Transmitting information using gravitational waves

[PeterDonis]

the equations of motion which are,
$$
\rho \ddot{u}_i = T_{ij,j}
$$

What is $u$?

$S_{i j}=0$ with $E_J \ne 0$ will generate stress.

But stress is an input, not an output. The question is whether an applied stress can result in a pure electric field with zero strain. I think you need to solve the equations of motion to know whether that is possible.

 

[PeterDonis]

Isn't that right at the extreme end of the scale for stellar mass BH?

What scale are you referring to?

 

[rootone]

What scale are you referring to?

A scale including all black holes that would have originated as the result of a very large stars collapsing, (or merging)
I guess that means all except the supermassive ones in galaxy cores.
I thought the typical estimated mass for those is usually in the region of 10x solar mass, and 20x or more would be very unusual.

 

[PeterDonis]

I thought the typical estimated mass for those is usually in the region of 10x solar mass, and 20x or more would be very unusual.

I've seen different estimates from different sources; I don't know that we have a very good understanding of what the expected range actually is.

 

[FactChecker]

Basically, $16\pi G/c^4=4.15\times 10^{-43}$ seems/is 43 powers of nope

Ha! -- "43 powers of nope". I like that.

 

[Paul Colby]

What is $u$?

$\ddot{u}$ is the $a$ in $F=ma$ for linear elastic materials. $S_{i j} = \frac{1}{2}(u_{i,j}+u_{j,i})$ in this case. The $T_{i j,j}$ is the divergence of the stress tensor. Note that $T_{i j,j}=0$ implies that $\ddot{u}=0$. This is at least in part the origin of all the wrong sounding off resonance comments I'm making.

 

[Paul Colby]

But stress is an input, not an output. The question is whether an applied stress can result in a pure electric field with zero strain. I think you need to solve the equations of motion to know whether that is possible.

I have solved the dynamical problem. If there is interest I can post it here. In the time harmonic case the strain, $S_{i j}$, and the stress, $T_{i j}$ and the circuit current, $I$ (yet to appear) are all proportional. The proportionality constants are all frequency dependent. So technically yes, there will not be zero strain for a given stress however, depending on the drive circuit and impedance match one may generate considerable stress with little resulting strain.

 

[Paul Colby]

AFAIK the usual coordinate choice for the LIGO scientists is one in which the speed of light is constant and the mirrors move.

My description is relative to this choice of coordinates. They are consistent with how LIDARs are modeled.

 

[PAllen]

In the case of a binary pulsar, for example, yes, it's the third time derivative of the quadrupole moment of the mass distribution, because that's all that's significant. But consider the case of a black hole merger: there is no mass present (the holes are both vacuum), so what drives the radiated power?

In my view there is mass, but not matter. A BH has adm mass, bondi mass, and mass according to the various formulations of quasi local mass. Especially the quasilocal formulations allow definition of quadrupole moment for the system.

 

[PeterDonis]

$\ddot{u}$ is the $a$ in $F=ma$ for linear elastic materials.

Ah, got it.

Note that $T_{i j,j}=0$ implies that $\ddot{u}=0$.

But $i$ and $j$ here range only over the spatial indices, not the time index. So even though conservation laws require that $T_{\mu \nu, \nu} = 0$ (in the limiting case where spacetime is flat, at least to a good enough approximation), where we are including all four indices (space and time), that does not mean that we must have $T_{ij, j} = 0$.

 

[PeterDonis]

In my view there is mass, but not matter. A BH has adm mass, bondi mass, and mass according to most variants of quasi local mass. Especially the quasilocal formulations allow definition of quadrupole moment for the system.

Ah, ok.

 

[PeterDonis]

My description is relative to this choice of coordinates.

But in these coordinates, the GW does work on the mirrors. The laser beams have no work done on them; their frequency does not change. All that changes is the distance they cover in each arm.

They are consistent with how LIDARs are modeled.

Do you have a reference? I'm not familiar with how LIDARs are modeled, so I am curious.

 

[Paul Colby]

Do you have a reference? I'm not familiar with how LIDARs are modeled, so I am curious.

Arguably any elementary book on interferometers should cover this material.

https://www.arm.gov/publications/tech_reports/handbooks/dl_handbook.pdf

 

[Paul Colby]

But in these coordinates, the GW does work on the mirrors. The laser beams have no work done on them; their frequency does not change. All that changes is the distance they cover in each arm.

Interesting. The phase difference between the arms changes as a function of time. How do you modulate the phase in time without changing the frequency of the light ever so slightly. Do you have a reference?

 

[Paul Colby]

that does not mean that we must have $T_{ij,j}=0$.

Don't recall saying that. However, when discussing illumination by GW isn't this the case when the wavelength is much larger than the crystal? The gravitational stress is nearly constant over the volume. When discussing transmission the stress and strain vary as sine and cosine through the crystal thickness. The stress term also has a component which is essentially constant arising from the applied field. The strain lacks this term.

 

[PeterDonis]

How do you modulate the phase in time without changing the frequency of the light ever so slightly

By changing the distance between the mirrors.

Do you have a reference?

A typical description is on the LIGO site at Caltech, here:

https://www.ligo.caltech.edu/page/what-is-interferometer

See the "How does it work?" section.

I thought I had links to more technical references that describe the math underlying the description given there, but I can't seem to find them at the moment. I know MTW discusses GW detection by interferometers and gives a simple model of this type. I can't remember whether Wald does.

 

[Paul Colby]

By changing the distance between the mirrors.

Wow, that seems to violate basic mathematical facts? In phase modulated radio transmissions the carrier develops side bands. One need only Fourier transform $cos(\omega t +\phi(t))$ to see this. No sidebands no information transmission. In optics it's very much the same situation or at least that's what I was lead to believe. Well, all snark aside it should be possible for me to dig this out of your reference, thanks. [edit] I think a more complete description would include what I'm discussing. I expect LIGO looks for modulation in the phase difference and that this is an important detail.

 

[Paul Colby]

I know MTW discusses GW detection by interferometers

I have a dent in my chest from where I rest this book when I read it in bed.

 

[PeterDonis]

that seems to violate basic mathematical facts?

I don't see how. The GW moves the mirrors. Moving the mirrors changes the round-trip travel time of the light beams, and does so differently in the two arms (stretch in one arm, squeeze in the other). Changing the relative round-trip travel time changes the relative phase of the beams when they come back together.

I expect LIGO looks for modulation in the phase difference

I think that's what I described just above, yes. [Edit: but see a caveat in my next post.]

 

[PeterDonis]

phase modulated radio transmissions

I don't think this is a good analogy, because the phase modulation in this case is at the source, whereas in the case of LIGO it's at the detector (and "phase modulation" might not even be a good term in the LIGO case if it connotes modulation at the source). Nothing at all is done to the laser beams themselves; their source emits them in an unchanging state.

 

[Paul Colby]

I don't see how.

"that" in my comment refers to time modulating the phase of light without developing sidebands (frequency shifts).

I don't think this is a good analogy

It's not an analogy. The light returned from one arm goes like $\cos(\omega t + \phi(t))$. This is a modulation and has sidebands quite independent of everything else that might be said.

 

[PeterDonis]

"that" in my comment refers to time modulating the phase of light

But that's not what's happening. The light itself is unchanged. Only the distance it has to cover changes. It's not the same thing as modulating the source of the light.

This is a modulation

Not in the sense you appear to be using the term.

and has sidebands

I don't think that's true; that would require modulating the source of the light. That's not happening here.

Consider a simpler situation: you have a laser light source that emits a constant frequency (no modulation at the source) and a detector and sit at rest at some location. I have a mirror and gradually move away from you. You repeatedly fire the laser at the mirror and watch the detector signal. The phase of the light at the detector changes as I move the mirror away. Does that mean the laser light is being modulated? Will someone else sitting between us be able to detect side bands?

 

[Paul Colby]

But that's not what's happening. The light itself is unchanged. Only the distance it has to cover changes. It's not the same thing as modulating the source of the light.

Well, for me this discussion is occurring in a frame where the mirrors are moving. Light reflected off a moving mirror experiences a doppler shift which depends on the velocity of the mirror. The reflected beam will be at a higher or lower frequency relative to the source. For very slow variations this is best described as a time dependent phase. Slow or fast the physics/arithmetic is always the same.

Will someone else sitting between us be able to detect side bands?

Hell yes. Remember that LIGO isn't pulsed it's CW not that this changes anything. The coherence length of their lasers is likely well over 4km. One crest looks very much the same as the next. The absolute phase has no meaning from the experimental point of view. This ties back to my original (wrong sounding) view, the energy in these sidebands you think I'm hallucinating is due to the work the GW (or moving mirrors) is doing on the detector.

 

[PeterDonis]

Light reflected off a moving mirror experiences a doppler shift which depends on the velocity of the mirror.

Yes, this is a fair point--the phase difference seen at the detector is partly due to this and partly due to the change in the round-trip distance traveled by the light.

LIGO isn't pulsed it's CW

Yes, and I should have specified CW in my simplified example.

The absolute phase has no meaning from the experimental point of view.

Yes, I was ignoring that complication in my simplified example. (Or, if you like, I was assuming that the detector contains a reference beam of some sort so that it is detecting relative phase, not absolute phase.)

these sidebands you think I'm hallucinating

I wasn't saying you were hallucinating the sidebands; I was trying to understand why you think varying the distance between the mirrors, which is how the standard model of LIGO interprets what is going on (as in the link I gave), "violates basic mathematical facts".

Googling on "ligo sidebands" quickly turned up some references, such as this one:

http://www.sjsu.edu/faculty/beyersdorf/Archive/Phys208F07/Sideband generation in LIGO.pdf

This reference indicates that there is some modulation done at the laser source (more precisely, by Pockels cells right after the laser source) in LIGO. This appears to be done in order to optimize the resonance of the lasers in the cavities in each arm (the cavities are there to increase the effective length of the arms). So I was incorrect in thinking that no source modulation is being done. It also makes clear that sidebands are in fact present in LIGO. So the presence of sidebands appears to be perfectly consistent with time-varying distance between the mirrors.

 

[PeterDonis]

the energy in these sidebands you think I'm hallucinating is due to the work the GW (or moving mirrors) is doing on the detector

I'm not sure that the reference I linked to in my previous post shows this. It looks to me like it's saying that the sidebands are due to the modulation that is done to keep the laser beams resonant in the cavities. But the reference doesn't actually appear to talk about the effects of GWs, so it's hard to be sure.

 

[Paul Colby]

I'm not sure that the reference I linked to in my previous post shows this.

There exist a level of understanding somewhere between the detail in the slides you linked to and the public relations level of understanding you seem to subscribe to. It's common in heterodyne radio design to use one or more IF frequencies to take advantage of the technology available at IF frequencies. This is done using frequency translation via mixers. This seems to be what the link in #54 is doing. It's a complication (very interesting though) to move the desired signal off base band and thus improve the S/N. This complexity is beside my point. My point is, in a frame in which the mirrors move, doppler shifts of the light generate sidebands. The slower the change the nearer these sidebands are to the carrier but they are still present.

 

[PeterDonis]

in a frame in which the mirrors move, doppler shifts of the light generate sidebands

Shouldn't the generation of sidebands be frame independent? (More precisely, the base frequency of the carrier is frame dependent, but the presence of frequencies other than the carrier frequency is not, correct?)

 

[PeterDonis]

doppler shifts of the light generate sidebands

There might be a terminology issue here as well. I am used to seeing the word "sideband" used to refer to frequencies other than the carrier frequency produced by modulation of the source. That's a more specific usage than the one in your quote above.

 

[Paul Colby]

Shouldn't the generation of sidebands be frame independent? (More precisely, the base frequency of the carrier is frame dependent, but the presence of frequencies other than the carrier frequency is not, correct?)

It's all the same thing. As I said the return beam goes like $\cos(\omega t + \phi(t))$ with any frame choice. This function is predominantly the carrier but has sidebands when it's spectrum is plotted. It also happens to be a carrier with a time dependent phase. The doppler shift explanation holds if the mirrors move. The explanation in the TT gauge where the coordinates of the mirrors are fixed is that of an optical cavity with a time dependent length. Pick your poison. It's the same physics either way.

 

[PeterDonis]

The doppler shift explanation holds if the mirrors move. The explanation in the TT gauge where the coordinates of the mirrors are fixed is that of an optical cavity with a time dependent length.

In other words, sidebands either way, but the specifics of how you explain why they are there are frame-dependent. Fair enough.