I LIGO light changes frequency not wavelength

[pervect]

I have some general comments about the discrepancies between the geodesic congruence and the Born rigid congruence as far as its impact on distance goes. Finding the exact expression for the congruence whose expansion is exactly zero should be theoretically possible but I didn't do it, it would be a lot of work. I settled for an expression that I believe should be accurate to the second order. To be specific, since f(t) has a magnitude of 10^-21, I counted the "order" of the discrepancy by the number of times f or any of its derivatives were multiplied together. The lowest order term in the expansion by this criterion was one proportional to $f \, df/dt$.

The geodesic congruence has an expansion of the first order in f, namely:
$$\frac{\frac{df}{dt}}{1 + 2\,f(t)}$$

Thus the expansion of the almost-rigid congruence is about 10^21 times lower than the expansion of the geodesic congruence, because it's proportional to $f df/dt$ and f is so small.

For the metric
$$-dt^2 + \left( 1 + 2 f(t) \right) ds^2 $$

the almost expansion-free congruence is given by:

$$\partial_t - s \frac{df}{dt} \partial_s$$
Actually one needs to normalize this first to calculate the expansion, but it's easier to write without the normalization.

Physically, we can interpret this as saying the geodesic congruence is approximately moving with some velocity $v = -s \frac{df}{dt}$ relative to the rigid congruence. Because of this relative motion, we expect discrepancies in second order in the relative velocity $v = s\,df/dt$ due to Lorentz contraction. We are assuming f is small, and we are assuming that s is small enough that s*f is still small, and we are assuming that f is changing slowly enough that df/dt is also small.

Because $f(t) \approx 10^{-21}$ in Ligo, agreement to the second order means that the two concepts of distance agree to about 1 part in 10^20 or so. (Possibly a little less, depending on how big s is, and how fast f changes - but still an excellent approximation).

 

[GeorgeDishman]

You can't ignore the small internal stresses and the strains they cause, because those are what keep the congruence Born rigid. A geodesic congruence in the presence of spacetime curvature will not, in general, be Born rigid, because tidal gravity will cause geodesics to deviate.

I don't intend to ignore them, quite the opposite in fact. A major part of my confusion is how we deal with the stresses in what was called the "Earth frame" earlier in the thread, but I need to explain why they're giving me a headache. The consensus here seemed to be that it was so obvious it didn't need to be discussed but I'm struggling with it.

You keep on talking as if the LIGO device is a single tube with test masses at each end. It isn't; it's an L-shaped pair of tubes with the laser/detector at the corner of the L.

As I have said several times, I appreciate that but I think it can be dealt with fairly easily by choosing to analyse the situation where one arm is in the orbital plane of the binary system and the other is parallel to the axis. In the orbital plane, there is only plus polarisation so the signals from the arms are then the same but anti-phase.

It is my intention to extend the analysis to cover the other cases too though I'm just keeping it simple to start with.

Even if you only consider one arm of LIGO, that arm does not have two test masses in it; it only has one, at one end of the arm, and the laser/detector at the other. Since you keep on bringing up diagrams and videos of LIGO, it is going to be a lot easier if you talk about LIGO's actual configuration.

Good idea. The first attached image is the actual layout from page 3 of http://arxiv.org/abs/1411.4547:

As you can see, there are two mirrors or 'test masses' labelled ITM and ETM in each arm. The light bounces between the ITM and ETM approximately 280 times (that's an average of course, a small fraction of the photons get through on each pass). A displacement of the ITM towards the ETM by dx reduces the path length within the cavity by 560dx but only increases the path from the beamsplitter to the ITM by 2dx (once on input and once on output). Relative to a fiducial point in the middle, the ETM would move by -dx which also reduces the path length by 560dx. Overall, for displacements of each by dx and -dx, the optical path length changes by 2*560-2 = 1118dx. Obviously that is dominated by the change within the 4km cavity between the ITM and ETM. Ignoring the short distance from the beam splitter to the ITM (tens of metres at most) only introduces a small scaling error, much less than 0.1%. In other words, LIGO measures the length of the two cavities which are shown in the diagram as "4km" long and containing laser at a power level of "750kW".

The second image shows the suspension system typical of one of the four ITM and ETM mirrors. As you can see, there is nothing to stop their motion along the beam direction for either mirror so taking a fiduial point in the centre of the tube is reasonable IMHO. Of course the location we choose cancels out in the calculations anyway so it is a moot point.

It depends on the coordinates. As I understand it, in the preferred coordinates of the LIGO team, light still travels at $c$ and light worldlines will still be 45 degree lines.

OK, that's interesting but I'll leave it for another day.

This is probably getting beyond what we can discuss in a PF thread, unless you have a reference that analyzes this configuration.

I don't. I looked around and couldn't find anything so when I found out about "PF Insights" last year, I suggested it there. There were no takers so I thought I'd try to work it out myself and as I have recently got the opportunity to start learning MATLAB, this seemed like an ideal training project. I was making good progress until I hit the problem of understanding what the "Earth frame" means.

 

[PeterDonis]

there is nothing to stop their motion along the beam direction for either mirror so taking a fiduial point in the centre of the tube is reasonable IMHO

The fiducial point for the LIGO apparatus is at the corner of the "L" (the beam splitter). It is not in the middle of either arm. If you want to analyze the actual signal output by LIGO, that is the fiducial point you need to use. If you analyze with respect to a fiducial point in the middle of an arm, you will not be analyzing the same signal that LIGO outputs.

A major part of my confusion is how we deal with the stresses in what was called the "Earth frame" earlier in the thread

The paper you linked to makes it clear that all of the key components of the LIGO detector are free to move in the "detector plane" (the plane defined by both arms, considered as two intersecting lines--we can think of this plane as being "horizontal", perpendicular to the local direction of "gravity"). So none of the components will be stressed in that plane. (The tunnel walls will be, but they play no role in determining the signal LIGO outputs.) There will obviously be stresses perpendicular to that plane, but those should not affect motion in that plane. I believe this is the basic argument used by the LIGO team to justify ignoring stresses when analyzing the signal (though of course they have to account for those stresses in the complicated apparatus that tries to keep all of the key components in the same plane and isolate them from outside disturbances).

 

[GeorgeDishman]

The fiducial point for the LIGO apparatus is at the corner of the "L" (the beam splitter). It is not in the middle of either arm. If you want to analyze the actual signal output by LIGO, that is the fiducial point you need to use. If you analyze with respect to a fiducial point in the middle of an arm, you will not be analyzing the same signal that LIGO outputs.

If you understand why the mirror on the ITM is present, it should be clear that it is the length of the Fabry-Perot cavity that is measured, not the distance to the corner of the L. In addition, the signal (i.e. the phase difference at the photodetector) should not depend on the choice of the fiducial point (or the point wouldn't be "fiducial" in the way I understand the word).

However, what you are missing is that I don't "want to analyze the actual signal output by LIGO", I want to produce a visualisation of the gravitational waves to which it is responding.

The paper you linked to makes it clear that all of the key components of the LIGO detector are free to move in the "detector plane" (the plane defined by both arms, considered as two intersecting lines--we can think of this plane as being "horizontal", perpendicular to the local direction of "gravity"). So none of the components will be stressed in that plane. (The tunnel walls will be, but they play no role in determining the signal LIGO outputs.) There will obviously be stresses perpendicular to that plane, but those should not affect motion in that plane. I believe this is the basic argument used by the LIGO team to justify ignoring stresses when analyzing the signal (though of course they have to account for those stresses in the complicated apparatus that tries to keep all of the key components in the same plane and isolate them from outside disturbances).

Right, so if you imagine a wave propagating in the direction perpendicular to the detector plane (i.e. vertical) and such that the orbital plane just happens to align with one arm, then you get the toy model I've been describing. We can consider the distortion that the GW would consider in that orientation. Then, for any realistic situation, the actual wave will have components of both plus and cross polarisation and that combination will need to be projected onto the orientation of the detector arms to take account of the effect of the suspension.

That's a separate part of the problem, all I am trying to do is the first part, to produce a visualisation of how the GW would influence a constellation of free-falling test masses whose only constraint is that they are suspend against (or we simply ignore) the basic gravitational acceleration towards the binary system. For realistic scenarios, a few dozen solar masses thousands or billions of light years away aren't going to be significant in terms of GM/r², right?

 

[PeterDonis]

it is the length of the Fabry-Perot cavity that is measured, not the distance to the corner of the L.

More precisely, because of the "magnification" effect of the cavity, the contribution to the overall distance measurement of the distance from the ITM to the beam splitter is very small (a fraction of a percent). That is because the length of the cavity is magnified by a factor of 1000 or so while the ITM-beam splitter distance is not.

the signal (i.e. the phase difference at the photodetector) should not depend on the choice of the fiducial point

It does for the LIGO team's analysis because their "fiducial" point is part of the overall construction of the coordinate chart they are using. It is true that, in principle, any choice of chart should give the same values for actual observables like the interference pattern at the photodetector, so in that sense the choice of fiducial point doesn't matter.

I don't "want to analyze the actual signal output by LIGO", I want to produce a visualisation of the gravitational waves to which it is responding.

Then that is off topic for this thread and you should start a new one.

 

[GeorgeDishman]

More precisely, because of the "magnification" effect of the cavity, the contribution to the overall distance measurement of the distance from the ITM to the beam splitter is very small (a fraction of a percent). That is because the length of the cavity is magnified by a factor of 1000 or so while the ITM-beam splitter distance is not.

Exactly.

It does for the LIGO team's analysis because their "fiducial" point is part of the overall construction of the coordinate chart they are using. It is true that, in principle, any choice of chart should give the same values for actual observables like the interference pattern at the photodetector, so in that sense the choice of fiducial point doesn't matter.

Thank you.

Then that is off topic for this thread and you should start a new one.

OK, I'll drop it at that then. However, I think the point that what is measured in practice is the phase delay resulting from the influence of the GW on the F-P cavity relates back to the original question. Thanks for your responses.