B How Does LIGO Detect Gravitational Waves Despite Changes in Spacetime?

[asca]

Just for the sake of completness, I found this article surfing the net :
https://pdfs.semanticscholar.org/393a/af6b1ced305ee40d175d5f3c3a2b6020348d.pdfHowever I'd just like to share the following: what happens to the clocks in the "x" arm of the example? Isn't their pace slower than the pace of clocks placed at the beam splitter? so for any observer in the x arm the time it takes for a crest to "reach the subsequent crest" (I hope you see what I mean by that) is shorter than the time measured by a clock placed at the beam splitter. But light speed is the same, so for any observer in the x arm the "distance" between two crests is shorter than the same distance measured by an observer placed at the beam splitter, in other word his ruler is longer than an identical ruler placed at the beam splitter. The solution to my puzzle maybe lies in the assumption the GW effect was and is different, actually opposite, in the two arms, and in the assumption that the observation point is somehow not (or less or more) affected by the GW passing by. So any obesrver in the x arm observing juts what happens in the x direction would not realize a GW is passing by, the same for any observer in the y arm, only the comparison by a third observer of the X and Y observations can deduct that a GW has passed by. That is probably the key.

 

[asca]

Sorry, I'm not so sure about what I just wrote: my question is what happens to the clocks in the X arm and to the clocks in the Y arm? I'm not so sure about the aswer I gave before, because the clock should not be affected by the direction of observation.

 

[Ibix]

That paper has specifically chosen a coordinate system where the passage of time is unaffected by the gravitational wave. Free floating clocks will remain synchronised throughout.

 

[asca]

Meanwhile I found an even clearer (at least to me) answer to my initial question.


Still I remain puzzled by the new question: what happens in general to the clocks placed in the X and Y arms? Why does Ibix say that their pace does not change?

 

[Ibix]

Why does Ibix say that their pace does not change?

Because that's what the maths says. For a free floating clock at rest in the coordinate system in use, $dx=dy=dz=0$, and the metric is diagonal with $g_{tt}=1$. Why do you think they should change?

 

[PeterDonis]

any obesrver in the x arm observing juts what happens in the x direction would not realize a GW is passing by, the same for any observer in the y arm, only the comparison by a third observer of the X and Y observations can deduct that a GW has passed by

That's not correct; observing just one arm can still tell you that a GW passed, because the round-trip travel time of light in the arm changes. However, it's much harder to measure that change in round-trip travel time in a single arm, then it is to measure the interference between the light in the two perpendicular arms. So an interferometer detector like LIGO is more sensitive than a single-arm round-trip travel time would be.

 

[PeterDonis]

what happens to the clocks in the "x" arm of the example?

Nothing. By the analogy the paper you linked to makes with cosmological models: as the universe expands, clocks are not affected, only distances are. Similarly, as the GW passes, clocks are not affected, only the lengths of the arms are.

 

[pervect]

Still I remain puzzled by the new question: what happens in general to the clocks placed in the X and Y arms? Why does Ibix say that their pace does not change?

Because that's an English language statement of the metric given in 2.1 in the paper you cite:

$$ds^2 = -c^2\,dt^2 + [1+h(t) ]\,dx^2 + [1-h(t)]\,dy^2 \quad [2.1]$$

In this equation, h(t) - the gravitational wave - doesn't modify the relationship between dt (coordinate time) and ds (which represents either distance or time intervals, which are unified in special relativity). So h(t) doesn't have any effect on time.

h(t) does modify the relationship between dx and dy (spatial coordinates) and ds, so the gravitatioanl wave does have an effect on space.

I'm not sure what pre-requisite knowledge you have, pessimestically I tend to assume you have little :(. Still, I'll take the risk of talking in ways that may go "over your head", as you seem to have some interest in the topic.

So, trying to thing about what you might need to know - you might start looking into what a metric is, as that's the mathematical key to answre your quesitons.

A good approach might be to start with understanding the pythagorean theoerm - the square of the hypotenuse is the sum of the square of the other two sides, then building upon this knowledge to understand what a purely spatial metric is. The translation of the pythagorean theorem in the language of metrics would be ds^2 = dx^2 + dy^2. s represents distance, x and y are coordinates, and d represents "a change in" or "differential". This would be hopefully familiar from calculus, if you have it. If you don't have calculus yet, that's another thing to add to your list of things you'd need to find out about.

At this point, you'd need to know what coordinates are. It's easy enough to state that coordinates are just lablels, but I've noticed in the past that sometimes this doesn't seem to be accepted by some PF posters, I'm not quiite sure what the difficult is.

Given that you know what coordinates are, and have some notion of what distances are, and the funamentals of calculus, then the metric allows you to take these coordinate labels (or rather their differentials), and computes differnetial distances from them.

After that, all you need to do is learn special relativity in general, and the metric approach to special relativity (the Lorentz interval), specifically.

After that, you'll have the needed bases to tackle this issue again with a firmer foundation. I've probably skipped a few steps of things you'd need to know along the way, but that's the short outline as I see it.

 

[asca]

Everything is clear now folks. When I wrote that stuff about the clocks I was momentarily carried away by the thought of a static gravitational field, but this is not the case when a wave is passing by, so I was really "off tune", sorry.

 

[pawprint]

Summary: How can an interferometer detect Gravitation waves, if the change in space time due to gravity affects all the rulers (and clocks) in that spot?

Some six or seven years ago I asked the question in here- "is detection based on (1) the length of the arm geodesics or (2) the movement of the test masses?". I received a single response who stated that it was (2). There was no further post on the thread so I have accepted this answer since. I concede that this is not the generally accepted point of view.

 

[Ibix]

the length of the arm geodesics

I'm not sure what you mean by "arm geodesics". There are many geodesics lying in the worldsheet of an arm - which ones do you mean? Perhaps you could link to your original question - you seem to have started five threads, none of which has fewer than two replies.

The point is, essentially, that there are many ways to describe "how a LIGO interferometer works". The test masses move, or the arms change length, or even that the (coordinate) speed of light changes. They're all different descriptions of the same thing - like calling a rectangular table either long and narrow or short and wide. It depends how you want to do it.

 

[Benish]

Would someone please provide a link to a spacetime diagram corresponding to the most cogent of the various explanations provided in this long discussion?

ascu may be convinced, but surely the question deserves a clear graphic answer to clinch it.

 

[Ibix]

I'm struggling to imagine what spacetime diagram you could draw that would be helpful. To be honest, I don't think you can really top the motion of free-falling test particles, as seen in this animation at Wikipedia. Simply imagine the beamsplitter in the middle and the mirrors attached to the 12 o'clock and 3 o'clock dots. You could take the frames from that gif and stack them on top of each other to get a spacetime diagram of sorts, one that regards the non-inertial LIGO tunnel walls as fixed references, but I'm not sure it adds anything.

Saulson's paper, linked in #51, is also good. It likens the stretch-and-squish to metric expansion in cosmology and examines a "step" gravitational wave to show that frequency changes are essentially transients and not relevant to the detection.

Edit: it's also worth noting that it's the mathematical description that's important, and no Euclidean graphical representation is going to completely express a 4d non-Euclidean structure.

 

[pervect]

Would someone please provide a link to a spacetime diagram corresponding to the most cogent of the various explanations provided in this long discussion?

ascu may be convinced, but surely the question deserves a clear graphic answer to clinch it.

It's hard to say what explanation will be "cognent" for you. I'll pick one that's easy to draw.

You can not draw a space-time diagram that's to scale on a flat sheet of paper. So we need to introduce the concept of a scale factor. We'll do that with a dashed line, labelled "scale factor"

The dashed line is of constant (proper) length.

So, on the space time diagram, the vertical lines represent the position of the test masses, which in the diagram have constant x coordinates, so they are just vertical lines. However, while the coordinates are constant, the coordinates do not represent distance in a uniform manner because of the time-varying scale factor. So the coordinates have no direct physical significance, they are convenient labels to describe the geometry.

The dashed lines on the diagram represent a constant proper distance. So they represent the scale factor, as one might see on a map. As you can see from the diagram, this scale factor changes with time. So while the free-floating test masses have constant x-coordinates, these coordinates on the diagram are not and cannot be "to scale". The distances on the diagram are represented by the dashed lines representing the scale factor, you can think of them as representing rulers of fixed proper length. So to recap, the diagram isn't to scale because it can't be, to understand the diagram one needs to understand the graphical representation of the time-varying scale factor.

 

[pervect]

I wanted to add a few things to my post. The previous post is OK as far as it goes, but it only shows one spatial dimension, not three. The actual GW has three spatial dimensions, two transverse spatial dimensions, plus the direction of propagation, and of course time. On refelction, I think that the "can't draw on a flat piece of paper" argument really applies only to the full GW. I don't currently see any reason why one couldn't draw the worldlines of the two test masses as below for the 1+1 slice.

The second choice as drawn above seems a bit more intuitive to me, though the first choice represents the usual usual math better. The two different diagrams represent two different descriptive approaches that yield the same experimental results. The first might be called "expanding space", the second could be interpreted as the test masses actually moving due to "gravitational forces".

 

[Ibix]

On refelction, I think that the "can't draw on a flat piece of paper" argument really applies only to the full GW. I don't currently see any reason why one couldn't draw the worldlines of the two test masses as below for the 1+1 slice.

This is the diagram I had in mind in my last post - if you stack up the frames of the animation from Wikipedia and then cut the stack vertically and look at the cut edge, you get your last diagram.

 

[Orodruin]

I don't currently see any reason why one couldn't draw the worldlines of the two test masses as below for the 1+1 slice.

As long as they do not cross and the metric is not specified, you can draw your world-lines however you want ...

 

[pawprint]

I'm not sure what you mean by "arm geodesics".

The geodesic which comes to mind is that traced by the laser beam within the arm.

The point is, essentially, that there are many ways to describe "how a LIGO interferometer works".

I agree. In an article by Lee Billings in the November 2019 Scientific American on Page 56/7 he says- "...a passing gravitational wave briefly stretches and shrinks spacetime, altering the chambers' lengths (and thus the total distance a beam of light travels.)". On page 59 the author of a caption for a related photograph takes the opposite view- "To ensure that KAGRA's lasers can accurately register the almost impercerptible distortions of its mirrors caused by gravitational waves...". It is the first view that is favoured by the majority of 'Popular Science' authors and reporters.

This thread began with...

Summary: How can an interferometer detect Gravitation waves, if the change in space time due to gravity affects all the rulers (and clocks) in that spot?

to which the answer is that the rulers and clocks are indeed affected; but in proportion to each other so
that they cancel each other out and C is conserved. Without the test masses' freedom to move toward and away from the laser source (i.e. along the axis of the arm) LIGO type instruments would be deaf to gravitational waves.