I How Many Gravitons Were Detected in Recent Gravity Wave Observations?

[snorkack]

Gravity waves are very weakly absorbed.
The gravity wave detectors have so far detected 3 events of gravity wave emission: 2 from black hole pairs and 1 from neutron star pair.

The absorption is weak but the detectors are very sensitive.

How much energy did the gravity wave detectors actually absorb from the gravity waves that propagated through Earth?
The frequency was low, peak something like 250 Hz, and therefore graviton energy low. How many quanta of gravity waves did the 3 detections consist of?

 

[jedishrfu]

Here's some stats on LIGO:

https://www.ligo.caltech.edu/page/facts

https://www.ligo.caltech.edu/page/ligos-ifo

http://www.kavlifoundation.org/how-ligo-works

They describe the sensitivity in terms of mirror movement not in terms of wave action or energy.

 

[snorkack]

Does the detector, as a matter of principle, make any alteration of the state of the gravity wave when itself undergoing an irreversible change of its state in a detection event?

 

[jedishrfu]

What are you trying to say?

As the gravitational wave passes through the Earth the arms of the detector shrink and expand a tiny amount. The detector doesn’t go into some irreversible state.

 

[Ibix]

You can have a machine that extracts energy from a gravitational wave - Feynman's sticky beads does it, at least in principle. And LIGO must generate a small amount of energy to be recorded.

I find it difficult to imagine that this doesn't affect the gravitational wave. You could stack up a sequence of sticky bead generators and produce arbitrarily high power outputs if it didn't - although I note that energy isn't always conserved in GR, so that's not as strong an argument as I'd like.

Note that they are called gravitational waves. Gravity waves are a kinfd of surface wave in water.

 

[Paul Colby]

How much energy did the gravity wave detectors actually absorb from the gravity waves that propagated through Earth?

Propagation through the Earth will effect the energy absorbed by the detector only at a level to which the GW are absorbed or scattered by the earth. I expect this is a very small effect when compared to the energy absorbed by the detector itself.

The interferometer mirrors in LIGO are suspended so as to allow inertial for movement along the interferometer arm. A high powered laser beam is then established in the between the mirrors. This beam will constantly impart a small pressure to the suspended mirrors. Because of this pressure, relative movement of the mirrors do work on the laser beam which shifts it's center frequency by a small amount relative to the light in the other interferometer leg. The work done by this relative movement is the energy absorbed by the detector.

 

[PeterDonis]

I find it difficult to imagine that this doesn't affect the gravitational wave.

It does. But for LIGO, the effect is very, very small. @Paul Colby gives a good description of the energy transfer process.

You could stack up a sequence of sticky bead generators and produce arbitrarily high power outputs if it didn't - although I note that energy isn't always conserved in GR, so that's not as strong an argument as I'd like.

It's strong enough. Yes, there are subtleties involved with energy conservation in GR, but TAANSTAFL still applies.

 

[Ibix]

It does. But for LIGO, the effect is very, very small.

Extending the analogy with Feynman's beads, it occurred to me that a back of the envelope estimate of the energy absorption would be the peak kinetic energy of the mirror assemblies. The displacement is around 10^-21 times 4km, and the frequency looks to top out around 50Hz, which gives a velocity on the order of 10^-16m/s. I don't know the mass of the mirror assemblies, but for any plausible answer the energy absorbed is going to be within an order of magnitude or two of 10^-32J.

It's strong enough. Yes, there are subtleties involved with energy conservation in GR, but TAANSTAFL still applies.

TANSTAAFL, unless there's a subtlety there I'm missing. I take it that a sphere of 100% efficient gravitational wave absorbers will generate no more than the mass loss of the emission event? So the upper bound on the energy available for absorbtion from the first event (mass loss of $3M_{sun}$, around 400Mpc away) is something like $(A/400^2)3M_{sun}$, where A is the effective area of the detector (order of $(4km)^2$ measured in megaparsecs squared?

 

[PeterDonis]

a back of the envelope estimate of the energy absorption would be the peak kinetic energy of the mirror assemblies

Kinetic energy relative to what? Remember that the mirror assemblies are moving inertially: no work is being done on them (see further comments below). The only work being done is on the laser beam.

"No work is being done on the mirrors" - this is probably the hardest think to grok about gravitational waves: they are waves of spacetime curvature, and spacetime curvature, by itself, does no work on anything. Things moving solely under the influence of spacetime curvature (e.g., gravity) are moving inertially, in free fall. That's why many people in the early days of gravitational wave studies (late 1950s ish) thought that GWs couldn't carry energy, because they couldn't see how they could do work.

In the Feynman's beads scenario, for example, the work isn't done directly on the beads by the GW; it's done by the inter-atomic forces between the beads. Think of the whole setup as a bunch of masses connected by springs, which all start out in equilibrium. The GW moves the masses, which stretches or compresses the springs, which then respond by pulling or pushing on the masses. The springs pulling and pushing are what directly do work. And all of this changes the stress-energy tensor of the mass-spring setup, which in turn changes the spacetime curvature--that's what reacts back on the GW and changes it to carry less energy than it was before (or, put more precisely, reduces its ability to induce stretching and compressing of springs in the next mass-spring setup it comes across).

I take it that a sphere of 100% efficient gravitational wave absorbers will generate no more than the mass loss of the emission event?

Yes.

 

[snorkack]

If a wave passes through detector changing neither wave's energy, momentum, angular momentum nor particle count, but does change wave's phase, can the detector change detector's state irreversibly, or is this illegal?

 

[Ibix]

Kinetic energy relative to what?

I was thinking of the rod in the sticky beads case - a non-inertial coordinate system. But I hadn't thought it all the way through, since the energy absorbed in that analysis is the difference between the kinetic energy of frictionless beads versus sticky beads. And in the case of LIGO there isn't (ideally) any interaction between the mirrors and the tunnel walls (which behave much like the rod in the sticky beads case), so that can't be the mechanism.

I'm a little confused where LIGO does absorb energy, then. If the work is done on the lasers, then to first order the energy increase in one beam is offset by the decrease in the other, I think. The arm lengths change by a factor of $\sqrt{1\pm h(t)}$, where $h(t)$ is the gravitational wave amplitude. That makes the energy in each arm change by one upon that factor - but $E/\sqrt{1\pm h(t)}\simeq E( 1\mp h(t)/2)$, so the increase in energy in one arm is the same as the decrease in the other. But surely it has to be absorbing something to make a detection? Are we really relying on the $h^2$ term?

In the Feynman's beads scenario, for example, the work isn't done directly on the beads by the GW; it's done by the inter-atomic forces between the beads. Think of the whole setup as a bunch of masses connected by springs, which all start out in equilibrium. The GW moves the masses, which stretches or compresses the springs, which then respond by pulling or pushing on the masses. The springs pulling and pushing are what directly do work.

That I can understand. The rod resists the length change, so its components are non-inertial, and the beads get dragged out of their inertial paths by friction leading to heating and usable energy.

And all of this changes the stress-energy tensor of the mass-spring setup, which in turn changes the spacetime curvature--that's what reacts back on the GW and changes it to carry less energy than it was before (or, put more precisely, reduces its ability to induce stretching and compressing of springs in the next mass-spring setup it comes across).

And this is similar to an analysis of EM antennae which regards the radiation after an antenna as the sum of the incident radiation and the (slightly out of phase) radiation from the induced dipole in the antenna. Gravitational waves are presumably more fun because they aren't linearly superposed, but I think the principle is the same.

grok

This is a rather jarringly different style of Martian to your avatar...

 

[PeterDonis]

If the work is done on the lasers, then to first order the energy increase in one beam is offset by the decrease in the other, I think.

Off the top of my head, I think either increase or decrease of the arm length should add energy; roughly speaking, the energy added should be proportional to the amplitude of the interference fringes in the detector. But I have not tried to take a detailed look at the math.

 

[Ibix]

Off the top of my head, I think either increase or decrease of the arm length should add energy; roughly speaking, the energy added should be proportional to the amplitude of the interference fringes in the detector. But I have not tried to take a detailed look at the math.

Does the "stretch and squish" apply to the tangent space as well? If so, it occurs to me that the EM wave becomes not quite an EM wave, as its component in the plane of the gravitational wave gets stretched-and-squished. That would obviously dissipate energy.

 

[PeterDonis]

Does the "stretch and squish" apply to the tangent space as well?

No. The tangent space at each point of spacetime is a flat Minkowski space.

 

[PeterDonis]

If so, it occurs to me that the EM wave becomes not quite an EM wave, as its component in the plane of the gravitational wave gets stretched-and-squished. That would obviously dissipate energy.

This has nothing to do with the tangent space. It happens in spacetime. I'm not sure what you mean by "not quite an EM wave", though. The frequency and wavelength can change, but the fact that it's in vacuum with no EM sources doesn't change.

 

[pervect]

Does the "stretch and squish" apply to the tangent space as well? If so, it occurs to me that the EM wave becomes not quite an EM wave, as its component in the plane of the gravitational wave gets stretched-and-squished. That would obviously dissipate energy.

I believe the "stretch and squish" is basically an artifact of the coordinate choice. By singling out a single point in space , which is a worldline in space-time, the point being an object "at rest in space", the worldline being the worldline of the object "at rest in space", one can eliminate the "stretch and squish" around a single point. The mathematical representation turns out to be less convenient, but it may be more physically intuitive in the new coordinates. Specifically, these are Fermi-Normal coordinates around that point in space (worldline in spacetime).

It's possible to talk about this without using coordinates. There is a local map called the "exponential map" <<wiki link>> that maps point from the tangent space to points on the manifold. Basically, distances in the tangent space from the central point in space to other nearby points are the same in the tangent space as they are in the manifold.

Of course one may need to define exactly what one means by "distances in the manifold". In this case by "distances in the manifold", I mean the length of a curve which is both a space-like geodesic in the associated space-time, and also is perpendicular to the reference worldline of the "stationary point" that is singled out.

Distances between points other than the central point are not necessarily quite the same in the tangent space and on the manifold, though. The best one can do is make the distances from the central point to nearby points match.

Creating such a map of the globe is an interesting thought experiment, one I've often wished I had a graphic of. But I don't have such a graphic. Technically speaking, this visual aid wouldn't be a Fermi-normal coordinate map, it'd just be Riemann normal coordinate map. The fermi-normal coordinates include time in the picture, the simplified graphic would not.. The idea with fermi-normal coordinates and the exponential map is similar though.

One imagines singling out a particular point on the globe (say the north pole), drawing great circles through this point (representing geodesics). Then one uses polar coordinates to define a map from the curved globe to a flat sheet of paper such that the polar coordinate "r" represents the distance along the great circle, and the polar coordinate "theta" represents the longitude of the great circle (which is constant).

This map would be the exponential map, the coordinates r and theta used in the construction would be Riemann normal coordinates. Fermi-normal coordinates are similar, but they also have to deal with time. This isn't all that complicated mathematically, but it may be difficult to visualize without some thought - and it's certainly harder to draw diagrams of.

Imagining such a map, one might see that it would be a good map near the North pole, but grow increasingly distorted way from the North pole. (Having a grahpic one would see this, but I don't have such a graphic, so one needs to imagine it).

Fermi-normal coordinates of a GW certainly exist, but they're a pain to work with. They do provide some physical insight though. The Fermi-normal coordinates have the property that near the reference point all the Christoffel symbols are zero, and their

Wiki's stub on "Fermi normal coordinates is" <<wiki link>>. It's worth noting that the metric near the reference worldline has a Minkowskii metric, and that the Christoffel symbols all vanish, which is what makes it fairly intuitive.

 

[PeterDonis]

I believe the "stretch and squish" is basically an artifact of the coordinate choice.

Not really. It's true that you can choose coordinates such that the coordinate distance between the mirrors in LIGO is constant as a GW passes through; but all that does is put all of the stretch and squish into the metric coefficients.

By singling out a single point in space , which is a worldline in space-time, the point being an object "at rest in space", the worldline being the worldline of the object "at rest in space", one can eliminate the "stretch and squish" around a single point. The mathematical representation turns out to be less convenient, but it may be more physically intuitive in the new coordinates. Specifically, these are Fermi-Normal coordinates around that point in space (worldline in spacetime).

You can do this around one of the LIGO mirrors, but the region covered by these coordinates (in which you can ignore tidal gravity, which is what the stretch and squish really is) will not be large enough to cover LIGO as a whole. So you cannot analyze LIGO as a whole using this method.

Distances between points other than the central point are not necessarily quite the same in the tangent space and on the manifold, though.

The tangent space, strictly speaking, has no distances. It is the space of all tangent vectors at a point. The vectors have lengths (norms), but their lengths do not correspond to distances.

Fermi-normal coordinates of a GW certainly exist

I disagree. The whole point of FN coordinates is to cover a patch on which tidal gravity can be ignored. GWs are waves of tidal gravity; that's what they're "made of". So within a single FN coordinate patch, GWs are not even detectable.

 

[pervect]

Not really. It's true that you can choose coordinates such that the coordinate distance between the mirrors in LIGO is constant as a GW passes through; but all that does is put all of the stretch and squish into the metric coefficients.

I can see that my remarks were not too clear, the ambiguity in my remarks is basically about what "stretch and squish" means.

In fact, I find it useful to think of the distance between the mirrors as varying with time when a gravitatioanl wave (GW) passes. This happens in coordinates where the metric coefficients are essentially very close to being Minkowskian - coordinats such as Fermi normal coordinates that make the metric Mikowskii at a point in space (in the neighborhood of a worldline of an observer in space-time). By also setting the first derivatives of the metric to be zero, this coordinate choice makes the metric close to Minkowskii over a rather broad region. Setting the first derivative of the metric coefficeitns to zero is equivalent to making the Christoffel symbols vanish.

The Riemann curvature tensor of the GW is non-zero, and the Riemann curvature tensor can be interpreted in the weak field case as a tidal force.

The opposing point of view has the metric varying , having the GW being a metric pertubation and the mirrors having constant coodinates. It's a valid point of view, but not the one I was espousing. One can also describe this point of view as having "stretch and squish", but in this point of view the stretch and squish happens to the metric coefficients.

 

[Paul Colby]

If a wave passes through detector changing neither wave's energy, momentum, angular momentum nor particle count, but does change wave's phase, can the detector change detector's state irreversibly, or is this illegal?

Some numbers are in order.

The power flux in a passing GW is

$P = \frac{c^3}{16\pi G}\omega^2 h^2$​

where $h$ is the amplitude of the wave in the transverse traceless gauge and $\omega$ is the angular frequency, say 200 Hz. $h \approx 10^{-21}$. I get 0.013 watt per square meter. Now the LIGO aperture is like 16 square kilometers so about 208,000 watts of gravitational wave energy are passing through the detector aperture. Clearly, only a tiny fraction of this power is converted into detectable signal.

Your central question I think is how much of this power is converted. Well, the power in the laser beam is $I$ watts at say 700 nm. The photons per second, $N$ are, (h is plank's constant when it is)

$N = \frac{I}{h \nu}$​

The momentum change of a photon bouncing off a mirror is,

$\delta p = 2 h \nu c^{-1}$​

so the radiation force is

$F = \delta p N = 2 I c^{-1}$​

$L$ being 4 km. In rough terms, the work done in changing the length of the arm is

$W = F L h = 2 I c^{-1} L h$​

What's I? I'm sure one can look it up, but let's assume 1kW. I get $W = 2.7\times 10^{-23}$ watts converted by the detector. The other stuff you're asking about such as radiation back reaction, is a small small small fraction of this number.

 

[Paul Colby]

Since the thread title is "How Many Gravitons Detected" I should do the final step and divide $W$ by $h f$ where $f = 200$ Hz to get the graviton rate. I get an estimated 200 million gravitons per second absorbed by the detector. The duration of the signal, again could be looked up, but I recall it's less than a second?

 

[PeterDonis]

I find it useful to think of the distance between the mirrors as varying with time when a gravitatioanl wave (GW) passes.

You are using "distance" to mean "coordinate distance". But this is coordinate-dependent. As you agree, you could choose coordinates in which the coordinate distance between the mirrors is constant and all of the stretch and squish is in the changes to the metric coefficients.

However, in both of these coordinate systems (and in any coordinate system), the GW passage will cause the LIGO detector to register interference between the beams in the two arms. The "distance" in the sense along each arm is changing in an invariant way.

This happens in coordinates where the metric coefficients are essentially very close to being Minkowskian - coordinats such as Fermi normal coordinates that make the metric Mikowskii at a point in space (in the neighborhood of a worldline of an observer in space-time). By also setting the first derivatives of the metric to be zero, this coordinate choice makes the metric close to Minkowskii over a rather broad region.

Not broad enough to cover all of LIGO.

Setting the first derivative of the metric coefficeitns to zero is equivalent to making the Christoffel symbols vanish.

You can only do this along a single worldline, because you cannot set all of the second derivatives of the metric coefficients equal to zero; those are curvatures and you can't make curvature go away by changing coordinates. And the region over which the Christoffel symbols are negligible (not zero, but small enough not to matter) will not be large enough to cover all of LIGO. (If it were, your equations would predict zero signal in the detector.)

 

[snorkack]

What was the effect of LIGO detection on particle count of gravitational waves it detected?

Photons can be and commonly are detected when they are absorbed. In which case the photon stops existing and its whole energy, momentum and angular momentum are transferred on the detector.

However, photons can also undergo Compton scattering. In which case the photon continues (in a different direction) and some of its energy and momentum are conferred on translational motion of the detecting particle. Also detectable.
And photons can also undergo Raman scattering. In which case the photon also continues in a different direction, but some of its energy is conferred on internal motion of the detecting system - or indeed received from the internal motion of the detecting system.

What is the mode by which Ligo detects gravitons? Absorption, Compton scattering, Raman scattering, some others?

 

[PeterDonis]

What was the effect of LIGO detection on particle count of gravitational waves it detected?

Nobody knows. The gravitational waves detected by LIGO were far too weak for us to detect any quantum aspects of them. So we have no way of counting the particles.

Also, you are assuming that the GWs detected by LIGO were in states that even have a definite particle count. We don't know that that's true (and I suspect it's not, just as most EM waves are not in states that have a definite photon number).

Photons can be and commonly are detected when they are absorbed.

Yes, and in fact the term "photon" is best used to refer to these discrete detection events, not to any property of the EM field before the detection. For example, detectors detect discrete events with laser light, but the state of the EM field in a laser is a coherent state, which does not have a definite photon number. (In fact it's an eigenstate of the annihilation operator, so "removing a photon" from it does not even change the state at all!) So the term "photon" can be highly misleading unless you discard virtually all of your normal intuitions about what it means. (Note that coherent states do not just model lasers; the kinds of EM field states we think of as classical EM waves, like radio waves, are also coherent states.)

What is the mode by which Ligo detects gravitons? Absorption, Compton scattering, Raman scattering, some others?

None of the above. LIGO does not detect gravitions. It detects gravitational waves--fluctuations in tidal gravity. LIGO is not the gravitational analogue of a "photon detector" that gives discrete events. It is the gravitational analogue of an antenna: the passage of a wave creates oscillations in the antenna that are driven by the oscillations in the wave. In short, LIGO is a classical detector, not a quantum one.

Note, btw, that if you put all the things I've said above together, the most likely answer to your original question, what is the effect of LIGO on the particle count of the GWs it detects, is "none" (because of my parenthetical notes about coherent states above).

 

[snorkack]

Nobody knows. The gravitational waves detected by LIGO were far too weak for us to detect any quantum aspects of them. So we have no way of counting the particles.

The angular momentum of the gravitational wave is quantized in units of h - no matter how small the ν, and thereby energy hν. (Actually, a supposed property of graviton is spin 2).
The quantum state of the Ligo detector is also quantized in the units of h.

Also, you are assuming that the GWs detected by LIGO were in states that even have a definite particle count. We don't know that that's true (and I suspect it's not, just as most EM waves are not in states that have a definite photon number).
Yes, and in fact the term "photon" is best used to refer to these discrete detection events, not to any property of the EM field before the detection. For example, detectors detect discrete events with laser light, but the state of the EM field in a laser is a coherent state, which does not have a definite photon number. (In fact it's an eigenstate of the annihilation operator, so "removing a photon" from it does not even change the state at all!) So the term "photon" can be highly misleading unless you discard virtually all of your normal intuitions about what it means. (Note that coherent states do not just model lasers; the kinds of EM field states we think of as classical EM waves, like radio waves, are also coherent states.)
None of the above. LIGO does not detect gravitions. It detects gravitational waves--fluctuations in tidal gravity. LIGO is not the gravitational analogue of a "photon detector" that gives discrete events. It is the gravitational analogue of an antenna: the passage of a wave creates oscillations in the antenna that are driven by the oscillations in the wave. In short, LIGO is a classical detector, not a quantum one.

Note, btw, that if you put all the things I've said above together, the most likely answer to your original question, what is the effect of LIGO on the particle count of the GWs it detects, is "none" (because of my parenthetical notes about coherent states above).

What are then the effects of the antenna on the coherent state of the laser radiation/radio waves/gravitational radiation?
If the perturbed wave state, post-detection, has the same frequency but lower amplitude, it also has a smaller particle count. That´s the mode of absorption.
Compton and Raman scattering have the same particle count post-detection, but energy is transferred to antenna because the perturbed wave ends up redshifted to a lower frequency.

So which of them does the Ligo antenna do?

 

[PeterDonis]

The angular momentum of the gravitational wave is quantized in units of h

No. The graviton is spin 2, but that does not mean a gravitational wave in free space must have quantized angular momentum, any more than the photon being spin 1 means an electromagnetic wave in free space must have quantized angular momentum.

The quantum state of the Ligo detector is also quantized in the units of h.

You appear to have a misconception that everything must be quantized in units of h. That's not correct. You need to improve your understanding of quantum mechanics.

What are then the effects of the antenna on the coherent state of the laser radiation/radio waves/gravitational radiation?

A very, very, very small reduction in amplitude.

If the perturbed wave state, post-detection, has the same frequency but lower amplitude, it also has a smaller particle count.

Wrong. A coherent state does not even have a definite particle count to begin with.

The state post-detection does have a very, very, very slightly smaller expectation value of energy and photon number. But that's not the same as "smaller particle count". (For one thing, the expectation value of photon number in a coherent state is not, in general, an integer.)

Again, you need to improve your understanding of quantum mechanics.

 

[PeterDonis]

The state post-detection does have a very, very, very slightly smaller expectation value of energy and photon number.

Note: This was talking about an EM wave, not a gravitational wave. We don't have a good theory of quantum gravity, so we don't know if there even are valid energy and "graviton number" operators for the quantum analysis of a GW. (The fact that gravitational wave energy cannot be localized strongly suggests, in fact, that the answer to both questions I just asked is "no".)

 

[Paul Colby]

What is the mode by which Ligo detects gravitons? Absorption, Compton scattering, Raman scattering, some others?

Well, can you answer the very same question replacing photon for graviton and AM radio for LIGO?

 

[snorkack]

Well, can you answer the very same question replacing photon for graviton and AM radio for LIGO?

Probably absorption, mostly. Because classical effects do not allow Compton and Raman scattering. (Rayleigh scattering is allowed... but does it create detection in the scatterer?)

But the energy detected by Ligo is not all energy of gravitational waves absorbed in it. The energy reaching detectors is of photons emitted by lasers inside Ligo. When not perturbed by the gravitational waves into interference minima, it just goes to interference maxima.

Ligo achieves detection by being very sensitive by weakly interacting gravitational waves through amplifying tiny effects. Is it sensitive enough to amplify to detectability quantum noise from the quantum nature of Ligo detector itself and of the gravitational waves detected?

 

[PeterDonis]

The energy reaching detectors is of photons emitted by lasers inside Ligo.

If you think of lasers as being made of photons, yes. But, as I've already said, a laser emits an EM field in a coherent state, which does not have a definite photon number and is not usefully thought of as "made of photons" unless you are using it with detectors that specifically detect discrete events. The LIGO detectors do not; they detect interference fringes. So the laser light in LIGO is best thought of as coherent EM waves, not photons.

It is true that LIGO does not directly detect the energy carried by GWs. The interference it detects is a side effect of the GW's depositing a very small amount of energy as @Paul Colby described.

When not perturbed by the gravitational waves into interference minima, it just goes to interference maxima.

No, when not perturbed by LIGO, there is no interference at all. Interference maxima and minima are both signs of a GW passing through.

Is it sensitive enough to amplify to detectability quantum noise from the quantum nature of Ligo detector itself and of the gravitational waves detected?

No, not by many orders of magnitude.

 

[romsofia]

I think this paper is very relevant to this thread:
https://arxiv.org/abs/1712.04437

 

[Paul Colby]

Probably absorption, mostly. Because classical effects do not allow Compton and Raman scattering. (Rayleigh scattering is allowed... but does it create detection in the scatterer?)

What is it you mean by absorption? Can you draw and sum the relevant Feynman diagrams for the radio? This would be somewhat of a challenge but possible, well in principle. I certainly couldn't do it nor would I try. It's not a great way to way to think about the problem.

So, you exclude Raman scattering? If I take an antenna and connect it to ground via a diode and then illuminate it with monochromatic radio waves, it will reradiate a small part of the energy at frequencies other than the illuminating wave because diodes are nonlinear. This is the cause of passive intermodulation or PIM a phenomena which plagues satellites if they put transmit antennas too close to receive antennas and aren't too careful about antenna construction details. Anyway, RF photons go in at $\omega$ and come out at $N \omega$ where $N$ is any integer. If one is broadcasting information then all sorts of different frequencies happen.

But the energy detected by Ligo is not all energy of gravitational waves absorbed in it.

So, I'm struggling with how is this any different than our radio example? In a radio the detected power is amplified to the point it can drive a speaker. In some radios this is done by mixing with local oscillators (AKA lasers for LIGO) and amplifying.

There is one major difference and this is a radio is converting electromagnetic power into more electromagnetic power. The GW detector must convert illuminating gravitational wave power ultimately into electromagnetic power. This is true of all GW detectors including resonate bar detectors since one must at some point detect the bar vibration which is done via electromagnetic means.

 

[Paul Colby]

I think this paper is very relevant to this thread:
https://arxiv.org/abs/1712.04437

This is a nice paper. I would question the assumption that the mirrors have a continuum spectra though. The LIGO mirrors are confined in a finite well so they will have a discrete energy spectrum. Not clear this matters but for tiny things like this it should be noted at least.

 

[Paul Colby]

This is a nice paper.

Starting to read this paper more closely. I think I may need to change my assessment. IMO equation (4), the "energy" of the mirrors, seems bogus. The metric strain given is in the TT gauge. In this gauge, the mirrors are not moving relative to the coordinate system so (4) hasn't its usual meaning as an energy. From here the argument goes down hill.

 

[snorkack]

So, you exclude Raman scattering? If I take an antenna and connect it to ground via a diode and then illuminate it with monochromatic radio waves, it will reradiate a small part of the energy at frequencies other than the illuminating wave because diodes are nonlinear. This is the cause of passive intermodulation or PIM a phenomena which plagues satellites if they put transmit antennas too close to receive antennas and aren't too careful about antenna construction details. Anyway, RF photons go in at $\omega$ and come out at $N \omega$ where $N$ is any integer. If one is broadcasting information then all sorts of different frequencies happen.

That´s overtone generation by multiphoton process, right?
Raman scattering is intermodulation where photons go in at ω and come out at ω-ω₀, where ω₀ is a frequency characteristic to the antenna.

So, I'm struggling with how is this any different than our radio example? In a radio the detected power is amplified to the point it can drive a speaker. In some radios this is done by mixing with local oscillators (AKA lasers for LIGO) and amplifying.

There is one major difference and this is a radio is converting electromagnetic power into more electromagnetic power. The GW detector must convert illuminating gravitational wave power ultimately into electromagnetic power. This is true of all GW detectors including resonate bar detectors since one must at some point detect the bar vibration which is done via electromagnetic means.

I was trying to track down what precisely the "detected power" is, and what the fundamental constraints are on the "detected power" that can be amplified to a detected signal.

Fundamentally, suppose an electromagnetic wave were incident on a laser/maser at a negative temperature, and were to cause stimulated emission.
Stimulated emission should, in principle, be just as detectable as absorption, and existence of stimulated emission under conditions where spontaneous emission is unlikely would be evidence of existence of incident radiation. Yet the "detected power" would be negative.

Could you detect an electromagnetic wave at precisely zero "detected power"?

The electromagnetic wave has conservative properties other than energy. These are momentum and angular momentum.

You might detect an electromagnetic wave by a detector that leaves the energy of the wave precisely unchanged but alters its momentum by altering its direction. The momentum must be received by the detector and become detectable, with amplifiable effects.

If an electromagnetic wave possesses circular polarization, it bears angular momentum. Therefore, a detector which leaves both the energy and direction of a wave perfectly unchanged, but changes its circular polarization, must receive angular momentum and be detectable, with amplifiable effect.

This leaves two other properties of a wave that do NOT carry conserved physical quantities.
These are phase and state of linear polarization.

Detection might, in principle, take place by rearranging conserved physical quantities inside the detector, without transferring any conserved quantities from wave to detector.
But I have a suspicion that there is some rule against this. Is this the case?

 

[Paul Colby]

I was trying to track down what precisely the "detected power" is, and what the fundamental constraints are on the "detected power" that can be amplified to a detected signal.

I feel your pain. This is a question well worth pondering. It has a simple answer for all the cases I know of. Any signal, electromagnetic or gravitational, will need to do work on the detector. The rate of this work is the detected power.

Could you detect an electromagnetic wave at precisely zero "detected power"?

No, never ever.

What you would find is a signal to noise ratio of 0 since nose power is never 0. To be detected this ratio needs to be big enough S/N (5 would be nice). This is why it's so very important to maximize the received power since the majority of noise comes from the first amplification stage and has nothing to do with your signal. In RF systems this is why impedance matching the antenna is important. One maximizes the power coupling from the antenna to the radio by matching the impedance. This means the most you get is half.

This is a question that comes up in every link calculation ever made. Basically, there is no free lunch.

Detection might, in principle, take place by rearranging conserved physical quantities inside the detector, without transferring any conserved quantities from wave to detector.
But I have a suspicion that there is some rule against this. Is this the case?

Interference fringes are like this, right. One is moving energy not creating or losing it. However, one must ask why (and how) have the beams changed. LIGO can be viewed as a parametric amplifier. The parameter in this case is the length, L, of the interferometer arms. As the GW passes it changes this length which takes work. BTW the work taken goes to 0 if the lasers are turned off. The work increases as the laser power in increased. The true problem for LIGO (the minor bit that took like 40 years and 1/3 of the physics community) is maximizing the signal to noise. This was not a simple task.

 

[Paul Colby]

That´s overtone generation by multiphoton process, right?
Raman scattering is intermodulation where photons go in at ω and come out at ω-ω0, where ω0 is a frequency characteristic to the antenna.

So Raman scattering is inelastic scattering off of atoms and molecules, so technically, no. My example was just to show the most primitive radio, a diode with an antenna, will involve harmonics which are, in a complete analysis, are multi photon processes but not technically Raman scattering as far as I know.

 

[PeterDonis]

The parameter in this case is the length, L, of the interferometer arms. As the GW passes it changes this length which takes work

I think this is oversimplifying. The arms are not solid objects. They are just mirrors in free fall (at least for the purposes of analysis here; I realize that the LIGO detectors are actually at rest on the surface of the Earth so they're not in free fall, although future space-based detectors will be). Changing the distance between them by changing tidal gravity, in and of itself, does no work. What does work is changing something that connects the mirrors or travels between them. In your earlier post, you said, correctly, that in the case of LIGO it's the light beams, reflecting off the mirrors, that have work done on them by the GW.

 

[Paul Colby]

I think this is oversimplifying.

Fair enough. At one time a read some papers on the design of LIGOs mirror mounts. They are supported on a rather complicated structure designed to isolate them from seismic noise. The laser mode existing between the mirror apply a force which is balanced by the detent of the support. Think of a pendulum bob. With no laser power the bob hangs vertically. With the laser it presses out till an equilibrium is reached, basically where the forces balance. This isn't a big distance. The total relative movement of the mirrors is less than a proton diameter after all. Basically laser + mirrors is an extended object upon which work is done changing it's length.

 

[Paul Colby]

Think of a pendulum bob. With no laser power the bob hangs vertically. With the laser it presses out till an equilibrium is reached, basically where the forces balance.

While I believe this is true in practice, it's kind of beside the real issue. Even if the mirrors were completely stationary in the TT frame, something I think is essentially true due to their mass, changing the length of the interferometer does work on the light. It's like standing up. My legs do work on my body but also a very very tiny amount of work on the earth. Tiny because I haven't moved the Earth very far from its original position.

 

[snorkack]

I feel your pain. This is a question well worth pondering. It has a simple answer for all the cases I know of. Any signal, electromagnetic or gravitational, will need to do work on the detector. The rate of this work is the detected power.
No, never ever.

What you would find is a signal to noise ratio of 0 since nose power is never 0.

This is a question that comes up in every link calculation ever made. Basically, there is no free lunch.
Interference fringes are like this, right. One is moving energy not creating or losing it. However, one must ask why (and how) have the beams changed.

Is it legal for a detector to receive pure linear momentum, unaccompanied by any power whatsoever, and amplify it into output power?

 

[PeterDonis]

pure linear momentum, unaccompanied by any power whatsoever

There is no such thing.

 

[snorkack]

There is no such thing.

True, because the detector has finite mass.
You can therefore choose a frame of reference where the input power is positive, a frame of reference where input power is negative, or frame of reference where input power is exactly zero.

 

[Paul Colby]

Is it legal for a detector to receive pure linear momentum, unaccompanied by any power whatsoever, and amplify it into output power?

No, momentum is the space component of a 4-vector. You aren't permitted to change the space part without the energy part also changing.

In simple terms, there is no free lunch. For something to be detected, something in the detector must change. For something in the detector to change, work must be done on the detector by some external "signal" or phenomena.

Angular momentum will be the same except it's a component of higher rank tensor and so more confusing.

 

[snorkack]

No, momentum is the space component of a 4-vector. You aren't permitted to change the space part without the energy part also changing.

In simple terms, there is no free lunch. For something to be detected, something in the detector must change. For something in the detector to change, work must be done on the detector by some external "signal" or phenomena.

What makes it illegal for a detection to happen in a frame of reference where the momentum of the detector changes direction, but the energy of the detector is precisely unchanged? Momentum is a vector, energy a scalar.

 

[PeterDonis]

True, because the detector has finite mass.

No, it's because there is no such thing as linear momentum with zero power, period. Even massless objects, like light, can't have linear momentum without delivering power.

You can therefore choose a frame of reference where the input power is positive, a frame of reference where input power is negative, or frame of reference where input power is exactly zero.

Please show your work.

 

[PeterDonis]

a frame of reference where the momentum of the detector changes direction, but the energy of the detector is precisely unchanged?

How do you know there is such a frame of reference? Remember we're not talking about a general possibility, but about the specific case of a gravitational wave detector.

 

[snorkack]

No, it's because there is no such thing as linear momentum with zero power, period. Even massless objects, like light, can't have linear momentum without delivering power.

They cannot have linear momentum without carrying power.
Carrying power is not the same as delivering it.
If a ray of light interacts with a mirror, what is the "power delivered"? The total power of the light which changes direction on reflection? Or only the power which the light loses by getting redshifted, and confers on the mirror?

Please show your work.

Easy.
Since incident and reflected light move in different direction, they undergo different Doppler shift on change of reference frame.
You can choose a reference frame where the reflected light is blueshifted and it is the mirror that is doing work against light pressure to blueshift the incident light.
Or a reference frame where the energy of incident and reflected light are exactly equal, and no work whatever is done by or to light.

 

[Paul Colby]

Momentum is a vector, energy a scalar.

Energy in this case is the fourth component of the energy-momentum 4-vector. This fact matters.

 

[PeterDonis]

Carrying power is not the same as delivering it.

Carrying linear momentum is not the same as delivering it either. An object cannot deliver linear momentum to a detector without delivering power as well.

If a ray of light interacts with a mirror, what is the "power delivered"?

The rate of change of energy transfer from the light to the mirror.

 

[Paul Colby]

What makes it illegal for a detection to happen in a frame of reference where the momentum of the detector changes direction, but the energy of the detector is precisely unchanged? Momentum is a vector, energy a scalar.

Well, the obvious answer is nothing observable changes without some work being done somewhere on the detector. Can you give one complete example where this ever happens? I can't because it would violate general well established principles as I understand them. So help me out here.