't calculated it in detail), the amount of Lorentz contraction is negligible.

I think that all those simplifications are totally unnecessary and personally I find them more confusing than educational.

The LIGO experiment demonstrates that 'near' the event of detection spacetime was non-stationary to a level of being detected. The fact is that light travel time fluctuated near this event, even when other factors where eliminated. This fluctuation in travel time I think is the key in understanding the phenomenon.

 

[RockyMarciano]

No, it isn't.

Yes, nevermind the amplitude-distance issue, it is actually irrelevant to what the LIGO interferometer actually measures. I got distracted by your bringing it up. The photodetector is simply the way the fringes are realized as images, there is no more to the intensity thing, you need some form to observe the actual interference pattern and obtain an interferogram.

So if you just took a look at the linked video of how a Michelson interferometer works, you can see that the relevant measure consists on the counting o fringes(cycles:N) that the formula that is shown there relates with displacement of the mirror arms and a constant wavelength. Again the cycle counting amounts to a displacement of the fringes measure between two interference patterns. That one of the interference patterns is used as the corresponding to no phase shift doesn't mean it shows no fringe pattern, it's just used as the origin or the zero of the displacement. Just look at how in the video linked the fringes are displaced as the cycles are counted and the micrometer advances.

The difference of the patterns is what is frame independent regardless of the particular value assigned to the length of the fringe separation in a particular frame.

 

[RockyMarciano]

We all agree, I hope, that the round-trip time is observer independent? And we are debating whether half the round-trip corresponds to a meaningful notion of length or not?

In order to define the length corresponding to half the round-trip-time, we need to define a frame, with an associated notion of simultaneity. There is a logical frame to use - this is the frame of the Earth, or rather the Frame attached to the Earth in the vicinity of Ligo, henceforth the "Ligo Frame". We routinely measure distances on the Earth, so it is meaningful to leverage this pre-existing notion of distance that we use everyday in our lives for this notion of distance in the Ligo frame.

We note in passing that the Ligo frame is not inertial. Hopefully we don't need to discuss that in depth. Basically, the point is that we can (and do) measure distances on the surface of the Earth, in spite of the fact that it's not an inertial frame (due to the presence of gravity, and even more potentially confusing, due to the fact that it's rotating).

Concerns were raised about the presence of gravity waves upsetting the usual notion of distance on the Earth. As I mentioned eariler, the appropriate mathematical notion we need to address these concerns is the notion of measuring the distances in the tangent space. Once we realize that we can measure the distances in the tangent space, we don't have to worry about whether the manifold is stationary or not - it's totally irrelevant once we've made this approximation.So we have a tangent four-space to our manifold, and we use the usual process of projection operators to create a notion of 3-space in which we can measure the distance. There are tricky aspects here, due to the rotation of the Earth, but those tricky aspects aren't unique to gravity waves, they're the usual confusion with respect to relativistic rotating frames. And they're not particularly relevant to Ligo, we could avoid them entirely if we analyzed a Ligo-alike that was floating out in space and not rotating and had zero proper acceleration.

There is one other approximation we need to make. The test masses on the Ligo interferometer are not quite at rest with respect to the Ligo frame. If they were at rest, they would maintain a constant distance from each other, as the Ligo frame, as we've defined it, is rigid.. The test masses DO move relative to each other, and hence they move relative to the Ligo frame. Because the test masses are moving relative to our frame, one may be concerned with the Lorentz contraction induced by their motion. SInce the velocity of the test masses with respect to the Ligo frame is less than a nanometer per second (it's probably much less, I haven't calculated it in detail), the amount of Lorentz contraction is negligible.

But isn't the OP arguing that all those approximations you mention give an error bigger than the effect that LIGO seeks to detect?

 

[RockyMarciano]

Words are defined however a group of people chooses to define them, and often different groups of people use the same word to mean different things.

My comment that it is a valid tensor is correct, as is your comment that it is a pseudotensor. We were just using the definitions of different groups of people.

Let's put it like this then: that group of people of yours has a funny handle of mathematical tools..

 

[PeterDonis]

We all agree, I hope, that the round-trip time is observer independent?

Yes, in the sense that, given a laser source/detector in a particular state of motion, all observers will agree on the round-trip travel time of a given laser beam down a given arm and back, as measured by the source/detector's clock.

And we are debating whether half the round-trip corresponds to a meaningful notion of length or not?

The debate, as I understand it, isn't about whether such a notion of length is "meaningful"; it's about whether such a notion of length is coordinate-dependent. Your post makes it clear that you agree (with me, at least) that it is.

Once we realize that we can measure the distances in the tangent space, we don't have to worry about whether the manifold is stationary or not

Yes, you do. The tangent space is only a meaningful notion within a single local inertial frame (strictly speaking, it's only meaningful at a single chosen event; but the concept of "local inertial frame" is really the same as "the tangent space at a chosen event"). Within a single local inertial frame, tidal gravity is negligible. But gravitational waves are made of tidal gravity (spacetime curvature); if tidal gravity is negligible, then gravitational waves are negligible. So the tangent space, i.e., approximating spacetime as flat, can't possibly be sufficient, by itself, to treat the detection of gravitational waves.

As I understand it, the LIGO team's preferred tool is what MTW calls "linearized GR". In this approximation, spacetime is not flat; it is what I would call "close to flat". The metric is modeled as a flat background metric, plus a small correction $h_{\mu \nu}$ which describes fluctuations in spacetime curvature around the flat background. (Note that the background metric does not necessarily have to be flat in this treatment; you can use, for example, the Schwarzschild metric as the background with this technique. As I understand it, this is not done for LIGO because it's more complicated and the differences are too small to matter for their analysis.) The small correction is what LIGO is detecting and calling its "gravitational wave signal".

This is not the same as the tangent space analysis you are describing; it looks similar at first glance, and I suspect that some statements the LIGO team has made can be mistaken as saying they are using a tangent space in a local inertial frame; but they aren't.

AFAIK, this technique does work for a non-stationary spacetime, as long as the deviations from the background metric are small; in the LIGO case, this is obvious since the correction terms are small and the background metric has unit coefficients on the diagonal. But, AFAIK, you could use the same technique with, say, FRW spacetime as the background, as long as the corrections to the time part were small compared to unity and the corrections to the space part were small compared to the scale factor.

we use the usual process of projection operators to create a notion of 3-space in which we can measure the distance.

I don't think this is what the LIGO team is doing. I think they are simply using the usual notion of distance in their chosen frame--the "linearized GR" frame. This is almost the same as distance in an inertial frame, but not quite because of the small corrections to the metric. Those corrections make the distance as measured in their chosen coordinates fluctuate by a small amount around the "inertial" distance. Or, to put it another way, they make the 3-surfaces of simultaneity, in their chosen coordinates, slightly different from what they would be if spacetime were exactly flat.

 

[PeterDonis]

the cycle counting amounts to a displacement of the fringes measure between two interference patterns.

I think you're conflating two different kinds of measurements that a Michelson interferometer can be used for.

In its original use--the original Michelson-Morley experiment and its more recent versions--the interferometer's orientation is changed during the experiment, and the idea is to look for shifts in the fringes during the orientation change. This is supposed to indicate the "absolute velocity" of the apparatus relative to the ether. Of course the actual experiment, when done like this, shows no fringes at all--except very small ones due to unavoidable imperfections in the instrument.

In the LIGO use, the interferometer is in a fixed orientation, and in its "usual" state--no gravitational wave passing--it shows no fringes (no interference). When a GW passes, fringes are detected, and they do shift during the detection because the GW's effect on the interferometer is not constant--it fluctuates, because that's what a GW is, a fluctuation in tidal gravity. Counting the number of times the pattern shifts, and the details of each shift in terms of the fringe spacing and other data, is what the LIGO team uses to produce the "signals" that it publishes (after a good deal of analysis and cleanup of the data).

However, in neither of these cases is the amplitude or intensity of the laser light being measured by the spacing between the fringes. The intensity of the light is just the intensity of the light--the lighter parts of the interference pattern have higher intensity, the darker parts have lower intensity. The intensity at a given point on the detector (how light or dark it is) at a given instant of time is an invariant, and has nothing to do with any distance measurement.

 

[GeorgeDishman]

Hi all. I've been pointed at this thread as it relates to a discussion I've been having elsewhere so I'd like to query some statements made earlier. I've been working on creating an animation of GW using MATLAB but I've hit a brick wall (because I don't know GR) so I'd like to find out where I'm going wrong in relation to the motion of the interferometer mirrors versus their supports versus the "Earth frame". However, I need to find out how to phrase the question first, it's complicated, so I'll come back to that in another post. In the meantime, I'd like to toss in an engineering comment here:

In the LIGO use, the interferometer is in a fixed orientation, and in its "usual" state--no gravitational wave passing--it shows no fringes (no interference). When a GW passes, fringes are detected, and they do shift during the detection because the GW's effect on the interferometer is not constant--it fluctuates, because that's what a GW is, a fluctuation in tidal gravity. Counting the number of times the pattern shifts, and the details of each shift in terms of the fringe spacing and other data, is what the LIGO team uses to produce the "signals" that it publishes (after a good deal of analysis and cleanup of the data).

There are always two beams in the equipment so there is always interference. However, if you think of the image posted previously, it is possible to set up the pattern so that the centre point is about 50% of a bright fringe. If anything causes 'movement' of the fringes, it will also move the phase of the fringe pattern at the centre, one way it will increase the brightness while moving the fringes the other way will reduce the brightness. In the absence of any disturbance, a photodiode at the centre would give some DC output. A passing GW then causes that to vary producing an AC signal about the 'default' DC level. What we see published, after filtering and amplification obviously, should be that AC signal. OK, the technical details are probably much more subtle but that I think is a simplified way of looking at how the detector works in principle that avoids any confusion about measuring movement. It also explains how the detector output can be sensitive to phase shifts of a small fraction of a wavelength rather than 'counting fringes'.

Sorry to butt in, I hope that helps.

 

[GeorgeDishman]

I'd like to start from what I think is the right way to look at this and then take it forward to explain a problem I have and see if someone can straighten out my thinking.

I think that possibly you do not realize that the mirrors on Ligo are attached to test masses that are basically "hung from strings", so that the test masses, and the attached mirrors, are free to move?

We'll call the thing that the test masses are suspended from "the frame".

I think that may be confusing as later posts talk about the "LIGO frame" as a "tangent four-space" which is somewhat different, so I'd rather say the mirrors are suspended from the ends of the beam tubes (but there's nothing in name).

https://www.ligo.caltech.edu/image/ligo20150519c

What I would like to visualise is that we place a ruler under each mirror with the zero point directly under the suspension. The 'string' holds the mirror from a point on the top of the tube, the ruler is bolted mounted horizontally and supported on a pillar directly under the same point. "Motion" of each mirror can then be considered relative to its adjacent ruler.

Everyone agrees that the test masses move relative to the frame.

That's not entirely true. I have seen a PhD paper where the statement was made that "the proper length between the mirrors varies but not the coordinate length" and therefore that the mirrors would not move relative to their adjacent rulers at the tube ends, and the support mechanism is only there to isolate them from seismic noise. However, it seems to me that if that were the case, the so-called "sticky bead argument" would fail as it would no longer be possible to extract energy from a GW.

https://en.wikipedia.org/wiki/Sticky_bead_argument

This frame is of no particular interest to the way the Ligo experiment works, so little effort is spent explaining what happens to it. If one did measure what happened to the frame , it wouldn't change measuarbly in length. The test masses, that are perfectly free to move at the slightest influence, require our most sensitive insturments to measure their motion. The frame moves even less.

Actually, understanding the behaviour of the frame is crucial to what I'm trying to do so I want to come back to this later but I need to explain the background first.

The viewpoint that needs expanding space is a viewpoint that is attached, not to the frame, but to the suspended test masses. One can regard each test mass as having a constant coordinate, a coordinate that does not change with time. In this view, there are no external forces acting on these test masses, so one regards them as not moving. When the gravity wave passes by these test masses, changing their separation, but one ascribes this change in distance to "expanding and contracting space", rather than to any real force. There is no real force according to this viewpoint, the test masses are regarded isolated from any non-gravitaitonal forces, and gravity is not regarded as a real force (according to this viewpoint, which is different from the Newtonian one). One might say that the test masses are in a state of "natural motion", like a body at rest in Newtonian physics.

OK.

In this viewpoint, it's the frame that is "moving". Since the test masses are "standing still", i.e. have constant coordinates, and the frame is moving relative to the test masses, the frame must be "moving". The reason the frame moves is that internal forces generated by the interaction of the atoms that make up the frame keep the distance between atoms nearly constant. Internal forces due to the interaction of the atoms that keep the length constant (or nearly constant) are what causes the pieces of the frame to move in this viewpoint.

Surely, in order to get a detector output in this view, the motion of the mirrors must be in opposite directions so would the tube not need to stretch and shrink rather than moving as a whole? The time-varying length would still be ascribed to the atomic forces though.

This viewpoint of expanding space also occurs in cosmology, and there are similar issues of (mis)understanding the popularizations in cosmology as well.

Exactly.

While I understand this alternative view (and that they are equivalent), it will be easier for what I want to do to stick with the more common interpretation but what is important to me first is to confirm that the effect of a GW would be to make the mirrors move relative to their respective adjacent rulers which I think is an coordinate independent question. Am I OK so far?

 

[pervect]

I don't think this is what the LIGO team is doing. I think they are simply using the usual notion of distance in their chosen frame--the "linearized GR" frame. This is almost the same as distance in an inertial frame, but not quite because of the small corrections to the metric. Those corrections make the distance as measured in their chosen coordinates fluctuate by a small amount around the "inertial" distance. Or, to put it another way, they make the 3-surfaces of simultaneity, in their chosen coordinates, slightly different from what they would be if spacetime were exactly flat.

I believe tried doing what I think you mean by "what the Ligo team is doing" in another thread. (Of course, I could be more sure if you had a reference of some sort, to make sure that what I think the Ligo team is doing is the same thing as what you think the Ligo team is doing). But rather than revisit that approach, I'll explain the approach I was using.

The essence is simple: we just apply the geodesic deviation equation. I don't think I ever saw "the Ligo team" mention the geodesic deviation equation, nor any popularization mention the geodesic equation. But I also think it's a good and reasonably simple approach - it is slightly advanced for an general audience, but it should be comprehensible to someone with some very basic knowledge of General Relativity.

In an additional effort to keep things simple, we will consider "Ligo in space". Moving Ligo into outer space gets rid of many complications that occur if we try to analyze it in situ on the Earth (such as the Earth's gravity, and the Earth's rotation), but captures the essence of the problem without introducing irrelevant details.

With that background let's proceed. The interferometer has two arms, we'll just analyze one arm. We have one fiducial test mass, where the interferometer is located, and one nearby test mass. Both our fiducial test mass and our nearby test mass are following geodesics, i.e. they are in free fall.

We let the separation between the fiducial observer and the nearby observer be represented by a vector $\xi$, as per MTW's remarks on pg 31 (and elsewhere).

We'll restrict the problem two two dimensions - time, and one spatial dimension. Via the geodesic deviation equation, we can write:

$$\frac{d^2 \xi}{d\tau^2} = R \xi$$

(See MTW 1.6).

R is some number, which is in our 2d case, the sole component of the Riemann curvature tensor.

We basically observe that in order to use the geodesic deviation equation to calculate the relative acceleration between our fiducial and our nearby observers, we needed to have some notion of the distance between our fiducial observer and our nearby observer, which we are representing by the vector $\xi$. If we didn't have some notion of distance, we couldn't calculate the second derivative of $\xi$ and call it a relative acceleration.

MTW remarks that the separation vector can be regarded as being measured in the local Lorentz frame of the fiducial observer. But the Local Lorentz frame of the fiducial observer is just the tangent space of the fiducial observer.

So there you have it, in a nutshell. The fiducial observer is following a geodesic, as is the nearby test observer. The distance between the two observers has a meaning when the fiducial observer is sufficiently close to the nearby observer, this meaning is represented by the vector $\xi$ which can be regarded as the distance in the Local Lorentz frame of the fiducial observer (i.e. the distance in the tangent space of the fiducial observer).

And - this distance is changing with time - the nearby observer is accelerating relative to the fiducial observer, due to the gravitational waves.

The only remaining issue might be to answer the question is "are the two mirrors in Ligo sufficiently close for this analysis to work". I believe the answer is yes - the residual discrepancies should be on the order of the Lorentz contraction due to velocities of nanometers per second, i.e. negligible for all practical purposes.

 

[PeterDonis]

we just apply the geodesic deviation equation.

Which cannot be done in any tangent space, because every tangent space is flat; there is no geodesic deviation.

I don't have a problem with the general strategy of analysis you are describing, at least on a first read. The only thing I have been objecting to is your use of the term "tangent space", and your claim that we can just use the usual notion of "distance" in a tangent space, when analyzing LIGO, or indeed any GW scenario. That can't be right, for the reasons I have already given (and which I just restated above).

MTW remarks that the separation vector can be regarded as being measured in the local Lorentz frame of the fiducial observer.

I'll have to look up this specific reference when I get a chance. But my previous remarks still stand: if you are analyzing geodesic deviation, you can't possibly be doing the analysis in any local Lorentz frame, because in any such frame there is no geodesic deviation at all, by definition. If your measurements are able to detect any geodesic deviation, they must cover a region of spacetime that is too large for a single local Lorentz frame.

It's possible that MTW are using the term "local Lorentz frame" somewhat inconsistently in this particular context, to mean something like Fermi normal coordinates centered on the worldline of the fiducial observer. In those coordinates, the metric is not flat and there is geodesic deviation, and I believe that deviation, expressed in terms of the separation vector $\eta$, obeys the equation you wrote down. But Fermi normal coordinates are not a tangent space, and this usage of "local Lorentz frame" would not make that term equivalent to the term "tangent space". Again, I will need to look up the specific reference to see the context.

 

[RockyMarciano]

Maybe this apparent discrepancy can be clarified(and we can recover the OP dealing with what LIGO measures since the starting point in linearized gravity is the 3+1 decomposition) by going back to a comment of PeterDonis:

it is not always possible to even find such a 3+1 foliation in a spacetime containing a given timelike congruence, such that every 3-surface in the foliation is orthogonal to every timelike curve in the congruence. It happens to be possible for the congruence of comoving observers in FRW spacetime, but you can't depend on it as a general property.

This can be further qualified in the pseudo-Riemannian case, because the Lorentzian signature introduces some caveats that don't come up in the purely Riemannian case with positive definite signature. It so happens that in the Lorentzian case the 3+1 foliation itself, by virtue of the Lorentzian inner product picks the relation of orthogonality between the spacelike and the timelike vectors at a point, and if one chooses spacelike hypersurfaces, with 3 spacelike directions tangent to it, a fourth tangent vector normal to them is necessarily timelike. So the 3+1 formalism(with simultaneity hypersurfaces, i.e. spacelike 3-surfaces) always allows to find the timelike congruence orthogonal to spacelike 3-sufaces. Another way to see this is that a manifold in GR is always locally minkowskian, so locally there is a natural 3+1 decomposition with minkowskian coordinates. Comoving observers can be found and therefore becomes a general property whenever we use the 3+1 formalism.

This issue doesn't come up in the Riemannian case where no distinction is made between timelike and spacelike vectors and coordinates, and a 3+1 foliation can't single out a special type of coordinate for the 1 in a 3+1 decomposition.

It has been debated for a long time whether the 3+1 formalism is coordinate dependent in the sense of generally covariant, i.e.:
-“The very foundation of general covariant physics is the idea that the notion of a simultaneity surface all over the universe is devoid of physical meaning”. C. Rovelli
in: H. Garcia-Compe`an (Ed.), Topics in Mathematical Physics. General Relativity and Cosmology, in honor of Jerzy Plebanski, Proceedings of 2002 International Conference (Cinvestav, Mexico City, 17-20 September, 2002);
-“the split into three spatial dimensions and one time dimension seems to be contrary to the whole spirit of relativity”, S.W. Hawking, In: S. W. Hawking and W. Israel (Eds.), General Relativity. An Einstein Centenary Survey (Cambridge University Press, Cambridge, 1979), p. 746
-“Being non-intrinsic, the 3+1 decomposition is somewhat at odds with a generally covariant formalism, and difficulties
arise for this reason” by Pon in J.M. Pons, Class. Quant. Grav. 20 (2003) 3279

 

[PeterDonis]

[Moderator's note: a subthread about expansion of the universe has been moved to the Cosmology forum. Please keep this thread focused on discussion of LIGO.]

 

[PeterDonis]

There are always two beams in the equipment so there is always interference.

No, this is not correct. If the two beams are exactly in phase when they come back after being reflected, there is no interference. This is the state of the LIGO detector when no gravitational wave is passing. Interference is only seen when a GW passes, since the passage of the GW causes the two beams to be out of phase when they return after being reflected.

 

[PeterDonis]

the 3+1 formalism(with simultaneity hypersurfaces, i.e. spacelike 3-surfaces) always allows to find the timelike congruence orthogonal to spacelike 3-sufaces.

No, this is not correct. There are spacetimes in which there is no way to foliate the spacetime with any set of spacelike hypersurfaces that is everywhere orthogonal to some timelike congruence. But you do not need the orthogonality condition in order to have a foliation; you can still foliate many of these spacetimes with a set of spacelike hypersurfaces, they just won't be orthogonal everywhere to any timelike congruence.

The point you appear to be missing is that, while any vector orthogonal to a set of 3 orthogonal spacelike vectors must be timelike, it is not the case that any timelike vector must be orthogonal to a set of 3 orthogonal spacelike vectors. The latter is what would need to be true for there to always be a foliation with the properties you have specified.

 

[PeterDonis]

What I would like to visualise is that we place a ruler under each mirror with the zero point directly under the suspension. The 'string' holds the mirror from a point on the top of the tube, the ruler is bolted mounted horizontally and supported on a pillar directly under the same point. "Motion" of each mirror can then be considered relative to its adjacent ruler.

Here you are constraining the motion of the ruler, so it will not be moving on a geodesic--it will be accelerated if a GW passes. (This will be true even in a space-based experiment where there is no gravity, from Earth or anything else, to complicate things.) So you can't assume that the ruler provides an "unchanging" standard of either position or length.

 

[RockyMarciano]

No, this is not correct. There are spacetimes in which there is no way to foliate the spacetime with any set of spacelike hypersurfaces that is everywhere orthogonal to some timelike congruence. But you do not need the orthogonality condition in order to have a foliation; you can still foliate many of these spacetimes with a set of spacelike hypersurfaces, they just won't be orthogonal everywhere to any timelike congruence.

The point you appear to be missing is that, while any vector orthogonal to a set of 3 orthogonal spacelike vectors must be timelike, it is not the case that any timelike vector must be orthogonal to a set of 3 orthogonal spacelike vectors. The latter is what would need to be true for there to always be a foliation with the properties you have specified.

Remember that I'm only addressing the general case in GR, not specific spacetimes where in order to model isolated objects time independence of the metric (a timelike Killing vector field) is assumed(like it is the case for instance in the stationary but not static Kerr rotational geometry that might be the example you had in mind of spacelike 3-surface that is not orthogonal to the timelike congruence).
In the general case in GR, with no timelike Killing vectors, it is indeed the case that the choice of a spacelike hypersurface with 3 spacelike vectors, i.e. a 3+1 foliation, determines having a fourth timelike vector orthogonal to them, and if the manifold is to be locally Minkowskian, a requirement of GR spacetimes, I don't think any other foliation deserves to be called a spacetime. By all means if you have in mind a physically plausible GR spacetime with no timelike Killing vector and with a 3+1 foliation that is not time-orthogonal,bring it up, but I'd be surprised.

 

[PeterDonis]

In the general case in GR, with no timelike Killing vectors, it is indeed the case that the choice of a spacelike hypersurface with 3 spacelike vectors, i.e. a 3+1 foliation, determines having a fourth timelike vector orthogonal to them

At a single event, yes, you can always find a timelike vector that is orthogonal to a chosen set of 3 orthogonal spacelike vectors. But your claim is much stronger: you are claiming that, in any spacetime, you can find a foliation by spacelike hypersurfaces covering the entire spacetime, and a timelike vector field covering the entire spacetime that is everywhere orthogonal to every hypersurface in the foliation. I don't think that claim is true for every possible spacetime. (For one thing, not every possible spacetime even admits a foliation to begin with; see below.)

It does happen to be true for Minkowski spacetime (obviously) and FRW spacetime. It is also true for Schwarzschild spacetime, but the "standard" foliation (using hypersurfaces of constant Schwarzschild coordinate time $t$) only covers the region outside the horizon. To cover the region inside the horizon, you need to use a non-standard foliation (such as the one implied by Kruskal coordinates), whose physical interpretation will be strained, to say the least.

I don't think any other foliation deserves to be called a spacetime.

A foliation is not the same thing as a spacetime. Hawking & Ellis is full of examples of spacetimes that cannot be foliated by a family of spacelike hypersurfaces.

 

[pervect]

Which cannot be done in any tangent space, because every tangent space is flat; there is no geodesic deviation.

I don't have a problem with the general strategy of analysis you are describing, at least on a first read. The only thing I have been objecting to is your use of the term "tangent space", and your claim that we can just use the usual notion of "distance" in a tangent space, when analyzing LIGO, or indeed any GW scenario. That can't be right, for the reasons I have already given (and which I just restated above).

I don't necessarily have a problem with your objection to the wording, except that I don't have a better wording at the moment (and I've looked for inspiration in MTW). Perhaps the following clarification will help, though. "Measuring distances in the tangent space" means that one first uses the exponential map https://en.wikipedia.org/wiki/Exponential_map_(Riemannian_geometry) to map points that are close to each other from the manifold to points in the tangent space. Then one measures the distance in the tangent space.

More generally, the point I'm trying to get across is that there is a well-known and coordinate-independent way of computing and measuring distance when objects are sufficiently close, and that the construction does not depend on whether the underlying manifold is stationary or not.

 

[MeJennifer]

More generally, the point I'm trying to get across is that there is a well-known and coordinate-independent way of computing and measuring distance ...

That is true but is such "distance" a physically measurable quantity in non-stationary spacetimes?
I think not!

 

[GeorgeDishman]

No, this is not correct. If the two beams are exactly in phase when they come back after being reflected, there is no interference. This is the state of the LIGO detector when no gravitational wave is passing. Interference is only seen when a GW passes, since the passage of the GW causes the two beams to be out of phase when they return after being reflected.

Can I suggest you have a look at fig 1 in this paper:

https://arxiv.org/ftp/arxiv/papers/1411/1411.4547.pdf

and then look at paragraph 2.3:

"2.3 Gravitational wave readout
Readout of the gravitational wave signal is accomplished using an output mode cleaner in conjunction with homodyne, or DC detection. In this scheme, a local oscillator field is generated by offsetting the arm cavities slightly from their resonance (typically a few picometers), thereby pulling the Michelson slightly off the dark fringe."

 

[GeorgeDishman]

Here you are constraining the motion of the ruler, so it will not be moving on a geodesic--it will be accelerated if a GW passes. (This will be true even in a space-based experiment where there is no gravity, from Earth or anything else, to complicate things.) So you can't assume that the ruler provides an "unchanging" standard of either position or length.

I agree, but my assumption (which may not be correct) is that the test mass is free to move in the direction long the tube (though constrained in other directions) in response to the GW so if the mass follows the geodesic but the ruler does not, then there should be a resulting relative motion between them.

My thought is that, if we treat the mid-point of the tube as a fiducial location, the motion of the masses at the ends will have the opposite sense, thus both moving away from or towards the tube centre as measured against the rulers.

The alternative that might apply is that the mirrors are constrained to remain fixed relative to the rulers other than not following local seismic noise which would couple to the rulers. However, this page seems to suggest the mirrors move in response to a GW:

https://www.ligo.caltech.edu/page/vibration-isolation

"Since gravitational waves will make themselves known through vibrations in LIGO's mirrors, the only way to make gravitational wave detection possible is to isolate LIGO's components from environmental vibrations to unprecedented levels. The change in distance between LIGO's mirrors (test masses) when a gravitational wave passes will be on the order of 10^-19 m"

"The goal is to keep our hands off the masses as much as possible so they will move only due to gravitational waves."

 

[PeterDonis]

"Measuring distances in the tangent space" means that one first uses the exponential map https://en.wikipedia.org/wiki/Exponential_map_(Riemannian_geometry) to map points that are close to each other from the manifold to points in the tangent space. Then one measures the distance in the tangent space.

Yes, I understand that. The question is how close the points in the manifold have to be for this to work. The answer as I understand it is that they have to be close enough that no tidal gravity effects are measurable. As I've said several times now, that obviously can't be true if we are analyzing a gravitational wave detector, since gravitational waves are waves of tidal gravity, so if tidal gravity is not measurable, neither are GWs.

there is a well-known and coordinate-independent way of computing and measuring distance when objects are sufficiently close

Yes, but the question is what "sufficiently close" means. I'm not sure the LIGO mirrors are "sufficiently close" to the LIGO detector in the necessary sense to make the distance from mirrors to detector coordinate-independent. Certainly the distance in linearized GR is not coordinate-independent; the linearized GR distance described in MTW requires a particular choice of coordinates (harmonic gauge plus an additional constraint to make the metric correction $h_{\mu \nu}$ traceless). So if the LIGO team is using linearized GR for their analysis, as I suspect, the "distance" they are describing is dependent on that particular coordinate choice.

 

[PeterDonis]

is such "distance" a physically measurable quantity in non-stationary spacetimes?

For objects that are close enough together, yes; you can construct a local inertial frame in any spacetime whatever, whether it's stationary or not. The issue is not that GWs make the spacetime non-stationary; the issue is that GWs are made of spacetime curvature, and spacetime curvature is not observable within a single local inertial frame, by definition.

 

[PeterDonis]

my assumption (which may not be correct) is that the test mass is free to move in the direction long the tube (though constrained in other directions)

This would certainly not be true in an idealized space-based GW detector, where the apparatus as a whole is in free fall. In such a detector, the test masses would move along geodesics; their motion could not be constrained in any way, since that would mean they would be moving on non-geodesic worldlines.

In LIGO, which is not in free fall, it is true that there is a constraint imposed on the motion of the test masses; they are not in free fall, because their average position has to lie on an accelerated worldline (since the apparatus as a whole is accelerated at 1 g, not in free fall). But I don't think their motion about that average position is constrained; it can be in any direction, not just the direction along the tube.

My thought is that, if we treat the mid-point of the tube as a fiducial location, the motion of the masses at the ends will have the opposite sense

The LIGO detector is not a single long tube. It's two tubes, perpendicular to each other, in the shape of an "L", with the detector at the junction of the two tubes (the corner of the "L"), and the two mirrors at the other ends of each of the tubes. The fiducial location is the detector; the motion of each mirror due to a GW is relative to the detector, not relative to the other mirror.

this page seems to suggest the mirrors move in response to a GW

That's correct; they do.

 

[MeJennifer]

For objects that are close enough together, yes; you can construct a local inertial frame in any spacetime whatever, whether it's stationary or not.

Well obviously!
If it is so close that we consider the region flat (and thus stationary) we basically have SR.

 

[PeterDonis]

a local oscillator field is generated by offsetting the arm cavities slightly from their resonance (typically a few picometers), thereby pulling the Michelson slightly off the dark fringe."

Ah, I see; it looks like LIGO's "default" state (when no GW is detected) has an artificially induced constant phase shift between the signals from the two arms. But this is an intentionally added feature; it is not something that has to be there whenever you have beams in both arms. If the arm cavities were not offset as this paragraph describes, there would be no fringes at all at the detector (i.e., no interference) in the absence of a GW, even with beams in both arms. That is why I objected to your earlier statement that there must be interference whenever there are beams in both arms.

 

[GeorgeDishman]

Ah, I see; it looks like LIGO's "default" state (when no GW is detected) has an artificially induced constant phase shift between the signals from the two arms. But this is an intentionally added feature; it is not something that has to be there whenever you have beams in both arms. If the arm cavities were not offset as this paragraph describes, there would be no fringes at all at the detector (i.e., no interference) in the absence of a GW, even with beams in both arms. That is why I objected to your earlier statement that there must be interference whenever there are beams in both arms.

OK, it's just a slight difference in the way we look at the situation, more semantics than physics. In the "default state" without the pulling, the article notes that the photodiode would be in a "dark fringe" which for me means the two beams arrive exactly 180 degrees out of phase and hence the "interference" between them results in the null.

The article states that the pulling is of the order of a few picometres or about 10^-5 of the laser wavelength (1064nm). A path length variation of say 10fm would be 10^-8 of a wavelength and alter the photodiode current by around 0.1% and the mirror motion would be much smaller due to the Fabry-Perot design of course.

 

[GeorgeDishman]

This would certainly not be true in an idealized space-based GW detector, where the apparatus as a whole is in free fall. In such a detector, the test masses would move along geodesics; their motion could not be constrained in any way, since that would mean they would be moving on non-geodesic worldlines.

Yes, the problem is that so is the surrounding satellite so there's no local comparison. This has been mentioned before in the thread in that we can idealise the situation by thinking of the LIGO beam tube "in space" to remove the annoyance of local Earth gravity and rotation, in other words like eLISA but with a long steel tube bolted between the satellites. Again, I would put small rulers (a nanometer long!) adjacent to each mirror but fixed to the tube to get a local indication of whether the mirrors move.

In LIGO, which is not in free fall, it is true that there is a constraint imposed on the motion of the test masses; they are not in free fall, because their average position has to lie on an accelerated worldline (since the apparatus as a whole is accelerated at 1 g, not in free fall). But I don't think their motion about that average position is constrained; it can be in any direction, not just the direction along the tube.

I think they are constrained against yaw, torque and pitch as well as vertically. I'm not sure about linear motion transverse to the beam but if you think of the mirror as a flat plane, that should have negligible consequences. (In fact there aren't quite planes as the beam diameter isn't the same at the near and far ends but that's a finer detail.)

The LIGO detector is not a single long tube. It's two tubes, perpendicular to each other, in the shape of an "L", with the detector at the junction of the two tubes (the corner of the "L"), and the two mirrors at the other ends of each of the tubes. The fiducial location is the detector; the motion of each mirror due to a GW is relative to the detector, not relative to the other mirror.

Yes, you're right to an extent but this is a bit more complex in practice and will become easier in the "LIGO in space" thought experiment I want to look at. The majority of the path length in each arm is between the two "test masses" as described on this page:

https://www.ligo.caltech.edu/page/ligos-ifo

"The Fabry Perot 'cavity' actually is the full 4 km length of each arm between the beam splitter and the end of each arm. Additional mirrors placed near the beam splitter are precisely aligned to reflect each laser beam back and forth along this 4 km length about 280 times before it finally merges with the beam from the other arm. These extra reflections serve two functions:

1. It stores the laser light within the interferometer for a longer period of time, which increases LIGO's sensitivity

2. It increases the distance traveled by each laser beam from 4 km to 1120 km

With Fabry Perot cavities, LIGO's interfereometer arms are not just 4 km long, they are essentially 1120 km long, making them 144,000 times bigger than Michelson's original instrument! This bit of 'mirror magic' greatly increases LIGO's sensitivity and makes it capable of detecting changes in arm-length thousands of times smaller than a proton, while keeping the physical size of the interferometer manageable."​

this page seems to suggest the mirrors move in response to a GW

That's correct; they do.

The way I look at it, although the beams are combined at the junction of the L, the signal is primarily a measure of the "distance" (in terms of phase change) between the test masses. The effect of the short distance from the ITM to the beamsplitter has an effect 280 less than the distance variations between the test masses.

Now since both masses are free to move but the beam tube is somewhat more rigid, we can think of the masses moving in opposition within the tube. In fact there will be a common motion as well (that's where my problem really lies but more on that later) but if you think of the LIGO in space version, the rigidity of the tube would sort of average out the motion of its parts so that it would all move along roughly the geodesic of the centre plus a small strain resulting from the stress within the metal. It therefore makes more sense to put the fiducial point in the centre. However, that's really a matter of preference and I like the symmetry. All I'm going to be asking about is the motion of the test masses and the shape of their geodesics, that's what I'm trying to model, i.e a collection of independent point masses which are in free fall other than not "dropping" due to gravity.

With regard to the L shape, a bit of thought-experiment simplification can make life easier. Consider a distant binary system and our system happens to lie in the orbital plane (and ignore spins of the binary components). The signal reaching us would have "plus" polarisation. Let's set the first arm in the plane and perpendicular to the line of sight to the binary. Set the second arm aligned with the orbital axis of the binary. Now whatever signal we have in the arm in the plane, the other arm will produce an exactly equal signal but 180 degrees out of phase. In that case I think we need only consider the arm in the plane. What I want to look at is extending a series of those arms and see how they add up. (The exception to free fall mentioned above is that I'm not worrying about the equipment falling towards the distant binary star system.)

Does that make sense so far?