Is stress a source of gravity?

[PAllen]

Let's say this is correct. It should be possible then to pinpoint where the balance between a varying Komar mass and non-varying ADM mass is taken up. Given motion is invalidating Komar, it must be because certain SET terms behave differently under radial motion, agreed? So what are these motion dependent terms that compensate in a spherical geometry? Can we at least drill down that far? It's what I've basically been asking from the start. If SET terms acting as suggested above cannot be identified, then it follows there really are extra SET terms de facto introduced. For instance, if time-rate-of-change of a 'standard' SET term becomes a source, that becomes a distinctly different SET term. I'm talking here about 'new' SET terms - clearly radial motion of mass constitutes an energy-momentum flow there, which is just a standard SET term. Rate of change of that would not be. Anyone say otherwise?

This doesn't make any sense to me. There is no concept of SET terms changing meaning that needs to be explained. There is just a specialized formula that can be used of none of the terms of T is time varying. Is this concept so hard to grasp? Instead, you can use ADM mass always - it applies to dynamic as well as stationary spacetimes. Any concept of directly relating terms of T to gravitational mass is wrong.

 

[Dale]

I would describe this by saying that the energy and momentum described by the tensor are conserved (or that the flow of energy and momentum locally obey continuity equations), not that the "stress energy tensor is conserved", which I consider potentially confusing.

That is fine by me. It is always difficult to put the math into words. I like the "locally obey continuity" one.

 

[Dale]

Instead, you can use ADM mass always - it applies to dynamic as well as stationary spacetimes.

AFAIK the ADM mass requires asymptotic flatness, so it cannot be used always, in particular not in the FRW spacetime with a nonzero cosmological constant.

 

[Dale]

Given motion is invalidating Komar, it must be because certain SET terms behave differently under radial motion, agreed?

It is invalid because the timelike Killing vectors do not exist.

 

[PeterDonis]

I agree the zero divergence holds at all times, including for example cases where a wave of sudden pressure change is moving through the object (causing brief accelerations and slight readjustments of positions). However, I would describe this by saying that the energy and momentum described by the tensor are conserved (or that the flow of energy and momentum locally obey continuity equations), not that the "stress energy tensor is conserved", which I consider potentially confusing.

I agree with this.

 

[PAllen]

AFAIK the ADM mass requires asymptotic flatness, so it cannot be used always, in particular not in the FRW spacetime with a nonzero cosmological constant.

I mention this many other posts. I got tired of always mentioning it. The 'always' here meant stationary or time varying.

 

[PeterDonis]

I thought I already explained that earlier, with a similar description, for the benefit of PeterDonis, who seems to be having a problem with it. That model describes how each component of momentum is conserved, and similarly the divergence of the energy-momentum row shows how energy is conserved. It does NOT say that the volume integral of the normal stress (which is what we are using in the Komar mass expression) is conserved, and my "pole" models illustrate clearly that it is not in fact conserved.

I agree that the volume integral of pressure (i.e., "normal stress") is not conserved; I didn't mean to imply that I was contesting that claim. I was only pointing out that the Komar integral is *not* just the volume integral of pressure; it includes the energy (T_00) in the integrand as well. However, we appear to agree that the Komar integral should not be expected to be conserved anyway in a non-stationary spacetime, so the point I was making is only a minor point.

 

[Dale]

It's finally dawning on me the level to which I am dealing with particular mindsets that simply cannot conceive of the possibility of any consequential flaw in GR. Just cannot be.

Actually, I not only can conceive of the possibility that GR is wrong, I completely expect it to be experimentally proven wrong at some point. The difference is that I recognize that GR cannot be attacked theoretically, only experimentally. The mathematical framework that defines GR ensures that it is a self-consistent theory. The only way to disprove GR is to show it to be inconsistent with experimental evidence.

show where there is some basic error in logic in #1,69. Point to precisely where and how they fail, and I might take some of your less pejorative comments above seriously.

Your basic error in logic in #1 is precisely when you use the Komar mass which is not defined in a non-stationary spacetime. You cannot possibly prove anything about GW's using the Komar mass because the first excludes the second.

You just don't get it. Any rigorous math proof acceptable to you and others here would entail working within a framework gauranteed to self-exhonerate GR.

I do get it, in fact, I agree 100%. That is precisely why so much effort goes into rigorously defining the mathematical framework of a theory. Once that has been done the theory is guaranteed to not have logical inconsistencies. If you would learn the math then you would understand that.

 

[Dale]

I mention this many other posts. I got tired of always mentioning it. The 'always' here meant stationary or time varying.

Understood. I figured that you were aware.

 

[PeterDonis]

You and Peter and others here at times freely use simple non-rigorous arguments where it suits

Just a brief comment: we are using non-rigorous arguments to counter similar non-rigorous arguments from you by casting a reasonable doubt on your premises. We are not using non-rigorous arguments as a basis for claiming we have *proved* anything. We're not the ones making positive claims; you are.

(Strictly speaking, that's not quite true; we have made some positive claims, for example I made the positive claim that Birkhoff's Theorem rules out the possibility of monopole GWs. But that positive claim is based on a rigorously proved theorem.)

 

[PeterDonis]

One other thought since the ADM mass has been mentioned. The Wikipedia article on "Mass in general relativity", here...

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

...has the following interesting statement:

"In a way, the ADM energy measures all of the energy contained in spacetime, while the Bondi energy excludes those parts carried off by gravitational waves to infinity."

Wald (1984) is referenced. I have seen statements like this elsewhere as well. Given the definition of ADM mass vs. Bondi mass, this makes sense: ADM mass involves picking a spatial 3-surface out of the spacetime, doing an integral over a 2-sphere in that 3-surface, and taking the limit as the 2-sphere goes to spatial infinity (or, equivalently, as the radius of the 2-sphere goes to infinity). That means that, even if a system is emitting gravitational waves, those waves are still somewhere on any given 3-surface, so they will eventually be contained within the 2-sphere of integration as the radius of the 2-sphere goes to infinity, and hence the energy carried by the waves will be "counted" in the ADM mass. (Since the ADM mass integrand involves the metric coefficients, not the stress-energy tensor components, the wave energy is unproblematically accounted for even though the waves are in vacuum, i.e., zero SET.)

The Bondi mass, on the other hand, evaluates a similar integral at future null infinity, so the gravitational waves will "escape" from the region that is being integrated over, and hence their energy will not be "counted" in the Bondi mass. So in order to determine whether a particular asymptotically flat spacetime is radiating GWs or not, one would compare the ADM mass to the Bondi mass and see if there is a difference.

This also helps clarify what Birkhoff's Theorem is saying: for Schwarzschild spacetime, the ADM mass and Bondi mass are equal, so any spacetime that is isometric to Schwarzschild spacetime outside some finite radius r (which applies to any spherically symmetric spacetime with an exterior vacuum region, by BT) will also have both masses equal, and therefore can't contain any GWs.

 

[PAllen]

One other thought since the ADM mass has been mentioned. The Wikipedia article on "Mass in general relativity", here...

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

...has the following interesting statement:

"In a way, the ADM energy measures all of the energy contained in spacetime, while the Bondi energy excludes those parts carried off by gravitational waves to infinity."

Wald (1984) is referenced. I have seen statements like this elsewhere as well. Given the definition of ADM mass vs. Bondi mass, this makes sense: ADM mass involves picking a spatial 3-surface out of the spacetime, doing an integral over a 2-sphere in that 3-surface, and taking the limit as the 2-sphere goes to spatial infinity (or, equivalently, as the radius of the 2-sphere goes to infinity). That means that, even if a system is emitting gravitational waves, those waves are still somewhere on any given 3-surface, so they will eventually be contained within the 2-sphere of integration as the radius of the 2-sphere goes to infinity, and hence the energy carried by the waves will be "counted" in the ADM mass. (Since the ADM mass integrand involves the metric coefficients, not the stress-energy tensor components, the wave energy is unproblematically accounted for even though the waves are in vacuum, i.e., zero SET.)

The Bondi mass, on the other hand, evaluates a similar integral at future null infinity, so the gravitational waves will "escape" from the region that is being integrated over, and hence their energy will not be "counted" in the Bondi mass. So in order to determine whether a particular asymptotically flat spacetime is radiating GWs or not, one would compare the ADM mass to the Bondi mass and see if there is a difference.

This also helps clarify what Birkhoff's Theorem is saying: for Schwarzschild spacetime, the ADM mass and Bondi mass are equal, so any spacetime that is isometric to Schwarzschild spacetime outside some finite radius r (which applies to any spherically symmetric spacetime with an exterior vacuum region, by BT) will also have both masses equal, and therefore can't contain any GWs.

This agrees with my understanding of all this.

 

[Q-reeus]

For your spherically symmetrical case, I don't have a problem with Birkhoff's result that a spherically symmetrical distribution of oscillation inwards and outwards momentum would give no overall effect on the external field, as the average motion over the whole spherical surface is zero, and similar symmetries probably apply to any stress terms.

First bit is just what I argued back in #1 - spherical symmetry means cancellation of momentum flow terms. I invited comment, none came. So one presumes that is accepted as true. Second part is surely far from correct - just look at the Komar expression in #1. Stress just adds scalar-like. Arbitrarily tiny radial motions cannot on any reasonable measure make stress disappear as source. But no-one else wants to tackle the matter in those similar terms for all contributions. And it seems evidently futile to persist, there just are no takers.

A similar scheme applies in electrostatics, where you can either view the energy distribution in terms of charges within potentials or in terms of the energy in the field, proportional to the square of the field locally. The two descriptions give equal results, but describe the energy as being differently located.

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.

 

[Q-reeus]

This doesn't make any sense to me. There is no concept of SET terms changing meaning that needs to be explained. There is just a specialized formula that can be used of none of the terms of T is time varying. Is this concept so hard to grasp?

Of course not, but what in turn is so hard to grasp with seeking to look at it in terms of individual SET contributions in the given time varying situations, all in the very weak field regime. None of you will have a bar of it and I can't see why. No such reticence to do the equivalent in EM exists afaik, and why should it.

Instead, you can use ADM mass always - it applies to dynamic as well as stationary spacetimes. Any concept of directly relating terms of T to gravitational mass is wrong.

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?

 

[Q-reeus]

Q-reeus: "Given motion is invalidating Komar, it must be because certain SET terms behave differently under radial motion, agreed?"
It is invalid because the timelike Killing vectors do not exist.

That is an explanation, or merely a statement begging further questions?

 

[Dale]

That is an explanation, or merely a statement begging further questions?

A non-stationary spacetime does not have any timelike Killing vectors, and the timelike Killing vector is part of the definition of the Komar mass. See the Wikipedia page that you linked to in the OP.

I.e. the problem with the Komar mass is not due to "certain SET terms", it is due to the missing Killing vector.

 

[Q-reeus]

Your basic error in logic in #1 is precisely when you use the Komar mass which is not defined in a non-stationary spacetime. You cannot possibly prove anything about GW's using the Komar mass because the first excludes the second.

As position statement that's now been said often enough. What is not said once is just where and how and how much it would fail for the case of e.g. vibrating shell.

I do get it, in fact, I agree 100%. That is precisely why so much effort goes into rigorously defining the mathematical framework of a theory. Once that has been done the theory is guaranteed to not have logical inconsistencies.

So it is generally believed for GR, but there are experts of a different opinion, even if in a small minority. But this is just sophistry either way.

 

[PeterDonis]

First bit is just what I argued back in #1 - spherical symmetry means cancellation of momentum flow terms. I invited comment, none came. So one presumes that is accepted as true.

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.

However, since you mention this specific point, I went back and took a look at what I think is the relevant portion of #1:

For the momentum-energy flux terms T_i0 = -T_0i, having radial acting velocity vector character, spherical symmetry implies net cancellation.

First of all, the SET is symmetric, not antisymmetric, so T_i0 = T_0i, with no minus sign. 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.

So I was right, silence did not imply consent on that point.

Since I'm already posting anyway, I'll respond briefly to your other statement as well:

Stress just adds scalar-like. Arbitrarily tiny radial motions cannot on any reasonable measure make stress disappear as source.

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.

 

[Q-reeus]

Just a brief comment: we are using non-rigorous arguments to counter similar non-rigorous arguments from you by casting a reasonable doubt on your premises. We are not using non-rigorous arguments as a basis for claiming we have *proved* anything. We're not the ones making positive claims; you are.

(Strictly speaking, that's not quite true; we have made some positive claims, for example I made the positive claim that Birkhoff's Theorem rules out the possibility of monopole GWs. But that positive claim is based on a rigorously proved theorem.)

From #1: "My contention is that if normal stresses truly are a source for gravitating mass m, it implies the following:"
No subsequent claim by me of a rigorous proof of anything, anywhere. Always cast as 'if such and such is true, it implies such and such. And I have invited all the way along to be picked up on any specific point of error - note the word specific. 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. Just 'can't use it - live with it - just accept BT is true - end of story'. Not terribly satisfactory imo. Why is it so hard to put the finger on precisely where it fails? Does it fail gracefully and in a highly predictable and quantifiable manner, or just implodes at the slightest sign of time variation? No-one it seems can say for sure - it's undefined and that's that.

 

[Q-reeus]

Wald (1984) is referenced. I have seen statements like this elsewhere as well. Given the definition of ADM mass vs. Bondi mass, this makes sense: ADM mass involves picking a spatial 3-surface out of the spacetime, doing an integral over a 2-sphere in that 3-surface, and taking the limit as the 2-sphere goes to spatial infinity (or, equivalently, as the radius of the 2-sphere goes to infinity). That means that, even if a system is emitting gravitational waves, those waves are still somewhere on any given 3-surface, so they will eventually be contained within the 2-sphere of integration as the radius of the 2-sphere goes to infinity, and hence the energy carried by the waves will be "counted" in the ADM mass. (Since the ADM mass integrand involves the metric coefficients, not the stress-energy tensor components, the wave energy is unproblematically accounted for even though the waves are in vacuum, i.e., zero SET.)

An interesting passage indeed Peter - 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." Too bad I couldn't quote it back then, not that appeals to authority are worth much anyway.

This also helps clarify what Birkhoff's Theorem is saying: for Schwarzschild spacetime, the ADM mass and Bondi mass are equal, so any spacetime that is isometric to Schwarzschild spacetime outside some finite radius r (which applies to any spherically symmetric spacetime with an exterior vacuum region, by BT) will also have both masses equal, and therefore can't contain any GWs.

Right but that seems to be top down definitions to me. 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.

 

[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 definition 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".