Is stress a source of gravity?

[PeterDonis]

The divergence zero condition Gij|j=0 for i=1, accounting for the many terms that are zero (including Christoffel symbols), ends up relating G22=G33 with G11,1 G41,4 G11 G41 G44. Of course, the same must be true for T.

Ok, this is pretty much what I suspected after walking through the components of the EFE several posts ago. Thanks!

 

[Dale]

if stress is not legitimately a source term, both SET and EFE's are wrong, and the correct pair will be a reduced SET and FE's.

OK, but that is an experimental question only, and no amount of hypothetical examples or scenarios can possibly shed any light on that question.

As for correct predictions - can you cite any conclusive observational evidence for stress as source?

No (although it appears that PeterDonis can). Can you cite any conclusive observational evidence against it?

I can cite conclusive evidence for many other aspects of GR. Can you cite any conclusive observational evidence against any aspect of GR? IMO, this is the only possible way to attack GR, as you seem inclined to do.

My intuitive explanation(s) above still not clear enough? Then I suggest repeat reads until it is.

This is such a useless refrain. Whenever you are asked for clarification on some point you always simply say to re-read what you wrote. And it is not just me, you do this to many other people. If what you wrote were sufficiently clear then we wouldn't need to ask for clarification in the first place, and since what you wrote wasn't sufficiently clear then re-reading it won't clarify anything.

 

[PAllen]

Note that in the general case, M can be a function of time; but if the spacetime is stationary or static, the metric is time-independent, so M is also. And in the stationary or static case, M is just the Komar mass integral (or the ADM mass, or the Bondi mass--they're all equal in this case). So SET components in the non-vacuum region affect the metric in the vacuum region only via their contribution to the Komar mass integral.

Note that if we are allowing pulsating shells or stars, the non-vaccuum region is not static or stationary, so Komar cannot be used (you are integrating over the matter region). Birkhoff still guarantees that an SC geometry with some constant M describes the vacuum region, and this M may obviously be taken to be the conserved mass/energy of the 'universe'. This M will agree with both the ADM and Bondi mass, since these must be the same with static exterior vacuum region.

 

[PeterDonis]

Birkhoff still guarantees that an SC geometry with some constant M describes the vacuum region, and this M may obviously be taken to be the conserved mass/energy of the 'universe'. This M will agree with both the ADM and Bondi mass, since these must be the same with static exterior vacuum region.

Yes, you're right, the metric in the exterior vacuum region has to be static if the spacetime is spherically symmetric, even if the non-vacuum region is not stationary or static. So I mis-stated things when I said that M could be a function of time; actually the "M" in the exterior Schwarzschild metric cannot vary with time regardless of what the interior non-vacuum region does.

 

[Q-reeus]

No (although it appears that PeterDonis can). Can you cite any conclusive observational evidence against it?
I can cite conclusive evidence for many other aspects of GR. Can you cite any conclusive observational evidence against any aspect of GR? IMO, this is the only possible way to attack GR, as you seem inclined to do.

No I can't but that's because at current levels there is insufficient sensitivity to distinguish between GR and certain rival theories that depart radically only at strong gravity regime (e.g. Baryshev, Yilmaz). My point was there was no experimental/observational compulsion to incorporate stress in the first place.

Q-reeus: "My intuitive explanation(s) above still not clear enough? Then I suggest repeat reads until it is."
This is such a useless refrain. Whenever you are asked for clarification on some point you always simply say to re-read what you wrote.

I won't say outright lie, just a massive distortion.

And it is not just me, you do this to many other people. If what you wrote were sufficiently clear then we wouldn't need to ask for clarification in the first place, and since what you wrote wasn't sufficiently clear then re-reading it won't clarify anything.

Rubbish on both counts. Go ahead and cite one example where I have asked others to go back and check something where it was not fully justified. Tip - make good and sure to get everything in context, to save yourself some embarrassment. As for not being sufficiently clear, why was it not bleeding obvious to you back in #1 that, given both the title and my examples strongly suggesting stress is not really a source, it automatically implies SET and therefore EFE's are not correct? You could not add 2+2 and get 4? Then why not raise it back about #2, rather than acting like this is something new, here more than 200 entries later. Just another example imo of your non-genuine attitude.

 

[Q-reeus]

So any relationship between the Weyl curvature in the vacuum region and individual SET components in the non-vacuum region is going to be very indirect: you first have to solve the EFE in the non-vacuum region, then determine from that what the boundary conditions are at the edge of the vacuum region, then solve the EFE *again* in the vacuum region subject to those boundary conditions. But in the case of spherical symmetry, even doing that is pointless, because we already know what the vacuum solution looks like: it is the Schwarzschild vacuum solution for any radius r > R, where R is the radius of the boundary between non-vacuum and vacuum. If the spacetime is spherically symmetric, this is true *regardless* of what's going on in the non-vacuum region; so there is *no* relationship between any particular SET component in the non-vacuum region and the metric in the vacuum region, except whatever contribution that SET component makes to the total mass M that appears in the Schwarzschild metric.

Allright, that at least clarifies the GR methodology. And it's obviously needed in strong gravity situations where everything is intertwined non-linear wise. Even in acoustics or EM non-linearity requires that sort of approach. But in the linear regime of those two, it is certainly possible to tease things apart and determine contributions separately. So just to be real clear on this, in the weak gravity limit of 'linear' gravity, is it merely not done as a matter of standard GR procedure, or does GR actually preclude in principle an independent summation procedure in that regime?

There are two regimes where stresses are significant compared to energy densities: neutron stars and cosmology. In both regimes including stress as a "source" on the RHS of the EFE is necessary to match observations.
In the case of neutron stars, without pressure acting as a "source", there would be no maximum mass for a neutron star;

Are you quite sure on that - can you refer to some article stating as such?

so given that we observe stars of 10 times or more the mass of the Sun, we would also expect to observe neutron stars of those masses.

I'm pretty sure in most type II supernova scenarios the star blows the greater proportion of its mass outward, so I don't see that as true.

But we don't; all the neutron stars we have observed are no more than 1.5 times the mass of the Sun or thereabouts.

I linked in #207 to the Wiki article on NS's because it makes it clear there is large uncertainty here. Again, is there some definitive statement in the literature claiming pressure as source is essential, against this backdrop of uncertainties?

In the case of cosmology, the current "best-fit" cosmological model requires stresses as a source to match observations. More precisely, except for the "matter-dominated" phase of the universe's expansion (which lasted roughly from 100,000 years after the Big Bang until a few billion years ago when the expansion started accelerating), the presence of nonzero pressure in the SET is required to get the correct dynamics. In the "radiation dominated" phase, the pressure is the radiation pressure of light;

If it can be proven that the pressure as source is essential here, I would have to concede. Given the enormously varied estimates I have seen just for when inflation phase ceased - from 'grapefruit sized universe' to many orders of magnitude larger than observable universe, impression is of a lot of fudge room in cosmology in general. Maybe BB phase is much more tightly constrained, but would still need a definitive statement from literature tying pressure as source to exact Helium abundance etc. Seem to recall initial conditions are adjusted to get the observational match, not the other way around.

in the current "vacuum dominated" phase (or in the early inflationary phase), the "pressure" is the cosmological constant. In fact, including the "pressure" due to the cosmological constant is the only way to get an accelerating expansion at all, so again we can link the consequence of pressure as a source to a simple observation.

This gets into the weirdness of 'exotic energy' having positive energy density coupled with negative pressure. Let's please stick to the relative knowns.

Now I will ask once again, is it so in GR that GW's follow an equipartition principle with equal division into PE/'electric' and KE/'magnetic' parts, as is the case in acoustics and EM? Again, if so there is a fundamental problem imo reconciling that with stress generated 'GW's', as per last part in #173.

 

[Dale]

No I can't but that's because at current levels there is insufficient sensitivity to distinguish between GR and certain rival theories that depart radically only at strong gravity regime (e.g. Baryshev, Yilmaz). My point was there was no experimental/observational compulsion to incorporate stress in the first place.

But that isn't the same thing as it being incorrect to do so. Furthermore, there are many aspects of GR for which there is sufficient sensitivity to distinguish between GR and other theories. Since that data supports GR, the theory as a whole has ample experimental support, and the use of the SET is an essential part of the theory as a whole. So the use of the EFE (including SET) is experimentally justified even though this specific aspect may be currently untested.

Go ahead and cite one example where I have asked others to go back and check something where it was not fully justified. Tip - make good and sure to get everything in context, to save yourself some embarrassment.

https://www.physicsforums.com/showpost.php?p=3816751&postcount=155
It is never fully justified. The point is that the writer can never be the judge of the clarity of their own writing. Of course the writer thinks that they were being clear, that is why they wrote it. But if the reader doesn't understand the writer's point and asks for clarification then by definition is was not as clear as the writer thought. Perhaps the reader has a different background, and so the writer has to be unusually clear, but only the writer can clarify the writer's intent to the point where even someone with a different background can understand.

As for not being sufficiently clear, why was it not bleeding obvious to you back in #1 that, given both the title and my examples strongly suggesting stress is not really a source, it automatically implies SET and therefore EFE's are not correct? You could not add 2+2 and get 4? Then why not raise it back about #2, rather than acting like this is something new, here more than 200 entries later. Just another example imo of your non-genuine attitude.

Because the only way to show that the EFE's are not correct (as I understand the term and as you intended the term) is through experimental evidence, which you did not present there nor since.

 

[Passionflower]

I do not want to complicate things in this discussion but I think that generally we cannot talk about a vacuum region and a non vacuum region. The only completely non-vacuum regions are those of elementary particles provided they are not point particles.

 

[PAllen]

I do not want to complicate things in this discussion but I think that generally we cannot talk about a vacuum region and a non vacuum region. The only completely non-vacuum regions are those of elementary particles provided they are not point particles.

Classically you can. Hydrodynamics is all about a continuum approximation, even though we now know (but it wasn't known in 1800) that everything is ultimately particles. Similarly, the literature is filled with solutions involving continuous stress energy tensor. Within the classical theory, this is completely valid. As an exact statement about our universe, obviously it not correct (though often a very good approximation).

 

[PAllen]

Rubbish on both counts. Go ahead and cite one example where I have asked others to go back and check something where it was not fully justified. Tip - make good and sure to get everything in context, to save yourself some embarrassment. As for not being sufficiently clear, why was it not bleeding obvious to you back in #1 that, given both the title and my examples strongly suggesting stress is not really a source, it automatically implies SET and therefore EFE's are not correct? You could not add 2+2 and get 4? Then why not raise it back about #2, rather than acting like this is something new, here more than 200 entries later. Just another example imo of your non-genuine attitude.

Throughout this and other threads, I always have an extremely difficult time figuring out your point, your examples, etc. In this thread, all main participants have asked for clarifications and being rudely referred back to a post that was unclear and ill-specified (to them) to begin with is not helpful.

 

[Q-reeus]

Throughout this and other threads, I always have an extremely difficult time figuring out your point, your examples, etc. In this thread, all main participants have asked for clarifications and being rudely referred back to a post that was unclear and ill-specified (to them) to begin with is not helpful.

Well I'm truly sorry you feel that way, especially your perception of rudeness on my part. I have never intended that, and any brusqueness was in reaction to perceived difficult attitudes by others. Time and again there has been agreement on some issue, only to have it raised again and again as though new each time, and that I find infuriating. As for finding my points extremely difficult to follow, that leaves me stunned. Different worlds may explain part of this. From Peter's #209 it's becoming clearer that my approach, my gedanken examples, are simply excluded from consideration it now seems more or less by definition of how GR operates. First paragraph in #216 is asking just that. But I still hold out for the static shell case results. And just maybe someone will answer the GW's matter raised at least three times recently but so far without response.

 

[Q-reeus]

"Go ahead and cite one example where I have asked others to go back and check something where it was not fully justified. Tip - make good and sure to get everything in context, to save yourself some embarrassment."

https://www.physicsforums.com/showpost.php?p=3816751&postcount=155
It is never fully justified. The point is that the writer can never be the judge of the clarity of their own writing. Of course the writer thinks that they were being clear, that is why they wrote it. But if the reader doesn't understand the writer's point and asks for clarification then by definition is was not as clear as the writer thought. Perhaps the reader has a different background, and so the writer has to be unusually clear, but only the writer can clarify the writer's intent to the point where even someone with a different background can understand.

Can't argue with that as a matter of general principle. Get's down to personal judgement on a case-by-case basis. In the example above, I did give a brief summary of the issues, and with that in mind yes, politely asked him to reread. And yes I did feel everything there was clear enough. People often skim through first up, and all that's needed is to go through again more slowly. I do personally detest having to repeat points made, especially if I consider they have been made clearly. Instead of repeating it all again and again, why not rather reference to where it was all said well enough in the first place? And I don't as you claimed, always do that. In your case, that bit in #206 being a minor example, strikes me as just mean-mindfed sniping - you knew my position perfectly well after so long.

So what was not perfectly clear, in general principle at least, with #1? Just where are the hard or obscure bits exactly for you? Which is a very different question to saying which bits are not accepted.
Now if we're done with personal issues for a while - I'd much rather discuss the matter of stress as source term.

 

[PeterDonis]

So just to be real clear on this, in the weak gravity limit of 'linear' gravity, is it merely not done as a matter of standard GR procedure, or does GR actually preclude in principle an independent summation procedure in that regime?

In the weak gravity limit, stresses are negligible compared to energy densities, so I doubt anyone has ever had any need to actually include the stress contributions when doing GR calculations in that limit. But someone with more knowledge of how these calculations are actually done in practice could give a better answer on that.

Are you quite sure on that - can you refer to some article stating as such?

If you want specific references, I can try to select some from among the many papers I have been perusing. But it's talked about in most of the GR textbooks I'm aware of, and even in some popular texts (for example, I believe Kip Thorne's book, Black Holes and Time Warps, talks about it). The short answer is yes, I'm quite sure.

I'm pretty sure in most type II supernova scenarios the star blows the greater proportion of its mass outward, so I don't see that as true.

Supernovas do typically eject a large fraction of the original star's mass. However, they would have to eject 90% of it or more to get stars above 10 solar masses under the limit. As I understand it, the typical fraction of original mass ejected is nowhere near *that* high.

Plus, the argument based on observed neutron star masses is not just based on assuming that supernovas don't always eject enough mass to get them below the limit. See below.

I linked in #207 to the Wiki article on NS's because it makes it clear there is large uncertainty here. Again, is there some definitive statement in the literature claiming pressure as source is essential, against this backdrop of uncertainties?

First of all, pressure as a source is not just required to compute the specific value of the mass limit; pressure as a source is required for there to be a mass limit *at all*. If pressure as a source is not included, there is *no* mass limit; a neutron star could in principle be of unlimited size. So if pressure were not a source, we would expect to see neutron stars of varying sizes, up to 2, 3, 5, maybe 10 times the Sun's mass, with some reasonably even distribution. We would *not* expect to see *all* neutron star masses observed clustering in the region around 1 to 1.5 times the Sun's mass, which is, as I understand it, what is actually observed. That observation makes sense only if there *is* a limit, and there can be a limit *only* if pressure *is* a source.

Given the enormously varied estimates I have seen just for when inflation phase ceased - from 'grapefruit sized universe' to many orders of magnitude larger than observable universe, impression is of a lot of fudge room in cosmology in general. Maybe BB phase is much more tightly constrained, but would still need a definitive statement from literature tying pressure as source to exact Helium abundance etc. Seem to recall initial conditions are adjusted to get the observational match, not the other way around.

Someone with more detailed knowledge of how the cosmological models are calculated could give better info on this.

This gets into the weirdness of 'exotic energy' having positive energy density coupled with negative pressure. Let's please stick to the relative knowns.

When I'm talking about the "vacuum dominated" phase, I am not talking about "exotic matter". That's something different. I'm talking about a simple cosmological constant such as Einstein originally proposed, which as a "source" of gravity is just a constant times the metric. It's the fact that the "vacuum dominated" SET is a constant times the metric that is crucial; that's what requires its pressure to be negative (given that its energy density is positive). And again, given that the universe's expansion is observed to be accelerating, whatever is causing it *has* to have negative pressure somehow--that's the only way to produce an accelerating expansion. The "vacuum dominated" SET, a constant times the metric, is just the simplest way to get negative pressure.

 

[PeterDonis]

The short answer is yes, I'm quite sure.

Just to expand on this a bit more, you can actually get a sense of how pressure as a source is needed for a maximum mass limit by looking at the GR vs. the Newtonian equations for hydrostatic equilibrium.

The Newtonian equation is:

\frac{dp}{dr} = - \rho \frac{m}{r^{2}}

Take the simplest case, where \rho is constant. In that case, m(r) = 4/3 \pi \rho r^{3}, and we have

\frac{dp}{dr} = - \frac{4}{3} \pi \rho^{2} r

Integrating this inward from a finite r will obviously always give a finite value of p at r = 0. It takes some more mathematical work to see that that conclusion continues to hold when \rho is not constant but is a function of r, but it can be done; the basic idea is that, because of the relationship between dm / dr and \rho, m(r) will always end up decreasing fast enough to keep the total integral finite, even if \rho(r) increases as you integrate inward from any finite r to r = 0.

Now consider the relativistic equation, with pressure included as a source; I've written it in a somewhat different form than I did in a previous post, to make the comparison with the Newtonian form clearer (also note this specific equation only applies to a perfect fluid, with isotropic pressure, but the general conclusion remains the same when that condition is relaxed):

\frac{dp}{dr} = - \rho \frac{m}{r^{2}} \left( 1 + \frac{p}{\rho} \right) \left( 1 + \frac{4 \pi r^{3} p}{m} \right) \left( 1 - \frac{2m}{r} \right)^{-1}

All three added factors on the RHS are greater than 1, and it is no longer clear that integrating this inward from a finite r will always give a finite result. Again, consider the idealized case where \rho is constant:

\frac{dp}{dr} = - \frac{4}{3} \pi \rho^{2} r \left( 1 + \frac{p}{\rho} \right) \left( 1 + \frac{3 p}{\rho} \right) \left( 1 - \frac{8}{3} \pi \rho r^{2} \right)^{-1}

The third factor approaches 1 as r -> 0, but the first two will increase, and they are nonlinear in p, so one has to actually dig into the math in more detail to see whether, and under what conditions, integrating it will give a finite answer. It turns out that there are definite regimes where it doesn't--where the integral, starting at a finite r and working inward, diverges--it predicts an infinite value of p at r = 0. Working through the details is what ends up resulting in a specific value for the maximum mass (the actual value depends somewhat on the specific equation of state that is used, i.e., on the specifics of how \rho and p are related) that will keep the integral finite. But the key is that the integral can *only* diverge if p is included as a source.

 

[PeterDonis]

Again, consider the idealized case where \rho is constant:

\frac{dp}{dr} = - \frac{4}{3} \pi \rho^{2} r \left( 1 + \frac{p}{\rho} \right) \left( 1 + \frac{3 p}{\rho} \right) \left( 1 - \frac{8}{3} \pi \rho r^{2} \right)^{-1}

The third factor approaches 1 as r -> 0...

Actually, on re-reading, this may be somewhat misleading, because in the general case dp/dr can diverge because of the third factor diverging, i.e., because the ratio 2m(r)/r goes to 1. I haven't had references handy for the last couple of posts, so take them with a grain of salt until I've had a chance to check myself.

 

[Q-reeus]

A rough way of looking at contributions of ρ vs p to central pressure is to note that the maximum relative gravitational mass contribution for p is when ρ = 3p (photon gas) - i.e. a 50:50 split. Ehlers paper cited before shows this in eq'n (3.3) there. At the center of a NS that value may be approached (not sure how closely), but must decline steadily outwards as pressure declines and is fractionally zero at the surface. So ball-park would expect maybe 1/4 to 1/3 at most of central pressure owes to gravitating effect of pressure itself, maybe quite a bit less. And there is surely no need of infinite pressure anywhere. As soon as the perfectly finite degeneracy pressure threshold is exceeded in the core center, don't we have a 9-11 WTC style dynamically driven collapse? This article gives a sense of the difficulties in pinning down NS parameters: www.slac.stanford.edu/econf/C0507252/papers/L007.PDF (4.2, 4.3, 4.5 and elsewhere there).

 

[Dale]

And yes I did feel everything there was clear enough.

Understood. Except in cases where I am rushed I feel the same about my posts.

In your case, that bit in #206 being a minor example, strikes me as just mean-mindfed sniping - you knew my position perfectly well after so long.

That is just the thing. I didn't know your position. If I had I wouldn't have asked. I knew what my position would be if I had used that language (EFE didn't match with experiment), but it seemed completely out of context (no discussion of experiment). And a reply involving specific references to six different posts in two different threads just didn't clarify the question for me.

If I ask you for clarification you assume mean-minded sniping, and if I assume that I understand you accuse me of deliberate misrepresentation.

So what was not perfectly clear, in general principle at least, with #1? Just where are the hard or obscure bits exactly for you? Which is a very different question to saying which bits are not accepted.

I followed [1] pretty well and addressed that already, but [2] was completely confusing to me.

First, what you describe as a scaling argument is not even remotely similar to any other scaling argument that I have ever seen. So I wasn't sure how you came to your scaling relations. I am not sure that they are wrong, but they are not clearly right either.

Second, it was not at all apparent from reading [2] that you were suggesting a perpetual motion machine. From what I read I saw that you were arguing that you could produce GWs with a G-clamp and that the amplitude of those GW's depends on the stiffness, neither of which seemed terribly controversial.

However, since I didn't see the conclusions of [2] as controversial I didn't bother arguing against it and focused instead on the obvious problem with [1] until much later when some of your intended conclusions from [2] were clarified.

Now if we're done with personal issues for a while - I'd much rather discuss the matter of stress as source term.

OK. It wasn't directly addressed to you, but did you read PeterDonis' post 152:
https://www.physicsforums.com/showpost.php?p=3815763&postcount=152

That pretty much says it all IMO. The stress is related to the curvature via the EFE. Some solutions to the EFE will have clearly identifiable contributions from the stress in the external region and in other solutions the effects from the various stresses will cancel out in the external region.

Finding one or more solutions of the canceling type doesn't mean that stress is not a source in general. Stress is clearly part of the EFE and the EFE are clearly self-consistent. The only legitimate avenue you have for questioning the validity of stress as a source of gravity is experimental evidence. Your point about the limited accuracy of current observations is reasonable, but hardly damning.

 

[PeterDonis]

A rough way of looking at contributions of ρ vs p to central pressure is to note that the maximum relative gravitational mass contribution for p is when ρ = 3p (photon gas) - i.e. a 50:50 split.

You're correct that this is a limit on a physically possible solution for a fluid, but that doesn't by itself guarantee that the p <= 1/3 rho requirement can be satisfied for an arbitrary mass. In fact, the existence of this limit makes the constraints on a solution tighter, because now the pressure at the center doesn't just have to remain finite for a valid stable equilibrium; it has to also be less than 1/3 of rho at the center. You're correct that the physical meaning of that constraint is that an apparent "solution" that computes a pressure exceeding 1/3 rho will not be a valid static equilibrium; it will be unstable against collapse.

If you look at the relativistic hydrostatic equilibrium equation I wrote down, in the form of the Newtonian value times multiplicative factors, the limit of p <= 1/3 rho constrains the first factor (it will max out at 4/3 rho if p remains within the limit), but not the second; if m(r) -> 0 faster than r^3 p as r -> 0, the second factor will diverge (as in, really diverge--go to infinity). But even if it doesn't quite diverge, it can still cause p to exceed 1/3 rho, making the solution unstable against collapse.

Of course the obvious next question for you to ask is: how does the *Newtonian* equation of hydrostatic equilibrium behave when the p <= 1/3 rho constraint is imposed? Does it now predict some "solutions" that exceed that constraint and are therefore unstable? Off the top of my head, I would have to answer "probably yes", simply because the "standard" Newtonian analysis, which does not take into account *any* relativistic limits (essentially, ignoring the p <= 1/3 rho limit allows the speed of sound in the material to exceed the speed of light--under strict Newtonian theory it can go to infinity), has to allow the pressure gradient to become arbitrarily high in order to maintain static equilibrium as the total mass of the star goes up.

However, if I were you I would not go breaking open any bottles of champagne just yet, because I would also say, off the top of my head, that imposing the p <= 1/3 rho constraint but still trying to use a Newtonian equation of hydrostatic equilibrium is not consistent. If you are really interested in developing your own alternative theory of gravity that doesn't require pressure to be a source , I would instead try the following "hydrostatic equilibrium" equation:

\frac{dp}{dr} = - \frac{\rho m}{r^{2}} \left( 1 - \frac{2m}{r} \right)^{-1}

As you can see, what I've done is taken the relativistic TOV equation and removed the pressure terms. The key is that the last multiplicative factor is left in; the physical argument for that is that it corrects for the fact that the radial coordinate r does not measure actual radial distance, so it at least respects the spacetime curvature aspects of relativity (but see below for a further caveat in the "extra credit" section ). If I'm right that the strict Newtonian equation will lead to *some* maximum mass, as I argued above, then this one should too, since the RHS is multiplied by a factor that is always at least 1 (if it were to become less than 1 the star would have collapsed to a black hole). [Edit: see correction in later post below.]

But the key question, of course, is *what* maximum mass? Can we actually obtain a neutron star mass limit similar to the standard one using this equation? Or does it lead to a limit that is way different (assuming it leads to a limit at all--I haven't proved that, just guessed it, and it's quite possible my off the cuff guess is wrong)? If you weren't allergic to math, this would be a great problem for you to tackle. (For extra credit, you could also show how to derive the above equation for hydrostatic equilibrium from the "field equations" of your alternative theory. This would be a key requirement in any case to make an argument for the theory's consistency, since you would need some sort of field equation to derive equations of motion, conservation of the source, etc. and show that everything fits together. For example, I suspect, off the top of my head, that the alternate hydrostatic equilibrium equation above violates local energy conservation. If I were going to challenge your alternative theory of gravity, that's probably where I would start. )

A final note: I fully expect you to say that the above is some kind of "admission" from me that since there *might* possibly be some alternate model that gives a mass limit for neutron stars without using pressure as a source, GR is somehow wrong, or inconsistent, or at least not proven. First, please bear in mind that I was careful to characterize what I said above as off the cuff, off the top of my head, etc. Second, please bear in mind that the standard GR model does not include pressure as a source just because physicists feel like it; the presence of the pressure terms in the standard TOV equation (which I gratuitously took out above) is *required* by the EFE and by the conservation law obeyed by the SET (covariant divergence = 0). Including the pressure is not optional in standard GR.

So what I did above is *not* legitimate physics; it's just hand-waving, of the same sort that I have complained about you doing. What I described in my "homework assignment" is some of the work that would have to be done to move the "model" I wrote down above at least some way in the direction of legitimate physics. But only some; there are whole piles of other data out there that I didn't even bother thinking about when I wrote down the off the cuff equation above. Is there a model that generates that equation and is also consistent with all of the other data? I have no idea. My personal judgment is that the off the cuff equation I wrote down is wrong: the correct equation of hydrostatic equilibrium is the standard GR one. Your judgment may differ, but judgment alone doesn't make either of us right or wrong.

 

[PeterDonis]

This article gives a sense of the difficulties in pinning down NS parameters: www.slac.stanford.edu/econf/C0507252/papers/L007.PDF (4.2, 4.3, 4.5 and elsewhere there).

Good article, it will take me some time to digest it fully. I see that there is more of a spread in known or estimated neutron star masses than I thought (it's been some time since I last looked); the 1.5 or so solar masses that I quoted is still the peak of the overall distribution, but there are some outliers now with higher masses.

 

[PeterDonis]

(if it were to become less than 1 the star would have collapsed to a black hole)

Oops, mis-stated this; I should have said that that factor is always at least 1 as long as it is positive; if it is *negative* (or diverges), the star will have already collapsed to a black hole.

 

[Q-reeus]

That is just the thing. I didn't know your position. If I had I wouldn't have asked. I knew what my position would be if I had used that language (EFE didn't match with experiment), but it seemed completely out of context (no discussion of experiment). And a reply involving specific references to six different posts in two different threads just didn't clarify the question for me.

If I ask you for clarification you assume mean-minded sniping, and if I assume that I understand you accuse me of deliberate misrepresentation.

Allright then, I don't like carrying grudges. Looks like I may have misread your intentions. Style comes into it somwhat, and also what seems blindingly obvious to me here evidently isn't necessarily so for others. So taking the position you were quite sincere and I just got the wrong slant, my apologies.

I followed [1] pretty well and addressed that already, but [2] was completely confusing to me.

First, what you describe as a scaling argument is not even remotely similar to any other scaling argument that I have ever seen. So I wasn't sure how you came to your scaling relations. I am not sure that they are wrong, but they are not clearly right either.

Second, it was not at all apparent from reading [2] that you were suggesting a perpetual motion machine. From what I read I saw that you were arguing that you could produce GWs with a G-clamp and that the amplitude of those GW's depends on the stiffness, neither of which seemed terribly controversial.

However, since I didn't see the conclusions of [2] as controversial I didn't bother arguing against it and focused instead on the obvious problem with [1] until much later when some of your intended conclusions from [2] were clarified.

I thought about putting in diagrams, but decided it was unnecessary. Maybe a mistake. Can't see the difficulty though. Just the stiffness scaling part is easy enough surely. Stress maximum amplitudes were specified as constant, so obviously changing Young's modulus E has no appreciable effect on GW amplitude owing to stress. Whereas that part owing to rest-mass flexure under stress drops, not directly as E^-1 (in direct proportion to strain) but as E^-2 (quadrupole moment formula). And so on there - because strain drops as E^-1, therefore also power drain from stress component of GW's. And faster again for other contributions - E^-4 for mass-flexure component (square of that component's GW amplitude, therefore as (E^-2)²). I'd say the problem is trying to take it all in in one go. Maybe equations would have helped, but given the simplicity of concept, i decided to leave them out. Mass density ρ was the other scaling factor but no need to recap here.

As for the 'perpetual motion' part, it is the obvious conclusion from taking the limit of infinite stiffness and assuming stress truly acts as SET source - nothing else but stress contributes in that limit and does so with zero flexure and therefore zero input power. And I was always careful to point out it was not necessary to assume infinite stiffness - just that the divergent relations necessitate in general an E-dependent power imbalance. But given the insistence this is all invalidated because the spacetime is non-stationary to however small a degree, I desist. Again though I draw attention to the decidedly strange nature of stress-as-source originated GW's - 'electric' part only! The non-stationary spacetime criticism was met in #162, but not good enough it seems so again I desist.

Finding one or more solutions of the canceling type doesn't mean that stress is not a source in general. Stress is clearly part of the EFE and the EFE are clearly self-consistent. The only legitimate avenue you have for questioning the validity of stress as a source of gravity is experimental evidence.

Perhaps but I will be most interested to see how the static shell thing comes out.

Your point about the limited accuracy of current observations is reasonable, but hardly damning.

That point wasn't even directly related to role of stress, as afaik none of the so far done tests of GR check on stress as source.

OK. It wasn't directly addressed to you, but did you read PeterDonis' post 152:
https://www.physicsforums.com/showpost.php?p=3815763&postcount=152

I admit to not studying that carefully as indeed it was addressed to someone else. The points there are all valid, but only if pressure truly belongs in SET in the first place. I have no real idea of Einstein's reason for doing so, and it might be interesting to know. Whatever, it must be surely true one can come up with a different T/FE's replete with it's own consistent conservation/divergence relations. Is that not in fact the case with numbers of rival theories, some at least still quite viable contenders?

 

[Q-reeus]

However, if I were you I would not go breaking open any bottles of champagne just yet,...

Breaking open is something best left for ship launches imo - popping the cork is more my style. Not that I'm ready for that by any means.

because I would also say, off the top of my head, that imposing the p <= 1/3 rho constraint but still trying to use a Newtonian equation of hydrostatic equilibrium is not consistent. If you are really interested in developing your own alternative theory of gravity that doesn't require pressure to be a source , I would instead try the following "hydrostatic equilibrium" equation:

dp/dr=−ρmr²(1−2mr)⁻¹

Yes that would makes sense. Making simplifying assumption of constant density then inserting m(r)=4/3πρr³ the inverse part blows up when average mass density reaches critical limit. Indeed just enough pressureless dust inside a certain volume and a la standard GR we have a BH. So I'd say we are gauranteed of maximum size limits using above. But I think we agree too that neutron stars are not the easiest barometer for testing pressure's role as source.

For example, I suspect, off the top of my head, that the alternate hydrostatic equilibrium equation above violates local energy conservation. If I were going to challenge your alternative theory of gravity, that's probably where I would start.

That would admittedly be a telling argument against any alternative proposal if so. Not seeing where violation of coe comes in though. My sense of it is that we would still have conservative behavour just a different EOS leading to different mass limits etc. but nothing drastic otherwise. Hopefully - but I am allergic to maths. :shy:

Mainly though, given that non-stationary spacetime arguments have torpedoed my use of examples in #1, it probably gets down now to how the static shell balancing act pans out.

 

[PeterDonis]

Indeed just enough pressureless dust inside a certain volume and a la standard GR we have a BH.

That's not quite the same, though, because in the pressureless dust scenario there is no static equilibrium possible at all, so the concept of "maximum possible mass that can sustain equilibrium" doesn't even apply.

The model I was talking about would still include pressure, so a hydrostatic equilibrium would still be possible; but it would not "count" the pressure as a "source" to determine the pressure gradient required to maintain equilibrium.

 

[Q-reeus]

That's not quite the same, though, because in the pressureless dust scenario there is no static equilibrium possible at all, so the concept of "maximum possible mass that can sustain equilibrium" doesn't even apply.

The model I was talking about would still include pressure, so a hydrostatic equilibrium would still be possible; but it would not "count" the pressure as a "source" to determine the pressure gradient required to maintain equilibrium.

Fair comment. Core collapse occurs past some given finite maximum pressure, so with enough mass piled on, it can always happen. I see from the modified TOV formula you provided, GR just ensures it happens faster than Newtonian case, with or without pressure as source.
I have been hoping to derive a 'direct' way of showing whether or not stress emerges as source by looking at the case of a charged particle in oscillatory motion under the action of linear electrostatic restoring forces, but best leave it for now.

 

[PeterDonis]

Not seeing where violation of coe comes in though.

Because the standard conservation law, covariant divergence of SET = 0, leads to the standard TOV equation with pressure included on the RHS. So if we remove the pressure terms on the RHS, we would also have to somehow change the conservation law and/or the effective SET. But changing those would require changing other stuff too: the standard conservation law is an automatic consequence of the standard EFE and the way the Einstein tensor on the LHS is built, which ultimately comes down to simple geometry (MTW has a whole chapter on the geometric meaning of the conservation law, "the boundary of a boundary is zero"; if I can find a decent reference online I'll post it). And the standard SET is a consequence of varying the standard action principle for a perfect fluid with respect to the metric; so you would also have to somehow find a *different* action principle that gave rise to the changed SET, without breaking something else in the process.

My off the cuff guess is that there's no way to come up with an alternate conservation law that somehow makes all this stuff match up again; the way it all fits in the standard theory simply leaves no "wiggle room" to adjust anything at all without breaking something else. That's one theoretical reason why standard GR has the status it does; there's simply no other "nearby" theory that works at all.

it probably gets down now to how the static shell balancing act pans out.

Still working on that, but I can at least give a preliminary report. The key item that was giving you pause was the g_rr metric coefficient (what I was calling the "K" factor in that other thread), which goes to 1 as r -> infinity, gets larger and larger as we move inward in the vacuum region, but then somehow has to get all the way back to 1 as we move through the shell to the interior vacuum region (where spacetime is flat, hence K = 1).

A few posts ago I said that one of the components of the EFE, the 0-0 component, involves only the derivative of g_rr and the energy density T_00. The usual procedure is to first integrate T_00 to obtain the "mass function" m(r), which gives the mass inside radial coordinate r (so if we designate the outer surface of the shell as r = b, and the inner surface as r = a, then we have m(r) = M for r >= b, and m(r) = 0 for r <= a). In terms of m(r), then, it turns out that, when we solve the 0-0 component of the EFE, we find that g_rr has exactly the same form as we're used to in the vacuum Schwarzschild metric, just with a variable m:

g_{rr} = \left( 1 - \frac{2 m(r)}{r} \right)^{-1}

This form is valid for all r, given the behavior of m(r) as given above. So for r >= b, g_rr has the standard Schwarzschild vacuum form, and for r <= a, where m(r) = 0, we have g_rr = 1. Basically, g_rr only "sees" the mass that's inside your radius.

The interesting thing about this, of course, is that it holds *regardless* of anything else. In other words, it holds regardless of the behavior of the radial or tangential stresses, of the exact way in which m(r) varies with r inside the shell, etc. All that stuff affects only the "potential", g_tt (what I was calling the "J" factor in that other thread), *not* g_rr. And the "potential" does not go to 1 inside the shell; it continues to get smaller (relative to the value of 1 "at infinity"), so that in the interior vacuum region, the "flow of time" is effectively re-scaled (slowed) compared to the time flow at infinity. And the exact difference in "rate of time flow" *does* depend on the details of stresses inside the shell, how m(r) decreases with r, etc. But none of that affects g_rr.

More details to follow, but that's the key point.

 

[Dale]

Allright then, I don't like carrying grudges. Looks like I may have misread your intentions. Style comes into it somwhat, and also what seems blindingly obvious to me here evidently isn't necessarily so for others. So taking the position you were quite sincere and I just got the wrong slant, my apologies.

Thanks, that is very generous. I will try to assume the best also.

Just the stiffness scaling part is easy enough surely. Stress maximum amplitudes were specified as constant, so obviously changing Young's modulus E has no appreciable effect on GW amplitude owing to stress. Whereas that part owing to rest-mass flexure under stress drops, not directly as E^-1 (in direct proportion to strain) but as E^-2 (quadrupole moment formula). And so on there - because strain drops as E^-1, therefore also power drain from stress component of GW's. And faster again for other contributions - E^-4 for mass-flexure component (square of that component's GW amplitude, therefore as (E^-2)²).

I guess I should clarify my lack of clarity , It was clear that this is your claim, I just don't understand why. For me a scaling argument would be of the form that I showed earlier in order to show the various scaling relationships you claimed here. None of them seem obvious to me (neither obviously right nor obviously wrong).

Maybe equations would have helped, but given the simplicity of concept, i decided to leave them out.

Yes, equations would have helped me anyway. However, given the non-controversial nature of the conclusions that I understood I didn't see the value in arguing the point.

As for the 'perpetual motion' part, it is the obvious conclusion from taking the limit of infinite stiffness and assuming stress truly acts as SET source - nothing else but stress contributes in that limit and does so with zero flexure and therefore zero input power. And I was always careful to point out it was not necessary to assume infinite stiffness - just that the divergent relations necessitate in general an E-dependent power imbalance.

Here it was not even clear to me that this was the claim that you were making. For me it is obvious that the conclusion is not justified by the argument (and, of course, it is contradicted by other arguments like the ADM). Basically, the scaling argument says that as stiffness increases both the energy input and the GW output go down, but at different rates. That is all.

As you were very careful to point out, infinite stiffness is non-physical, so there is always going to be some energy input and some GW output. Because scaling arguments, by their nature, cannot shed any light on the overall coefficient (κ in my previous post on the topic) there is simply no way for a scaling argument to make the claim that the energy input ever becomes smaller than the GW output.

But given the insistence this is all invalidated because the spacetime is non-stationary to however small a degree, I desist.

Only if you use the Komar mass, which there is no need to use in [2].

The points there are all valid, but only if pressure truly belongs in SET in the first place. I have no real idea of Einstein's reason for doing so, and it might be interesting to know.

Basically, Einstein wanted something that would reduce to Poisson's equation (\nabla \cdot g = -4 \pi G \rho) in the Newtonian limit, but would be compatible with relativity. Unfortunately, energy density (ρ) is not a tensor. However, it is the time-time component of the stress-energy tensor. So, in the Newtonian limit all of the other terms go to 0 and only the time-time component is non-zero and the EFE reduce to Poisson's equation, as desired.

Whatever, it must be surely true one can come up with a different T/FE's replete with it's own consistent conservation/divergence relations. Is that not in fact the case with numbers of rival theories, some at least still quite viable contenders?

Certainly, although the only one that I know anything about is Brans-Dicke Gravity.

 

[Q-reeus]

Because the standard conservation law, covariant divergence of SET = 0, leads to the standard TOV equation with pressure included on the RHS. So if we
remove the pressure terms on the RHS, we would also have to somehow change the conservation law and/or the effective SET. But changing those would require changing other stuff too: the standard conservation law is an automatic consequence of the standard EFE and the way the Einstein tensor on the LHS is built, which ultimately comes down to simple geometry (MTW has a whole chapter on the geometric meaning of the conservation law, "the boundary of a boundary is zero"; if I can find a decent reference online I'll post it). And the standard SET is a consequence of varying the standard action
principle for a perfect fluid with respect to the metric; so you would also have to somehow find a *different* action principle that gave rise to the changed SET, without breaking something else in the process.

Hmm - as per comments in #1 I'm aware of how in SR the notion of pressure as source of added momentum density arises from non-simultaneity. Maybe it ties in somehow in the derivation of SET. Geometry argument would be interesting to follow.

My off the cuff guess is that there's no way to come up with an alternate conservation law that somehow makes all this stuff match up again; the way it all fits in the standard theory simply leaves no "wiggle room" to adjust anything at all without breaking something else. That's one theoretical reason why standard GR has the status it does; there's simply no other "nearby" theory that works at all.

Admittedly this sounds persuasive, and may be correct, but have been thinking about what happens when we chase the nature of stress all the way down to the atomic/molecular level. In particular for an ionic solid, it becomes evident that stress is there simply manifest as the forces arising from the slight redistrbution of interatomic electrostatic fields and electronic orbital motion. A rearrangement of PE and KE. Static EM fields, have no stress-as-source character - just T₀₀ energy. Nature of so-called 'Maxwell stresses' - equally 'tensile stress' along, and 'compressive stress' normal to field lines, assure of that. For the electronic KE - as elementary particle there is no internal degrees of freedom so no place for stress-in-particle. Further, there is no net spacetime averaged T_i0, T_0i energy-momentum flux, just KE contribution to T₀₀. So stress as simply rearrangement of T₀₀ distributions seems to chase stress-as-irreducible-primitive T_ii out of the picture - at least for normal solids and fluids. When it comes to white dwarfs and neutron stars, PEP (Pauli exclusion principle) typically does the holding up against gravity and things are less clear as to how to interpret stress. Certainly particle KE rises with stress, but what constitutes 'PE' and it's redistribution is not so clear, to me anyway. I know most of QM crowd are adament repulsion owing to PEP is not a real force, but what's in a name.

g_{rr} = \left( 1 - \frac{2 m(r)}{r} \right)^{-1}
This form is valid for all r, given the behavior of m(r) as given above. So for r >= b, g_rr has the standard Schwarzschild vacuum form, and for r <= a, where m(r) = 0, we have g_rr = 1. Basically, g_rr only "sees" the mass that's inside your radius. The interesting thing about this, of course, is that it holds *regardless* of anything else. In other words, it holds regardless of the behavior of the radial or tangential stresses, of the exact way in which m(r) varies with r inside the shell, etc. All that stuff affects only the "potential", g_tt (what I was calling the "J" factor in that other thread), *not* g_rr. And the "potential" does not go to 1 inside the shell; it continues to get smaller (relative to the value of 1 "at infinity"), so that in the interior vacuum region, the "flow of time" is effectively re-scaled (slowed) compared to the time flow at infinity. And the exact difference in "rate of time flow" *does* depend on the details of stresses inside the shell, how m(r) decreases with r, etc. But none of that affects g_rr.

Thanks for that interesting recap. Yes I recall finally appreciating g_rr is not synonymous with 1/(g_tt), only 'numerically equal' when external to SET region (r >= b). Thinking of some vague analogues in EM, e.g. curl B is only nonzero inside a region of finite current density J, so both curl B and J 'stop at the border'. Likewise for div E and charge density ρ. However the fields B and E do not stop at the border. So I'm wondering what is the specific influence of shell stress components in that r >= b region.
But anyway that's an aside and will be most interested to see the stress distributions required for matching metrics.

 

[Q-reeus]

Thanks, that is very generous. I will try to assume the best also.

No problems - we just need to all keep working at it.

I guess I should clarify my lack of clarity , It was clear that this is your claim, I just don't understand why. For me a scaling argument would be of the form that I showed earlier in order to show the various scaling relationships you claimed here. None of them seem obvious to me (neither obviously right nor obviously wrong).

Realise now it would have helped to express things more formally. Never mind.

Here it was not even clear to me that this was the claim that you were making. For me it is obvious that the conclusion is not justified by the argument (and, of course, it is contradicted by other arguments like the ADM). Basically, the scaling argument says that as stiffness increases both the energy input and the GW output go down, but at different rates. That is all.

'That is all' is actually a lot I would suggest. Since in this idealized system the sole net loss is via GW's, that there is no general match between input and GW output is synonymous with general failure of coe. But non-stationary spacetime to the rescue - somehow. I posed the question earlier as to whether GR admits in the weak gravity limit to being able to apply superposition principle (not quite in those words). The non-stationary spacetime criticism is tantamount to saying no, there is never a sufficiently linear regime where one can apply my scaling arguments, which merely requires superposition holding well enough. That I find incredible. In any other discipline we must have that linearity is approached arbitrarily closely, either in some typically low amplitude limit, or over some small enough interval. But I desist.

Will just point to one further example, where geometry rather than material parameter scaling, leads to interesting considerations. A rod set vibrating at some frequency f in fundamental axial mode. Motion is in and out along the rod major axis - generating harmonic axial compression and tension. Matter motion generates a mass quadrupole moment. Axial stress, spatially distributed sinusoidally along the rod, has the same sign at any instant, and thus contributes mostly a monopole moment. True for both stress as T_ii source, and as elastic energy T₀₀ source. However the latter fluctuates at frequency 2f and decouples from the other two just on that basis. As does the KE contribution from rod motion, which interplays with the elastic component and is in phase quadruture to it. One can either say well it doesn't really matter it must work out 'normally', or seriously take some time to ponder just what will be going on. Today's puzzle if you will.

As you were very careful to point out, infinite stiffness is non-physical, so there is always going to be some energy input and some GW output. Because scaling arguments, by their nature, cannot shed any light on the overall coefficient (κ in my previous post on the topic) there is simply no way for a scaling argument to make the claim that the energy input ever becomes smaller than the GW output.

See above.

Only if you use the Komar mass, which there is no need to use in [2].

Then the situation gets simpler in one way but more complex in another. No time variation to argue over, but the sign dependence of stress contributions means looking closely at effects from varying stress distributions owing to varying structure geometry - the biasing I mentioned in earlier posts.

Basically, Einstein wanted something that would reduce to Poisson's equation (∇⋅g=−4πGρ) in the Newtonian limit, but would be compatible with relativity. Unfortunately, energy density (ρ) is not a tensor. However, it is the time-time component of the stress-energy tensor. So, in the Newtonian limit all of the other terms go to 0 and only the time-time component is non-zero and the EFE reduce to Poisson's equation, as desired.

Sure that property in the linear regime adds up no problems.

 

[PeterDonis]

Static EM fields, have no stress-as-source character - just T₀₀ energy.

This is not correct. The SET of a static EM field still has nonzero components other than T_00. See the Wikpedia page here:

http://en.wikipedia.org/wiki/Electromagnetic_stress–energy_tensor

Nature of so-called 'Maxwell stresses' - equally 'tensile stress' along, and 'compressive stress' normal to field lines, assure of that.

This reasoning does not show that all components except T_00 are zero. It does show that there must be definite relationships between the various components (as are shown in the SET given on the Wiki page).

For the electronic KE - as elementary particle there is no internal degrees of freedom so no place for stress-in-particle.

The stress-energy tensor assumes a continuous model of matter. Once you start talking about individual particles, you have to have some model of interactions between the particles, or between the particles and something external like a container, and then do some kind of statistical averaging to obtain the sort of SET you can use in the EFE. See, for example, here:

http://en.wikipedia.org/wiki/Kinetic_theory

I see what you are getting at with this general line of thought, and it's a valid line of thought, but it goes beyond what GR as a theory is intended to do. From the viewpoint of elementary particle physics, GR is an emergent theory (or an "effective field theory", which seems to be a popular term--Steven Weinberg, for example, has talked about this in a number of books and papers, including I believe his 1972 text on GR); it is not supposed to be fundamental. So when you try to dig down to the fundamental level, many of the concepts used in GR, like "stress" or "stress-energy", simply don't apply any more. You've gone "underneath" them to see a deeper level.

When it comes to white dwarfs and neutron stars, PEP (Pauli exclusion principle) typically does the holding up against gravity and things are less clear as to how to interpret stress. Certainly particle KE rises with stress, but what constitutes 'PE' and it's redistribution is not so clear, to me anyway. I know most of QM crowd are adament repulsion owing to PEP is not a real force, but what's in a name.

From the standpoint of GR, "degeneracy pressure" due to the PEP is simply pressure that arises from a different equation of state than the usual one. The usual EOS for a fluid relates pressure to temperature, but degeneracy pressure can exist at zero temperature. But as far as the SET and the EFE are concerned, pressure is pressure; those equations don't "care" how the pressure is produced--what the specific EOS is--except in so far as the EOS affects the overall structure and dynamics of the system, which is usually captured as some kind of relationship between the T_ii and T_00 components of the SET.

So I'm wondering what is the specific influence of shell stress components in that r >= b region.

For the case of spherical symmetry, none, as I've said before. The only thing that affects the metric in the exterior vacuum region, in the spherically symmetric case, is the total mass M that's contained in the non-vacuum region.

 

[Dale]

Since in this idealized system the sole net loss is via GW's, that there is no general match between input and GW output is synonymous with general failure of coe.

How so? You put in some quantity of energy, some small fraction radiates off as GW, another fraction goes off as mechanical waves, some fraction is returned to the input device if you have it set up to do so and the material is elastic, and the rest goes to heat to radiate off that way. I certainly don't think that your scaling argument justifies overthrowing something as fundamental as COE, particularly since the ADM energy conservation holds. It just means that one of these other mechanisms cannot be neglected even in the idealized case if you wish to consider energy balance.

In any other discipline we must have that linearity is approached arbitrarily closely, either in some typically low amplitude limit, or over some small enough interval.

If you want to study GW's that is usually how it is done. By linearizing the EFE and doing a perturbative analysis about some background metric. Since you are approximating things you introduce errors, but you can also analyze those errors to put an upper bound on their magnitude. That is essentially what I was recommending earlier.