How Does LIGO Detect Gravitational Waves Despite Changes in Spacetime?

[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.