Is stress a source of gravity?

[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:

 

[Dale]

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.

The fact that it is not defined in a non-stationary spacetime invalidates it completely. Any assertion that the error is small needs to be justified mathematically. Frankly, I don't think that it is even possible to do since the error from the actual quantity and an undefined quantity is undefined.

 

[PeterDonis]

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?

Ah, ok. As the masses fall onto the new pole, the new pole will compress. As it compresses, pressure will build up inside it, in an inverse process to the one by which the old pole expanded as pressure was relieved. During this process, there will be some motion of the new pole (no net motion of its center of mass, but motion of parts of the pole, which counts as well).

 

[PeterDonis]

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

Which is? Obviously if I could have told from #1 whether you intended the shell to have vacuum inside and outside, or something else, I wouldn't have had to ask about it.

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.

I agree a proper model for a solid should not assume that the pressure is isotropic; so perhaps a better term would be "quasi-perfect fluid", where the SET is diagonal (no shear stresses--if the system is spherically symmetric that is certainly going to need to be the case), but the radial pressure can be different than the tangential pressure.

And simple approximations, like assuming uniform tangent stress for a thin shell in this setting are fine.

Yes, I would agree with this approximation.

We want the essentials - mechanical oscillation that entails sinusoidal exchange between motion/momentum and stresses/elastic energy.

But again we come back to the question: what drives the oscillation? But I'll defer that until I get a definite answer to which scenario you want to discuss, as I asked above.

 

[PeterDonis]

First time that bolded bit has been presented

Actually, pervect mentioned it way back in post #18. He gave the definition of a stationary metric there, and said that oscillating shells do not meet that definition. Why they do not should be obvious from what he said, but just to make sure, I stated it explicitly in my post.

As far as treating the Komar mass integral as an approximation for a spacetime that is "almost stationary", I personally don't have any problem with that in principle (though some others may), but DaleSpam is right that approximations need to be justified. Since you are the one who is claiming that GR is wrong, as he said and as I have said before, it is up to you to justify whatever approximations you are making, and to justify the claim that the Komar mass should be conserved to whatever level of approximation you are using.

 

[pervect]

I haven't seen anything on using the Komar formalism with quasi-stationary spacetimes, unless you count the approach that derives the Bondi mass as a consequence of asymptotic time translations (which is in Wald).

If you can find some $\xi^b$ such that $\nabla_a \xi_b$ "small" in in the vacuum regions of the space-time, you should have an approximately conserved flux integral.

Something else that MIGHT work, is to say that if the actual pressure is negligibly different from the equilibrium pressure anywhere, the approach using Komar mass could give sensible approximate results. It's at least clear from your arguments (or a similar case, thinking about what happens if a shell under tension containing a very high pressure cracks and the contents explode) is that if the actual pressure is considerably different from the equilibrium pressure, the approach does NOT give sensible results. That doesn't really prove that it WILL give snesible results if the conditions are met, but it seems intuitively promising. Which may or may not mean it actually works.

 

[PeterDonis]

In looking around for references about gravitational waves and how they are generated, I came across this thread from PF from 2005:

https://www.physicsforums.com/showthread.php?t=60805

In it pervect gives the simple reason why monopole GWs are prohibited: Birkhoff's Theorem, which states (at least this is one way of stating it) that the metric in an exterior vacuum region of any spherically symmetric spacetime must be the Schwarzschild metric. See here:

http://en.wikipedia.org/wiki/Birkhoff's_theorem_(relativity )

Since the Schwarzschild metric contains no GWs, there can be no spherically symmetric (monopole) GWs. Since this hasn't been explicitly mentioned in this thread, I thought I'd mention it.

 

[Jonathan Scott]

Ah, ok. As the masses fall onto the new pole, the new pole will compress. As it compresses, pressure will build up inside it, in an inverse process to the one by which the old pole expanded as pressure was relieved. During this process, there will be some motion of the new pole (no net motion of its center of mass, but motion of parts of the pole, which counts as well).

True, but that's not the point of the example, and as I mentioned before the amount of internal potential energy is minimized for a sufficiently light and stiff pole and can certainly be much smaller than the gravitational potential energy of the system.

My point is to show that in a dynamic situation the stress part of the Komar mass is something which can vanish from one object and reappear later in another, so it isn't even approximately like a conserved quantity.

 

[TrickyDicky]

My point is to show that in a dynamic situation the stress part of the Komar mass is something which can vanish from one object and reappear later in another, so it isn't even approximately like a conserved quantity.

I think this is pretty evident (I don't know why peter donis keeps saying otherwise) and yes it is strange but it seems to be consistent with other well known and "weird" facts of GR.
We all agree (I think) that first of all the komar mass is not a valid concep in non-stationary situations and therefore we don't necessarily expect it to be conserved in those situations.
Second, we all agree (I think) that in GR what is conserved strictly is energy-momentum, not necessarily energy or mass by themselves (only in spacetimes with timelike killing vector it is energy strictly conserved).
So what you are describing about komar mass is what is expected according to what we agree about.
Maybe what is more difficult to explain is that in the setting you describe energy seems to be approximately conserved in the dynamical case unlike the komar mass.

 

[Q-reeus]

I haven't seen anything on using the Komar formalism with quasi-stationary spacetimes, unless you count the approach that derives the Bondi mass as a consequence of asymptotic time translations (which is in Wald).

If you can find some $\xi^b$ such that $\nabla_a \xi_b$ "small" in in the vacuum regions of the space-time, you should have an approximately conserved flux integral.

Something else that MIGHT work, is to say that if the actual pressure is negligibly different from the equilibrium pressure anywhere, the approach using Komar mass could give sensible approximate results. It's at least clear from your arguments (or a similar case, thinking about what happens if a shell under tension containing a very high pressure cracks and the contents explode) is that if the actual pressure is considerably different from the equilibrium pressure, the approach does NOT give sensible results. That doesn't really prove that it WILL give snesible results if the conditions are met, but it seems intuitively promising. Which may or may not mean it actually works.

Pervect, thanks for stepping back in with some interesting observations. Unfortunately there are sufficient caveats there to make it essentially impossible for me to absolutely defend using Komar expression (or presumably any similar one like ADM or Bondi). This leaves me in an invidious position. As a layman I am being required by some to take on the role of supreme GR expert in order to prove that Komar expression doesn't fail badly enough to throw out the basic argument of #1. Of course that I have long acknowledged I cannot do, yet not doing just that will gaurantee my 'failure' on this issue. I have special respect for how you handle matters in general. so I invite you please to consider the following scaling argument.

Take the case in #1 - and specifically we make it gravitationally small - basketball sized, and all in vacuo. As a typical mechanical oscillator, we know from basic mechanics it will have a natural frequency scaling as (E/ρ)^1/2, with those quantities defined in #1. Let's suppose at some specific value of E/ρ, whatever it is that puportedly ensures pressure is exactly canceled out as contribution to time varying gravitating mass m is actualy so. Now change just one parameter. Say E is made n times higher. Frequency of oscillation f rises by a factor n^1/2, and specifying amplitude of pressure oscillation is kept the same, radial displacement amplitude drops in the ratio 1/n. So radial velocity amplitude is a factor n^-1/2 smaller. If Komar redshift were somehow ever important as factor here, it has now been reduced owing to the reduced displacement amplitude (fluctuations in gravitational potential, dependent on radius R). Similarly for anything relating to velocity of motion - reduced as a factor. We notice that pressure is solely unaffected here. In the limit as E goes very high, every other physically reasonable contributor tends to zero. The graphs can all intersect at one point at most. If cancellation is a general principle, those graphs must match at all points, an obvious absurdity.

Get my point here? And the same kind of thing comes up if mass density ρ is altered. Or size scale (radius R and shell thickness δ grow/shrink in same proportion). It was all related in #1, but keeps getting buried under recycled issues here. Can there be any way around the above? I think not. Making things overly complex won't change the basic scaling arguments one iota imo. Now please no-one else jump in here first, I'm asking for a response from Pervect. Other subsequent responses are then welcome - in principle.

 

[Dale]

This leaves me in an invidious position. As a layman I am being required by some to take on the role of supreme GR expert in order to prove that Komar somehow doesn't fail badly enough to throw out the basic argument of #1.

You are putting yourself in the role of the supreme GR expert by your claims that the recognized experts are wrong. If you don't have the wherewithal to back up your claims that you are smarter than they are then don't make the claims. You cannot have it both ways. If you are expert enough to find these subtle flaws that have been overlooked for decades then you are expert enough to produce a valid exposition of the errors you have found.

Did you really expect to make a major theoretical breakthrough without doing some math?