Is stress a source of gravity?

[PeterDonis]

From #1: "My contention is that if normal stresses truly are a source for gravitating mass m, it implies the following:"

And I have invited all the way along to be picked up on any specific point of error - note the word specific.

Yes, your *contention*. But it seems that nobody else in this thread can understand your specific arguments for that contention. That makes it hard to make specific criticisms. We have pointed out some specific items that look questionable, but that has not seemed to lead to a fruitful discussion.

OK use of Komar mass came up, but no attempt to put a finger on where in that expression things were going wrong or why, or to what degree.

Pervect addressed that in post #65; if you can find a timelike vector field that is "almost conserved", then you can use it to define the "redshift factor" in the Komar integral and that should make the integral "almost conserved" as well. He also made other suggestions.

I have also said several times now that in principle I have no problem with trying to look at "approximate conservation" of the Komar integral. And so far, every time I've worked an example, "approximate conservation" has appeared to hold reasonably well.

However:

Just 'can't use it - live with it - just accept BT is true - end of story'. Not terribly satisfactory imo.

Wanting a better understanding of whether and under what circumstances a particular approximation scheme might work is reasonable. Thinking that you will be able to find *any* approximation scheme that will justify results that contradict an exact, rigorous theorem about spherically symmetric spacetimes is not, imo, reasonable.

So if you had approached this issue by phrasing your question as "it seems like the Komar mass integral ought to be almost conserved in spacetimes that are almost stationary; can anyone give a more precise definition of how that works?", you might well have gotten some response. However, since you insisted on taking the position "GR is wrong, monopole GWs can exist, and I'll keep shouting that at the top of my lungs unless you can show me exactly how and why the Komar mass integral isn't conserved", people might quite reasonably think, "look, monopole GWs are ruled out by BT, regardless of how the Komar mass integral works, so what's the point?" And the result will be...pretty much what it has been in this thread.

Why is it so hard to put the finger on precisely where it fails?

Pervect made some good comments that may relate to this in post #65 as well.

 

[Dale]

As position statement that's now been said often enough.

Do you now agree with it? If not, then it has apparently not yet been said often enough.

What is not said once is just where

Where it fails is anywhere that the metric is not stationary. That includes but is not limited to anywhere that gravitational waves exist, so it must fail somewhere in any example with GWs. In your example there are no GWs (per Birkhoff's theorem) but the metric is not stationary at the location of the vibrating matter, so the Komar mass fails there.

and how

How it fails is that there is no timelike Killing vector and so the Komar mass is undefined. See your own Wikipedia link.

and how much it would fail for the case of e.g. vibrating shell.

This is your argument to make, not mine. I only claim that it fails and therefore the argument is invalid. I am not making any claims about the amount of failure.

 

[Q-reeus]

I don't think that presumption is justified. Many of your assertions have not been responded to, but I think the other commenters in this thread would agree with me that silence does *not* imply consent.

Have to agree with that as principle. Only wish it's application had been some 100+ entries earlier.
First of all, the SET is symmetric, not antisymmetric, so T_i0 = T_0i, with no minus sign.
Yes I made a blue there. Good thing it didn't change anything of substance re argument.

Second, I don't see how spherical symmetry implies net cancellation of *all* such terms. Spherical symmetry would imply that there is no net *tangential* momentum flow, yes, but spherical symmetry imposes no such constraint on *radial* momentum flow; that does not have to cancel.

But is it not true there will be overall cancellation given symmetry of radial flow? We're talking about net contribution re externally observed mass, not local quantities.

Not sure what you mean by this. Radial momentum flow can certainly contribute to *changing* stress, which does change how much stress is present to be a source.

Naturally stress and motion go hand in hand of necessity for mechanical vibration. What I meant was stress as source will be just as good whether generated dynamically as in oscillating sphere, as opposed to statically (self-gravitation or whatever).

 

[PAllen]

So is there some accessible version of ADM that can be simply applied to the shell case - one where the difference to Komar expression is readily apparent?

I gave you a link to a paper describing a simplified way to calculate ADM mass (wikipedia gives no specific formula, as I recall).

However, before you can compute ADM mass, you have to have the complete metric. So then, what is the metric outside a pulsating spherical shell? You can do this the hard way - write an expression for the SET of a pulsating shell that satisfied e.g. the dominant energy condition (see wikipdedia, for example, for the dominant energy condition). Then solve the EFE for metric.

However, no one in their right mind would do this, because if the SET is zero outside a closed spatial 2 surface, the defintion of ADM mass says nothing inside matters. So all you need to know is the vacuum solution. Unless you believe that a spherically symmetric SET can produce a non-spherically symmetric vacuum, then you simply need to know the most general spherically symmetric vacuum solution to the EFE. And that is where Birkhoff comes in - it says this solution is unique. And that unique solution is static. Repeat: it is a mathematical fact that there is no such thing as non-static spherically symmetric vacuum solution of the EFE. And that implies that implies two things: the ADM mass is constant and is equal to the Bondi mass, and there are no GW.

 

[Jonathan Scott]

And here is my problem. I keep asking for evaluation via the 'charge/potentials' route - SET contributions for specific geometry and motions etc. All I get back is - we only use the approved 'field approach' formula which doesn't look at it in those terms.

The answer is that it seems no-one really knows how to map GR to "intuitive" terms, and this is at least partly because some concepts, such as the physical location of gravitational energy, cannot be unambiguously defined. There are also obviously good reasons for this, in that one person's gravitational acceleration is free fall from another point of view, but I feel it should be possible to get some idea for conventional cases.

It seems that the most obvious approximate model in which total energy is conserved and continuous is to assume that there is an energy density in the field of $+g^2/(8 \pi G)$ where $g$ is the Newtonian field, and that the energy of any rest mass is time-dilated by the potential in which it resides (as in the Komar mass expression). In this case the total energy of the rest mass is effectively decreased by twice the potential energy by time-dilation, but the energy of the field adds an amount equal to the potential energy back in, so the overall energy is as expected.

When I came across the "Landau-Lifgarbagez" pseudotensor, which is designed to satisfy a conservation law for gravitational energy, I expected it to match this scheme in the trivial case (in a weak approximation), but I have recently confirmed that the t_00 gravitational energy density term in that pseudotensor seems to be $-7g^2/(8\pi G)$, which differs from the value I expected by $-4g^2/(4\pi G)$, for which I don't yet have any sort of "intuitive" explanation.

 

[Q-reeus]

Sorry all but burnt out and must run. Thanks for a lot (like avalanche) of interesting feedback. Can I leave you with a request to just consider what I said in #88 - last paragraph. There Komar should hold, so my scaling arguments, though in a slightly messy configuration (has to be if static quadrupole stresses are to be generated), is imo valid. :zzz:

 

[PeterDonis]

What would really impress is knowing what BT enforces about the specific behavour of SET terms for say the shell of #1. Knowing that would clear up much, but it seems beyond the reach of anyone.

BT doesn't say anything specific about the SET terms in the "interior" region; the whole point is that as long as there is an exterior *vacuum* region, and as long as the spacetime is spherically symmetric, the exterior vacuum *must* be Schwarzschild. The whole reason the theorem is so general and powerful is that it makes *no* assumptions whatsoever about the interior region, other than spherical symmetry.

Part of the problem here may be that you have not considered just how restrictive the assumption of *exact* spherical symmetry is. It is really that assumption, all by itself, that is enforcing restrictions on the SET terms. Think about what has to be constrained to ensure exact spherical symmetry: all motions must be radial, and radial motions cannot vary *at all* with angular coordinates. Basically, the whole problem is reduced to two dimensions from four; t and r are the only coordinates of interest, and energy density, radial momentum density, radial pressure, and tangential pressure are the only other variables of interest. That is a huge reduction in complexity from the general problem, and a huge restriction on possible solutions.

Also, if you look at the conservation laws for the SET, you see something else: tangential stress is completely uncoupled from the other variables. Here's the generic conservation equation again:

$$\nabla_{b} T^{ab} = 0$$

I.e., the covariant divergence of the SET is zero. But this is really four equations, one for each coordinate (t, r, theta, phi) (the range of the index a in the above; the index b is summed over all four coordinates). So the above expands to:

$$\nabla_{0} T^{00} + \nabla_{1} T^{01} + \nabla_{2} T^{02} + \nabla_{3} T^{03} = 0$$

$$\nabla_{0} T^{10} + \nabla_{1} T^{11} + \nabla_{2} T^{12} + \nabla_{3} T^{13} = 0$$

$$\nabla_{0} T^{20} + \nabla_{1} T^{21} + \nabla_{2} T^{22} + \nabla_{3} T^{23} = 0$$

$$\nabla_{0} T^{30} + \nabla_{1} T^{31} + \nabla_{2} T^{32} + \nabla_{3} T^{33} = 0$$

Spherical symmetry forces many of these components to be zero; what we are left with is the following, making the substitutions $T^{00} = \rho$ (energy density), $T^{01} =\mu$ (radial momentum density), $T^{11} = p$ (radial pressure/stress), $T^{22} = T^{33} = t$ (tangential pressure/stress):

$$\nabla_{t} \rho + \nabla_{r} \mu = 0$$

$$\nabla_{t} \mu + \nabla_{r} p = 0$$

$$\nabla_{\theta} t = 0$$

$$\nabla_{\phi} t = 0$$

As you can see, there are *no* equations relating t to any other variables. (The last two equations simply confirm our prescription that there are no tangential variations in stress.) What this means is that *no* changes in any other SET components can be driven by changes in t. But again, it is exact spherical symmetry that drives that constraint (the fact that no tangential momentum flow can exist--since by the above equations, you can see that tangential momentum flow is what would be required to "exchange" tangential stress with energy density, as radial momentum flow allows "exchange" between radial pressure and energy density by the first two equations).

 

[PeterDonis]

But is it not true there will be overall cancellation given symmetry of radial flow? We're talking about net contribution re externally observed mass, not local quantities.

If by "overall cancellation" you mean "time-averaged cancellation over a complete cycle of oscillation", then yes, that's what I would expect. But radial momentum flow is stil needed to understand the details of the dynamics of the oscillation.

Naturally stress and motion go hand in hand of necessity for mechanical vibration. What I meant was stress as source will be just as good whether generated dynamically as in oscillating sphere, as opposed to statically (self-gravitation or whatever).

If by "just as good" you mean "generates the same value for the relevant SET component at a given event", then yes; the SET components are "instantaneous" values, and at a given event it doesn't matter whether the component is changing dynamically or is static, if the value is the same at that event then it has the same effect at that event.

 

[PeterDonis]

Can I leave you with a request to just consider what I said in #88 - last paragraph. There Komar should hold, so my scaling arguments, though in a slightly messy configuration (has to be if static quadrupole stresses are to be generated), is imo valid. :zzz:

The situation in #88 last paragraph is static, so yes, the Komar mass integral should hold *once it is static*.

However, your "scaling argument", as far as I can understand its point, appears to be intended to support this claim...

there can be no parameter (e.g. Young's modulus E) independent match between work in stressing, and field energy resulting

...which has no meaning in a static situation, since no work can be done statically. The scaling argument would only apply to the time-varying portion of the spacetime, while the stress was being applied; and the Komar mass integral would *not* apply to that portion.

As PAllen suggested, the ADM mass would be a better one to use anyway, since it applies to any asymptotically flat spacetime. To even tackle the clamp problem, however, would be difficult because your scenario is not very symmetric. It would seem like a better "warmup" exercise would be to consider something like this: a spherical ball of matter is compressed perfectly spherically symmetrically by its own gravity, until it reaches equilibrium. Evaluate the ADM mass "before" and "after" compression; they should be identical by BT. Then evaluate the Komar mass "before" and "after" compression to see how the contributions to the integrand are "redistributed" by the compression. (Assume the "before" and "after" states are both stationary.)

Then, after the "warmup", you could try to extend the same type of analysis to cases which are not spherically symmetric; for example, to an axisymmetric matter distribution. *Then* extend it to a distribution with a nonzero quadrupole moment.

 

[PeterDonis]

as an aside here reminds me of our discussions in another thread over 'gravity gravitating' or not. There I mentioned Clifford Will was on record saying that 'gravity is a source of further gravitation', but couldn't then find the reference. Did subsequently, it's in sect. 4.3, 3rd para. at http://relativity.livingreviews.org/Articles/lrr-2006-3/fulltext.html
"In GR, the gravitational field itself generates gravity, a reflection of the nonlinearity of Einstein’s equations, and in contrast to the linearity of Maxwell’s equations."

I think someone else may have pointed out that passage later on in that thread (or maybe it was another thread--there have been a number of them recently all hovering around this same subject). Yes, with the definition of "gravity" Will is using in that passage, he is correct: "gravity gravitates" according to that definition. And also, according to that definition, the "source" of gravity is not conserved, and can't be localized. That's why that definition is not always used; for some purposes, it's very important that the "source" be conserved, or that the "source" be localizable. That's why the only real answer to the question "does gravity gravitate?" is "it depends".

 

[pervect]

And here is my problem. I keep asking for evaluation via the 'charge/potentials' route - SET contributions for specific geometry and motions etc. All I get back is - we only use the approved 'field approach' formula which doesn't look at it in those terms.

This is rather like the old joke:

Questioner: "I'd like to take a plane from Yamagata to Gifu" (two Japaneese cities, without airports, which are rather scare in Japan).

Ticket agent. "I see."

Questioner: "Is there some problem?"

Ticket agent. "Would you rather take a train?"

Questioner: "No, I want to take a plane!"

Ticket agent. "I see."

Unfortunately, while you can view GR as a perfectly consistent theory , based on an action formalism, it doesn't have any known GENERAL expression in terms of "charges" and "potentials". Which is more or less what we've been trying to say all along.

Note that this is different from saying that the theory is incosistent or malformed. It's consistent in its own way (and there are some proofs here and there on the details, for instance Wald has a proof that GR fits into the framework of a well-posed initial value problem). While it's self consistent, just doesn't fit into this particular form that you're requesting. It will fit nicely into an action formalism, or as a set of nonlinear differential equations - and you can solve for test particle oribts and whatnot as well, so it's not really a "field" issue per se.

While I can appreciate that it's frustrating to try and drive the square peg (GR) into the round hole (charge and potential formalims), it's just not fitting. And the problem isn't the square peg - it's a perfectly fine peg! It's just square, and the hole you're trying to pound it into is round.

There isn't really any reason why the source of gravity has to be a simple scalar quantity, and as far as anyone knows there is no such scalar quantity in general, only in special cases.

There are some important special cases which can be understood in this manner, but in general what you're looking for doesn't exist in classical GR.

There are occasional arguments about whether such a thing as a conserved charge (energy) exists in the FTG formalism or not, but there are also arguments about whether this is the same theory as GR or not. In classical GR, such a thing doesn't exist, though. Or if it does exist, it's not known, even after about 100 years of people looking for it.

 

[PeterDonis]

I don't know what the actual geometric effect of the stress term is on the shape of space-time as described by the LHS of the Einstein Field Equations, and I don't have the patience to try to work it out at the moment

I meant to respond to this before but it got lost in the shuffle. This part is actually pretty easy for the idealized cases we have been considering. John Baez gives a good exposition here:

http://math.ucr.edu/home/baez/einstein/node3.html

Quick summary: imagine a ball of "test particles" of initial volume V embedded in a perfect fluid, with diagonal SET (rho, p, p, p). Then the acceleration of the ball due to the gravity of the fluid is given by (in units where G = c = 1):

$$\frac{1}{V}\frac{d^{2} V}{dt^{2}} = - \frac{1}{2} \left( \rho + 3 p \right)$$

In other words, pressure *does* contribute to the local inward "acceleration due to gravity" of test particles *within the fluid*.

The question of whether pressure contributes to the *external* field (in the vacuum region outside the fluid) is a bit more complex. Q-reeus linked earlier to a paper by Ehlers et al. that considers the case of a spherically symmetric ball of gas enclosed in a container, with self-gravity of the gas assumed negligible. In that case, the (positive) contribution of the gas pressure to the externally observed mass of the system (which was defined as the Komar mass in the article since the system was static) is exactly canceled by the (negative) contribution of the tensile stress in the container; so pressure appears to make no contribution to the external field. However, note that in order to arrive at this conclusion, we had to include pressure in the calculation; we couldn't just ignore it or ignore the fact that it contributes to the mass integral.

Now consider a different case, a spherically symmetric ball of matter in hydrostatic equilibrium, such as a neutron star. Here the solution for the star's structure actually requires several equations: the Tolman-Oppenheimer-Volkoff Equation for relativistic hydrostatic equilibrium, an equation of state for the matter, and an equation relating the gravitational "potential" to the other parameters. The first and third of these are given in multiple sources, such as MTW; here they are in the simplest form I have found (again, in units where G = c = 1):

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

$$\frac{d \phi}{dr} = \frac{m + 4 \pi r^{3} p}{r \left( r - 2m \right)}$$

$$\frac{dm}{dr} = 4 \pi r^{2} \rho$$

where $m$ is the "mass function", the total mass inside radius r, and the third equation given defines how it is calculated.

(These forms, btw, are from Kip Thorne's text on stellar structure; see here:

http://www.its.caltech.edu/~kip/scripts/PubScans/II-12.pdf

Page 185 has these equations plus a bunch of others, which he needs because he's considering a much wider class of stellar models.)

Looking at the above, we can see several things:

(1) Pressure is not *directly* included in the equation for m, the mass function; we can determine the total mass M of the star, which is what will determine its external field, simply by integrating dm/dr from r = 0 to r= R, the star's surface.

(2) However, the integral in question will have to include the potential in the integrand, to adjust for the fact that the radial coordinate r does not directly measure physical radial distance. And the potential *is* affected by the pressure; so the pressure does affect the value of the mass integral indirectly, through the potential.

(3) Also, since pressure appears on the RHS of the TOV equation (the first equation above), increasing pressure has a "snowball" effect; each increase in pressure forces a further increase in pressure just to maintain the balance, so a given quantity of matter will end up with a higher pressure at its center, and hence a deeper potential well, than would be expected from a purely "Newtonian" standpoint.

I think the upshot of all this is that we can't single out one particular aspect of pressure's effects and say that this is "the pressure contribution to the mass", and then decide whether it is zero or not. Pressure has multiple effects on "gravity", and has to be taken into account; that's the bottom line.

 

[Q-reeus]

The situation in #88 last paragraph is static, so yes, the Komar mass integral should hold *once it is static*.

However, your "scaling argument", as far as I can understand its point, appears to be intended to support this claim...

Q-reeus: "there can be no parameter (e.g. Young's modulus E) independent match between work in stressing, and field energy resulting"

...which has no meaning in a static situation, since no work can be done statically. The scaling argument would only apply to the time-varying portion of the spacetime, while the stress was being applied; and the Komar mass integral would *not* apply to that portion.

Oh dear. I will once again reproduce an excerpt from a much earlier post here:

...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...

We need to establish a firm idea of what is reasonable and what is not in dealing with this squeezed G-clamps matter and similar problems with shell.
Take a look at eqn's (34), (35), under 'Sources of Gravitational Waves' at http://elfweb.mine.nu/Me/CV/Projects/incl/GW/GW.html#eqEnOut for GW output of a massive steel rod 20m long x 1m dia., rotating at the point of centrifugal force rupture. Result: 2.2*10^-29J/s. Note the ω⁶ dependence. Need I do a comparison between that 'huge' output and the one-off quarter-cycle GW loss of an arbitrarily slowly squeezed G-clamp pair? Einstein's use of trains was unbelievably sloppy compared to my assumption of negligible perturbation from GW's during the squeezing phase for G-clamps! Dispute that?
If so, it amounts to saying Komar expression has no validity under any circumstances whatever. So why is it still part of the GR pantheon? I shall anticipate a concession that it is screemingly unreasonable to invalidate Komar because ca 10^-50 or so (who cares about a dozen orders of magnitude either way here) Joules drained from the clamps just wrecked the use of Komar. What I wrote in #139 is again apt: "Does it fail gracefully and in a highly predictable and quantifiable manner, or just implodes at the slightest sign of time variation?" Can the latter be a resonable proposition in the slightest? You know my answer.

As PAllen suggested, the ADM mass would be a better one to use anyway, since it applies to any asymptotically flat spacetime. To even tackle the clamp problem, however, would be difficult because your scenario is not very symmetric.

As I noted in #146, it has to be less than perfectly symmetric by definition of having a Q moment. It's the most symmetric general configuration giving rise to a static Q moment I can conceive of. You might think a single G-clamp would be better but you would be mistaken.

It would seem like a better "warmup" exercise would be to consider something like this: a spherical ball of matter is compressed perfectly spherically symmetrically by its own gravity, until it reaches equilibrium. Evaluate the ADM mass "before" and "after" compression; they should be identical by BT. Then evaluate the Komar mass "before" and "after" compression to see how the contributions to the integrand are "redistributed" by the compression. (Assume the "before" and "after" states are both stationary.)

No because significant enough gravity to be source of compression makes everything more complex and is a different regime entirely. Clamps are the best possible for what I want to show. I defy anyone to do better re elastic self-stressed body. [a rotating hoop could qualify as centrifugal stress generated monopole source, but then we have 'magnetic' issues etc.]

Then, after the "warmup", you could try to extend the same type of analysis to cases which are not spherically symmetric; for example, to an axisymmetric matter distribution. *Then* extend it to a distribution with a nonzero quadrupole moment.

Similar remarks apply. Let's just see if there is any *reasonable* counter to the scaling argument for my example [2] in #1, modified as per #88, where GW's are not an issue at all (hope you agree on that last bit now).

It's important to get this one right before going back and looking at shell case, which is why I'm not replying for now to some recent postings dealing only with the shell/GW's issue. Patience please from such posters.

 

[Jonathan Scott]

We need to establish a firm idea of what is reasonable and what is not in dealing with this squeezed G-clamps matter and similar problems with shell. ...

Sorry, but I never understood what you are trying to show with your G-clamps example. Can you clarify please?

Note that the total force (equal to the integral of the normal stress) across any plane of a static object is zero apart from any term needed to counteract gravitational forces (even though it may vary from a large positive force to a large negative force locally).

 

[Q-reeus]

Sorry, but I never understood what you are trying to show with your G-clamps example. Can you clarify please?

Note that the total force (equal to the integral of the normal stress) across any plane of a static object is zero apart from any term needed to counteract gravitational forces (even though it may vary from a large positive force to a large negative force locally).

Last bt necesarily true by reason of static force balance. But that in no way precludes a quadrupole (or higher moment) distribution. Please reread that bit in #1, #88. Hopefully clear enough - it's all about how things scale re parameters, and simpler gain in 'static' case. The problem getting across just how really simple and effective this line of argument is as always a total faith in GR.

 

[PeterDonis]

Please reread that bit in #1, #88. Hopefully clear enough - it's all about how things scale re parameters, and simpler gain in 'static' case. The problem getting across just how really simple and effective this line of argument is as always a total faith in GR.

The problem, at least for me, is that I've read through your posts #1 and #69 (the latter is where you actually lay out your "scaling argument"; #88 just refers to other content and doesn't really give the argument itself) several times now, and I still can't figure out exactly what your argument is or what you think it proves (other than that it somehow proves GR is wrong).

Here is what I take to be the relevant part of #69:

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.

I don't understand what this is supposed to mean. The "general principle" that ensures that pressure does not contribute to the Komar mass integral in this case only holds *in static equilibrium*; it says that the (positive) contribution from the pressure of the gas inside the container is exactly canceled by the (negative) contribution from the tensile stress in the container. That's what the Ehlers paper you referenced shows. But that cancellation only holds in static equilibrium; it was never claimed to hold dynamically at all times while the system is oscillating.

What is missing from the above quote, and from your other attempts to describe this scenario, is, as I said before, that you have not answered the fundamental question: *why* does the system oscillate? Oscillation requires that as the system moves away from equilibrium, a restoring force is created that drives it back towards equilibrium. For small oscillations, one would expect the restoring force to be linear in the displacement ("displacement" in this case being, roughly speaking, the difference between actual radius of the system and its equilibrium radius). So what, specifically, is the restoring force in this system, and how do we know it is linear in the displacement for small amplitudes?

To me, the fundamentals of the dynamics of the oscillation should come first, before any other claims about what this scenario does or does not show. If we don't all agree on the fundamental dynamics, we're not going to even be able to talk about the other stuff.

 

[Dale]

Need I do a comparison between that 'huge' output and the one-off quarter-cycle GW loss of an arbitrarily slowly squeezed G-clamp pair? Einstein's use of trains was unbelievably sloppy compared to my assumption of negligible perturbation from GW's during the squeezing phase for G-clamps! Dispute that?
If so, it amounts to saying Komar expression has no validity under any circumstances whatever. So why is it still part of the GR pantheon?

I don't understand your comments here. In your OP the argument [2] re: G clamps did not mention Komar mass at all. Are you trying to add it in now? If so, why and how?

Btw, I have no problem with your conclusion [2] in the OP. It didn't seem to indicate that the divergence of the SET was non-zero, and that is all that is claimed by GR. That GWs carry energy and that some spacetimes don't have a globally conserved energy is well known.

 

[PAllen]

I don't understand your comments here. In your OP the argument [2] re: G clamps did not mention Komar mass at all. Are you trying to add it in now? If so, why and how?

Btw, I have no problem with your conclusion [2] in the OP. It didn't seem to indicate that the divergence of the SET was non-zero, and that is all that is claimed by GR. That GWs carry energy and that some spacetimes don't have a globally conserved energy is well known.

I'll slightly amend Dalespam's observation from my point of view. By default, I would assume asymptotic flatness for the G-clamp case, which says ADM energy is consderved. However, there is nothing remotely to suggest non-conservation. GW have energy. The amount they carry would be incomprehensibly small. There is periodic source of energy implied: motors, etc. All you need is to assume, e.g. 1 part in 10^50th or smaller of the periodically applied energy (to the clamps) is converted to GW energy. ADM energy includes GW energy. So I see nothing in this argument that implies its conclusion at all

Note: you cannot assume no work done on the bar: for pressure to increase, bar must give; also screws must move. I would think thermal radiation from this cycling process would dwarf GW by 10s of orders of magnitude, and present no conservation problem because of whatever energy source is needed to power the clamps.

[EDIT: One key claim being made is that the energy of the GW is somehow proportional to the stress on the bar, and 'this is too big'. Problem is there is no direct relationship between amplitude of variation of terms of T and energy carried by propagating metric disturbances. In fact, for binary stars, we know that rapid (enormous) oscillations in mass terms in T produce GW energy 10s of orders of magnitude smaller that the variation in T terms. Thus, as presented, I see less than nothing to the whole scenario.]

 

[Q-reeus]

Q-reeus: "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."

I don't understand what this is supposed to mean. The "general principle" that ensures that pressure does not contribute to the Komar mass integral in this case only holds *in static equilibrium*; it says that the (positive) contribution from the pressure of the gas inside the container is exactly canceled by the (negative) contribution from the tensile stress in the container. That's what the Ehlers paper you referenced shows. But that cancellation only holds in static equilibrium; it was never claimed to hold dynamically at all times while the system is oscillating.

The problem imo here Peter is your's not mine. You are confusing the situation applying with my self-supporting oscillating elastic shell to that of the static, basically gas filled balloon model in Ehlers paper. I advise to go back and carefully reread #51, which clarified the conditions applying for my shell - one being in-vacuo, and also gave the proper relevance of referring to Ehlers paper - simply demonstrating that your claim of shell stability against radial applied forces (static or dynamic in origin) via opposing radial shell stresses is an impossibility. And in #58 there was some further discussion on those things.
The cancellation of pressure contribution(s) (that added (s) is important!) to Komar mass in Ehlers example: It's saying that in respect of externally observed m, contribution from internal gas pressure is exactly compensated by the *tangent* shell stress contribution. Which is no more profound or relevant to my case than saying 1+(-1) = 0. I won't quibble with the exactness of Ehler's argument, since it discounts entirely any modification to that simple conclusion from a radially varying redshift modifier. But that's ok by me - I adopt a reasonable attitude and appreciate this was a gravitationally insignificant scenario. Please - get what I have actually said and argued right! Conflating chalk and cheese scenarios doesn't help.

What is missing from the above quote, and from your other attempts to describe this scenario, is, as I said before, that you have not answered the fundamental question: *why* does the system oscillate?

(a) Because it's elastic, and elastic bodies in general freely vibrate under the dual actions of inertia and restoring elestic stress.
(b) Because obviously we assume an initial (spherically symmetric) impulse was applied in the first place. Say from an air cannon radial array - or anything else that would provide a uniform radial impulse. Normally one doesn't argue of such things - assume a given initial state (radial oscillation) and proceed from there. But things are decidedly not normal here.

Oscillation requires that as the system moves away from equilibrium, a restoring force is created that drives it back towards equilibrium. For small oscillations, one would expect the restoring force to be linear in the displacement ("displacement" in this case being, roughly speaking, the difference between actual radius of the system and its equilibrium radius). So what, specifically, is the restoring force in this system, and how do we know it is linear in the displacement for small amplitudes?

I refer you back to relevance of #51 - what in that argument do you specifically dispute? I asked your opinion back then, and got none. Time to say. Either you will continue to defend radial elastic restoring forces, or concede only tangent elastic stresses provide the restoring forces needed. And I can tell you now, any mechanical engineer of sound and sober mind would laugh out loud at the suggestion radial eleastic stresses could work - and I mean specifically in vibrating shell case (also applies to Ehlers example as part of a general situation here). My $1000 offer still stands! Not trying to be disparaging in saying that. You have a reason for your position, but it just can't work. Think I've figured pretty well why you so doggedly stick with radial stresses - because of the SET impositions in GR re stress contributions to curvature. Pointing to the inconsistencies in GR again - imho.

To me, the fundamentals of the dynamics of the oscillation should come first, before any other claims about what this scenario does or does not show. If we don't all agree on the fundamental dynamics, we're not going to even be able to talk about the other stuff.

Absolutely, and I've said my bit above. Give me a detailed point-by-pointer rebuttal if you can. But my appeal is - get what I actually say and mean right, and save me the continued effort of searching back through a very long thread in order to quote from previous entries. Exhausting.

 

[Q-reeus]

I don't understand your comments here. In your OP the argument [2] re: G clamps did not mention Komar mass at all. Are you trying to add it in now? If so, why and how?

Didn't repeat the explicit mentioning of Komar re [2] there, but was then any need? Recall it was only from later crticisms I got the drift Komar was somehow invalidated completely for arbitrarily tiny motions. Could never have anticipated such a line of attack.

Btw, I have no problem with your conclusion [2] in the OP. It didn't seem to indicate that the divergence of the SET was non-zero, and that is all that is claimed by GR. That GWs carry energy and that some spacetimes don't have a globally conserved energy is well known.

In saying that you are in effect agreeing to it acting as a perpetual motion machine capable of churning out an excess (more generally - a mismatch) of power endlessly. This is a cyclical process, set in an arbitrarily flat background spacetime. And I would assume the purported failures of coe in GR all relate to non-cyclic processes - e.g. expanding universe. Very different scenarios and implications imo. And just for the record unless I am again misrepresented on my stand - read *carefully* my commentary in last main paragraph in #1!

 

[PAllen]

A few more thoughts on this not so static case, and the issue "how wrong is Komar mass?"

First, I will augment an argument Dalespam has given several times. The Komar formula is written in terms of killing vectors. If there is GW, there does not exist killing vectors - GW is time variation of the metric, so it precludes killing vectors. Going a step further, how wrong would it be to apply the formula with 'approximate killing vectors' ? Well an OP type scaling argument would suggest the error is on the order of deviation of 'best possible killing vectors' from being true killing vectors. How big is this error? At least of the same order as time variation of the metric. Thus the error is at least of the same order as what the OP argument (both cases) is trying to analyze - GW. An approximation whose scale of error is inherently of the same order as what you are trying to analyze is simply completely inapplicable.

So, there is simply no choice at all but to look at the problem using ADM energy in asymptotically flat spacetime (or worse, use numerical relativity if you want to consider manifolds where ADM energy is not defined). Then we hit the mathematical theorems the OP wants to distrust. The spherically symmetric case can't produce GW at all. The case with quadrupole moment could, but the conservation theorem for ADM energy guarantees exact conservation of mass/energy when the energy of the GW is accounted for.

As for sense of scale on GW for the quadropole case, I would argue that GW would be a vanishingly small fraction of thermal radiation fromm oscillating compression, the energy for both would come from whatever is powering the clamps.

[Edit: reading post #162, again I see the only argument for GW being 'large' is based on relation between terms of T and Komar mass, which I argue is completely inaccurate for this purpose. A corrected analysis might well show that Qs contributions to GW are scaled by other terms such as to become vanishing. I don't know and am unconcerned, because I don't find it constructive to dispute mathematical theorems with unfounded approximations. The ADM theorems guarantee that the GW will not violate conservation (assuming asymptotic flatness - equivalently, isolating this setup sufficiently from its surroundings and operating on time scale infinitesimal compared to cosmological expansion.)]

 

[Q-reeus]

I'll slightly amend Dalespam's observation from my point of view. By default, I would assume asymptotic flatness for the G-clamp case, which says ADM energy is consderved. However, there is nothing remotely to suggest non-conservation. GW have energy. The amount they carry would be incomprehensibly small. There is periodic source of energy implied: motors, etc. All you need is to assume, e.g. 1 part in 10^50th or smaller of the periodically applied energy (to the clamps) is converted to GW energy. ADM energy includes GW energy. So I see nothing in this argument that implies its conclusion at all.

So you take a very different tack to another poster - demanding conservation of energy strictly hold here. Fine. I agree that's what GR would say - but then contradicts itself in certain scenarios as I shall attempt to demonstrate again below. But in saying you see nothing in my argument means you have failed to understand the nature of parameter scaling/not scaling for various contributors. I'll repeat briefy, but ask you to go back and read it again and again, then the same with #88, until it sinks in. Here is the #88 scenario (assumes Komar is applicable) in some detail:

We start with a G-clamp pair + enegizing/driving source - in an initial state where clamps are unstressed and no energy drain from batteries. There is a certain overwhelmingly monopolar gravitational field and associated total system energy.

Next we throw a switch. The drive motors whirr into action - driving the screwed legs into compressive stress, and the opposing sides - which being welded together forms a central column - into tensile stress. Shear and bending moments in the top and bottom cantilever arms either have no contribution at all (shear stresses) or merely add a relatively and by careful design an arbitrarily small essentially quadrupolar stress distribution - orthogonal to that applying to the horizontal linear Qs owing to stress in the main vertical columns. Assume zero losses to friction or electrical resistance etc. So this is a nominally closed system in the sense of including all the static field energy out to a sufficiently large bounding surface. We do *not* bother to account for a vanishingly small GW pulse owing to the squeeze-up process. Is the gravitating mass m and system total energy W constant, for arbitrary material parameters? Let's see.

Specify that final stress level is held to be constant, while we look at the effect of varying parameters E (Young's modulus of elasticity), and ρ (material density). Let's first look at E made n times higher. Has no effect on contribution to field from Q_s since final stress is by definition held constant. But strain has gone down by the factor n^-1. So assuming the clamps flex predominantly in the vertical direction by some value d, the vertical oriented linear quadrupole moment Q_m so produced, orthogonal to that of Q_s and thus non-interacting, drops in magnitude by factor n^-2 (square of flexure displacement d). We cannot simply use a direct comparison between Q_s and Q_m to net field energy density contributions, owing to dominant cross-interaction terms, not between each other, but with that from the dominant rest mass m₀. However there is net cancellation owing to symmetries when integrated over all space, so that it is safe to simply treat each contribution seperately. The trend is obvious; in the limit as 1/E -> 0, only the contribution from Q_s to changed field survives. No need to specify a physically impossible infinite E, the trend is entirely enough.

What about the energetics in generating this Q_s contributed field? That ties in with any contribution to field from shifting non-field energy around in the stressing process (apart from gross rest-mass motion considered above). Note the batteries can be placed anywhere - including positions that either completely or almost completely eliminate any quadrupole moment Q_e owing to shift from chemical to elastic energy. Any remaining higher order moments will be negligible wrt dominant Q_e. Further though, as elastic energy density is a product of stress and strain, it drops as E^-1. Hence also vanishes as 1/E -> 0, though more slowly than Q_m does. Trend is clear. As E goes high, all other contributors to field other than stress plunge towards zero. At the same time, strain also drops in proportion to E^-1, and so the energy drain to generate a stress-only Q_s contribution drops accordingly. All with a fixed final Q_s and field so produced. This is not conservation of energy in action. Total system gravitating mass m and total energy W rises simply because energy expended in generating Q_s becomes vanishingly small.

As a second example we might instead make material density rho n times smaller. Again, stress contribution Q_s is indifferent, but Q_m drops in direct proportion, as does the monopole and higher order contributions from gross rest mass m₀. Energy expended in strain here drops only slightly - *almost* just to the extent the change in net field energy is reduced. That *almost* is an important caveat - it acknowledges the odd-man-out bahavour of stress as assumed field source.
That odd behavour is here laid bare if one cares to acknowledge. In the instance given above, we see that any proper conservative coupling between energy input and field generated is just not there. Stress as linear source of field obeys no conservation principle, no divergence relation that makes sense. Not if you want to hold onto conservation of energy/momentum, especially in flat spacetime setting.

Note: you cannot assume no work done on the bar: for pressure to increase, bar must give; also screws must move. I would think thermal radiation from this cycling process would dwarf GW by 10s of orders of magnitude, and present no conservation problem because of whatever energy source is needed to power the clamps.

You obviously never read #1 and #88 carefully. Hope it's all clearer to you now.

[EDIT: One key claim being made is that the energy of the GW is somehow proportional to the stress on the bar, and 'this is too big'. Problem is there is no direct relationship between amplitude of variation of terms of T and energy carried by propagating metric disturbances. In fact, for binary stars, we know that rapid (enormous) oscillations in mass terms in T produce GW energy 10s of orders of magnitude smaller that the variation in T terms. Thus, as presented, I see less than nothing to the whole scenario.]

And I see no specific connections or relevance here to what I have presented. Never claimed a direct proportionality you suggest I did - that is your invention. Again I ask you; make quite sure to have read and understood just what I do say, not what you think or vaguely recollect I said. It saves much headache.

I'm engaged for next day or so, so I urge any respondents to these last three postings to take their time and check carefully before responding.

 

[PAllen]

In light of #160, I more strongly claim this whole exercise is logically equivalent to the following:

- On my high precision IEEE floating point implementation, if I compute 1 - 1/7 - 1/7 - 1/7 - 1/7 - 1/7 - 1/7 - 1/7 I don't get zero. I can repeat this as often as possible, accumulating an ever larger discrepancy in the laws of mathematics. Thus there is something fundamentally inconsistent in mathematics.

 

[Dale]

In saying that you are in effect agreeing to it acting as a perpetual motion machine capable of churning out an excess (more generally - a mismatch) of power endlessly. This is a cyclical process, set in an arbitrarily flat background spacetime. And I would assume the purported failures of coe in GR all relate to non-cyclic processes - e.g. expanding universe. Very different scenarios and implications imo. And just for the record unless I am again misrepresented on my stand - read *carefully* my commentary in last main paragraph in #1!

I am agreeing to the conclusions that a quadrupolar stress could in principle generate GW's and that the GW's carry energy. I didn't see that your scaling arguments justified any other claims. In particular, a scaling argument is not capable of making the perpetual motion claim.

A scaling argument has the following form. From first principles or physical intuition we assume that some quantity q is related to some other quantities a and b as follows:
$q = \kappa a^{\alpha} b^{\beta}$
Where κ, α, and β are all dimensionless. Then, by analyzing the units of q, a, and b we can solve for α and β.

A scaling argument can never give you any information about κ. In particular, it cannot tell you the sign of κ. So it cannot give you the difference between losing energy to the GW produced and gaining energy from the GW produced. All a scaling argument can do is tell you what α and β must be.

In your argument I believe that q is energy, a is Youngs modulus, and b is density. Correct?

 

[PeterDonis]

You are confusing the situation applying with my self-supporting oscillating elastic shell to that of the static, basically gas filled balloon model in Ehlers paper.

Ah, so now I finally have a definite answer (at least I think I do): your scenario was supposed to be the shell with vacuum inside and outside. (I did go back and look at #51, and yes, it does appear to say that; but it would have been nice to have confirmation when I asked for it a number of posts ago.) But that still leaves me confused about some claims you have made. See below.

the proper relevance of referring to Ehlers paper - simply demonstrating that your claim of shell stability against radial applied forces (static or dynamic in origin) via opposing radial shell stresses is an impossibility.

But the two scenarios have different static equilibrium conditions. The gas-filled balloon's static equilibrium is determined, as the Ehlers paper says, by a balance between the pressure of the gas and the tension in the container. But the vacuum shell's static equilibrium can't be determined by that, because there is no gas pressure inside. In fact, the equilibrium conditions for the vacuum shell are these:

(1) Do a similar force balance as is given in the Ehlers paper: cut a plane through the center of the shell (i.e., the center of the spherical vacuum region inside) and look at the force balance across that plane. Since there is no gas inside the shell, pressure contributes nothing to this force balance; the only forces acting perpendicular to this plane are the tensile stresses in the shell itself. That implies that the tensile stresses must sum to zero through the shell--i.e., when I integrate tensile stress from the shell's inner radius to its outer radius, the result must be zero (as I said way back in an early post).

(2) Radially, since there is vacuum inside the shell, there is nothing "holding the shell up" against its own gravity. True, the shell's gravity can be very small, but it is not zero; so if there is to be any radial force balance at all, gravity is the only thing that can balance radial pressure. In the limit, of course, you could say that radial pressure is negligible throughout the shell because gravity is too weak to give the shell any significant weight that needs to be supported; but then, I'm not sure what the point is of the scenario in the first place since it's completely non-relativistic--see further comments below.

(a) Because it's elastic, and elastic bodies in general freely vibrate under the dual actions of inertia and restoring elestic stress.
(b) Because obviously we assume an initial (spherically symmetric) impulse was applied in the first place. Say from an air cannon radial array - or anything else that would provide a uniform radial impulse. Normally one doesn't argue of such things - assume a given initial state (radial oscillation) and proceed from there.

Yes--proceed from there. Once the shell is perturbed out of equilibrium, what, specifically, is the restoring force? And how does that contradict GR in any way? GR does not dispute that elastic bodies vibrate when deformed out of equilibrium.

You do answer the first question I just asked, sort of:

I refer you back to relevance of #51 - what in that argument do you specifically dispute? I asked your opinion back then, and got none. Time to say. Either you will continue to defend radial elastic restoring forces, or concede only tangent elastic stresses provide the restoring forces needed.

Not just "tangent elastic stresses". You have to be very specific about how those stresses provide a restoring force *while maintaining spherical symmetry*. I assume you are thinking along the lines of a force imbalance across a plane cut through the center of the shell (similar to the first equilibrium condition above): but however "standard" you think this may be from mechanics textbooks (and I'm not disputing that it is, btw), I would still like to see you explain, in *your* words, how, specifically, the dynamics works. I don't want to know what the mechanics textbook authors think, or what analysis they have done; I want to know what *you* think and what analysis *you* have done, because the mechanics textbooks don't claim that standard stress theory of materials contradicts GR, but you do.

Think I've figured pretty well why you so doggedly stick with radial stresses - because of the SET impositions in GR re stress contributions to curvature.

All stresses contribute to curvature, not just radial ones. I was originally thinking of a scenario similar to, say, a pulsating star, where the shell was heavy enough that its self-gravity made a significant contribution to the dynamics. You have insisted on neglecting self-gravity altogether, which IMO makes it kind of pointless to discuss a comparison with GR in the first place (if you're neglecting self-gravity altogether, then plain old Newtonian stress theory works just fine, and that is simply the extreme weak field limit of GR, so it's manifestly consistent with GR). But it's your scenario.

Absolutely, and I've said my bit above.

And I've asked for more detail; what you've given isn't remotely enough to even begin to talk about whether or not there is any problem with GR.

In saying that you are in effect agreeing to it acting as a perpetual motion machine capable of churning out an excess (more generally - a mismatch) of power endlessly.

No, he isn't. In your clamp scenario, something has to provide the force that compresses the clamp. That something contains energy, and for a fully consistent solution, you have to include that energy. Obviously that power source is the source of the energy that is being put into GWs (as well as, in any real case, lots more energy being put into dissipative modes in the clamp itself, which heat it up, and which heat then gets radiated as non-gravitational radiation, etc., etc.--as PAllen has already pointed out). There is no perpetual motion machine.

 

[PeterDonis]

Here is the #88 scenario (assumes Komar is applicable) in some detail:

<snipped lots and lots of words>

I've read through this several times, and it looks to me like you are using an awful lot of words (and in multiple posts, to boot) to make what seems to be a very simple argument:

Consider a scenario where we exert force on a material, initially unstressed, to achieve a final state with a fixed stress. (The stress must be one of the diagonal stress components in order to be relevant to the Komar mass integral, which is the standard you are using for "energy conservation". It doesn't really matter which component, so we won't bother specifying.) As the stiffness of the material goes up, I can produce the same fixed final stress by expending less and less energy. So the initial and final Komar mass integrals will look like this ("rest mass" here is the rest mass of all parts of the system, the material and the "engine" that is applying the stress to it):

initial: rest mass + energy to be expended (gets smaller and smaller)

final: rest mass + final stress (fixed)

Since the system is completely closed, these two results should be the same (assuming the initial and final states are both stationary, which seems reasonable). But they aren't. So what gives?

Is this the argument you are making?

(This sounds similar to what Jonathan Scott was saying in several posts a while ago, btw, although he didn't claim this showed any problem with GR, just with taking the Komar mass integral as the standard of "energy conservation" for non-stationary systems.)

(Edit: I should also point out that I don't think Jonathan Scott was claiming that the above summary of the initial and final Komar mass integrals would be correct. They aren't; but I'll save that for a separate post.)

 

[Jonathan Scott]

(Edit: I should also point out that I don't think Jonathan Scott was claiming that the above summary of the initial and final Komar mass integrals would be correct. They aren't; but I'll save that for a separate post.)

I'll just reiterate that for a static system, the Komar mass integral has nothing to do with internal stresses built up in the system, nor the material of which anything is made, nor its elasticity. All stresses except those balancing gravitational forces cancel out, and those balancing gravitational forces add up exactly to the same value as the potential energy.

 

[PeterDonis]

I'll just reiterate that for a static system, the Komar mass integral has nothing to do with internal stresses built up in the system, nor the material of which anything is made, nor its elasticity. All stresses except those balancing gravitational forces cancel out, and those balancing gravitational forces add up exactly to the same value as the potential energy.

I agree, and that's why I said my summary of what the Komar mass integral should look like in Q-reeus' example was not correct. There are two things left out:

First of all, the energy stored in the "battery" originally (which gets smaller and smaller as the stiffness of the material to be stressed goes up) ends up as an extra contribution to T_00 in the stressed material (which gets smaller and smaller as the stiffness of the material goes up).

Second, as you point out, all the stress contributions to the integral have to cancel in static equilibrium; in Q-reeus' example, the stress contribution from the material has to be canceled by an equal and opposite stress contribution from the "engine" (whatever it is that is applying the force to the material). This is just the argument from the Ehlers paper that Q-reeus linked to, generalized. (Q-reeus is saying gravity should be neglected in his scenario.)

So the correct Komar mass integrals look like this:

initial: rest mass + energy to be expended

final: rest mass + energy stored in material + final stress in material - final stress in "engine"

The additional "energy" terms are equal, and the stress terms cancel, so both integrals give the same final answer. But I'm not sure Q-reeus has grasped that point yet. I am hoping that by giving a much simpler description of the scenario, we can clear away a lot of underbrush.

 

[Dale]

I agree that's what GR would say - but then contradicts itself in certain scenarios as I shall attempt to demonstrate again below.

This is certainly not demonstrated by your scaling argument. Are you claiming now to have shown by your scaling argument that the SET diverges?

I will remind you of your previous correct comment.

Any rigorous math proof acceptable to you and others here would entail working within a framework gauranteed to self-exhonerate GR.

If you actually work through the math using GRs framework then you cannot get a contradiction. By design it is "guaranteed to self exonerate" GR.

Here is the #88 scenario (assumes Komar is applicable) in some detail:

Komar is not applicable, the spacetime is not stationary.

 

[PeterDonis]

Komar is not applicable, the spacetime is not stationary.

Just to clarify where I was coming from in my last couple of posts, I agree that the Komar mass integral is not applicable when the spacetime is not stationary. I was only looking at the "initial" and "final" states, which are stationary, not the transition period, which isn't.

 

[Q-reeus]

I am agreeing to the conclusions that a quadrupolar stress could in principle generate GW's and that the GW's carry energy. I didn't see that your scaling arguments justified any other claims. In particular, a scaling argument is not capable of making the perpetual motion claim.
I am agreeing to the conclusions that a quadrupolar stress could in principle generate GW's and that the GW's carry energy. I didn't see that your scaling arguments justified any other claims. In particular, a scaling argument is not capable of making the perpetual motion claim...

Do you expect me to believe the above is consistent with that actually said in #157? You also failed to admit to fluffing it wrt resolution of shear stress into orthogonal equal and opposite sign principal stresses in your #2 - when I challenged in #3,#24, you never responded. If you still doubt, go check last sentence, p111 here: http://books.google.com.au/books?id=y4WalY4ZptAC&pg=PA111&lpg=PA111&dq=pure+shear+stress+as+equal+and+opposite+principal +stresses&source=bl&ots=rakU9jxhol&sig=nZEBpYwpyuF0-mR-KaE_1uKoweA&hl=en&sa=X&ei=tclpT_b0KZHjrAfYsZH_Bw&ved=0CDoQ6AEwBQ
Are you prepared to own up on these two matters now? If not, don't enter into discussion with me again. For the sake of cleaning up will deal with the rest of your #164 below.

A scaling argument has the following form. From first principles or physical intuition we assume that some quantity q is related to some other quantities a and b as follows:
$q = \kappa a^{\alpha} b^{\beta}$
Where κ, α, and β are all dimensionless. Then, by analyzing the units of q, a, and b we can solve for α and β.
A scaling argument can never give you any information about κ. In particular, it cannot tell you the sign of κ. So it cannot give you the difference between losing energy to the GW produced and gaining energy from the GW produced. All a scaling argument can do is tell you what α and β must be.

That scaling relation looks pretty restrictive for a general statement - only a product of two exponentiated parameters. And just one equation - if you want to make it really general then why not a set of simultaneous eqn's, allowing thereby to tie down sign of K etc. Still, just based on your restrictive definition, applied to what you are angling at below, sure all by itself one can't determine the sign of K. But if that is essential we necessarily supplement pure scaling argument with e.g.
a) A sensible assumption about the sign (Experience sugests rubber makes more sense than 'flubber' so assume a + sign).
b) A derivation based on known relations between known quantities. A simple example: https://www.physicsforums.com/showpost.php?p=3785574&postcount=45, supplemented with #54 there - which in turn led to this thread!
c) Working from quantities where the signs are all known by experiment/observation.

In your argument I believe that q is energy, a is Youngs modulus, and b is density. Correct?

The last sentence is applicable if specifically applied to analyzing how summed gravitational energy density dq/dv (dv a volume element containing an elemental dq) varies with variation of a and b in the weak gravity regime where non-linearity is insignificant wrt effect on the parameters of interest. And where gravitational energy is determined on a g² basis, where g is the magnitude of gravitational acceleration experienced by a hovering observer. There is an important caveat - relating to including gravitational depression of rest mass-energy, dealt with in a following posting.

 

[Q-reeus]

Q-reeus: "the proper relevance of referring to Ehlers paper - simply demonstrating that your claim of shell stability against radial applied forces (static or dynamic in origin) via opposing radial shell stresses is an impossibility."

But the two scenarios have different static equilibrium conditions. The gas-filled balloon's static equilibrium is determined, as the Ehlers paper says, by a balance between the pressure of the gas and the tension in the container. But the vacuum shell's static equilibrium can't be determined by that, because there is no gas pressure inside. In fact, the equilibrium conditions for the vacuum shell are these:

(1) Do a similar force balance as is given in the Ehlers paper: cut a plane through the center of the shell (i.e., the center of the spherical vacuum region inside) and look at the force balance across that plane. Since there is no gas inside the shell, pressure contributes nothing to this force balance; the only forces acting perpendicular to this plane are the tensile stresses in the shell itself. That implies that the tensile stresses must sum to zero through the shell--i.e., when I integrate tensile stress from the shell's inner radius to its outer radius, the result must be zero (as I said way back in an early post).

(2) Radially, since there is vacuum inside the shell, there is nothing "holding the shell up" against its own gravity. True, the shell's gravity can be very small, but it is not zero; so if there is to be any radial force balance at all, gravity is the only thing that can balance radial pressure. In the limit, of course, you could say that radial pressure is negligible throughout the shell because gravity is too weak to give the shell any significant weight that needs to be supported; but then, I'm not sure what the point is of the scenario in the first place since it's completely non-relativistic--see further comments below.

First, what is the relevance of either (1) or (2) to a gravitationally tiny, elastic shell subject to breathing mode harmonic stresses of inertial origin? Zero. You sneek in 'static equilibrium' in (1), maybe hoping I won't notice that has nothing to do with the dynamic balance needed for a vibrating shell - what we are supposed to be discussing. In particular though, what is said in (2) is just dumbfounding. In saying there is nothing holding up the shell, what on Earth is your shell composed of? A perfect fluid again - the one I dealt with in #58? Hope you're not trying to resurrect that one. Can we stick with an elastic shell? Or do you believe elastic stresses cannot hold an elastic shell up against self-gravity? I'm lost either way as to what you are arguing re (2). Please clarify here.

Yes--proceed from there. Once the shell is perturbed out of equilibrium, what, specifically, is the restoring force?

Same answer as always from me - tangent stresses, acting across any given shell curved area element. Evidently from below comments you reject standard engineering derivation of that known fact (engineers put strain gauges across such strucures, or use more sophisticated techniques like photoelasticity or laser holography). It is established fact. Not like some theory driven assumption afaik yet to be observationally verified. Someone in an earlier entry claimed collapse of protostellar gas clouds as solid evidence of pressure as gravitating source in GR. What!? The corrections to Newtonian gravity are how small at that level again? And I get accused of making wild assertions.

And how does that contradict GR in any way? GR does not dispute that elastic bodies vibrate when deformed out of equilibrium.

There is supposed to be a sensible connection here? Your line of attack really is 'GR is True, and role of stress in SET/EFE's demand radial shell stresses really do the balancing' - you specifically claimed so in #17. Uh uh; engineering wins here - tangent stresses do the balancing. No pulling up by one's boot straps allowed.

Q-reeus: "I refer you back to relevance of #51 - what in that argument do you specifically dispute? I asked your opinion back then, and got none. Time to say. Either you will continue to defend radial elastic restoring forces, or concede only tangent elastic stresses provide the restoring forces needed."

Not just "tangent elastic stresses". You have to be very specific about how those stresses provide a restoring force *while maintaining spherical symmetry*. I assume you are thinking along the lines of a force imbalance across a plane cut through the center of the shell (similar to the first equilibrium condition above): but however "standard" you think this may be from mechanics textbooks (and I'm not disputing that it is, btw), I would still like to see you explain, in *your* words, how, specifically, the dynamics works. I don't want to know what the mechanics textbook authors think, or what analysis they have done; I want to know what *you* think and what analysis *you* have done, because the mechanics textbooks don't claim that standard stress theory of materials contradicts GR, but you do.

That piece is as good as a frank acknowledgment that shell equilibrium, via tangent not radial shell stresses, threatens the integrity of GR, at least for one particular GR devotee - you. Very well, your response is to duck my request to justify your position, and throw the burden back at me, knowing there is no way you can get balance your way. How good would my derivation need to be I wonder, if standard engineering textbook accounts aren't? Oh well, applying the KISS principle in the spirit of reasonableness, here goes just a bit.

Our thin elastic shell has a static mid-radius R and thickness δ << R, so outer, inner radii are R+1/2δ, R-1/2δ. Assume an initial impulse induces spherically symmetric sinusoidal radial vibration, at angular frequency ω, of very small amplitude h << R (hence non-linearity is insignificant). Instantaneous position of mid-radius r is then r = R+h*sinωt. Differentiate twice wrt time to give d²r/dt² = -h*ω²*sinωt. Any small area element dA of shell will have a volume and mass dA*δ, dA*δ*ρ respectively (ρ the material volume density). From F=ma applied to the foregoing we find there is a radial applied area force density ('pressure') of -dF/dA = -(dA*δ*ρ)*-h*ω²*sinωt)/(dA) = δ*ρ*(h*ω²*sinωt). This is the force density exerted by the accelerated shell matter. It must be opposed by tangent elastic stresses, which will naturally arise as a consequence of shell compression/dilation yielding the requisite biaxial stresses. Equate the two to determine the value of ω that gives such equality. We simply substitute this dynamic origin force density instead of gas pressure, and obtain hoop stresses as per Ehlers. From this point I hand over to:
A step-by-step derivation of that a radial acting applied force density is countered solely by tangent hoop stresses, begin at lower part of p7 here:
http://www.google.com.au/url?sa=t&rct=j&q=hoop%20stresses&source=web&cd=11&ved=0CF8QFjAK&url=http%3A%2F%2Fwww.engr.colostate.edu%2F~dga %2Fmech325%2Fhandouts%2Fpressure_vessels.pdf&ei=TtZpT9yQKaOViAe2rbmTCg&usg=AFQjCNEWq0OsySkPxIEA1yMxzSkCMYfNZQ&cad=rja
First part is as for Ehlers. When you get down to 'slides' 61, 62 on p16, notice a derivation for ratio of maximum radial elastic stress to tangent stress - proving my contention radial stresses are negligible for a thin shell. Further though, it should be evident that if tangent stresses failed to act as shown there, the only 'balance' possible via radial elastic stress would be from acceleration of that element away from the pressure source - the shell explodes. And the only function of radial stress is to share the reception of surface acting gas pressure throughout the volume. Tangent stresses provide 100% of static resistance against gas pressure. Basic stuff. Yet GR it seems denies this as possible. Amazing.

In that specific analysis applied force acts on just the inner surface, yet response is across the whole width of shell, as expected from basic principle. If you specifically dispute that the hoop stresses will be near uniform for a thin shell, then you dispute the load-shedding principle in mechanics that derives from energy minimization principle - e.g. http://en.wikipedia.org/wiki/Minimum_total_potential_energy_principle
In dynamic case, applied force is a body force from F=ma acting radially throughout the shell, so opposing shell stresses would be expected to be even more uniformly distributed than in gas pressure case. And just remember my comments from #27; zero tractive forces on the shell inner/outer surfaces in vibrating case. If you demand more detail, try here: http://books.google.com.au/books?id=ViebCriF-ssC&pg=PA281&source=gbs_toc_r&cad=4#v=onepage&q&f=false
Look at pages 286-7, 512-514(not online), 538-549 (partly online), 583-586(not online). Still referred to as the bible of elasticity theory. And pretty hard going, even if written in 1920.

Q-reeus: "Think I've figured pretty well why you so doggedly stick with radial stresses - because of the SET impositions in GR re stress contributions to curvature."

All stresses contribute to curvature, not just radial ones. I was originally thinking of a scenario similar to, say, a pulsating star, where the shell was heavy enough that its self-gravity made a significant contribution to the dynamics. You have insisted on neglecting self-gravity altogether, which IMO makes it kind of pointless to discuss a comparison with GR in the first place (if you're neglecting self-gravity altogether, then plain old Newtonian stress theory works just fine, and that is simply the extreme weak field limit of GR, so it's manifestly consistent with GR). But it's your scenario.

Not pointless at all. The issue is role of stress, and it makes perfectly good sense to isolate that as far as possible from complicating factors. My scenarios were, as stated over and over, intentionally completely dominated by inertial-elastic or just elastic exchange - self-gravity inconsequential to the dynamics and stresses. And as stated in #27, redshift fluctuation is also insignificant as factor there. Scaling.

Q-reeus: " Absolutely, and I've said my bit above."
And I've asked for more detail; what you've given isn't remotely enough to even begin to talk about whether or not there is any problem with GR.

Maybe above bit will. At least it points to just where the crisis of confidence re pressure as source will be revealed. I can smell it. You have no way of reconciling standard engineering approach to shell stresses with that demanded by SET/EFE's. Your resorting to an inherently unstable, impossible-to-assemble model of shell comprised of perfect fluid as admitted in #54, is one such 'clue'. But now it's your turn to prove me wrong.

Q-reeus: "In saying that you are in effect agreeing to it acting as a perpetual motion machine capable of churning out an excess (more generally - a mismatch) of power endlessly."
No, he isn't.

Yes, in #157 he was, but too proud to own up personally. Instead, shifting ground without admitting to.

In your clamp scenario, something has to provide the force that compresses the clamp. That something contains energy, and for a fully consistent solution, you have to include that energy. Obviously that power source is the source of the energy that is being put into GWs (as well as, in any real case, lots more energy being put into dissipative modes in the clamp itself, which heat it up, and which heat then gets radiated as non-gravitational
radiation, etc., etc.--as PAllen has already pointed out). There is no perpetual motion machine.

Read #1 again - carefully. I specifically addressed and dealt with via scaling argument and otherwise the only legitimately relevant part you imply I ignored (energy input from source going to elastic stress/strain energy). The dissipation bit is bogus - we assume an idealization as is common in such situations. Without that, needless complexity clouds things. By now anticipating such attacks, it was specifically addressed in #162 (irrelevant in that respect that static fields the issue there - not GW's, despite #161 ignoring that and only referencing to GW's scenario as per #1).

 

[Q-reeus]

Best to show some personal honesty in demanding it of others. After posting #162 dealing with stressed G-clamps scenario, realized there were two important non-fatal errors I had missed.

One is that 'magically' adding gravity for free via stress, in the limit of negligible strain, will thereby effect the redshift factor of mass-energy involved. Result is that net system energy change is not simply from the field, but field + changed potential of mass-energy. The bizarre nature of stress as source - no conserved currents - no concommitant mass-energy flow involved in changing the field, makes it difficult to justify a split based on e.g. a partial gravitational collapse scenario as usually done (I referred to my use of that in #171). Giving any credence at all to stress as source, seems reasonable nevertheless to stick to a 1:1:2 split rule: dW_s = dW_g+dW_m = -dW_g, (since dW_m = -2dW_g), where
dW_s is the net change in system mass - the binding energy
dW_g is the net change in energy in the gravitational field
dW_m is the net change in rest mass-energy as per redshift factor.
On that basis, it means only that the sign of any net system change is opposite to what I had assumed on the basis of field energy alone.

This leads into the second issue. I had taken for granted that a qudrupole distribution of stress gives rise automatically to a net positive gravitational energy. Not necessarily. Again, the bizarre character of stress as source has that negative stress generates a negative curvature 'anti-gravity' field contribution. While I assume positive energy condition means there can be no net negative curvature anywhere, negative stress component will subtract wrt that from rest mass-energy. There is a tendency for the positive and negative stress distributions to cancel overall, despite the finite 'quadrupole' distribution. But only a tendency. Suppose complete cancellation applied to some specific G-clamp geometry. Just changing the cross-sectional area of one of the stressed columns will then remove that perfect cancellation. Reason being the same as when one radially compresses a charged wire - net volume integral of field energy has been increased. In stressed column case, increased stress density owing to reduced cross-section translates to increased net field energy relative to wider columns. And there would be other means to bias the situation. For instance surround one column with say a sleeve of unstressed material. Interaction terms between stress contribution from column and that of rest mass in sleeve would in general drastically alter the overall balance, were the sleeve absent.

Last point here is the nature of any GW's resulting from periodically stressed G-clamps as per #1. Assuming a periodically time varying quadrupole-like distribution of stress leads to regular quadrupolar GW's is wrong. A true time-varying quadrupole source has mass currents flowing - hence both 'electric' and 'magnetic' components in accordance with the equipartition of energy rule surely applying for any periodic physically real wave. The
absence of any 'stress current' rules out any 'magnetic' component for G-clamp scenario. Just the superposition of purely 'electric' monopole sources spatially displaced to look like a real quadrupole source. Is this consideration alone not fishy enough to rule out stress as genuine source? Or is 'electric' only GW's actually the case in GR?

[PS - to PeterDonis; re your question in #166, the answer is yes, provided one qualifies that by taking into account everything stated above and other recent postings!]

 

[PeterDonis]

[PS - to PeterDonis; re your question in #166, the answer is yes, provided one qualifies that by taking into account everything stated above and other recent postings!]

Q-reeus, I'll go ahead and post this while I'm trying to digest the rest of your recent posts. Given your answer to my question in #166, what is your response to my #168, where I show that the supposed comparison of "initial" and "final" Komar mass integrals in #166, which you have just said, in effect, that you are claiming is a correct comparison, is in fact *not* correct?

 

[Dale]

Do you expect me to believe the above is consistent with that actually said in #157?

Yes.

You also failed to admit to fluffing it wrt resolution of shear stress into orthogonal equal and opposite sign principal stresses in your #2 - when I challenged in #3,#24, you never responded.

I don't know what you mean by "fluffing it", but I didn't respond because I don't think that it is a point of disagreement. It is hard to tell with your wording, but I didn't see a disagreement, if you think it is a point of disagreement then please be explicit and clear, because I don't see it. It seems silly to argue on a point where we agree just out of habit.

Are you prepared to own up on these two matters now?

Which two matters?

That scaling relation looks pretty restrictive for a general statement - only a product of two exponentiated parameters.

Sure, I only put two because I thought that was the argument that you were making, but in general you can have an arbitrary number of parameters with unknown exponents.

And just one equation - if you want to make it really general then why not a set of simultaneous eqn's, allowing thereby to tie down sign of K etc.

I have never seen a scaling argument with a system of equations, but I think it makes sense, particularly if you are looking to see how multiple things scale with the same parameters. What system of equations are you proposing here?

Still, just based on your restrictive definition, applied to what you are angling at below, sure all by itself one can't determine the sign of K. But if that is essential we necessarily supplement pure scaling argument with e.g.
a) A sensible assumption about the sign (Experience sugests rubber makes more sense than 'flubber' so assume a + sign).

I agree, and a sensible assumption would be that any GW's would radiate energy off, not generate perpetual energy.

b) A derivation based on known relations between known quantities. A simple example: https://www.physicsforums.com/showpost.php?p=3785574&postcount=45, supplemented with #54 there - which in turn led to this thread!

Or better yet, the known relations of the ADM energy, which is conserved in this scenario.

c) Working from quantities where the signs are all known by experiment/observation.

Sounds good, if you have any mainstream experiments or observations reporting perpetual energy by squeezing a pair of G clamps I would be very interested.

I agree with all 3 of your points raised here.