I LIGO and light speed constancy

[timmdeeg]

When a GW wavefront hits a hunk of matter it suddenly finds itself squished or stretched. It responds, of course, but only on a time scale commensurate with the velocity of sound in the material. In this limit the ends of the bar and the mirrors are in free fall.

Hm, interesting, but wouldn't that mean that in this case the sensitivity is comparable to LIGO's suspended mirrors? We are talking here about a non-resonant case.

 

[PeterDonis]

It responds, of course, but only on a time scale commensurate with the velocity of sound in the material.

This is true if you are talking about collective motion or vibration of the entire bar. But if you are talking about the motion of atoms at the end of the bar in response to the stretch or squeeze coming from a GW, the relevant response time is the time it takes for them to move enough away from their equilibrium positions with respect to neighboring atoms, to have those neighboring atoms exert a restoring force on them. That time scale, for a typical GW with a frequency of a few tens or hundreds of Hz, will be much faster than the time scale for the GW to stretch or squeeze the atoms. So the response of either end of the bar to the GW will be damped, compared to the response of the LIGO mirrors, which, as you note, are designed specifically to have no forces acting on them at all (to the greatest accuracy possible).

 

[Paul Colby]

Hm, interesting, but wouldn't that mean that in this case the sensitivity is comparable to LIGO's suspended mirrors? We are talking here about a non-resonant case.

Well, there are two limiting cases both non-resonant, both quite different. In case 1) the frequency of GW is well below the lowest mechanical resonance, while in case 2) the GW frequency is well above the lowest mechanical resonance. By well above and well below let's say 3 orders of magnitude in each case. I was referring to case 2, not case 1, in the quoted text. In case 1) the ends of the bar can and will move to compensate for spatial distance changes due to the slowly varying metric. In case 2 the bar ends simply can't move fast enough to compensate for the distance change due to the GW. Clearly, much depends on the size of the bar.

For measuring very high frequency GW one might well mount mirrors on a ridged mount because at 100MHz GW the mount is essentially provides an inertial mirror. This, however, is not what I would do.

 

[Paul Colby]

This is true if you are talking about collective motion or vibration of the entire bar. But if you are talking about the motion of atoms at the end of the bar in response to the stretch or squeeze coming from a GW, the relevant response time is the time it takes for them to move enough away from their equilibrium positions with respect to neighboring atoms, to have those neighboring atoms exert a restoring force on them. That time scale, for a typical GW with a frequency of a few tens or hundreds of Hz, will be much faster than the time scale for the GW to stretch or squeeze the atoms. So the response of either end of the bar to the GW will be damped, compared to the response of the LIGO mirrors, which, as you note, are designed specifically to have no forces acting on them at all (to the greatest accuracy possible).

The equations of motion given in post #58 are relevant to the discussion. For an atom in the interior of the bar the forces do to metric strain are symmetric and cancel. This is the meaning of, $T_{jk,j}=0$. There is no change in an interior atom's reference position due directly to GW. For atoms on the boundary this is not the case since there are no atoms exterior to the bar to compensate for the change in interatomic distances. For an atom on the boundary the distance change to it's neighbor is all that matters as far as the force goes. This distance change is $~10^{-21} \delta$ where $\delta$ is the interatomic distance. Typical interatomic distances are on the order of angstroms so $\delta ~ 10^{-31}$ meters which isn't large. So in case 2) as defined in #63 the distance between mirrors will be roughly $10^{-21} L$ where $L$ is the length of the bar. In case 1) it will be much smaller than this.

 

[PeterDonis]

the distance between mirrors will be roughly $10^{-21} L$

I think you mean the change in the distance between mirrors. More precisely, it's the amplitude of the oscillation in the change in the distance between mirrors. This is true for the LIGO mirrors (where the applicable $L$ is the length of the LIGO arms), but I don't think it's true for mirrors mounted on a bar, because, as I said, the inter-atomic forces on the atoms at the ends of the bar, which are the ones, as you say, that will be displaced from their equilibrium positions by the GW, can respond much faster than the time scale of the GW vibrations. That means the vibrations will be damped; the amplitude of the oscillation in the change in bar length will be smaller than $10^{-21} L$, which is the amplitude of the GW oscillations in the distance between hypothetical freely falling test particles that start out at the equilibrium positions of the atoms at opposite ends of the bar.

 

[Paul Colby]

This is true for the LIGO mirrors (where the applicable LLL is the length of the LIGO arms), but I don't think it's true for mirrors mounted on a bar, because, as I said, the inter-atomic forces on the atoms at the ends of the bar, which are the ones, as you say, that will be displaced from their equilibrium positions by the GW, can respond much faster than the time scale of the GW vibrations.

I discussed two limiting cases which are quite distinct. One in which your statement is correct and one in which it is not. If one were to magically remove all interatomic forces between atoms (and neglect their thermal motion) no change from their reference position (reference position as defined in the theory of linear elasticity) would occur independent of the frequency or time scale of the GW. The distance between atoms is changed by GW but not the reference position as defined by the equations of motion given in #58. For LIGO frequencies the answer depends on the length of the bar and the speed of sound in said bar.

 

[DanMP]

@PeterDonis and @Ibix - many thanks for your replies.

 

[PeterDonis]

The distance between atoms is changed by GW but not the reference position as defined by the equations of motion given in #58.

This is quibbling over terminology. The observed interference pattern at the LIGO detector has a given amplitude in response to a given gravitational wave. If the mirrors were mounted on bars instead, the amplitude of the observed interference pattern at the detector, for the same gravitational wave, would be smaller. That is what I am saying, and it's true regardless of how you define your "reference position" (which basically means how you choose your coordinates).

 

[timmdeeg]

In case 2 the bar ends simply can't move fast enough to compensate for the distance change due to the GW.

For measuring very high frequency GW one might well mount mirrors on a ridged mount because at 100MHz GW the mount is essentially provides an inertial mirror. This, however, is not what I would do.

Lets consider case 2: "the GW frequency is well above the lowest mechanical resonance." Which seems to fit with "very high frequency GW" (second sentence). If the mirrors are inertial here then why aren't they inertial in your case 2 description "can't move fast enough to compensate for the distance change due to the GW."?

 

[Paul Colby]

Lets consider case 2: "the GW frequency is well above the lowest mechanical resonance." Which seems to fit with "very high frequency GW" (second sentence). If the mirrors are inertial here then why aren't they inertial in your case 2 description "can't move fast enough to compensate for the distance change due to the GW."?

It all depends on how one defines the word "move". For an extended object, like a bar, one must carefully define what this means to avoid confusion. First we pick a reference frame. The reference frame I choose is the one in which the bar is at rest prior to the arrival of the GW. I further specify the coordinate system known as the Transverse Traceless one. This is a coordinate system in which inertial mass points at rest with respect to the coordinates prior to the GW arrival, remain at rest relative to the coordinates during and after the departure of the GW. Remember, the metric is time dependent so distances between points are still changing even between points at rest with respect to these coordinates.

I select this coordinate system because the equation of motion of an elastic solid, specifically what I was asked to comment on, are valid in their classical form. In this coordinate system "can't move fast enough" is referring to points within the bar which remain inertial or unaccelerated relative to the coordinates for some period of time after the arrival of the GW. By the equations of motion the surrounding matter exerts no net force on interior points of the bar. The passing GW applies a metric strain to the bar. This strain results in a stress field by Hooks law. However, this stress field has 0 divergence. By the material equations of motion, the effect of the GW on the bar is only felt through traction[1] forces which only appear on the bar boundary.

Now, let's take a bar 10m long and a GW pulse short enough in duration sound generated by the forces on the boundary only travels 1 mm. The GW will have a spatial wavelength dictated by the speed of light and so will be much greater the 10m. Such a GW pulse will act uniformly over the entire bar volume while the acoustic induced mechanical vibration barely has time to propagate 1 mm into the bar volume. In this limit the bar's overall length change will be very nearly the inertial limit. The difference between truly inertial mirrors and ones constrained by the bar will be small, on the order of 1mm/10m as the wave passes.

[1] Forces may be tangential as well as normal to the surface.

 

[PeterDonis]

a GW pulse short enough in duration sound generated by the forces on the boundary only travels 1 mm

So, if the speed of sound in the bar is about 5000 m/s (a roughly correct value for steel), this means the GW pulse duration is 200 nanoseconds, or the time it takes light to travel about 200 feet or 61 meters.

The GW will have a spatial wavelength dictated by the speed of light and so will be much greater the 10m.

This is true, but it doesn't mean what you appear to think it means. The "spatial wavelength" here is longitudinal, along the direction of propagation of the GW--it basically tells you how far apart surfaces of constant phase are along the direction of propagation along the GW. But that is not the same as the transverse amplitude of the GW, which is the relevant comparison with 10m. The tranverse amplitude, by assumption, will be $10^{-21}$, which is a dimensionless amplitude: it means that fraction of whatever transverse distance we are talking about. So, for example, if the bar is 10 m long and is oriented exactly transverse to the GW, then, in the absence of inter-atomic forces in the bar, the amplitude of the length variation in the bar will be $10^{-21}$ times 10m, or about $10^{-20}$ meters.

However, inter-atomic forces are not absent. Here is how I would analyze your scenario. In your chosen coordinate chart, the GW pulse arrives at both ends of the bar simultaneously. Since sound can travel in the bar only 1mm during the length of the pulse, the two ends of the bar, and in fact any two pieces of the bar separated by well over 1mm, move independently during the pulse. (Note that I didn't have to say anything about the GW's spatial wavelength in order to make that statement.)

However, atoms in the bar are much closer than 1mm apart, so we cannot assume that atoms are unaffected by neighboring atoms during the passage of the pulse. If we assume that atoms in the bar are roughly 1 nm apart (which is probably an overestimate), and that forces between neighboring atoms propagate at the sound speed in the bar (which is a substantial underestimate, since the macroscopic sound speed is the collective effect of many inter-atomic interactions and is slower than the individual interactions are), then neighboring atoms will affect each other's motion on a timescale of 0.0002 nanoseconds (200 femtoseconds), or a million times as fast as the GW pulse time. So we should assume that an atom at the end of the bar will not be able to move inertially in response to the GW pulse; its motion will be constrained by inter-atomic forces, so the amplitude of its vibration in response to the GW will be much smaller than that of an inertially moving particle. In your chosen coordinates, the coordinates of an atom at the end of the bar will change in response to the GW, while the coordinates of an inertially moving particle (such as a LIGO mirror) would stay the same.

 

[Paul Colby]

Well, I did caution I would try to answer. I've explained my understanding in terms of the actual equations of motion for an elastic solid in the weak GW limit. As far as I can see everything is in order with that explanation. I suggest people interest in this subject (it's kind of far afield of the original topic) ask it in a new thread. I also recommend reading up on continuum mechanics which itself I find complicated subject even without the help of GR. I don't see how these types of questions can be answered without referring to the basic equations of motion to guide the discussion.

 

[PeterDonis]

I don't see how these types of questions can be answered without referring to the basic equations of motion

Sure, but when you say the ends of the bar move inertially, you are ignoring the equations of motion, because the equations of motion, which give a linear stress-strain relationship (Hooke's Law), require that the motion of any small piece of the bar is not inertial--it is subject to forces from neighboring pieces. The forces in the interior of the bar have zero divergence, but that is not true at the boundary of the bar, which is why I specifically talked about atoms at the ends of the bar in my previous post.

 

[Paul Colby]

Sure, but when you say the ends of the bar move inertially, you are ignoring the equations of motion, because the equations of motion, which give a linear stress-strain relationship (Hooke's Law), require that the motion of any small piece of the bar is not inertial--it is subject to forces from neighboring pieces.

No, I took the equations into account and provided an estimate of the error involved. The fractional error is of order 1mm/10m ~ $10^{-4}$. So the non-inertial distance error incurred is very roughly $10^{-28}m$ which is very close to negligible compared with the purely inertial value $10^{-20}m$.

 

[PeterDonis]

I took the equations into account and provided an estimate of the error involved.

I don't see your analysis as doing that. I see your analysis as assuming that the motion of atoms at the ends of the bar are inertial, and then you waving your hands and saying the error involved should be the ratio 1mm/10m, which, as I explained in my previous post, is not relevant for what you are using it for.

I'm also a little unclear about your zero divergence condition. You write it as $T_{ij,j} = 0$. Are you ranging $i, j$ over all four spacetime indexes, or just over the three spatial indexes in your chosen coordinate chart?

 

[Paul Colby]

I don't see your analysis as doing that. I see your analysis as assuming that the motion of atoms at the ends of the bar are inertial, and then you waving your hands and saying the error involved should be the ratio 1mm/10m, which, as I explained in my previous post, is not relevant for what you are using it for.

I'm also a little unclear about your zero divergence condition. You write it as $T_{ij,j} = 0$. Are you ranging $i, j$ over all four spacetime indexes, or just over the three spatial indexes in your chosen coordinate chart?

Sorry, I assume the "analysis" was obvious. I'll settle for correct . Clearly double counted in my haste.

The bar is of length $L$. We've established (okay, you seem to partially admit to it) that the influence of a GW on an isotropic linear elastic material is through the traction forces applied to the boundary because the induced stress field is divergence free. This I take as a true statement independent of time scale. Let us take $L=10m$ with a circular cross section of 10cm. All traction forces applied by the GW tangent (shear) or normal to the bar's sides do not contribute to the bars length change. Only the normal forces on the bar ends do. The short GW pulse hits resulting in a force on the end caps of the bar. This force then starts accelerating the atoms on the bar ends in the opposite direction of the GW induced length change. The effect of this acceleration propagates in 1mm over the duration of the wave. I consider it established that the $10m - 2mm$ of the bar is moving inertially by virtue of the equations of motion. The inertial change of the bar is $10^{-21}L$ except for the 1mm on each end. The motion (length change of these) is of the order of $10^{-21}\times 2mm$ on the outside, or $2\times 10^{-24}m$ which is still negligible compared to the over all bar length change of $10^{-20}m$.

The indices all run over just the space coordinates.

 

[PeterDonis]

the induced stress field is divergence free.

I'm not sure that's correct as you are defining "divergence free". See below.

The short GW pulse hits resulting in a force on the end caps of the bar.

No, you have it backwards. The "motion" induced by the GW pulse itself is inertial--it's changing the metric of spacetime, such that inertial motion, instead of leaving all of the atoms in the bar at rest relative to one another, makes them move relative to one another ("move" in the sense of "the round-trip travel time of light signals between them changes"). The GW pulse itself exerts no force on anything.

The only force, in the GR sense, that is present is the force exerted by atoms in the bar on neighboring atoms. This force is induced by the relative motion (in the sense I defined it above--changing round-trip light travel times) between those atoms and neighboring atoms. That is the force that prevents the atoms at the edge of the bar from moving inertially in response to the GW pulse. That force is not present for the LIGO mirrors, because they are designed to eliminate all such forces.

The indices all run over just the space coordinates.

Then I'm not sure that your divergence free expression is correct. It's correct if you run over all 4 spacetime coordinates, because that's enforced by the Einstein Field Equation and the Bianchi identities. But I'm not sure it's correct for just the space coordinates. For that to be true, all of the terms in the Bianchi identities involving the time coordinate would have to vanish identically, and I'm not sure they do.

 

[PeterDonis]

I consider it established that the 10m - 2mm of the bar is moving inertially by virtue of the equations of motion.

I disagree. As far as I can tell, the "equations of motion" you refer to are Hooke's Law, but Hooke's Law is correctly applied to the inter-atomic forces, not to any "force" applied by the GW pulse itself (since there is no such force, per my previous post). And Hooke's Law tells you that the motion of atoms at the edge of the bar is not inertial, because those atoms are subjected to forces from the neighboring atoms, which are not canceled by anything, and which, per an earlier post of mine, act on a much faster timescale than the GW pulse does.

(As I noted in my previous post, I'm also not sure about the divergence-free condition as you state it. But that's a separate question from what happens to the atoms at the end of the bar, since it's obvious that the forces on those atoms from other atoms in the bar cannot all cancel out.)

 

[Paul Colby]

No, you have it backwards. The "motion" induced by the GW pulse itself is inertial--it's changing the metric of spacetime, such that inertial motion, instead of leaving all of the atoms in the bar at rest relative to one another, makes them move relative to one another ("move" in the sense of "the round-trip travel time of light signals between them changes"). The GW pulse itself exerts no force on anything.

Well, this confusion is why I chose the coordinate frame as indicated. Let me ask this, let the bar be along an axis parallel to the GW wavefront. Let the GW be a square wave such that at $t=0$ the distance between the coordinate points located at ends, $x_1$ and $x_2$, of the bar suddenly decrease by $10^{-21}(x_1 - x_2)$. Does the instantaneous length of the bar remain unchanged in your world view or does it shrink leaving the endpoints at the same coordinate value they were initially? Continuity suggest (actually demands but what ever) that the ends remain at there original coordinate location the instant the wavefront hits. If you're suggesting that the bar length is continuous then the endpoint would have to jump to a new location with infinite acceleration.

 

[PeterDonis]

Let the GW be a square wave such that at $t=0$ the distance between the coordinate points located at ends, $x_1$ and $x_2$, of the bar suddenly decrease by $10^{-21}(x_1 - x_2)$

That's not what will happen. The GW causes the metric coefficients to change continuously. It doesn't instantaneously change them by a discrete amount. And, as I've already shown, the timescale on which the GW changes the metric coefficients is much slower than the time scale on which atoms exert forces on neighboring atoms.

Continuity suggest (actually demands but what ever) that the ends remain at there original coordinate location the instant the wavefront hits.

No, continuity demands that the instantaneous change you have postulated is impossible. Of course your model will give you wrong answers if you start with impossible premises.

 

[Paul Colby]

That's not what will happen. The GW causes the metric coefficients to change continuously. It doesn't instantaneously change them by a discrete amount. And, as I've already shown, the timescale on which the GW changes the metric coefficients is much slower than the time scale on which atoms exert forces on neighboring atoms.

True in all practical situations, however, the GW I describe exists as a limit for GR as well as EM so your point seems moot. I'm happy to replace infinite acceleration with physically unbounded in the limit. The point is the stress field induced in the bar is $T_{ij} = \gamma h_{ij}$ where $\gamma$ is the shear modulus. We are working in the TT gauge so the time components of $h_{ij}$ are zero in this frame. The traction force is therefore quite bounded (actually rather small at that) and depends only on the magnitude of the step not it's derivative. Typical values for $\gamma$ are like 50 to 100 GPa.

No, continuity demands that the instantaneous change you have postulated is impossible. Of course your model will give you wrong answers if you start with impossible premises.

Again, there is no numerical limit to it's rate of change so what is your point? A general plane wave solution is $h_{xx}(t,z) = f(t-z)e_{xx}$ where $f(s)$ is any function. A step function works just fine here.

 

[PeterDonis]

the GW I describe exists as a limit for GR as well as EM so your point seems moot.

What seems moot to me is making an obviously unrealistic assumption when we are discussing an actual device (LIGO) and the actual GWs that it detects, which are very, very different from your unrealistic instantaneous impulse GW. However, it is true (though IMO irrelevant to this thread) that my comments in previous posts do not apply to your unrealistic instantaneous impulse GW, since by hypothesis the GW acts faster than anything else in the entire universe.

there is no numerical limit to it's rate of change so what is your point?

Um, that in this thread we are talking about actual real GWs detected by LIGO, whose rate of change is what it is regardless of the lack of a "numerical limit" in your unrealistic model? See above.

 

[PeterDonis]

there is no numerical limit to it's rate of change so what is your point?

Perhaps it's worth replying to this along different lines in addition to my previous post. While it is mathematically true that you can construct the model you propose, just having a mathematical model is not physics. You have to pick the right mathematical model for the physics you are trying to describe. AFAIK the GWs we expect to detect in our actual universe are not instantaneous GW pulses, nor anything even close to them. So while an instantaneous pulse model might be appropriate for some situations, it is not, IMO, appropriate for any discussion of GWs.

Of course this is a judgment call, and your judgment might differ from mine. I would be interested, though, to see if you have any examples of GWs, even theoretical ones that have not been observed but only hypothesized as possibly produced somewhere in our universe, for which you think your model is a realistic description. I'm not aware of any.

 

[PeterDonis]

there is no numerical limit to it's rate of change so what is your point?

And yet one more comment from a different point of view. The metric has to be continuous, and a step function $f(s)$ is not continuous. So as you state it your model is not actually correct even mathematically. You would need to replace the step function with a continuous one that had an appropriately rapid rate of change (much more rapid than any other rate of change in the problem you are modeling).

 

[Paul Colby]

Um, that we are talking about actual real GWs detected by LIGO, whose rate of change is what it is regardless of the lack of a "numerical limit" in your unrealistic model? See above.

I was responding to a question asked directly to me in this thread. What I've said on the matter is correct and germane as far as the question posed. GW have been established and are not limited in theory to just LIGO per say. To suggest optical frequency GW can't exist would be inconsistent with what is currently known IMO even if an astronomical source of sufficient intensity for detection do not exist.

And yet one more comment from a different point of view. The metric has to be continuous, and a step function $f(s)$ is not continuous. So as you state it your model is not actually correct even mathematically. You would need to replace the step function with a continuous one that had an appropriately rapid rate of change (much more rapid than any other rate of change in the problem you are modeling).

No, the step function obeys the wave equation just fine. Any practical band limiting of the step is clearly completely beside the point being made. The reason to look at such limiting cases is the clarity and understanding of the phenomena gained. Clearly needed in this case apparently. I can only hope someone out there is illuminated by the discussion.

 

[PeterDonis]

To suggest optical frequency GW can't exist would be inconsistent with what is currently known

I have said no such thing. An optical frequency GW is not the same as an instantaneous pulse GW such as you have modeled--unless you know that all other relevant timescales are much slower.

Also, the GW you postulated when you gave a specific model had a timescale of 200ns, which is much slower than optical frequency (for which the relevant timescale is femtoseconds). For the case of optical frequency, you would be getting to the point where the forces between neighboring atoms in a steel bar would probably be slower--but I don't think they'd be a lot slower (since the estimate I gave earlier was very conservative). So I'm still not sure that your instantaneous model would work well for that case. You would need a GW with frequency well above optical.

 

[PeterDonis]

the step function obeys the wave equation just fine.

Whether it obeys the wave equation or not is beside the point. The metric has to be continuous in GR, regardless of what wave equation you write down.

 

[PeterDonis]

The reason to look at such limiting cases is the clarity and understanding of the phenomena gained.

That's fine as long as whatever clarity and understanding you gain carries over to other cases. That does not apply here, since what you are saying about your instantaneous GW model is no longer true as soon as the GW timescale becomes comparable to the timescale of forces between neighboring atoms in the material.

 

[Paul Colby]

That's fine as long as whatever clarity and understanding you gain carries over to other cases. That does not apply here, since what you are saying about your instantaneous GW model is no longer true as soon as the GW timescale becomes comparable to the timescale of forces between neighboring atoms in the material.

Well, look back through the discussion. Three limiting cases are evident, two of which I've attempted illuminate with limited success gauging by your rebuttals. The basic interaction of GW with linear elastic materials is given by the equations I've quoted and are well known in the literature on bar detectors. Once the traction forces are known from the GW induced stress, the problem reduces to solving the acoustic wave equation in the solid subject to the prescribed boundary conditions. This is the case independent of what you or I may say on the matter.

 

[Paul Colby]

Of course this is a judgment call, and your judgment might differ from mine. I would be interested, though, to see if you have any examples of GWs, even theoretical ones that have not been observed but only hypothesized as possibly produced somewhere in our universe, for which you think your model is a realistic description. I'm not aware of any.

Cases of interest to me are GW in the HF to VHF radio frequency bands. I would expect very broadband signals which would be difficult to detect using an interferometric approach because of noise consideration. Possible astrophysical sources are not strictly required and may very well not exist. This should prevent one from thinking about potential detection schemes IMO.