Is stress a source of gravity?

[PeterDonis]

For the scenario as actually given in #1 yes, but as stated there, it doesn't matter. Your whole conception of that model evidently is badly mistaken.

I guess I was not pessimistic enough when I said I had been too pessimistic. :sigh: Apparently I was right the first time; I won't be able to say anything useful, because your idea of what it takes to actually specify a model is too different from mine. I have nothing more to add to what DaleSpam has already pointed out in response to your post.

 

[Dale]

That in turn is your claim without any evidence. Looks like a dead heat.

This is an excellent point, and I agree. I rescind my claim as to the amount of violation and stick only with the claim that the Komar mass is not defined for non-stationary spacetimes. That one I certainly can provide evidence for in the form of references if desired, but since you even mentioned it in your OP then I guess it is not a point of contention.

The fact that the key formula in your argument is not defined for the scenario contemplated completely invalidates your argument. I will just stop there because my claims about the amount of error are unnecessary and, as you point out, not backed up with any evidence.

So you can't specify either just why the spacetime will be non-stationary

Sure, I can, I just thought it was obvious.

As pervect mentioned, the technical definition of a stationary spacetime has to do with the existence of a timelike Killing vector, but the practical result is that in stationary spacetimes you can write the metric in a set of coordinates such that all of the components of the metric tensor are independent of the time coordinate. A metric with gravitational waves is, by definition, a function of time, and is therefore non-stationary.

Well how about at least a link to the correct mass formula?

I don't think that there is a single "correct" mass formula in GR which applies for all spacetimes. Other common mass formulas are the ADM mass, and the Bondi mass, but those also only apply for a subset of possible spacetimes.

 

[Q-reeus]

I guess I was not pessimistic enough when I said I had been too pessimistic. :sigh: Apparently I was right the first time; I won't be able to say anything useful, because your idea of what it takes to actually specify a model is too different from mine. I have nothing more to add to what DaleSpam has already pointed out in response to your post.

You prefer to finish it here as is? Very well. Just one final request though if you don't mind. Do you accept that given my clarification of what governs the dynamics of the shell in #1, elastic/inertial not gravitational, Komar redshift cannot be invoked to cancel out pressure as source?

 

[PeterDonis]

I don't see how we can get an instantaneous redistribution of something which is behaving (a) like energy for purposes of conservation of overall energy and (b) like energy for purposes of acting as a gravitational source.

You keep talking about "instantaneous" redistribution; it isn't. The conservation equation (covariant divergence of SET = 0) relates *rates of change* of the different SET components, such as pressure and momentum flux. If you are going to adopt a model coarse enough that one changes "instantaneously", then so must the other.

For example, consider your scenario of two masses held apart by a pole. You have stipulated that there is significant stress in the pole--i.e., that the pole's pressure makes a significant contribution to the Komar mass integral. That means that the pressure in any infinitesimal element of the pole *cannot* simply go to zero "instantaneously", unless that fluid element also "instantaneously" acquires a nonzero momentum flux that is "equivalent" to the pressure it had an instant before.

Here's a more "continuous" way to think about it: suppose at some instant of time we cut the supporting pole exactly in half and put the two halves slightly out of alignment. Consider the infinitesimal element of either half of the pole right at the location of the cut. What will be the immediate effect of the cut on its pressure? Answer: *none*. What will change "instantaneously" is the *rate of change* of its pressure--before the cut, that rate of change was zero; now it is negative. And the rate of change of the momentum of that infinitesimal element will also become nonzero, since it will start to fall.

Why is there still pressure on that element? And why will it start to fall? Because the pole as a whole was compressed, like a spring; and removing the constraint on the pole does not remove the compressive stress inside it. It just allows the pole to start re-expanding to its "normal" unstressed length. As it does so, the infinitesimal elements closest to the cut in the pole will start falling, then the ones further up, etc., etc. As each infinitesimal element starts to move, the pressure felt by that element starts to decrease. The *rates of change* of the momentum and the pressure are what are related by the conservation equation.

 

[Q-reeus]

This is an excellent point, and I agree. I rescind my claim as to the amount of violation and stick only with the claim that the Komar mass is not defined for non-stationary spacetimes. That one I certainly can provide evidence for in the form of references if desired, but since you even mentioned it in your OP then I guess it is not a point of contention.

I respect those comments, thanks.

The fact that the key formula in your argument is not defined for the scenario contemplated completely invalidates your argument. I will just stop there because my claims about the amount of error are unnecessary and, as you point out, not backed up with any evidence.

An odd mix of words there, but I guess it's a case of take it or leave it on that matter.

Sure, I can, I just thought it was obvious.
The technical definition of a stationary spacetime has to do with the existence of a timelike Killing vector, but the practical result is that in stationary spacetimes you can write the metric in a set of coordinates such that all of the components of the metric tensor are independent of the time coordinate. A metric with gravitational waves is, by definition, a function of time, and is therefore non-stationary.

There is not an obvious contradiction in that? Komar mass invalidated because of a non-stationary spacetime (monopole GW's), whilst simultaneously agreeing to claims there can be no such GW's, and hence no non-stationary spacetime to invalidate Komar expression! Food for thought maybe.

I don't think that there is a single "correct" mass formula in GR which applies for all spacetimes. Other common mass formulas are the ADM mass, and the Bondi mass, but those also only apply for a subset of possible spacetimes.

Accept that it's horses for courses in that respect, but I thought maybe a better model fitting the shell scenario. Anyway it has clicked for me when answering, and frequently editing my #27 - pressure will easily dominate any 'correcting' redshift factor, and it's easy to prove via a simple scaling argument. But nobody wants to know it seems so too bad.

 

[PeterDonis]

Do you accept that given my clarification of what governs the dynamics of the shell in #1, elastic/inertial not gravitational, Komar redshift cannot be invoked to cancel out pressure as source?

No. Your explanation of the dynamics of the shell is incorrect. Read again the "Interlude" in my previous post. If the shell's oscillations are spherically symmetric, then tangential stresses cannot play any part in its dynamics.

 

[Jonathan Scott]

You keep talking about "instantaneous" redistribution; it isn't. The conservation equation (covariant divergence of SET = 0) relates *rates of change* of the different SET components, such as pressure and momentum flux. If you are going to adopt a model coarse enough that one changes "instantaneously", then so must the other.

For example, consider your scenario of two masses held apart by a pole. You have stipulated that there is significant stress in the pole--i.e., that the pole's pressure makes a significant contribution to the Komar mass integral. That means that the pressure in any infinitesimal element of the pole *cannot* simply go to zero "instantaneously", unless that fluid element also "instantaneously" acquires a nonzero momentum flux that is "equivalent" to the pressure it had an instant before.

Here's a more "continuous" way to think about it: suppose at some instant of time we cut the supporting pole exactly in half and put the two halves slightly out of alignment. Consider the infinitesimal element of either half of the pole right at the location of the cut. What will be the immediate effect of the cut on its pressure? Answer: *none*. What will change "instantaneously" is the *rate of change* of its pressure--before the cut, that rate of change was zero; now it is negative. And the rate of change of the momentum of that infinitesimal element will also become nonzero, since it will start to fall.

Why is there still pressure on that element? And why will it start to fall? Because the pole as a whole was compressed, like a spring; and removing the constraint on the pole does not remove the compressive stress inside it. It just allows the pole to start re-expanding to its "normal" unstressed length. As it does so, the infinitesimal elements closest to the cut in the pole will start falling, then the ones further up, etc., etc. As each infinitesimal element starts to move, the pressure felt by that element starts to decrease. The *rates of change* of the momentum and the pressure are what are related by the conservation equation.

Your model of the split pole is exactly the one I'd use. And in my first post on the pole I specifically mentioned that the change would propagate at the speed of sound in the material.

I also agree that the changes add up correctly; we get a brief pressure wave and then a gradient that will induce acceleration, and that gradient will immediately start to cause a change in momentum flow. Throughout the process, the overall four-vector energy and momentum terms are conserved for each infinitesimal part and no immediate change occurs to the energy or momentum, yet afterwards the pressure has dropped to zero.

By the time the pressure drop has propagated to the end I would not expect any significant change to have occurred in overall momentum, especially if the pole is light and rigid so it stores very little internal energy. Where did the "energy" go that was previously assumed to be described by the Komar mass pressure term?

 

[Q-reeus]

No. Your explanation of the dynamics of the shell is incorrect. Read again the "Interlude" in my previous post. If the shell's oscillations are spherically symmetric, then tangential stresses cannot play any part in its dynamics. That means there can be *no* energy exchange between tangential stresses and any other parts of the SET.

I know you want out, but that claim is, well, too controversial to let pass. Provide just one link to any reputable source dealing with shell dynamics that backs your position above and as per interlude in #20, and I will concede unreservedly. In fact I will personally promise to wire you $100 to your nominated account.

 

[Dale]

An odd mix of words there, but I guess it's a case of take it or leave it on that matter.

Considering that using an inapplicable formula completely invalidates your whole argument, it is a matter that you cannot "take or leave" without conceeding the argument.

There is not an obvious contradiction in that? Komar mass invalidated because of a non-stationary spacetime (monopole GW's), whilst simultaneously agreeing to claims there can be no such GW's, and hence no non-stationary spacetime to invalidate Komar expression! Food for thought maybe.

I agree, there is a very obvious contradiction, but the contradiction is all yours, not mine. You claim that you have found GWs in some spacetime. Without any further details we know that for your claim to be correct the spacetime must be non-stationary. Therefore we know that the Komar mass is not defined for your spacetime. You then proceed to calculate the Komar mass, contradicting your own claim that the spacetime is non-stationary.

 

[PeterDonis]

Provide just one link to any reputable source dealing with shell dynamics that backs your position above and as per interlude in #20, and I will concede unreservedly. In fact I will personally promise to wire you $100 to your nominated account.

Since you're the one claiming to refute GR, the burden of proof is on you. If you think that tangential stresses can drive the dynamics of a spherically symmetric oscillation, then *you* show how.

 

[Q-reeus]

I agree, there is a very obvious contradiction, but the contradiction is all yours, not mine. You claim that you have found GWs in some spacetime. Without any further details we know that for your claim to be correct the spacetime must be non-stationary. Therefore we know that the Komar mass is not defined for your spacetime. You then proceed to calculate the Komar mass, contradicting your own claim that the spacetime is non-stationary.

Not so. If you say there will be no non-stationary spacetime for oscillating shell, by that same token I should be perfectly correct in applying said Komar expression. Any consequent finding of GW's using that expression points to an internal GR problem, or at least that assumptions in Komar are invalid. I gave a 4-point, rehash of #1 list on that in #13. Note though Pervect has previously said oscillating shell implies non-stationary spacetime, or at least that's my understanding from #18. Can the vibrating shell generate a non-stationary spacetime that simultaneosly generates no GW's? A no-man's land here imo.

 

[Dale]

Not so. If you say there will be no non-stationary spacetime for oscillating shell

I am not saying that. I am saying there is no stationary spacetime for GWs.

Can the vibrating shell generate a non-stationary spacetime that simultaneosly generates no GW's? A no-man's land here imo.

Almost missed this. This is correct, a vibrating shell is non stationary, but does not generate GWs. All GW space times are non stationary, but not all non stationary space times have GWs. The FRW metric is a common example of a spacetime that is not stationary but doesn't have GWs.

 

[Q-reeus]

Since you're the one claiming to refute GR, the burden of proof is on you. If you think that tangential stresses can drive the dynamics of a spherically symmetric oscillation, then *you* show how.

Seems that by now you are too deeply committed to back down, so best I will do is remind of the link http://arxiv.org/abs/gr-qc/0505040 (part 5), already given in #27 (last link), where Elhers & co derive very simply the result for stability of a thin shell under static internal gas pressure. The extension to the dynamic case of radial vibration should be blindingly obvious. And I'm raising that offer in #38 to $1000. Not interested in some easy money?

 

[Q-reeus]

...a vibrating shell is non stationary, but does not generate GWs.

That's not the essence of what was claimed by yourself and others. It was that a vibrating shell generates a non-stationary spacetime. You then need to explain how this *periodically* varying spacetime can simultaneously be GW free.

...All GW space times are non stationary, but not all non stationary space times have GWs. The FRW metric is a common example of a spacetime that is not stationary but doesn't have GWs.

I have no problem with such a trivial example. Periodic variation is a very different beast. As per above.

 

[PeterDonis]

By the time the pressure drop has propagated to the end I would not expect any significant change to have occurred in overall momentum, especially if the pole is light and rigid so it stores very little internal energy.

By the time the pressure has dropped to zero throughout the pole, the energy and momentum *have* changed. They have to, by the conservation law. The "topmost" part of the pole (furthest away from the cut) may still be (instantaneously) at rest when the "pressure wave" reaches it, but the rest of the pole will already be moving. Remember that the pole is not infinitely rigid; the "bottom" part (closest to the cut) will be moving faster than the top part (in fact the momentum of the pole's substance will gradually decrease, continuously, from bottom to top) because the pole is stretching back out from its compressed to its "normal" length.

If the pole is in fact storing very little "internal energy", that's not because it's light and rigid; it's because it's not compressed very much. That may be partly because it's very rigid, but it will also be because the weight of the masses it is supporting is not very large, in which case the gravitational attraction between them is also not very large. In that case, yes, the pole will have acquired very little overall momentum by the time the pressure drops to zero. But it will still have *some* momentum; the momentum won't be zero.

Where did the "energy" go that was previously assumed to be described by the Komar mass pressure term?

Into the kinetic energy and momentum of the pieces of the pole, as above.

At this point, though, the spacetime is no longer stationary, so the Komar mass is no longer conserved anyway.

 

[PeterDonis]

stability of a thin shell under static internal gas pressure

Which is completely irrelevant to the scenario we are discussing; at least I thought it was. Your scenario stipulates that the shell is "self-supporting"; that means there must be vacuum inside and outside the shell. (You also state, later on, that the radial pressure on the inner surface of the shell is zero; that will be true only if there is vacuum inside the shell.) A shell with internal gas pressure is not "self-supporting" and both its static equilibrium configuration and the dynamics of its small oscillations about that equilibrium are different. So which case are we talking about?

 

[Jonathan Scott]

By the time the pressure has dropped to zero throughout the pole, the energy and momentum *have* changed. They have to, by the conservation law. The "topmost" part of the pole (furthest away from the cut) may still be (instantaneously) at rest when the "pressure wave" reaches it, but the rest of the pole will already be moving. Remember that the pole is not infinitely rigid; the "bottom" part (closest to the cut) will be moving faster than the top part (in fact the momentum of the pole's substance will gradually decrease, continuously, from bottom to top) because the pole is stretching back out from its compressed to its "normal" length.

If the pole is in fact storing very little "internal energy", that's not because it's light and rigid; it's because it's not compressed very much. That may be partly because it's very rigid, but it will also be because the weight of the masses it is supporting is not very large, in which case the gravitational attraction between them is also not very large. In that case, yes, the pole will have acquired very little overall momentum by the time the pressure drops to zero. But it will still have *some* momentum; the momentum won't be zero.

Sorry, nice try, but this whole thing doesn't work. Consider instead a pole which is moved out of line at both ends simultaneously.

Then, just to rub it in, replace it moments later with a pole that is just a tiny bit shorter.

Most of the "whatever-it-is" due to stress that was in the original pole has then magically jumped to the new one.

Also, I'm sure that the more light and rigid the pole is, the less energy (in the sense of mechanical potential energy of a compressed spring) is stored in the pole; that quantity is related to the properties of the pole, not the configuration.

In contrast, the opposing tension between particles through space due to the gravitational field (and proportional to its square locally) is equal and opposite to the Komar stress terms in the static case but keeps the same value even in the dynamic case, and when it is combined with the potential energy the result is mathematically consistent with the flow of conserved energy and momentum.

 

[PeterDonis]

Consider instead a pole which is moved out of line at both ends simultaneously.

Then, just to rub it in, replace it moments later with a pole that is just a tiny bit shorter.

How do you propose to do this in a way that's consistent with the Einstein Field Equation and the conservation law that goes with it?

 

[Jonathan Scott]

How do you propose to do this in a way that's consistent with the Einstein Field Equation and the conservation law that goes with it?

No significant amount of energy or momentum (at least compared with the potential energy, which is what we are talking about) is required for example to knock out a pole sideways which has a clean frictionless surface at the ends.

As I said before, the conservation law applies to the total energy-momentum, not to integrals of stress.

(The original of this quote has now been deleted after the author spotted the mistake, so I'm removing the quote here as well)

No, this is wrong by basic mechanics! When the same force moves the spring through a smaller distance, it does less work. If the spring is compressed a distance x by force F, then the average force throughout the compression is F/2 so the stored energy is Fx/2. A stiffer spring therefore stores less energy.

 

[Dale]

That's not the essence of what was claimed by yourself and others.

Then you have been misunderstanding my principal claim. From the begninning my principal claim is that your Komar-mass argument is invalid because any spacetime with GW's is non-stationary and the Komar mass is not defined on such spacetimes. My argument is very general has nothing whatsoever to do with vibrating shells nor any of the other irrelevant details of the specific scenario.

Do you now agree with that or not?

If you do not agree, then which part do you disagree with? Do you disagree that any spacetime with GW's is non-stationary, or do you disagree that the Komar mass is only defined on stationary spacetimes?

If you do agree, then we can proceed to discuss details.

 

[Q-reeus]

Q-reeus: "stability of a thin shell under static internal gas pressure"
Which is completely irrelevant to the scenario we are discussing; at least I thought it was.

Not really, but more below.

...Your scenario stipulates that the shell is "self-supporting"; that means there must be vacuum inside and outside the shell. (You also state, later on, that the radial pressure on the inner surface of the shell is zero; that will be true only if there is vacuum inside the shell.)

While I had not explicitly stated in #1 a fully evacuated environment, it was implied. So in essence, yes to the above. And further, surface radial pressure is zero at all instants at both inner and outer surfaces, more or less by definition of the model used.

A shell with internal gas pressure is not "self-supporting"

It can be. A balloon isn't, but a glass light bulb continues to be so (in that case negative relative internal-to-external pressure generally applies).

and both its static equilibrium configuration and the dynamics of its small oscillations about that equilibrium are different. So which case are we talking about?

Relevance of Ehlers model is this: replace static gas pressure with inertial forces of inward or outward radial acceleration. It represents a per unit area of shell radial acting 'pressure' of the same vectorial nature as gas pressure. One is a static thing, the other dynamic, but otherwise the same character. The balancing forces from elastic shell stress don't 'care' which it is. The Ehlers model shows tangent stresses do the balancing. It is impossible in that setting for radial elastic stresses to provide any balance. Spent several hours trawling for online material specifically stating the stress distributions for the breathing shell mode. Unfortunately the references were all oblique - overwhelmingly the focus is on mode patterns and frequencies. Hence the Elhers ref.

Now I probably got your jack up on this issue by using some emotive wording. Pardon please my personal failing that way - it's a habit hard to break. I want to keep this discussion, which imo is quite important, civil and pleasant as possible. So I will just venture a guess here (can't even say educated guess, as I have no professional qualifications or background of any kind) that the model you used was based on equilibrium conditions for a mathematically excised shell within a self-gravitating perfect fluid sphere. Then I can see how your findings would make good sense re force balances. Would that be about right?

 

[Q-reeus]

Then you have been misunderstanding my principal claim. From the begninning my principal claim is that your Komar-mass argument is invalid because any spacetime with GW's is non-stationary and the Komar mass is not defined on such spacetimes.

OK but this is a chicken-and-egg thing I have been trying to get across repeatedly. If as all you folks insist there will be no GW's for the shell scenario given, the Komar model should be valid to use! Can't have it both ways imo.

My argument is very general has nothing whatsoever to do with vibrating shells nor any of the other irrelevant details of the specific scenario.
Do you now agree with that or not?

Apart from my earlier comment, I have obviously to agree that a disturbance (GW's) means non-stationary spacetime. But that is not saying much apart from stating that motion = movement.

If you do not agree, then which part do you disagree with? Do you disagree that any spacetime with GW's is non-stationary,

How could I? As per above, by definition, GW's = non-stationary spacetime. Hardly the issue.

or do you disagree that the Komar mass is only defined on stationary spacetimes?

And this is where it get's to be slippery eel wrestling territory. Here's an excised piece from #9:

Honestly, there are truckloads of gedanken experiments accepted as valid that regularly fail to include every single possible factor and detail. How could Einstein get away with his use of trains and lights in SR setting when 'clearly' the masses involved are warping spacetime thus invalidating the flat spacetime postulated in SR. But of course we use reasonableness and accept such warping is of no real consequence.

Apply that to this case of a basketball sized shell vibrating in breathing mode. Just on the assumption pressure really does provide an uncompensated gravitating mass m_s as per #1, the fluctuation in m_s will be gravitationally minute. Any resulting monopole GW's so generated will be vastly smaller again in magnitude. Are we being remotely reasonable in saying such tiny perturbations (and again, recall they 'officially' can't exist anyhow) will seriously throw out the Komar expression. Yet once again, my appeal goes out to all, including you silent onlooker GR pros. Provide a sensible, qualitative and order of magnitude quantitative justification for the implied claim here that *any* non-stationary spacetime generated, no matter how exceedingly feeble, invalidates use of Komar expression.

If you do agree, then we can proceed to discuss details.

Up to you on that. I've said my piece above, for the umpteenth time really.

 

[PeterDonis]

No significant amount of energy or momentum (at least compared with the potential energy, which is what we are talking about) is required for example to knock out a pole sideways which has a clean frictionless surface at the ends.

Sorry, posting too early in the morning. I meant the part about "magically" replacing the pole with a slightly shorter one. The part about knocking the pole sideways is fine; in that case the pole will expand outward at both ends, with the "wave" of expansion propagating inward from both ends towards the center. As it does so, the internal stresses in the pole will gradually be relieved, starting at the ends and working in towards the center.

No, this is wrong by basic mechanics!

Yes, I realized that after I posted; that's why I deleted that part. You type fast.

 

[PeterDonis]

It can be. A balloon isn't, but a glass light bulb continues to be so (in that case negative relative internal-to-external pressure generally applies).

Relevance of Ehlers model is this: ...

Let's first get clear about just what scenario we are discussing. See below for the model I've been using; is that the scenario you want to discuss, or is it something else?

So I will just venture a guess here (can't even say educated guess, as I have no professional qualifications or background of any kind) that the model you used was based on equilibrium conditions for a mathematically excised shell within a self-gravitating perfect fluid sphere. Then I can see how your findings would make good sense re force balances. Would that be about right?

The model I have been basing my posts on is a thin spherical shell made of perfect fluid-type matter with vacuum inside and outside the shell. The only caveat to such a model is that a shell made of an actual perfect (or near-perfect) fluid, like air, would not be self-supporting; it would collapse under its own gravity. I'm assuming that a typical solid material (like metal or plastic or wood) which *can* support itself under its own gravity can still be described by a perfect fluid-style stress-energy tensor. That seems reasonable for the kind of scenario we're discussing.

 

[PeterDonis]

OK but this is a chicken-and-egg thing I have been trying to get across repeatedly. If as all you folks insist there will be no GW's for the shell scenario given, the Komar model should be valid to use! Can't have it both ways imo.

The Komar mass doesn't apply to your scenario, strictly speaking, because it's non-stationary; that's true whether or not GWs are emitted. What makes it non-stationary is the oscillation of the shell; the metric is time-dependent in the region in which the shell oscillates.

 

[Jonathan Scott]

I meant the part about "magically" replacing the pole with a slightly shorter one. The part about knocking the pole sideways is fine; in that case the pole will expand outward at both ends, with the "wave" of expansion propagating inward from both ends towards the center. As it does so, the internal stresses in the pole will gradually be relieved, starting at the ends and working in towards the center.

Shortly after that, one can push a replacement pole sideways into the gap (or even have it standing by the original so that when the original has been removed, the masses will fall together onto the shorter pole). I have specified the new pole as being slightly shorter to allow for the masses having fallen slightly closer together. The new pole will then take up the same stress at the original (apart from tiny corrections for being closer together), and will contain whatever "something" the original pole contained, as if it had transferred from the old pole by "magic". Certainly the Komar mass stress term would be approximately the same.

Is that clearer now?

 

[Dale]

How could I? As per above, by definition, GW's = non-stationary spacetime. Hardly the issue.

OK, so then the only possible point of disagreement is the issue of whether or not the Komar mass is defined in a non-stationary spacetime:

I can, of course, provide several references that state explicitly the the Komar mass is only defined on stationary spacetimes, including your OP. So I don't think that is actually the issue. I think you understand quite clearly that it is not defined in non-stationary spacetimes. From the above it seems that the issue is that you believe that, even though it is not defined, it is a good approximation:

And this is where it get's to be slippery eel wrestling territory. Here's an excised piece from #9:

Honestly, there are truckloads of gedanken experiments accepted as valid that regularly fail to include every single possible factor and detail. How could Einstein get away with his use of trains and lights in SR setting when 'clearly' the masses involved are warping spacetime thus invalidating the flat spacetime postulated in SR. But of course we use reasonableness and accept such warping is of no real consequence.

There are several ways that you can justify an approximation.

1) You can do a full non-approximated calculation of your quantity of interest and demonstrate that the approximated calculation is close.
2) You can expand the quantity of interest as an infinite series with terms of strictly decreasing magnitude and stop when the next term gets small enough.
3) You can expand the quantity of interest as the approximated plus some error term and determine some upper bound on the error term or expand the error term as in 2. (Btw, this approach is very common in the analysis of GWs, called linearized EFE or perturbative analysis. If you want to pursue your analysis this is the approach I would recommend.)
4) You can parameterize your degree of approximation and establish a maximum value for the parameter based on your measurement errors.

The gedanken experiments that I am aware of can be justified by one or more of those above methods. You have not justified your approximation in any of those ways nor provided any other justification besides your unsubstantiated assertion that it is small, such as:

Apply that to this case of a basketball sized shell vibrating in breathing mode. Just on the assumption pressure really does provide an uncompensated gravitating mass m_s as per #1, the fluctuation in m_s will be gravitationally minute. Any resulting monopole GW's so generated will be vastly smaller again in magnitude.

You certainly haven't demonstrated that. Just as you challenged my claim that the magnitude of the errors were equal to the magnitude of the GWs, so I challenge your claim that the magnitude of the errors are small. And just as I had to drop my claim since I wouldn't justify it, so I expect you to drop your claim if you won't justify it. A repeated assertion that they are small is not a justification.

 

[Q-reeus]

Let's first get clear about just what scenario we are discussing. See below for the model I've been using; is that the scenario you want to discuss, or is it something else?

The one presented in #1. It has the nice advantage of being physically realizable as is.

The model I have been basing my posts on is a thin spherical shell made of perfect fluid-type matter with vacuum inside and outside the shell. The only caveat to such a model is that a shell made of an actual perfect (or near-perfect) fluid, like air, would not be self-supporting; it would collapse under its own gravity.

Yes and that's a big caveat. In order to prevent collapse a fluid shell must be enclosed within some other supporting structure, itself solid. Hence the total system is now more complex and extended.

I'm assuming that a typical solid material (like metal or plastic or wood) which *can* support itself under its own gravity can still be described by a perfect fluid-style stress-energy tensor. That seems reasonable for the kind of scenario we're discussing.

No it can't for the reason above. Solids can bear static shear, uniaxial, and biaxial stresses impossible for a fluid. As the Ehlers bit demonstrates, tangent stresses supply all the support against any radial acting forces, whether static (gas, gravity) or inertial (vibratory motion). They have to, because radial surface forces are nonexistent. Any small interior elastic radial forces are providing internal balancing, they can do no more than that.

And simple approximations, like assuming uniform tangent stress for a thin shell in this setting are fine. We want the essentials - mechanical oscillation that entails sinusoidal exchange between motion/momentum and stresses/elastic energy. But all that was laid out in #1. Only thing missing there was a specific model setting stresses against motion. And imo entirely superfluous. We know the shell is a mechanical oscillator exhibiting the usual dynamics. Scaling behaviour for various parameters are what matters, and they are readily enough determinable from basic mechanics. I see no need to go beyond #1 re any further modelling. One unimportant error there was to miss a factor of 2 relating to biaxial stress contribution, but that's it.

 

[Q-reeus]

The Komar mass doesn't apply to your scenario, strictly speaking, because it's non-stationary; that's true whether or not GWs are emitted. What makes it non-stationary is the oscillation of the shell; the metric is time-dependent in the region in which the shell oscillates.

First time that bolded bit has been presented - previously I was being told it was the presence of ultra-feeble GW's. There was mention of shell oscillation invalidating, but no real explanation how. And nobody bothered to explain exactly what region(s) non-stationary spacetime referred to.
So what does this translate at exactly? Is it referring specifically to there being motion in the non-zero SET region (shell wall)? If so, in what way does this specifically impact on Komar expression? And how does it get around the constancy of system total energy? Just saying it invalidates is no real answer here.

 

[Q-reeus]

You certainly haven't demonstrated that. Just as you challenged my claim that the magnitude of the errors were equal to the magnitude of the GWs, so I challenge your claim that the magnitude of the errors are small. And just as I had to drop my claim since I wouldn't justify it, so I expect you to drop your claim if you won't justify it. A repeated assertion that they are small is not a justification.

I'm running late as usual, but will repeat in essence what I just wrote in #59 - some expert needs to set out just what is it that invalidates and exactly how and to what degree. when that is presented in comprehensible form, sensible positions can be taken. Till then, further disputing over higher moral ground on this is counterproductive. :zzz: