Is There Really a Strictly Conserved Stress-Energy Tensor in GR?

[PAllen]

Just to be clear, the complete action in question is:

$$S = \int \left[ \frac{R}{16 \pi} + L_{M} \right] \sqrt{-g} d^4 x$$

where $R$ is the Ricci scalar, $L_{M}$ is the Lagrangian due to matter fields, and $g$ is the determinant of the metric tensor. The variation of the first term with respect to the metric gives the Einstein tensor, and the variation of the second term gives (minus) the SET. The total variation must be zero, which yields the Einstein Field Equation.

It's true that the variation of the second term with respect to the metric will include the metric. But it includes no *derivatives* of the metric, so it contains no information about the curvature (or even about the connection, which is first derivatives of the metric--curvature is second derivatives).

Is it so simple as that?

y'' = y * g(x)

has very different solutions than:

y'' = g(x)

The metric, which describes aspects of geometry directly, is buried in the source term.

 

[PeterDonis]

y'' = y * g(x)

has very different solutions than:

y'' = g(x)

Yes, that's true. The possible solutions of the EFE are certainly affected by the fact that the metric is contained in the SET. But the fact remains that the equation itself does not have curvature on the RHS, and the interpretation of "does gravity gravitate?" that was under discussion was whether *curvature* is a "source" of further curvature in the EFE. Perhaps there's yet another interpretation of the question "does gravity gravitate?" that would turn on the presence of the metric in the SET.

 

[PAllen]

Yes, that's true. The possible solutions of the EFE are certainly affected by the fact that the metric is contained in the SET. But the fact remains that the equation itself does not have curvature on the RHS, and the interpretation of "does gravity gravitate?" that was under discussion was whether *curvature* is a "source" of further curvature in the EFE. Perhaps there's yet another interpretation of the question "does gravity gravitate?" that would turn on the presence of the metric in the SET.

Some people consider the metric an analog of gravitational potential - the 'field' comes from its derivatives. Then one can say gravitational potential is included in the source term.

I don't know how far one could go with this analogy; I think it is more useful to get at the issue via non-linearity and the observation that SET being the only source of Ricci curvature does not mean (even close) that SET can be directly related to effective gravity at a distance (except in special cases, e.g. where Komar volume integral is valid).

 

[PeterDonis]

Some people consider the metric an analog of gravitational potential - the 'field' comes from its derivatives. Then one can say gravitational potential is included in the source term.

Well, that would be yet another interpretation of "does gravity gravitate?", wouldn't it?

SET being the only source of Ricci curvature does not mean (even close) that SET can be directly related to effective gravity at a distance (except in special cases, e.g. where Komar volume integral is valid).

I think this is a good way to look at it, yes.

[Edit: To forestall a potential question from Q-reeus, who remembers me talking in previous threads about the field at a given event being due to nonzero SET somewhere in the past light cone of that event: the words "directly related" in the above are key. Ultimately, wherever there is Weyl curvature, there must have been a nonzero SET "source" somewhere in the past light cone. The Weyl curvature in the Schwarzschild exterior vacuum region around a gravitating body is ultimately due to the nonzero SET inside the body. (And if the "body" is a black hole, the Weyl curvature of the hole is ultimately due to the nonzero SET inside the body that collapsed to form the hole.) But that's only "ultimately"; the connection is indirect, since there may be a lot of "empty" spacetime in between, so to speak, and so the properties of "empty" spacetime--i.e., of solutions of the vacuum EFE--play a role in determining what the Weyl curvature is.]

 

[PeterDonis]

Unfortunately, I ran up against the PF post length limit and still had more to cover, so there will be a second follow-up post, hopefully soon!

And now the third (and final) post in the series is up:

https://www.physicsforums.com/blog.php?b=4293

 

[Q-reeus]

And now the third (and final) post in the series is up:
https://www.physicsforums.com/blog.php?b=4293

Peter - great effort in producing those two well-argued follow-up blogs Does Gravity Gravitate: The Sequel , Does Gravity Gravitate: The Wave
In respect of the first one. Find in difficult to avoid concluding that ADM mass as you have derived it, starting with EFE's and culling out an expression that corresponds to gravitating mass M, there is not here a de facto recognition that curvature explicitly contributes to that M - as directly part of the source and not just modifier of T_ab (which √(g_tt) is). So my impression is GR is made 'consistent' by way of a rather cunning and circuitous route, to put it diplomatically.
In respect of the second one. The vexed issue of non-localizability has it seems a majority consensus 'yes' (localization of gravitational field energy is impossible). But there are those who say 'no' - that this is not a consistent or http://en.wikipedia.org/wiki/Cooperstock%27s_Energy_Localization_Hypothesis. That article also brings in Feynman's sticky bead argument which you also refer to in that 3rd and final blog in the series. Quite frankly the more I try and make sense of the sticky bead argument, the less sense it seems to make. This is probably an issue for a separate thread, but since it has been used here as justification for energy in GW's, and thus sensibility of ADM mass, shall here briefly outline the problem as I see it. From that Wiki article:

The thought experiment was first described by Feynman (under the pseudonym "Mr. Smith") in 1957, at a conference at Chapel Hill, North Carolina.[2][3] His insight was that a passing gravitational wave should in principle cause a bead on a stick (with the stick parallel to the wave velocity) to slide back and forth, thus heating the bead and the stick by friction. A gravitational wave pulse will stretch spacetime behind the bead, pushing the bead forward; after the wave passes through the bead the stretching will occur in front of the bead, accelerating the bead in the opposite direction. This heating, said Feynman, showed that the wave did indeed impart energy to the bead and stick system, so it must indeed transport energy.

Two basic issues. First, as I understand it a GW involves purely transverse shear deformations of just spatial components of metric (zero dilational component). How can that even in principle allow induced motion of a bead along the propagation axis? Makes no sense imo, even if there is an unstated assumption stick length is long wrt, or at least appreciable fraction of, GW wavelength. Second, even when orienting stick orthogonal to propagation axis, induced motion of bead on stick seems nonsensical. Do not these shear deformations have as analogy the orthogonal stretching and un-stretching of a rubber sheet? Then the stick and bead and anything else gravitationally small existing in this 'rubber sheet' act as just figures drawn on it, hence must co-deform with the rubber. Thus would be undergoing motions (or rather deformations) only relative to an undetectable background flat metric. Hence no detectable relative motion of bead wrt stick, making any kind of local detection or energy absorption impossible in principle. Evidently Eddington adopted the lifelong view that along this or similar line of argument, GW's were merely coordinate artifacts - 'ripples in the coordinates' and thus unphysical. Considered now antiquated thinking, was he wrong?

The only way one could posit relative motion imo is to interpret the metric stretching as giving rise to tidal 'g' accelerations everywhere in the transverse plane. That seems like a geometrical impossibility for plane wave situation - to me only for something like spherically symmetric Schwarzschild geometry would everywhere transverse tidal 'g' make physical sense. But that is always there accompanied by comparably sized radial component too, and diminishes rapidly at large r no matter how strong the proper acceleration of a stationary observer is there (say for super-massive BH). One cannot have in a plane wave (strictly spherical but we are dealing with GW's at very, very large r from source) the necessary diverging radial vectors that apply in SG case. I'm wondering whether Hulse-Taylor binary-pulsar results might actually indicate a non-conservative process - orbital decay purely owing to field retardation effects. Yet another way conservation of energy can fail in GR?
Just when you thought it was all done.

 

[PeterDonis]

Find in difficult to avoid concluding that ADM mass as you have derived it, starting with EFE's and culling out an expression that corresponds to gravitating mass M, there is not here a de facto recognition that curvature explicitly contributes to that M - as directly part of the source and not just modifier of T_ab (which √(g_tt) is).

Once again, it depends on what you mean by "source". You are trying to fix one unique meaning for words that don't have one unique meaning. Curvature does not appear in the SET. That's the fact. How you connect that to the word "source" is a matter of terminology, not physics.

Also, don't confuse curvature with the metric. The ADM mass depends on the metric; curvature does not appear in it. Of course if the metric is something other than Minkowski, curvature is present, but that's not the same as curvature explicitly appearing in the ADM integral. It doesn't.

So my impression is GR is made 'consistent' by way of a rather cunning and circuitous route, to put it diplomatically.

By "consistent" here you can only mean "consistent with my intuitions". There's no point in arguing about that. The "route" by which GR is shown to be consistent mathematically is not circuitous at all.

Quite frankly the more I try and make sense of the sticky bead argument, the less sense it seems to make. This is probably an issue for a separate thread

Yes, it probably is, but I'll comment briefly below since it is, as you say, relevant to the question of whether GWs carry energy.

First, as I understand it a GW involves purely transverse shear deformations of just spatial components of metric (zero dilational component).

Yes.

How can that even in principle allow induced motion of a bead along the propagation axis?

It doesn't. It induces motion of a bead *transverse* to the propagation axis. I apologize if that wasn't clear; in the blog post I didn't really describe the scenario in detail (and I'll go back and try to fix that). Feynman's thought experiment had beads strung along a stick that was placed *transverse* to the propagation direction of the GW, so the motion of the beads is induced by the transverse GW oscillations.

[Edit: I see the Wiki article describes this wrong; it says "parallel to the wave velocity". AFAIK Feynman proposed the thought experiment as I have described it just above. But I'll check some sources to confirm.]

Second, even when orienting stick orthogonal to propagation axis, induced motion of bead on stick seems nonsensical. Do not these shear deformations have as analogy the orthogonal stretching and un-stretching of a rubber sheet? Then the stick and bead and anything else gravitationally small existing in this 'rubber sheet' act as just figures drawn on it, hence must co-deform with the rubber.

No, they will deform differently. The beads are not connected to each other, so they can move independently in response to the changes in the metric. The stick is one object with internal forces between its parts, so the relative motion of the parts will be different because of those internal forces. That means there will be relative motion between a given bead and the part of the stick that it was originally in contact with.

Evidently Eddington adopted the lifelong view that along this or similar line of argument, GW's were merely coordinate artifacts - 'ripples in the coordinates' and thus unphysical. Considered now antiquated thinking, was he wrong?

Yes. He wasn't the only one; all those physicists I referred to in the blog post, who thought that GWs couldn't carry energy, made the same kinds of arguments.

The only way one could posit relative motion imo is to interpret the metric stretching as giving rise to tidal 'g' accelerations everywhere in the transverse plane.

Yes, you can look at it this way (another way of stating it would be to say that GWs are oscillations in Weyl curvature), but remember that these "tidal accelerations" *vary in time* as the wave passes. That's the key difference between this case and a static case like the Schwarzschild geometry. The oscillations are quadrupole, so roughly speaking, first there is tidal expansion along the N-S and E-W axes and compression along the NW-SE and NE-SW axes, then there is expansion NW-SE and NE-SW and compression N-S and E-W, and it keeps going back and forth. No longitudinal oscillations at all.

 

[Q-reeus]

[Edit: I see the Wiki article describes this wrong; it says "parallel to the wave velocity". AFAIK Feynman proposed the thought experiment as I have described it just above. But I'll check some sources to confirm.]

If there is mutually orthogonal transverse stretch and shrink of metric spatial components, that stretch and shrink must have somewhere to go - nominally transverse spherical wavefronts must have accompanying radial motions. An apt analogy here is shear waves induced in surface of a spherical elastic shell. Only for pure axial shear symmetry can radial deformations be avoided. In 3D GW propagation case it implies a puckered wavefront in general - phase of orthogonal component wavefronts cannot be uniform. Stretch component accompanied by radial phase advance, compression component accompanied by radial phase retardation. Implying a transverse dilational component exists owing to this partial out-of-phase situation, and doubtless harmonics too. And further implies a radial dilational component - maybe this is what Feynman was thinking? Hmm.

No, they will deform differently. The beads are not connected to each other, so they can move independently in response to the changes in the metric. The stick is one object with internal forces between its parts, so the relative motion of the parts will be different because of those internal forces. That means there will be relative motion between a given bead and the part of the stick that it was originally in contact with.

There is imo a serious problem with this. More below.

Yes, you can look at it this way (another way of stating it would be to say that GWs are oscillations in Weyl curvature), but remember that these "tidal accelerations" *vary in time* as the wave passes. That's the key difference between this case and a static case like the Schwarzschild geometry. The oscillations are quadrupole, so roughly speaking, first there is tidal expansion along the N-S and E-W axes and compression along the NW-SE and NE-SW axes, then there is expansion NW-SE and NE-SW and compression N-S and E-W, and it keeps going back and forth. No longitudinal oscillations at all.

Have touched on last bit above - if transverse stretch/contraction occurs in a spherical wavefront, accompanying radial advance/retardation is needed to make geometric sense.
However let's ignore for now the matter of radial motions. Consider as example where two non-spinning neutron stars collide head on. Resulting in predominantly axial quadrupolar ring-down. This should give, in equatorial plane, harmonic GW stretch/compression along polar and azimuthal directions - N-S and E-W. But is this logically consistent with sticky-bead argument? Consider at large r from source we have a circumferential hoop (stick joined onto itself) centred about polar axis and lying in equatorial plane. With a uniformly dispersed array of beads strung out along the hoop. So there are ring-down GW's passing through. Clearly we need consider only the azimuthal E-W GW component. At the point of maximal azimuthal metric change - half-way between maximum dilation and compression, let's suppose your pov is correct and rigidity of hoop prevents any appreciable azimuthal stretch and thus any accompanying radial motion of hoop. Now please explain how each and every sticky bead decides which way to move in this situation - east or west. I'll save you the trouble - by symmetry there can be no such motions.

But this is unfair you may say - a continuous hoop is different to a straight stick. OK then, let's fix that by cutting up the hoop into equal pieces, which at large r, each such 'stick' well approximates to a straight stick. Further, our cutting up introduces a small gap between each 'stick' to allow interference-free radial 'breathing' in and out. Does this make a whit of difference to whether the beads, now strung out on an azimuthal array of sticks, will know to move left or right? Seems clear the answer is no different to before; not at all! Stretch/compression along lines of longitude (polar axis here) logically should follow essentially the same - but not quite. Maximum deformation amplitude along lines of longitude at equator, goes to zero at the poles. And this weak second order stretch deformation gradient implies a translational force on beads - so then motion along a hoop so placed and oriented. But this weak gradient will die off as 1/r² with distance and thus cannot be considered a true wave property. Also, for a short stick there is essentially the same motion induced as for beads thus no relative motion.

For me this illustrates there is something nonsensical with the sticky-bead argument - it has become unstuck. And with it a famous traditional argument for physically real GW's. Notice we have stuck to the original assumption that only transverse relative bead-stick motions can in principle exist. Two topics now I guess but best to have this thrashed out here as it all ties together.

 

[PeterDonis]

If there is mutually orthogonal transverse stretch and shrink of metric spatial components, that stretch and shrink must have somewhere to go - nominally transverse spherical wavefronts must have accompanying radial motions.

If we're modeling them as spherical wavefronts, then we're talking about a more complicated model than the one Feynman was using (AFAIK--see note below), and that I was using. (Furthermore, there are problems with such a model as you are constructing it--see below.) I was modeling the GWs as pure plane waves, with only transverse components. This is the typical way that weak GWs are modeled in GR; it is an approximation, but since gravity is so weak it is a very good one for all cases of any practical interest for direct GW detection (though not, AFAIK, for cases like the binary pulsar, where the evidence of GWs is indirect).

In the case of pure plane waves, there are no tidal changes in the longitudinal direction at all. Put another way, if I have two thin, flat objects both placed transverse to the waves and very close together, and initially at rest relative to each other, there will be no relative motion between them longitudinally; they will simply undergo the same transverse oscillations, but slightly out of phase.

(A note: if there is a single flat object but its thickness is significant relative to the wavelength, there will be shear stress induced in the object because the transverse vibrations at the front surface will be slightly out of phase with those at the back surface. I suppose this could lead to radial relative motion because of internal forces within the object, as long as the net radial momentum was zero. I was not intending to talk about that case since it's more complicated, and we're only trying to answer the question of whether GWs can heat up an object at all, not investigate the details of various ways it could do so.)

(I should also note that I haven't been able to confirm what model Feynman actually had in mind; it's possible that he *was* thinking of a more complicated case than pure plane waves. More to come on that if I can find a reference.)

There is imo a serious problem with this. More below.

I don't see that you've raised a "serious problem" with the very simple case of a pure transverse plane wave. All I see is that you've constructed several more complicated scenarios and are having trouble seeing how they fit in. We've been here before, I believe. Can we please stick to the simple case of a pure transverse plane wave first, before dragging in more complicated ones? Do you have any argument for why a pure transverse plane wave can't heat up a stick with beads placed purely transverse to the wave direction?

Now please explain how each and every sticky bead decides which way to move in this situation - east or west. I'll save you the trouble - by symmetry there can be no such motions.

GWs are quadrupole, so there is no such thing as a true "spherical wavefront" from a GW source. GWs are impossible with perfect spherical symmetry--and only perfect spherical symmetry would support your argument. Approximate spherical symmetry leaves plenty of room for individual beads to be induced to move east or west by local quadrupole oscillations. (As I said above, this case is more complicated than the pure plane wave case because we can see the curvature of the wave fronts; but that doesn't mean it is spherically symmetric.)

 

[Q-reeus]

Peter - I have just time for short comment. Any real GW source produces at large r a spherical wavefront - by spherical it is only implied the wave phase is a function of r and not of θ or phi (spherical polar coord's). My arguments are correct re need for radial motions - just try imagine stretching a balloon without it's radius growing! A nonsense. And btw no matter how great the radius from source (so it all looks like plane-wave situation), easy to find that relative phase differential between stretch and compression components is constant. Please give this more thought. Must go.

 

[PeterDonis]

Peter - I have just time for short comment. Any real GW source produces at large r a spherical wavefront - by spherical it is only implied the wave phase is a function of r and not of θ or phi (spherical polar coord's).

And this is only true approximately, not exactly. It can't be true exactly for quadrupole radiation. And the fact that it is approximately true is not enough to support your argument.

My arguments are correct re need for radial motions - just try imagine stretching a balloon without it's radius growing! A nonsense.

That's not what pure plane transverse GWs are doing. Can we please stick to the simple case?

And btw no matter how great the radius from source (so it all looks like plane-wave situation), easy to find that relative phase differential between stretch and compression components is constant.

For the *transverse* stretch and compression, yes, it is. So what?

 

[Q-reeus]

...That's not what pure plane transverse GWs are doing. Can we please stick to the simple case?...

No. Vital to treat situation for what it is - spherical wave. When I get the chance, will continue this in a new thread. No more on it here please.

 

[PeterDonis]

No. Vital to treat situation for what it is - spherical wave.

So you believe you can exhibit a quadrupole wave which has spherical symmetry? This should be interesting. I eagerly await the new thread.

 

[Dale]

No. Vital to treat situation for what it is - spherical wave. When I get the chance, will continue this in a new thread. No more on it here please.

In the new thread, be sure to cite some solid evidence that supports your claim that a GW actually is a spherical wave. Good luck.

 

[Dale]

And this is only true approximately, not exactly. It can't be true exactly for quadrupole radiation. And the fact that it is approximately true is not enough to support your argument.

I actually disagree here. I don't think that it is even approximately true, at least not globally. If it were approximately true then that would mean that you could get a GW which was spherical plus some small higher order terms. However, the spherical and dipole terms are identically 0. You can write a realistic GW as a quadrupole term plus some small higher order terms, but no lower than quadrupole.

Of course, you can do local approximations, but then there is no advantage to expanding as local spherical waves rather than local plane waves. The symmetry argument doesn't apply, and you add needless complication to your approximation terms without adding accuracy.

 

[PeterDonis]

I actually disagree here. I don't think that it is even approximately true, at least not globally. If it were approximately true then that would mean that you could get a GW which was spherical plus some small higher order terms. However, the spherical and dipole terms are identically 0. You can write a realistic GW as a quadrupole term plus some small higher order terms, but no lower than quadrupole.

If you're going to actually try to model the wave fields directly as spherical harmonics, yes, I agree; the l = 0 and l = 1 terms are identically zero.

However, I can see doing a geometric optics approximation where we model the GWs, globally, as expanding spherical wavefronts of "graviton pulses", similar to the way "photon" wavefronts are modeled as spherical in SR as an approximation, even though they're really not (the lowest-order EM radiation is dipole so the spherical term is 0 for that as well). Of course the GW wavelength has to be much, much smaller than the size of the spheres for this to work, i.e., the GWs have to be high frequency. I suspect that the paper Q-reeus linked to about HFGWs was doing something along those lines. But that is still only an approximation.

Furthermore, it's a useless approximation for trying to decide if GWs carry energy, because a "yes" answer to that question is built into the geometric optics approximation in the first place. That approximation assumes that the "gravitons" are massless particles carrying some finite amount of energy and momentum (if they carried zero energy and momentum they would have infinite wavelength, which obviously violates the small-wavelength assumption).

Of course, you can do local approximations, but then there is no advantage to expanding as local spherical waves rather than local plane waves. The symmetry argument doesn't apply, and you add needless complication to your approximation terms without adding accuracy.

Exactly.

 

[Q-reeus]

So you believe you can exhibit a quadrupole wave which has spherical symmetry? This should be interesting. I eagerly await the new thread.

Sorry to have to say both yourself and DaleSpam are attacking a straw man in posts #83-86. Did I not make it clear in #78 I was talking about spherical wavefronts? You are both sadly uninformed about common terminology here. Spherical wavefront (often just the term 'spherical wave' is used - without confusion by those in the know) simply means that at large r (i.e. well into radiation zone) wavefronts of constant phase have spherical symmetry. And that much I clarified for you in #80 - so you are both without excuse for attacking this straw man of your own creation. From http://en.wikipedia.org/wiki/Antenna_measurement#Compact_range

The CATR uses a source antenna which radiates a spherical wavefront and one or more secondary reflectors to collimate the radiated spherical wavefront into a planar wavefront within the desired test zone.

(emphasis added)
See "eeweb.poly.edu/faculty/bertoni/docs/04sphericalwaves.pdf" (perhaps you should inform author of gross ignorance in using the term 'spherical wave' in respect of antenna radiation! What an ignoramus!)
"galileo.phys.virginia.edu/classes/312/notes/antenna.pdf"

Although the wave emitted by the oscillating dipole is a spherical wave, it does not have the same intensity in all directions.

(between (4.16) and (4.16')) Gees - yet another ignoramus! Must be crawling with em out there.

Get used to it folks - spherical wave simply refers to phase of wavefront, and need have no bearing on angular dependence of field strength or direction - savvy?! I never once used the term spherically uniform field or monopole field or monopole moment - that all came from inside your heads.
Now, assuming your bonfire for the straw man has burnt out, listen up. Have been feeling my way on this issue - beginning with #76. Some statements made in #78 I now see are wrong (phase differential bit and what flowed from that), but stand by the overall thrust. It needs considerably more refinement and better presentation, and that I intend to do, but hands are tied up at the moment elsewhere. Sufficient to say I'm now sure GW's are a phantom. One more thing. Since you and DaleSpam have not heeded my request to leave it all for now - you might as well make good on that undertaking to provide reference material for Feynman's 'strange' sticky-bead argument that had stick pointing along propagation axis. Have you found one yet?

 

[Dale]

Get used to it folks - spherical wave simply refers to phase of wavefront, and need have no bearing on angular dependence of field strength or direction - savvy?! I never once used the term spherically uniform field or monopole field or monopole moment - that all came from inside your heads.

Oh, you are correct. I was indeed thinking you were referring to complete spherical symmetry in every aspect of the wave instead of simply a spherical phase distribution. I stand corrected.

Of course, since you didn't intend to imply anything about anything other than the phase then your argument becomes a non sequiter. A GW has more than just phase, so these other components need not be spherically symmetric as your argument requires:

Now please explain how each and every sticky bead decides which way to move in this situation - east or west. I'll save you the trouble - by symmetry there can be no such motions.

I was mistakenly thinking that you were making a valid argument from an incorrect premise, when you were actually making an invalid argument from a correct premise.

 

[Q-reeus]

Oh, you are correct. I was indeed thinking you were referring to complete spherical symmetry in every aspect of the wave instead of simply a spherical phase distribution. I stand corrected.

Thanks for at least admitting that - there's hope yet for you DS.

Of course, since you didn't intend to imply anything about anything other than the phase then your argument becomes a non sequiter.

Actually it's that statement that is the non sequiter - more below.

A GW has more than just phase, so these other components need not be spherically symmetric as your argument requires:

Wrong on last bit. My particular argument you presumably are referring to - in last section of #78, does not at all require spherical symmetry of field - merely axial symmetry in equatorial plane. And that was correctly applied.

Q-reeus: "Now please explain how each and every sticky bead decides which way to move in this situation - east or west. I'll save you the trouble - by symmetry there can be no such motions."

I was mistakenly thinking that you were making a valid argument from an incorrect premise, when you were actually making an invalid argument from a correct premise.

Actually it is you hastily making an invalid judgement. What I wrote there is just basic fact and cannot be sensibly denied. Maybe you simply have not grasped what was being said there. Perhaps you have the basic geometry confused. Are you cognizant of the arrangement: A very large circular hoop encircling at large r an axial quadrupole oscillator, with latter's axis of oscillation normal to plane of hoop? Hoop lying in equatorial plane of oscillator. I certainly described it plainly enough as such, but it never fails to amaze me how readily some folks can still misinterpret. If you did understand arrangement, how can you criticize the bit you quoted? It necessarily is true.

 

[Vanadium 50]

This thread, like the ones before it, has degenerated to the personal. It's closed.

Q-reeus, please do not start another one.