I LIGO light changes frequency not wavelength

[exponent137]

Gravitational wave stretches and shrinks space. Why Laser light in Ligo arms changes frequency, and not wavelength.Is it some clear explanation?

If even frequency would not be changed than spacetime is not changed, I suppose. What cannot be measured, cannot exist.

 

[PeterDonis]

Why Laser light in Ligo arms changes frequency, and not wavelength.

This is only true in one particular coordinate chart. One can choose other coordinates in which the light changes wavelength but not frequency. Or one can choose coordinates in which both change.

 

[exponent137]

This is only true in one particular coordinate chart. One can choose other coordinates in which the light changes wavelength but not frequency. Or one can choose coordinates in which both change.

I suppose that length of static objects (arm) is proportionaly cnanged with space. In another case length of this arm is not part of this spacetime, isn't it?

 

[PeterDonis]

I suppose that length of static objects (arm) is proportionaly cnanged with space.

If you mean that the length of an object depends on your choice of coordinates, yes, that is true.

In another case length of this arm is not part of this spacetime, isn't it?

I don't understand what you mean by this.

 

[exponent137]

I will ask differently. I suppose that arbitrary strong gravitational wave cannot stretch LIGO arm so much that it can break it. This is because i suppose that stretch of the arm follows to stretch of space. Let us neglect tidal forces.

Of course this is a theoretical question, because very strong gravitational waves probably will not happen in our history.

I think that my preffered coordinate system is in rest at LIGO, thus I do not move with a rocket, or that I am not in strong gravitational field, thus as an observer I do not feel gravitational field of gravitational wave. I think that this is the simplest situation?.

I ask because I wish to understand what is stretching of spacetime.

 

[PeterDonis]

I suppose that arbitrary strong gravitational wave cannot stretch LIGO arm so much that it can break it.

The actual "arms" of LIGO are not solid structures; they are just the paths, through empty space inside underground tunnels, between the detector and the two mirrors. So there is nothing to "break".

However, a strong enough gravitational wave (far stronger than any we expect to observe on Earth) could move the mirrors enough that they would hit one of the tunnel walls. That would "break" LIGO in the sense that its measurements would no longer be accurate.

i suppose that stretch of the arm follows to stretch of space

"Stretch of space" is also coordinate-dependent.

Let us neglect tidal forces.

You can't neglect tidal forces. Tidal gravity is what gravitational waves are made of. Tidal gravity is just another name for spacetime curvature, and gravitational waves are waves of spacetime curvature.

my preffered coordinate system is in rest at LIGO

More precisely, coordinates in which the detector at the junction of the two arms of LIGO is at rest--yes, these are probably the most convenient coordinates to use for analyzing LIGO measurements, and the LIGO team uses these coordinates to describe their results. In these coordinates, the lengths of the two LIGO arms fluctuate, and the fluctuations are out of phase with each other, so the round-trip travel times of the laser beams going down the two arms are slightly different. That causes interference fringes to appear in the detector.

I wish to understand what is stretching of spacetime.

Curvature of spacetime is a better term; curvature of spacetime, as above, is just tidal gravity. So gravitational waves, which are fluctuations in the curvature of spacetime, are fluctuations in tidal gravity.

 

[exponent137]

As I understand you,
at the junctions of two arms,both frequencies of laser light are changed and both wavelengths are not changed.

What is/are coordinate charts where both frequencies of laser light are not changed and both wavelengths are changed or.
what is/are coordinate charts where both frequencies of laser light are changed and both wavelengths are changed.

 

[PeterDonis]

As I understand you,
at the junctions of two arms,both frequencies of laser light are changed and both wavelengths are not changed.

Not necessarily. For interference to be present, all that needs to change is the round-trip travel time of wave crests in one arm vs. the other. That can change if the length of the LIGO arms changes, even if the frequency and wavelength of the laser light does not change. As I understand it, that is how the LIGO team's model (the one they use to analyze their results) works.

As for other coordinate charts that would show different things changing, I don't have specific ones to point to. I am simply stating a general fact about GR, that things like distance, frequency, wavelength are coordinate-dependent. The interference pattern at LIGO's detector is the only real invariant (i.e., quantity independent of coordinates) in this scenario.

 

[exponent137]

I thought this:

If a gravitational wave stretches the distance between the LIGO mirrors, doesn't it also stretch the wavelength of the laser light?

A gravitational wave does stretch and squeeze the wavelength of the light in the arms. But the interference pattern doesn't come about because of the difference between the length of the arm and the wavelength of the light. Instead it's caused by the different arrival time of the light wave's "crests and troughs" from one arm with the arrival time of the light that traveled in the other arm. To get how this works, it is also important to know that gravitational waves do NOT change the speed of light.
https://www.ligo.caltech.edu/page/faq

This means, wavelength is changed, frequency is changed, length is changed and c is not changed,
(If c would be also changed, frequency would not be changed and intefrence pattern would not exist. But c is not changed.)
For this example I asked how it is with this.
Maybe I did not give a clear question.

 

[PeterDonis]

If a gravitational wave stretches the distance between the LIGO mirrors, doesn't it also stretch the wavelength of the laser light?

Ah, yes, you're correct, in the LIGO team's preferred coordinates, the light wavelength does change, for the same reason the arm lengths change.

If c would be also changed, frequency would not be changed and intefrence pattern would not exist.

This is not quite right. The interference pattern is a physical observable; it does not change regardless of what coordinates you adopt. But it would be possible to choose coordinates such that the coordinate speed of light changed while the frequency of the laser light did not.

 

[exponent137]

So I ask again:
If a gravitational wave stretches the distance between the LIGO mirrors, and proportionally stretches wave length, does it cause elastic force of the LIGO arm? (In the opposite case, atoms are all the time in stable positions, although stretched.)

 

[PeterDonis]

If a gravitational wave stretches the distance between the LIGO mirrors, and proportionally stretches wave length, does it cause elastic force of the LIGO arm?

The actual "arm" of LIGO is not a material thing; it's just the distance in empty space from the detector to one of the mirrors. So there isn't anything to be subjected to an elastic force.

If you consider the underground tunnel in which LIGO is placed, then yes, a passing gravitational wave will cause very tiny strains in the walls of the tunnel (and more generally in the material of the Earth). But these strains are direct observables, so they are present regardless of what coordinates you choose--including if you choose coordinates in which the "space" in which the LIGO apparatus sits is unchanging, not being "stretched" or "compressed" at all.

 

[exponent137]

OK, strains of the walls are present, (otherwise physical effects of gravitational wave are almost unobservable.)

How we can choose coordinates in which the LIGO apparatus sits is unchanging, not being "stretched" or "compressed" at all?
Do you think that observer is moving with relativistic speed according to LIGO, or do you think that stronger gravitational field is present and this means another coordinates, or anything of both?

 

[PeterDonis]

How we can choose coordinates in which the LIGO apparatus sits is unchanging, not being "stretched" or "compressed" at all?

Suppose that each LIGO arm were a single row of atoms. We could assign each atom a space coordinate (say, based on its measured distance from the reference atom at the detector at some instant of time), and then use that space coordinate to label that atom for all time. Coordinates are arbitrary; they don't have to have any kind of physical meaning.

 

[pervect]

I will ask differently. I suppose that arbitrary strong gravitational wave cannot stretch LIGO arm so much that it can break it. This is because i suppose that stretch of the arm follows to stretch of space. Let us neglect tidal forces.

Of course this is a theoretical question, because very strong gravitational waves probably will not happen in our history.

I think that my preffered coordinate system is in rest at LIGO, thus I do not move with a rocket, or that I am not in strong gravitational field, thus as an observer I do not feel gravitational field of gravitational wave. I think that this is the simplest situation?.

I ask because I wish to understand what is stretching of spacetime.

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?

(Rather than give references, I'll assume for now that the skeptical reader will look up this point in detail, and if the issue needs further clarification it will be addressed as needed.)

We'll call the thing that the test masses are suspended from "the frame". Everyone agrees that the test masses move relative to the frame. 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.

I suspect this is a common misunderstanding of this point, due to the popularization of gravitational waves as "stretching and shrink space. But I'm not sure how to clear up this misunderstanding. I will try though.

Everyone agrees that the test masses move relative to the frame. If one's default viewpoint is based on the frame (which I rather suspect is the default viewpoint for nearly everyone), there is no such thing as expanding space, and no need to understand it.

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.

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.

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

Why do people keep using these popularizations if so many people misunderstand them? I have no idea, really, it's partly a social phenomenon. It is true that a correct understanding of what the popularization are trying to say is useful, the issue as I see it is that the popularizations practically invite misunderstanding , and that there appears to be little concerted effort to address the common misunderstandings induced by the well-intentioned popularizations.

 

[exponent137]

Maybe a simpler example is expanding of the universe. Because the universe is growing wee see red shift, speed of light is the same, this means that the wavelenght is not the same. If universe would expand and shrink very fast this would be a LIGO, a one arm LIGO. We would see blue and red shift is short succession.

This is easier to me to imagine.

If we would have one wire, which would connect two rest planets, this wire would expand and shrink, this would cause tension in the wire, One problem with understanding of expanding of the spacetime is, if we can feel some forces. I think we feel in some situation.As I understand Peter Donis, these forces are oly tidal forces.

Is it correct?

 

[Orodruin]

Maybe a simpler example is expanding of the universe. Because the universe is growing wee see red shift, speed of light is the same, this means that the wavelenght is not the same.

This is only true in compving coordinates. In general, wavelength is not a property of the light itself, but also depends on the observer. Different coordinates will give you a different view of the physics, but all invariant quantities, such as the frequency a particular observer measures, remain the same.

If universe would expand and shrink very fast this would be a LIGO, a one arm LIGO. We would see blue and red shift is short succession.

This is incorrect. The gravitational waves observed by LIGO has a frequency such that the arm length does not change significantly during the time it takes light to pass it so the light will not change its frequency ar wavelength during this time. The point is that the arm length changing with time (let us adopt a coordinate system where this is the interpretation) the interference pattern of subsequent light pulses change.

 

[exponent137]

This is only true in compving coordinates.
I thought only rest source and rest observer, but expanding and shrinking universe.

The gravitational waves observed by LIGO has a frequency such that the arm length does not change significantly during the time it takes light to pass it so the light will not change its frequency ar wavelength during this time.
I thought such exagerrated example that light will significantly change its frequency or wavelength during this time.

 

[RockyMarciano]

If you mean that the length of an object depends on your choice of coordinates, yes, that is true.

The interference pattern is a physical observable; it does not change regardless of what coordinates you adopt.

Maybe you could elaborate a little on how these two assertions can coexist, it is not evident to me. It is my understanding that the interference pattern is basically a highly magnified length measurement of the extent of the discrepance between no displacement of the fringes in the default case versus the positive case with measurable displacement. So the difference you draw between length depending on the choice of coordinates in some cases and independent of it in others looks arbitrary.

 

[Ibix]

An interference pattern is a measure of a phase difference between two waves. That there is a phase difference and what its magnitude is are unarguable and invariant. Why there's a phase difference (change in length, change in speed of light, both, other) is coordinate dependant, if I understood Peter correctly.

 

[RockyMarciano]

An interference pattern is a measure of a phase difference

And this phase difference measure is itself operationally performed as a measure of length(position difference in the fringes), which is argued by PeterDonis in the quote above to be coordinate dependent.

 

[Ibix]

No. It's a measure of phase difference, which might be induced by length difference, or refractive index difference, or wave speed difference or various other things. Which coordinates you choose determines which interpretation you put on the source of the phase difference.

 

[RockyMarciano]

No, no, I don't mean what does one interpret the phase difference to mean or to be induced by. You are missing the word "operationally" in my last post. I'm talking about how is phase difference itself measured, which is clearly explained in the technical papers by LIGO. It is a measure of difference in position of the fringes with respect to a standard.

 

[MeJennifer]

Everyone agrees that the test masses move relative to the frame. If one's default viewpoint is based on the frame (which I rather suspect is the default viewpoint for nearly everyone), there is no such thing as expanding space, and no need to understand it.

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.

The truth is that gravitational waves are non-stationary spacetime phenomena, all (comformally flat) 3 + 1 formalisms are necessarily incapable of fully explaining them.

 

[Ibix]

No, no, I don't mean what does one interpret the phase difference to mean or to be induced by. You are missing the word "operationally" in my last post. I'm talking about how is phase difference itself measured, which is clearly explained in the technical papers by LIGO. It is a measure of difference in position of the fringes with respect to a standard.

I see what you're getting at.

The standard is coordinate dependant. The motion of the fringes is coordinate dependant. The two dependencies cancel and the phase shift derived from the fringe shift is coordinate independent.

 

[pervect]

The truth is that gravitational waves are non-stationary spacetime phenomena, all (comformally flat) 3 + 1 formalisms are necessarily incapable of fully explaining them.

That is certainly a true statement, however it is totally irrelevant to the points I was attempting to make. I think you're too busy talking to listen to what I have to say - and have already said. Since nobody else is commentin (at this point), it appears to be rather difficult to proceed with a useful discussiion.

 

[pervect]

An interference pattern is a measure of a phase difference between two waves. That there is a phase difference and what its magnitude is are unarguable and invariant. Why there's a phase difference (change in length, change in speed of light, both, other) is coordinate dependant, if I understood Peter correctly.

While it's certainly possible to describe a coordinate independent result in a coordinate-independent manner, if one adopts a coordinate dependent approach, one can arrive at the coordinate-independent result in a coordinate dependent manner.

From a coordinate dependency point, the point is that any explanation based on expanding space will be coordinate dependent. "Expanding space" is a cordinate dependent idea.

To give a specific example, if we have a space-time that is empty of mass, the flat space-time of special relativity, we can regard the space-time as either being non-expanding space-time, or, if we prefer, we can regard it as an empty, but expanding, Milne universe. https://en.wikipedia.org/wiki/Milne_model.

To give a famous example, we can say that "Brooklyn is not expanding", this being a reference to Woody Allen's "Annie Hall". This is a coordinate indepenent fact. We can also say "the universe is expanding". This is also a coordinate independent fact. The ultimate goal is to understand that Brooklyn is not expanding, but the universe is expanding. "Expanding space" is an attempt at a tool to explain these facts, but sometimes I think it causes more confusion than enlightenment.

 

[PeterDonis]

Why there's a phase difference (change in length, change in speed of light, both, other) is coordinate dependant, if I understood Peter correctly.

This is not quite what I meant, no. The phase difference, which is invariant, is caused by the gravitational wave, i.e., by a fluctuation in spacetime curvature, which is also invariant. Something that causes something else cannot be coordinate dependent. The only thing that is coordinate dependent is how you choose to interpret the fluctuation in spacetime curvature. But that is just an interpretation; it's a crutch to allow our limited cognitive abilities to try to grasp what is going on. It doesn't change the physics.

 

[RockyMarciano]

The expansion tensor is a perfectly valid tensor, both theoretically and operationally.

This seems to be a simple terminology confusion, because of the dependence on sign convention of the timelike vector fields that define congruences in GR, the object referred to as expansion tensor is technically a pseudotensor.

Incorrect. The definition of the expansion tensor uses the projection tensor $h_{ab} = g_{ab} + X_a X_b$ which projects out the part of an arbitrary vector or tensor field that is orthogonal to the vector field $X$. But such projected vectors, tensors, etc. are still 4-vectors, 4-tensors, etc.

Right but see above.
Also, could you please address #19?In any case, would you please address #19?

 

[PeterDonis]

the object referred to as expansion tensor is technically a pseudotensor.

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.

would you please address #19?

I will in a separate post.

 

[PeterDonis]

Maybe you could elaborate a little on how these two assertions can coexist

Because the interference pattern is an invariant, while the length of an object is not.

It is my understanding that the interference pattern is basically a highly magnified length measurement of the extent of the discrepance between no displacement of the fringes in the default case versus the positive case with measurable displacement.

No, it isn't. This is an interpretation, not the actual physics.

The actual physics is that an interference pattern is a pattern of wave intensities on a detector. Wave intensity is an invariant, independent of coordinates, hence the interference pattern is an invariant. The wave intensities are the result of interference between incoming waves from the two arms, hence the term "interference pattern". What you are calling the "default case" is the case of no interference--in this case the wave intensity is constant everywhere on the detector. In the case of interference, the wave intensity varies from point to point on the detector; this variation in intensity is what the term "interference fringes" describes (because the characteristic pattern of variation appears as light and dark fringes). There are no such fringes in the default case because there is no variation in intensity on the detector.

 

[pervect]

Maybe you could elaborate a little on how these two assertions can coexist, it is not evident to me. It is my understanding that the interference pattern is basically a highly magnified length measurement of the extent of the discrepance between no displacement of the fringes in the default case versus the positive case with measurable displacement. So the difference you draw between length depending on the choice of coordinates in some cases and independent of it in others looks arbitrary.

Excuse me for butting in, but I think this is fairly easy to explain.

1) The interference fringes are a result of round-trip times. Round trip times are a measure of "proper length", via the current SI definition of the meter as "The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second."

2) Proper length is not coordinate or observer dependent quantity, it's a coordinate and observer independent quantity.

3) "Length", as opposed to proper length, can be an observer dependent quantity, as evidenced by the existence of "length contraction".

So the main confusion arises from words having multiple meanings - the interference fringes are caused by the sort of "length", proper length, that's observer independent.

 

[MeJennifer]

2) Proper length is not coordinate or observer dependent quantity, it's a coordinate and observer independent quantity.

I would caution everybody thinking of length in the abstract in non-stationary spacetimes (and that is what we are talking about when we deal with gravitational waves). In such spacetimes there is no such thing as an observer and coordinate independent length because there does not exist an observer and coordinate independent moment of integration.

 

[PeterDonis]

Round trip times are a measure of "proper length", via the current SI definition of the meter as "The meter is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second."

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.]

 

[RockyMarciano]

Because the interference pattern is an invariant, while the length of an object is not.

No, it isn't. This is an interpretation, not the actual physics.

The actual physics is that an interference pattern is a pattern of wave intensities on a detector. Wave intensity is an invariant, independent of coordinates, hence the interference pattern is an invariant. The wave intensities are the result of interference between incoming waves from the two arms, hence the term "interference pattern". What you are calling the "default case" is the case of no interference--in this case the wave intensity is constant everywhere on the detector. In the case of interference, the wave intensity varies from point to point on the detector; this variation in intensity is what the term "interference fringes" describes (because the characteristic pattern of variation appears as light and dark fringes). There are no such fringes in the default case because there is no variation in intensity on the detector.

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). Interferometry is actually used in defining and calibrating length standards and gauges.

 

[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?