Oscillating masses and gravitational waves

[DaTario]

Let me ask one simple question. For less than a thousand dollars one can buy an apparatus which can provide experimental evidence for the gravitational attraction between two masses of one kilogram each, placed at distances of the order of $10^{-1} m$. By making one of these masses to oscillate one time with a very small amplitude ($\approx 0.005 m$) and with a period T of, say, 10 seconds, we will see a change in the gravitational force's direction on the other body. Assuming that the change in gravitational force has propagated according to the predictions of GR, this experiment will also correspond to one which exhibits one GW.

Is it correct?
DaTario

 

[Vanadium 50]

No, that's not radiation. That's near-field.

 

[DaTario]

But isn't the difference between these two cathegories just pure formality? Near the source the field also propagates, isn't it?
It seems to me that this names refer to some practical relevance due to the morphology of the waves or the way the propagation occurs, not due to something that has an intrinsically physical distinction. It seems that this difference has more to do with the mathematical multipole expansion of the field in the two regions than with the essencial physical principles involved.

Please, correct me if I am wrong.

 

[Vanadium 50]

But isn't the difference between these two cathegories just pure formality?

No. Near-field goes as 1/r^2. Radiation goes as 1/r.

 

[DaTario]

I am aware of this fact, but this seems to me as just a mathematical decomposition of the signal. Both sum up to being the complete signal. Near the source we find both.

Are you saying that near my oscilatting one kilogram there is no GW ?

 

[PAllen]

I am aware of this fact, but this seems to me as just a mathematical decomposition of the signal. Both sum up to being the complete signal. Near the source we find both.

Are you saying that near my oscilatting one kilogram there is no GW ?

There is GW, but it is like 20 orders of magnitude smaller than the near field effect. This is even true for the inspiralling BH (the factor might be different). The GW become significant at great distance because the near field effect decreases so much faster. Further, the affect of GW on matter is completely different than the effect of pushing a source back and forth.

 

[DaTario]

Ok. But when you say "Further, the affect of GW on matter is completely different than the effect of pushing a source back and forth." are you saying that the physics are totally different? Or is it only a difference in the function F(t) which will act on the body. By the way I manage to convert my question in one gracious figure.

 

[Vanadium 50]

I am aware of this fact, but this seems to me as just a mathematical decomposition of the signal.

Nevertheless, in the interest of communication, doesn't it make sense to use the words "near-field" and "radiation" the way everyone else does? Otherwise we're in Humpty-Dumpty Land, where "[a word] means just what I choose it to mean, neither more nor less!"

There is also the issue of quadrupole radiation. If you move a mass back and forth in only one direction, there is no gravitational radiation. It needs a changing quadrupole moment, which requires motion in two dimensions. In this respect, gravitational radiation is unlike electromagnetic radiation.

 

[PeterDonis]

I manage to convert my question in one gracious figure.

What is the waveform graph in this picture supposed to be describing? Do you know how the waveforms of GWs are described?

 

[DaTario]

Of course. No problem in using this terminology.
Now considering the figure I posted, it seems to be, in principle, possible to produce the same effect of Strain vesus Time, on those interferometers by artificially moving a considerable piece of mass in the vicinity of the apparatus.

My question address particularly the possibility of imitating the radiation field through the near field produced by the use of a well controlled gravitational source.

best wishes,

DaTario

 

[lpetrich]

Nevertheless, in the interest of communication, doesn't it make sense to use the words "near-field" and "radiation" the way everyone else does? Otherwise we're in Humpty-Dumpty Land, where "[a word] means just what I choose it to mean, neither more nor less!"

Yes indeed. Solar-System tests of general relativity are all near-field tests.

There is also the issue of quadrupole radiation. If you move a mass back and forth in only one direction, there is no gravitational radiation. It needs a changing quadrupole moment, which requires motion in two dimensions. In this respect, gravitational radiation is unlike electromagnetic radiation.

That's not correct. It will still have a quadrupole moment relative to some point, as it does a dipole moment. Monopole (M), dipole (D), and quadrupole (Q):
$$M = \sum m $$
$$D_i = \sum m x_i$$
$$Q_{ij} = \sum m (x_i x_j - (1/3) \delta_{ij} x^2)$$
The lowest-order mass-monopole effect is Newtonian gravity, much like the electrostatic force in electromagnetism. It is quasi-static, meaning that the field seems to propagate instantaneously.

The lowest-order source for electromagnetic waves is the electric dipole. Its gravitational counterpart is the offset of the barycenter, and it doesn't radiate.

The lowest-order source for gravitational waves is the mass quadrupole.

I'll now consider the mass quadrupole moment of an object with mass m displaced in only one direction: {x,0,0}. It is m*x²*{{2/3,0,0},{0,-1/3,0},{0,0,-1/3}}. So it's nonzero.

To lowest order, the energy output of gravitational waves is proportional to the square of the third derivative of the mass quadrupole moment. To vanish, that quadrupole moment must be a polynomial in time with a degree of at most 2. This in turn implies that a single source will have a constant velocity. So a single source accelerating will make gravitational waves, though multiple sources can produce waves that cancel each other out, as with electromagnetism.

 

[DaTario]

What is the waveform graph in this picture supposed to be describing? Do you know how the waveforms of GWs are described?

Sorry for having inverted the function. I only tried to express the fact that a pendulum tends to stop and it seemed to be reasonable to propose that this would also happen to the oscillations.
Trying to answer your question, I suppose these oscillations are produced due to a time dependent interference pattern caused by local metric oscillations which change differently the arms of the interferometer.

 

[PeterDonis]

My question address particularly the possibility of imitating the radiation field through the near field produced by the use of a well controlled gravitational source.

Before tackling this question with regard to gravity and gravitational waves, it's helpful to consider it for the simpler case of electromagnetism and electromagnetic waves. Suppose we have a source charge that we are wiggling back and forth. The wiggling produces changes in the EM field that propagate outward as wave fronts. But if we have a test charge a fairly short distance away, the field seen by that test charge will not be just the radiation field; instead it will be a combination of the radiation field and the "average" Coulomb field of the source charge. ("Average" here means, heuristically, the Coulomb field that would be produced by the source charge if it were stationary at its average position.) The Coulomb field will be much stronger if we are close enough to the source charge; but it falls off as $1/r^2$ while the radiation field falls off as $1/r$; so as we get farther away from the source charge, the radiation field becomes larger relative to the Coulomb field, and once we are very far away, the Coulomb field is negligible and the radiation field is all that we can detect.

The different dependences on distance ($1/r^2$ vs. $1/r$) are why the two effects, "Coulomb" and "radiation", are physically different; one can't "imitate" the other. Similar remarks apply to gravitational waves vs. the Newtonian gravitational "force".

 

[DaTario]

Yes indeed. Solar-System tests of general relativity are all near-field tests.

That's not correct. It will still have a quadrupole moment relative to some point, as it does a dipole moment. Monopole (M), dipole (D), and quadrupole (Q):
$$M = \sum m $$
$$D_i = \sum m x_i$$
$$Q_{ij} = \sum m (x_i x_j - (1/3) \delta_{ij} x^2)$$
The lowest-order mass-monopole effect is Newtonian gravity, much like the electrostatic force in electromagnetism. It is quasi-static, meaning that the field seems to propagate instantaneously.

The lowest-order source for electromagnetic waves is the electric dipole. Its gravitational counterpart is the offset of the barycenter, and it doesn't radiate.

The lowest-order source for gravitational waves is the mass quadrupole.

I'll now consider the mass quadrupole moment of an object with mass m displaced in only one direction: {x,0,0}. It is m*x²*{{2/3,0,0},{0,-1/3,0},{0,0,-1/3}}. So it's nonzero.

To lowest order, the energy output of gravitational waves is proportional to the square of the third derivative of the mass quadrupole moment. To vanish, that quadrupole moment must be a polynomial in time with a degree of at most 2. This in turn implies that a single source will have a constant velocity. So a single source accelerating will make gravitational waves, though multiple sources can produce waves that cancel each other out, as with electromagnetism.

I am not quite sure about these calculations, but the term which is instantaneous may be seen as an artificiallity of the formalism, once what is measured is always the sum of the terms. And by summing up all the terms, that one which is instantaneous cancels out with other terms, in such a way that the gravitational field cannot be understood as being instantaneous, except if you want to add and take out this term.

 

[DaTario]

Before tackling this question with regard to gravity and gravitational waves, it's helpful to consider it for the simpler case of electromagnetism and electromagnetic waves. Suppose we have a source charge that we are wiggling back and forth. The wiggling produces changes in the EM field that propagate outward as wave fronts. But if we have a test charge a fairly short distance away, the field seen by that test charge will not be just the radiation field; instead it will be a combination of the radiation field and the "average" Coulomb field of the source charge. ("Average" here means, heuristically, the Coulomb field that would be produced by the source charge if it were stationary at its average position.) The Coulomb field will be much stronger if we are close enough to the source charge; but it falls off as $1/r^2$ while the radiation field falls off as $1/r$; so as we get farther away from the source charge, the radiation field becomes larger relative to the Coulomb field, and once we are very far away, the Coulomb field is negligible and the radiation field is all that we can detect.

The different dependences on distance ($1/r^2$ vs. $1/r$) are why the two effects, "Coulomb" and "radiation", are physically different; one can't "imitate" the other. Similar remarks apply to gravitational waves vs. the Newtonian gravitational "force".

I understand this point, but note that I have used a malicious argument with the word "artificially" with respect to the movement.

 

[PeterDonis]

I suppose these oscillations are produced due to a time dependent interference pattern caused by local metric oscillations which change differently the arms of the interferometer.

Basically, yes. But one waveform can't really describe a GW. GWs have two possible polarizations, as shown here:

https://en.wikipedia.org/wiki/Gravitational_wave#Effects_of_passing

So to fully describe a GW, you need two waveforms, one for each of the polarizations; there is no necessary connection between them. (I don't know if LIGO was able to measure both waveforms; they may only have been able to measure one, since one waveform from each detector is all that I've seen in published info.) Also, you have to bear in mind that even if you have both waveforms, that is not a complete description of spacetime curvature, i.e., it's not a complete description of the effects of gravity; see my previous post.

 

[PeterDonis]

I have used a malicious argument with the word "artificially" with respect to the movement.

You're missing the point. The point isn't that one piece of the field is instantaneous and the other is time delayed. Any changes in the field can only propagate at the speed of light. So what looks like an "instantaneous" Coulomb or Newtonian field (depending on whether we're talking about EM or gravity) isn't really; the observed field isn't responding to the source "right now", it's responding to the source on the past light cone of the event at which the field is being measured. But for the portion of the source which can be considered static (for example, the "average" position of the source in my EM example), the source on the past light cone is the same as the source "right now", so the field looks like an instantaneous field because the source didn't change during the light propagation time.

However, you are ignoring the real point (even though you say you understand it), which is that one piece of the field falls off as $1/r^2$ and the other falls off as $1/r$; in other words, we have two different aspects of the field with two different distance dependences. That is why we separate them conceptually. Of course nature doesn't care how we write the equations; but the conceptual separation of different distance dependences is not "artificial". It isn't put in by hand; it pops right out of the equations.

 

[DaTario]

Basically, yes. But one waveform can't really describe a GW. GWs have two possible polarizations, as shown here:

https://en.wikipedia.org/wiki/Gravitational_wave#Effects_of_passing

So to fully describe a GW, you need two waveforms, one for each of the polarizations; there is no necessary connection between them. (I don't know if LIGO was able to measure both waveforms; they may only have been able to measure one, since one waveform from each detector is all that I've seen in published info.) Also, you have to bear in mind that even if you have both waveforms, that is not a complete description of spacetime curvature, i.e., it's not a complete description of the effects of gravity; see my previous post.

Ok with this part. The complete information about a wave, in general, rarely can be inferred by monitoring the pertubation in one or two locations.

 

[DaTario]

You're missing the point. The point isn't that one piece of the field is instantaneous and the other is time delayed. Any changes in the field can only propagate at the speed of light. So what looks like an "instantaneous" Coulomb or Newtonian field (depending on whether we're talking about EM or gravity) isn't really; the observed field isn't responding to the source "right now", it's responding to the source on the past light cone of the event at which the field is being measured. But for the portion of the source which can be considered static (for example, the "average" position of the source in my EM example), the source on the past light cone is the same as the source "right now", so the field looks like an instantaneous field because the source didn't change during the light propagation time.

However, you are ignoring the real point (even though you say you understand it), which is that one piece of the field falls off as $1/r^2$ and the other falls off as $1/r$; in other words, we have two different aspects of the field with two different distance dependences. That is why we separate them conceptually. Of course nature doesn't care how we write the equations; but the conceptual separation of different distance dependences is not "artificial". It isn't put in by hand; it pops right out of the equations.

Ok, so these two terms have important physical differences. But both can be understood as propagating terms, which do not respond to the source " right now", as you said.
What is the relevant difference between them? Specially with respect to this matter, GW?

 

[PeterDonis]

the conceptual separation of different distance dependences is not "artificial"

To elaborate on this: if you wiggle a mass around (with a quadrupole moment with a nonzero third time derivative), you will, in principle, generate GWs, i.e., fluctuations in spacetime curvature. But those fluctuations will be superimposed on the static field of the static part of the source (heuristically, the average position of the mass, such as the pendulum in your example). Only the fluctuations will produce changes in things like the arm lengths of a GW interferometer; the interferometer won't detect the static part of the field. And if the static part of the field is strong enough (i.e., if you're close enough to the source), it might overwhelm the fluctuating part--heuristically, your interferometer might have to be under such a large strain just to sit at rest in the field that the tiny changes in strain produced by the fluctuating GWs become unmeasurable. (Consider trying to measure small fluctuations in the position of a large electric charge using an antenna that is very close to the charge--the large static electric field might make the small fluctuations in position of the charges in the antenna due to the EM wave unmeasurable.)

So when you talk about "imitating" a radiation field with a near field, that doesn't really make sense. Wiggling a mass produces a radiation field, period. The question is whether that radiation field is detectable against the background field of the source. The further you get from the source, the weaker that background field becomes, and the easier it becomes to detect the radiation field.

 

[PeterDonis]

both can be understood as propagating terms, which do not respond to the source " right now", as you said.

Not responding to the source "right now" is not the same as "propagating". (The usual word for "not responding to the source right now" is "retarded", which I have never liked; I usually use "time delayed".) "Propagating" means something about the source is changing, causing changes in the field, which can only travel at the speed of light. If nothing about the source is changing, the field is still time delayed, but since it's not changing, the time delay makes no difference, because there is no change that needs to travel anywhere. And if something about the source is changing, but only by a small amount compared to its average value, then the time delay makes only a very little difference.

The relative magnitudes of the $1/r^2$ and $1/r$ terms tell you how much of a difference the time delay makes; and obviously those relative magnitudes change a lot as you go farther and farther from the source. That is why it is useful to distinguish the "near field" region (where the $1/r^2$ term is dominant and the effects of time delay are small) from the "radiation field" region (where the $1/r$ term is dominant and the effects of time delay are large and obvious).

 

[DaTario]

To elaborate on this: if you wiggle a mass around (with a quadrupole moment with a nonzero third time derivative), you will, in principle, generate GWs, i.e., fluctuations in spacetime curvature. But those fluctuations will be superimposed on the static field of the static part of the source (heuristically, the average position of the mass, such as the pendulum in your example). Only the fluctuations will produce changes in things like the arm lengths of a GW interferometer; the interferometer won't detect the static part of the field. And if the static part of the field is strong enough (i.e., if you're close enough to the source), it might overwhelm the fluctuating part--heuristically, your interferometer might have to be under such a large strain just to sit at rest in the field that the tiny changes in strain produced by the fluctuating GWs become unmeasurable. (Consider trying to measure small fluctuations in the position of a large electric charge using an antenna that is very close to the charge--the large static electric field might make the small fluctuations in position of the charges in the antenna due to the EM wave unmeasurable.)

So when you talk about "imitating" a radiation field with a near field, that doesn't really make sense. Wiggling a mass produces a radiation field, period. The question is whether that radiation field is detectable against the background field of the source. The further you get from the source, the weaker that background field becomes, and the easier it becomes to detect the radiation field.

I guess I have understood your point, but let me add one thing. By being placed on the Earth surface, aren't those interferometers suffering already such high gravitational near fields?

 

[DaTario]

Not responding to the source "right now" is not the same as "propagating". (The usual word for "not responding to the source right now" is "retarded", which I have never liked; I usually use "time delayed".) "Propagating" means something about the source is changing, causing changes in the field, which can only travel at the speed of light. If nothing about the source is changing, the field is still time delayed, but since it's not changing, the time delay makes no difference, because there is no change that needs to travel anywhere. And if something about the source is changing, but only by a small amount compared to its average value, then the time delay makes only a very little difference.

The relative magnitudes of the $1/r^2$ and $1/r$ terms tell you how much of a difference the time delay makes; and obviously those relative magnitudes change a lot as you go farther and farther from the source. That is why it is useful to distinguish the "near field" region (where the $1/r^2$ term is dominant and the effects of time delay are small) from the "radiation field" region (where the $1/r$ term is dominant and the effects of time delay are large and obvious).

Don't you think that "time delayed" and "propagating" are aspects of the same idea?

 

[PeterDonis]

By being placed on the Earth surface, aren't those interferometers suffering already such high gravitational near fields?

Certainly not. Earth's gravity is extremely weak. One way to see how weak is to estimate the ratio of the strain in an object sitting at rest on the Earth's surface to its rest energy density. Say we have a cube of steel 1 meter on a side weighing 10 tons, or 10,000 kg. The strain will be the pressure required to support the cube, which will be 980,000 Newtons per square meter or Joules per cubic meter (the units of pressure are the same as the units of energy density). The rest energy density of the cube is its mass density times $c^2$, or $9 \times 10^{20}$ Joules per cubic meter; so the ratio of strain to energy density is about $10^{-15}$.

 

[PeterDonis]

Don't you think that "time delayed" and "propagating" are aspects of the same idea?

Obviously not, since I took the trouble to distinguish them. One is an effect of the causal structure of spacetime; the other is an effect of changes in the source. They are related, since changes in the source must obey the causal structure of spacetime; but they're not the same thing.

 

[DaTario]

Certainly not. Earth's gravity is extremely weak. One way to see how weak is to estimate the ratio of the strain in an object sitting at rest on the Earth's surface to its rest energy density. Say we have a cube of steel 1 meter on a side weighing 10 tons, or 10,000 kg. The strain will be the pressure required to support the cube, which will be 980,000 Newtons per square meter or Joules per cubic meter (the units of pressure are the same as the units of energy density). The rest energy density of the cube is its mass density times $c^2$, or $9 \times 10^{20}$ Joules per cubic meter; so the ratio of strain to energy density is about $10^{-15}$.

If I got your point: if we start wiggling the earth, even having a very small near field the radiation field effect would be much much smaller.

Thank you for the beautiful example!

 

[DaTario]

Obviously not, since I took the trouble to distinguish them. One is an effect of the causal structure of spacetime; the other is an effect of changes in the source. They are related, since changes in the source must obey the causal structure of spacetime; but they're not the same thing.

Exemplifying: A static massive body is "updating" its static gravitational field on a time delayed basis. Once it is bumped by a comet, its position changes suddenly and now a propagating term of correction starts traveling in space in all directions. I am sorry, but although the circunstances are different, these two situations appear to me as having the same working mechanism. It may well be that the first one has a $1/r^2$ spatial dependence, while the second exhibits a $1/r$ dependence. But they seem to be members of the same family. Faces of the same phenomenon.

 

[PeterDonis]

if we start wiggling the earth, even having a very small near field the radiation field effect would be much much smaller.

If you're close to the Earth, yes. Far enough away, the $1/r^2$ field will become negligible compared to the $1/r$ field.

 

[PeterDonis]

A static massive body is "updating" its static gravitational field on a time delayed basis.

Only if "updating" means "not changing".

I am sorry, but although the circunstances are different, these two situations appear to me as having the same working mechanism.

So you think "not changing" is "the same working mechanism" as "changing". Since it's only a matter of choice of words, I can't say you're wrong, but it seems like an unusual use of words.

 

[DaTario]

Only if "updating" means "not changing".

Yes, it was my intention to express this.

So you think "not changing" is "the same working mechanism" as "changing". Since it's only a matter of choice of words, I can't say you're wrong, but it seems like an unusual use of words.

Put in these words it sounds strange, I agree. But what I have in mind is that no matter if the field is changing or not, the "time delayed basis" is there, doing its work. updating sequentially the field values.

 

[PeterDonis]

what I have in mind is that no matter if the field is changing or not, the "time delayed basis" is there, doing its work. updating sequentially the field values.

As I said, since this is just a matter of words, not physics, I can't say you're wrong. But since the only way to test the "time delayed basis" is to change something and watch the change propagate, there's no way to get evidence about how lack of change "updates" anything.

 

[DaTario]

As I said, since this is just a matter of words, not physics, I can't say you're wrong. But since the only way to test the "time delayed basis" is to change something and watch the change propagate, there's no way to get evidence about how lack of change "updates" anything.

Ok, we are done. Sorry for this drifting away from the OP. I would like to thank you, for your words helped me to shake some ideas which were rather static in my head.

Best wishes,

DaTario

 

[mfb]

Basically, yes. But one waveform can't really describe a GW. GWs have two possible polarizations, as shown here:

https://en.wikipedia.org/wiki/Gravitational_wave#Effects_of_passing

So to fully describe a GW, you need two waveforms, one for each of the polarizations; there is no necessary connection between them. (I don't know if LIGO was able to measure both waveforms; they may only have been able to measure one, since one waveform from each detector is all that I've seen in published info.) Also, you have to bear in mind that even if you have both waveforms, that is not a complete description of spacetime curvature, i.e., it's not a complete description of the effects of gravity; see my previous post.

LIGO is sensitive to relative changes in the lengths of the arms only, so each detector is limited to one polarization direction. A gravitational wave that changes both arms with the same amplitude is completely invisible to this detector. Two two LIGO observatories have different orientations for the arms, combining both they can get more data about the polarization. The Livingston site has roughly SSE/WSW orientation, while the Hanford site is SW/NW. The curvature of Earth has to be taken into account as well as the detectors are not at the same place.

 

[PAllen]

One thing to point out that is that for an oscillating mass, a test body's attractive response is to the quadratically extrapolated retarded position. This is different from EM, where the coulomb response is to the linearly extrapolated retarded position. In both cases, this velocity/acceleration dependent effect mimics responding to the instantaneous position of the source, but the mimicry is much more 'precise' for gravity. This is precisely why solar system motions, up to very small effects, seem consistent with instant action at a distance NOT delayed propagation. This is also related to why gravitational radiation has no dipole component; quadrupole is the lowed order.

So, in simple physical terms, the near field effect of an oscillating mass would be change in the center of attraction that seems to have almost no propagation delay because of the quadratic position extrapolation. The effect of the GW is not even approximately a change in center of attraction - it is compression in one direction, expansion in another. This distinction is in addition to the points about completely different distance dependence of strength of the effect.