B How to measure Gravitational waves

[Number 42]

Why are clocks not used to measure gravity waves?

Spacetime wriggle and time must also vary. It is possible to measure time very accurately and with a high resolution. Så why not use clocks to detect gravity waves?

 

[Vanadium 50]

What exactly do you intend to time?
How good a clock do you need?
How do you compare two clocks far away from each other?

 

[Ibix]

time must also vary

Not necessarily. Gravitational waves (not gravity waves - that's a kind of surface wave on water) can affect the spatial part of the metric only. I don't think they can ever affect the time part of the metric only.

 

[PeterDonis]

Gravitational waves (not gravity waves - that's a kind of surface wave on water) can affect the spatial part of the metric only.

In appropriately chosen coordinates, yes. Those coordinates (the transverse-traceless gauge) are the ones standardly used to analyze gravitational wave detectors like LIGO, but it's worth keeping in mind that it's still a choice of coordinates.

I don't think they can ever affect the time part of the metric only.

That's correct; to state it more precisely, there is no possible choice of coordinates for which a GW only affects $g_{00}$ and not any other metric coefficient.

 

[vanhees71]

One should emphasize however that the deformation of LIGO's arms due to the gravitational wave is of coarse gauge independent since it's a real physical effect of gravity. The choice of coordinates is completely arbitrary. Any choice will lead to the same outcome of the measurable physics, including the deformation of LIGO's arms and the change of the interference pattern of the laser light finally measured.

 

[Ibix]

In appropriately chosen coordinates, yes. Those coordinates (the transverse-traceless gauge) are the ones standardly used to analyze gravitational wave detectors like LIGO, but it's worth keeping in mind that it's still a choice of coordinates.

Understood. But whatever you choose to mean by "time" shouldn't there be a way to pick a gravitational wave so that it doesn't have any component in the time-like direction?

 

[vanhees71]

You need to define some physical quantity to get a sensible answer to the question about the physical meaning of a particular solution of Einstein's equations (here gravitational waves in the usual approximation of the linearized theory). That's analogous to the solutions of Maxwell's equations in terms of the potentials. The potentials have no physical meaning a priori, but you have to calculate physically measurable quantities (like the electromagnetic field components) or evaluate how particles move in the so defined electromagnetic field (which of course also only depends on the field components but not the potentials). For sure, any "gauge dependent" quantity in gauge theories is not directly physically interpretable but you have to use gauge independent measureable quantities to get physics sense from it.

 

[PeterDonis]

whatever you choose to mean by "time" shouldn't there be a way to pick a gravitational wave so that it doesn't have any component in the time-like direction?

For weak (i.e., linear) gravitational waves, yes, they are purely transverse, so you can always choose coordinates (the transverse traceless ones) that express all of the spacetime curvature due to the waves as purely spatial. But of course that choice will be different for different waves (i.e., waves going in different directions).

 

[Ibix]

For weak (i.e., linear) gravitational waves, yes, they are purely transverse, so you can always choose coordinates (the transverse traceless ones) that express all of the spacetime curvature due to the waves as purely spatial. But of course that choice will be different for different waves (i.e., waves going in different directions).

Right. So doesn't that answer why you wouldn't use clocks to detect gravitational waves? There are a whole family of waves you just can't detect without sensors in relative motion.

 

[PeterDonis]

doesn't that answer why you wouldn't use clocks to detect gravitational waves?

I would say it shows why you can't use a set of clocks all at rest relative to each other to detect gravitational waves. (The OP's question probably had the part I put in italics as an unstated assumption, but it's still an assumption.)

 

[Number 42]

I would think that the clocks should be spaced so that some are far enough away to be in different part of the wave. Say 1/4 of a wavelength.
Clocks close together will ofcourse not show any difference.
Transmitting the clock signal would not be a problem , but are clockes accurate enough?

 

[timmdeeg]

Lets assume LIGO's mirrors carry synchronized clocks. Aren't they no more synchronous in case a GW passes by?

 

[Vanadium 50]

I repeat:

What exactly do you intend to time?
How good a clock do you need?
How do you compare two clocks far away from each other?

 

[PeterDonis]

Lets assume LIGO's mirrors carry synchronized clocks.

This is not something you can just assume. You have to describe what "synchronized" means and how it will be accomplished. (For example, Google "Einstein clock synchronization".)

 

[pervect]

This is not something you can just assume. You have to describe what "synchronized" means and how it will be accomplished. (For example, Google "Einstein clock synchronization".)

Yes. On a practical note, one can define clock synchronization is to choose some coordinates. With defined coordinates, clocks can be defined as synchronized if they have the same time coordinate. But in general, the results will depend on one's choice of coordinates.

Clock synchronization is presented as a human choice, and to a large extent it is. However, if one wants decent compatibility with Newtonian phsyics, so one can use most Newtonian formulae, one needs to use something like Fermi normal coordinates and make sure that the region one is analyzing is "small enough". I've seen a lot of posters who deal with the clock synchnoziation issue by ignoring them :(, this almost always leads to confusion in the end.

Conventional wisdom is more along the lines of "don't use coordinate dependent methods at all". This is OK, if one can get the necessary physical intuition that way, but if one is seeking some intuitive insight, using the coordinate methods and the right set of coordinates can be helpful. Contrawise, using the wrong set of coordinates can lead to more confusion.

The biggest problem with using Fermi normal coordinates in this context is converting the GW solution to these coordinates - while the GW has a nice closed form solution in the TT coordinates, attempts to convert them to Fermi Normal coordinates quicly results in a real mess. I do recall vaguely seeing some paper on the topic that talked about this issue and (I believe) they managed to deal with the mess, somehow. BUt I don't know how.

The approach of "Fermi coordinates are too hard so I'll just use TT coordinates and interpret them in a Newtonian manner is just wrong :(.

 

[timmdeeg]

This is not something you can just assume. You have to describe what "synchronized" means and how it will be accomplished. (For example, Google "Einstein clock synchronization".)

I've read that but am thinking about a more-straight forward approach to see how the proper time is affected by the so-called ripples in spacetime. The proper time of a free clock in flat spacetime is recorded during the passage of a gravitational wave. The clock frequency shall be much higher than the frequency of the gravitational wave. After the wave has passed by the recorded ticks are compared with the ticks of the clock. Are the recorded ticks slower and faster according to the phase of the gravitational wave?

 

[PeterDonis]

After the wave has passed by the recorded ticks are compared with the ticks of the clock.

What "recorded ticks"? Other than the clock itself, what else is "ticking"?

 

[timmdeeg]

What "recorded ticks"? Other than the clock itself, what else is "ticking"?

The ticks which have been recorded by the clock during the passage of the GW will then be compared with the ticks of the same clock in flat spacetime however without a GW passing by.

 

[PeterDonis]

The ticks which have been recorded by the clock during the passage of the GW will then be compared with the ticks of the same clock in flat spacetime however without a GW passing by.

How would you make such a comparison? Think carefully.

 

[timmdeeg]

How would you make such a comparison? Think carefully.

I would check if the frequency of the recording is periodically lower and higher than the frequency of the the clock. But how to do this technically, I would have to ask my technician.

 

[PeterDonis]

I would check if the frequency of the recording is periodically lower and higher than the frequency of the the clock. But how to do this technically, I would have to ask my technician.

Your technician will tell you that there is no way to make this comparison, because the recording does not tell you at what speed to play it back, relative to the frequency of the ticks you are observing now. You have to have some independent way of telling when you are playing back the recording at "the right speed", and the only such way is to use clocks that are ticking right now. There is no independent standard of time.

 

[pervect]

The ticks which have been recorded by the clock during the passage of the GW will then be compared with the ticks of the same clock in flat spacetime however without a GW passing by.

I think you need to adopt a coordinate system, otherwise all you'll find out is that clocks tick at a constant rate as measured by clocks. That's what clocks do. The idea of clocks changing it's rate requires that one have a coordinate system that one compares the proper time reading of the clock (which doesn't depend on the coordinates) with. At least I can't imagine what else you'd be comparing the clock to. Then we back to asking - what coordinates are you talking about. If you define a set of coordinates, then we just need the metric associated with that set of coordinates, and what you're asking about is basically $g_{00}$ for that specific metric. But that isn't defined until you define the coordinates.

 

[Ibix]

I have the clock and the recording of this clock at the same place. Then I start play back and adjust the speed until the phase difference between the ticks of clock and recording is stable. My guess is that a recorded GW signal should shift the phase difference forth and back. But I'm not sure at all regarding this reasoning and appreciate your help.

Let’s assume that your recording is a standard CD quality digital one that takes 44k samples per second. If the clock being recorded slows down in some sense, what's going to happen to the clock driving the sampler?

 

[timmdeeg]

Let’s assume that your recording is a standard CD quality digital one that takes 44k samples per second. If the clock being recorded slows down in some sense, what's going to happen to the clock driving the sampler?

Oh, you have been very fast. I've seen your post after having deleted mine, because I started to see my misconception. What I described wasn't "an independent way of telling when you are playing back the recording at "the right speed", as PeterDonis mentioned in #21.
Thanks for taking care.

So would the only reasonable way be to compare the clock rates of the free and the fastened clock by means of Einstein synchronisation?

 

[timmdeeg]

I think you need to adopt a coordinate system, otherwise all you'll find out is that clocks tick at a constant rate as measured by clocks. That's what clocks do. The idea of clocks changing it's rate requires that one have a coordinate system that one compares the proper time reading of the clock (which doesn't depend on the coordinates) with.

Yes, thanks for your answer.

 

[PeterDonis]

would the only reasonable way be to compare the clock rates of the free and the fastened clock by means of Einstein synchronisation?

You can't do that either, because you're not talking about two different clocks, you're talking about the same clock during two different segments of its worldline (one segment in which it's traveling through flat spacetime, the other, later segment in which it's traveling through the curved spacetime of a gravitational wave). You can't Einstein synchronize the same clock with itself in its own past or future.

 

[timmdeeg]

You can't do that either, because you're not talking about two different clocks, you're talking about the same clock during two different segments of its worldline (one segment in which it's traveling through flat spacetime, the other, later segment in which it's traveling through the curved spacetime of a gravitational wave). You can't Einstein synchronize the same clock with itself in its own past or future.

Perhaps my wording was unclear. I've changed the scenario. One clock is freely suspended the other fastened to the earth.

 

[PeterDonis]

Perhaps my wording was unclear.

It certainly was (and is--see below) to me.

I've changed the scenario.

Then I didn't understand what scenario you were describing to start with.

One clock is freely suspended the other fastened to the earth.

This doesn't help. Please start from scratch and clearly describe what scenario you have in mind.

 

[timmdeeg]

This doesn't help. Please start from scratch and clearly describe what scenario you have in mind.

I'm sorry this was certainly too hasty. Hopefully the next attempt is an improvement.

A and B are identical clocks. A is freely suspended like the mirrors of LIGO, B is fastened to the Earth close to A. The light travel time between A and B is negligible.

A emits a light pulse each tick. B receives A's light pulses and compares them to its own ticks.

So what happens if a gravitational wave passes by? Originally I thought that the comparison would yield $dt/d\tau$. But on the other side it seems that B just measures A's periodical slowing down due to the the relative velocity between A and B, because special relativity holds locally in curved spacetime.
Would you kindly clarify?

 

[pervect]

The Earth is too big and the speed of sound in it is too low for it to be very rigid, so the Earth's shape will distort a lot. It'd probably be similar to the response of the long bar that was mentioned in another therad, but I haven't calculated it - or the long bar response, for that matter. I believe the long bar result was that one could essentially ingnore the internal forces of the long bar which try to make it hold it's shape, and treat it approximately as if it were a bunch of free-floating test masses. But I could have misread or misremebered.

 

[Vanadium 50]

Finally! A system described precisely enough to be analyzed.

Basically, you have a clock on a pedestal next to a clock hanging from a string. I don't see how a passing gravitational wave will change the reading on those clocks. Why do you think it would?

 

[timmdeeg]

Basically, you have a clock on a pedestal next to a clock hanging from a string. I don't see how a passing gravitational wave will change the reading on those clocks. Why do you think it would?

Because the hanging clock is in principle moving in the horizontal plane like a particle of the ring of particles shown here relativ to the other clock which is fastened. So there should be a frequency shift according to special relativity. I understand the arguments of @pervect but would like to discuss the idealized case.

 

[Vanadium 50]

is in principle moving in the horizontal plane

OK, work out the time dilation from this movement. Does there exist any clock this good?

 

[timmdeeg]

OK, work out the time dilation from this movement. Does there exist any clock this good?

I guess no, just trying to improve my understanding by means of such scenarios.

 

[Ibix]

Because the hanging clock is in principle moving in the horizontal plane like a particle of the ring of particles shown here relativ to the other clock which is fastened. So there should be a frequency shift according to special relativity. I understand the arguments of @perfect but would like to discuss the idealized case.

I'm struggling to convince myself that this will detect anything even in principle. The expansion and contraction of the ring is like the expansion of the universe. For any pair of the dots we can add an inertial dot halfway between, which will see the pair moving symmetrically about it. That dot will argue that there can be no difference in the clock readings between its pair, from symmetry. So I think they'll all show the same time.

This may not be the case for a ring of particles in motion with respect to this conveniently symmetric case.

 

[timmdeeg]

For any pair of the dots we can add an inertial dot halfway between, which will see the pair moving symmetrically about it. That dot will argue that there can be no difference in the clock readings between its pair, from symmetry. So I think they'll all show the same time.

But we don't have two inertial clocks. One is, the other isn't.

 

[Ibix]

But we don't have two inertial clocks. One is, the other isn't.

But @pervect pointed out in #30 that the Earth isn't rigid enough to be treated as anything significantly different from a cloud of free-floating masses.

 

[PeterDonis]

A is freely suspended like the mirrors of LIGO, B is fastened to the Earth close to A.

As @pervect pointed out, this won't make much difference in terms of the response of A and B to a gravitational wave, because for this purpose the Earth has negligible rigidity. So if we just consider the gravitational wave, without taking into account any other effects, A and B will move the same.

The reason why A is isolated in LIGO is because there are other effects--all kinds of other things that make the Earth vibrate, such as earthquake tremors and other seismic effects, vibrations due to passing cars and trucks, etc., etc. All of these things will make B move in ways that have nothing to do with a gravitational wave; but if A is isolated well enough, it won't respond to any of these things so the signal we obtain from it will be much closer to the "pure" signal we would get from a gravitational wave alone.

 

[timmdeeg]

@pervect, @Ibix and @PeterDonis , thanks for clarifying this question. I was underestimating the rigidity issue totally, so I should forget the idealization I mentioned above.

 

[pervect]

I think I've said this before, but we can rephrase the question in terms of a rigid framework rather than the Earth, if that's your intent. Our first question must be - does a rigid framework actually exist? I believe the answer is "yes, to a reasonable degree of approxiation", though I haven't seen discussed in a peer reviewed paper. When I talk about 'reasonable degree of approximation", one of the things I'm assuming is that if first order effects are on the order of 10^-20, then second order effects are on the order of 10^-40, so we can ignore them. So we're talking about a weak field. The second question we have to ask is "how big is our framework"? It turns out according to my analsyis (and this seems to match with other staetments I've read) that the ciritical dimension of the framework is how long it is in the direction of propagation of the GW. The dimension that's perpendicular to the direction of propagation can be quite large by my calculations, but not infinite. (Again, these are my personal alculations, not peer reviewed).

The techniques I used to convince myself that a rigid framework exists were to consider whether a Born-rigid congruence existed within the desired (first-order) accuracy limits. I used the criterion of vanishing shear and expansion to determine rigidity (but one could also use techniques based on the Lie derivative of the spatial projection of the metric tensor). In a plane perpendicular to the GW, we can cover a very large region before we see signficant deviations from rigidity - but if we consider a 3d spatial volume, we start to see effects of linear order in the direction of propagation of the GW, so the approach requires us to consider a "thin" plane whose thickness is negligible compared to the wavelength of the GW.

Probably the most convenient way of creating said framework is to use Fermi normal coordinates around a specified, special wordline representing a " reference" observer". I would imagine said reference observer would be following a geodesic, but one could define it in any manner one likes. The distance along a space-like geodesic to other observers in the congruence defined by constant Fermi-normal coordinates will be constant by the construction of the coordinates. The distance as measured by round-trip travel time (which is the SI definition of distance) will be approximately equal to the round-trip light light travel time when one is sufficiently close to the reference observer.

This ties in with what I've been saying before about "which coordinates".

This then reduces to a straightforwards question, what is the metric tensor for a plane GW space-time in Fermi-normal coordinates about some particular geodesic, specifically what is the $g_{00}$ term of the tensor. Unfortunately, I don't know the answer to that. There was a paper I glanced at that did discuss Fermi-normal coordinates and GW's, so there may be an answer in the literature if you care to dig for it, if the revised question is of interest. Google finds https://arxiv.org/abs/1409.4648 which is what I was recalling.

 

[Vanadium 50]

This then reduces to a straightforwards question, what is the metric tensor for a plane GW space-time in Fermi-normal coordinates about some particular geodesic, specifically what is the g00g_{00} term of the tensor.

Can you B-level-ify this?

 

[PeterDonis]

I would imagine said reference observer would be following a geodeosic

I don't think you can assume that, because a Born rigid congruence in the presence of a gravitational wave cannot possibly be a geodesic congruence; if it were, then LIGO would give a null result and would not be able to detect GWs.

 

[pervect]

I don't think you can assume that, because a Born rigid congruence in the presence of a gravitational wave cannot possibly be a geodesic congruence; if it were, then LIGO would give a null result and would not be able to detect GWs.

Most of the curves in the congruence won't be geodesics. But the reference congruence can be anything that makes sense, it represents the worldline of the "observer".

 

[pervect]

Can you B-level-ify this?

Probably not all the way to B=level, but - let's take an example of the globe. The 3-d analogy to Fermi normal coordinates would be to pick a particular point on the globe, and measure your bearing (which picks the direction you start out in) and the distance you move in that direction. Then you basically use the distance and bearing as polar coordinates on a sheet of paper (which, unlike the globe, is not curved.) The flat paper has whatever additional coordinates you like on it (say, a square grid of cartesian coordinates). The purpose of the construction is to make a map from the curved manifold of the surface of the Earth to a flat manifold (the piece of paper). The bearing/angle is an intermediate step in the contruction, not necessarily the end result. Another non-B level remark, mathematicians call this the exponential map, https://en.wikipedia.org/wiki/Exponential_map_(Riemannian_geometry), it's a mapping from the tangent space at a point to the manifold itself. The tangent space is the "flat sheet of paper", the manifold is the curved surface of the Earth.

In 2-space + 1 time, rather than a reference point, you have a reference worldline. At any instant in time (event on the worldline), you can construct an infinite number of spatial geodesics that radiate away from that point. You choose the subset of those that start out orthogonal to the reference worldline, that is a 2d space. Then you use the same principle of distance/bearing to map points on the space-time manifold to points on a flat plane. The time coordinate for any point on any of the curves reached via the radiating geodesics is the proper time reading of the clock on the observer on the reference worldline. The space coordinate are given again by the distance and bearing.

Time dilation is given by the usual procedure of the ratio of proper time to coordinate time. I'm not sure how to B-level that, but that's how time dilation in , say, Schwarzschild coordinates works.

There's several non-B level papers you can find with google, and MTW has a discussion of these coordinates.

 

[PeterDonis]

the reference congruence can be anything that makes sense

I think you mean the reference worldline. Yes, the reference worldline can be a geodesic, but no other worldlines in the congruence will be.

 

[l0st]

... So if we just consider the gravitational wave, without taking into account any other effects, A and B will move the same...

What if instead of putting A and B close to each other we set them up on the opposite sides of Earth? Would not we be able to detect a wave passing through by noticing their signals temporarily going out of sync due to different distance to the source?

 

[Ibix]

What if instead of putting A and B close to each other we set them up on the opposite sides of Earth? Would not we be able to detect a wave passing through by noticing their signals temporarily going out of sync due to different distance to the source?

Depends what you're trying to do. If you're just putting mirrors a really long way apart and bouncing signals between them then yes that would be more sensitive. It's just an upscaled LIGO, with a few minor engineering challenges to do with tunnelling through the core. You might be better to put the mirrors out in space, which is a future design I believe.

If you are planning to use the Earth as a rigid bar sensor, however, then moving the clocks further apart makes the problem worse, not better. The speed of sound is so low in rock that one side of the Earth doesn't "know" the other side is in motion until hours after the gravitational wave has passed. So your clocks would just acts like free-floating clocks and you don't get any accumulated time difference from it. Just the transients that a LIGO-style detector sees.

 

[l0st]

Transients is what I'd be looking for.
Seems like the idea is theoretically viable. Feasibility of practical application requires extensive research though.

 

[Ibix]

Transients is what I'd be looking for.
Seems like the idea is theoretically viable. Feasibility of practical application requires extensive research though.

By transients, I mean you could in principle, replace LIGO mirrors with clocks. Due to the distance change associated with the gravitational wave each clock would see the other's rate wobble. LIGO does this much more precisely with interferometry, and the practical difficulties of using clocks instead are, as V50 pointed out, way, way beyond anything you are imagining. Whatever it is you are imagining.

 

[pervect]

OK, here is my best shot at a truly B-level explanation of my remarks, sorry for the delay.

General Relativity can be understood as drawing a "map" of space-time. It's a 4 dimensional map. Sometimes this is called "the block universe". GR has a mathematical entity called the metric. This metric can be understood as representing whether or not the map is to scale - it also tells about other sorts of distortion of the map.

If you draw your map to scale, it's easier to interpret it.

Fermi-Normal coordinates are a technique for drawing maps that are to scale and not distorted in a small region around some specific observer (who is represented on the map by a worldine, the worldine is the timeline of the observer)., Because it is drawn to scale and undistorted in a small region, the "map" is easy to interpret. . There's more to it than that (and the language is rather vague) but that's the executive B-level summary and motivation for using Fermi-Normal coordinates.

Note that it's impossible to draw a map of a curved 2-d surface on a flat sheet of paper that is everywhere to scale (for instance, making a map of the surface of the Earth that is undistorted). Similar remarks apply to drawing a map of a curved 4-d space-time on a "flat" 4-d manifold.