Is stress a source of gravity?

[Dale]

A chain is as strong as it's weakest link

An invalid equation is an extremely weak link. Btw, in a non-stationary spacetime \xi^a doesn't even exist, so \sqrt{\xi^a \xi_a}\ne\sqrt{g_{tt}}. You can prove anything from a false premise.

 

[PAllen]

To even begin to make an argument here, you need to specify a stress energy tensor satisfying physical requirements (e.g. an energy condition) for the system under consideration. Then, if it is not stationary, and you want a conserved mass under asymptotic flatness (not true of our universe as a whole, but I believe adequate for a large empty region around some mass for a cosmologically short time), you should use ADM mass. I found the following for a simplified way to calculate it:

http://arxiv.org/abs/gr-qc/0609079

This is the only way to claim non-conservation of energy, because Komar mass is not a conserved quantity, while ADM mass is strictly conserved in asymptotically flat spacetime.

As for GW, the only way to claim this, is to show that the metric satisfying G = 8π T has periodic terms in the vacuum region (where T=0). I know you claim you can't solve this and should be 'excused' for this, but the fact is, neither can we. I did a fair amount of searching and I find not only no known exact solution but not even a high precision approximation that is known for pulsating spherical shell.

Instead of this, you insist someone should respond you your arguments in #1 or #69. I don't know about others, but I find these arguments simply incoherent. I don't find systematic reasoning at all, so I have nothing to respond to.

 

[Q-reeus]

What issues do you consider already supposedly settled in this thread that I am recycling? As far as I can see the only settled issues are that we both agree that a spacetime with GW's is not stationary and I have dropped the claim that the magnitude of the error is equal to the magnitude of the purported GWs, and those haven't been recycled since they were settled. None of the other issues have been settled.

Wow, you are really bent out of shape about this. I haven't made any claims whatsoever about your scaling argument, so I don't even know what I am supposed to "put up or shut up" about. You are the one with unsubstantiated claims that need to be backed up with some justification.

Here you are claiming that Birchoff's theorem is wrong without even looking at or referencing Birchoff's math to show where he made his error. Instead your "proof" that Birchoff's theorem is wrong is a rough calculation based exclusively on a quantity that is not even defined in the domain of the calculation. When called out on that you not only cannot defend your calculation rigorously you get offended that anyone would even expect you to be able to do so.

You simply cannot make major theoretical advances in this slipshod manner. You are complaining that I am not making detailed rebuttals to your minor details while you still have not justified your overall approach. I understand your frustration, but you are the one claiming the major breakthrough so the burden of proof is on your shoulders.

If you have enough math to actually find an error in Birchoff's theorem then you have enough math to prove it rigorously. If you do not have enough math to prove it righorously then you do not have enough enough math to actually find an error in Brichoff's theorem.

It seems you didn't get the message from my comments in #71. I deny not only the objective validity of every point(scoring) you make above, I despise the attitude behind them. You make it a habit not just in this thread but on numbers of others of continually raising false representations of what I both have said and mean - over and over in a deliberate campaign of psychological warfare by attrition. I'm thoroughly sick of having to trawl back through previous entries, just in order to show this or that statement of yours is bogus. And the longer the thread becomes, the more emotionally and physically enervating that becomes. Which I believe is your deliberate intent - get me to give up out of sheer exasperation. And that approach has at times been successful - I walked away from at least two previous threads for that reason. Not here. Unless you arrange for my permanent ban here at PF - and I wouldn't put that past you. So here's my message to you DaleSpam: Draw up a list of persons you vow never to respond to - and make sure my name is at the top of the list. OK! (and I won't, out of reciprocity on that arrangement, bother to answer your #101).

 

[Q-reeus]

Looking back over some recent and not so recent entries here, a pattern emerges. Beginning with myself then Jonathan Scott (and TrickyDicky with acute observations), Specific arguments of principle are raised via some simple gedanken experiments. The response in general (not by all) is to refer to the inapplicability of say Komar expression, without offering a viable alternative expression that is applicable. That or just saying that mass or energy or whatever is ill defined in GR in general - meaning the specific points raised are unresolvable in principle. An amazing stance from my pov. If it's the case that mass/energy-momentum etc is so ill-defined, then pray tell how is it that Birkhoff's theorem is not by that outlook also subject to uncertainty?. Strikes me as faintly rediculous to argue that Komar falls over because some ultra-tiny perturbation of spacetime is present somewhere. No qualitative or quantitative justification for showing any such tiny pertubations should be treated no differently than in other disciplines (EM, mechanics), as something rightly ignored in context. Not adding up imo. And of course I've had it continually thrown in my face that it's up to me to provide a rigorously mathematical proof.

Again I will say that is wholly unreasonable in the circumstances. What is wrong with me as laymen to offer two well enough reasoned specific scenarios that strongly suggest a problem for the standard position in GR? All that is being asked here is to tackle the specific claims of what's been presented in #1, #69 - show the *internal inconsistencies* of those symmetry and scaling arguments. Within the context of what GR claims - SET is wholly and solely the source for gravitating mass. I've yet to see it done, after more than 100 entries And why are matters specifically addressed back in #1, such as that pressure as contributor to the T₀₀ rest-energy term is completely distinct from it's purported action as source all by itself, continuing to be discussed so far down the line? Endless recycling of points and issues is called going around in circles, folks. Not productive.

Will someone do what I asked back in #1 - point to which SET terms, for either example [1] or [2], can be shown to offer parameter independent cancellation of pressure, so as to specifically justify Birkhoff's theorem? And please take note of a point raised time and again - I for one do not accept as valid overthrowing a counterexample by the very theorem that counterexample is calling into question. Please, someone out there in PF land - deal with the specifics of the two examples given back in #1 + #69. Show just specifically how it all comes out right for GR - or not! I don't enjoy continually repeating on this - tackle the specifics. And if it's felt that ADM or whatever is a better model to use, go ahead and work from that - justifying it's use. Failure to show how any other terms can reasonably act to cancel pressure should be a sign something is wrong, not with my offering, but GR.

 

[Q-reeus]

To even begin to make an argument here, you need to specify a stress energy tensor satisfying physical requirements (e.g. an energy condition) for the system under consideration. Then, if it is not stationary, and you want a conserved mass under asymptotic flatness (not true of our universe as a whole, but I believe adequate for a large empty region around some mass for a cosmologically short time), you should use ADM mass. I found the following for a simplified way to calculate it:

http://arxiv.org/abs/gr-qc/0609079

This is the only way to claim non-conservation of energy, because Komar mass is not a conserved quantity, while ADM mass is strictly conserved in asymptotically flat spacetime.

As for GW, the only way to claim this, is to show that the metric satisfying G = 8π T has periodic terms in the vacuum region (where T=0). I know you claim you can't solve this and should be 'excused' for this, but the fact is, neither can we. I did a fair amount of searching and I find not only no known exact solution but not even a high precision approximation that is known for pulsating spherical shell.

Instead of this, you insist someone should respond you your arguments in #1 or #69. I don't know about others, but I find these arguments simply incoherent. I don't find systematic reasoning at all, so I have nothing to respond to.

PAllen - wrote my piece in #104 before noticing your #102. OK so at least you are giving reasons in a general way for why my request is beyond resolution. I still make the point - the particular spherical geometry in example [1] in #1 was chosen for a number of reasons. One important reason being it implies complete cancellation of certain SET contributions - The T_i0 & T_0i energy-momentum flow density terms in particular, that in other situations makes argumentation messy. I think it not unreasonable that claims along those symmetry cancellation, and parameter scaling, lines should not be easily adressed in an in-principle manner by experts like yourself. Either those claims are valid or not in basic principle. To much to expect?!
If it cannot be shown there are any other, non-self-cancelling terms in principle capable of completely cancelling pressure, while still holding to Birkhoff's theorem as striclty correct, my conclusion can only be new, additional contributions to the SET are being snuck in under the door. That itself would be real news imo.

(Looked at Wiki on ADM : http://en.wikipedia.org/wiki/ADM_mass#ADM_Energy, but too obscure mathematically for me to make sense of the reasoning behind it.)
[as for your point about specifying an energy condition - why is my stipulation that total energy, as per integration over T₀₀ term, is constant, not an energy condition? If it fails in Komar/ADM, how significant is that failure in the limit of a small shell? Even roughly. Further on your last comments about incoherency, what specifically in #1? That we have a perfectly elastic spherical shell (later in #69 and before specified as gravitationally small). That it is set vibrating in fundamental breathing mode? That owing to spherical symmetry the T_i0 & T_0i energy-momentum flow density terms self-cancel? That SET contributions scale wrt parameters as per #1 and #69? Any of that particularly incoherent or difficult to grasp, really? Maybe someone finds those and other points made in #1 and #69 actually quite coherent - if not mathematically dense enough to impress. Only hope this doesn't get it all going around in circles again]

 

[Jonathan Scott]

It's true that *pressure* is not flowing from one place to another in your examples; but *stress-energy* is. The fact that the stress-energy changes form, so to speak, from pressure to something else and then back to pressure again, does not invalidate the applicable conservation laws.

As far as "source" goes, with respect to the Komar mass integral, once again, since the spacetime is not stationary, we can't expect that integral to be conserved. However, I think there's a fairly simple approximate picture of "where the source goes" in your scenario. I'll use the example with the two poles, and describe the key steps in the process:
...

Sorry, but you've completely missed some of the points in my earlier posts. The energy stored in the poles due to elasticity is not the same as the Komar "stress-energy", which is nominally equal to the potential energy. If it were, the pole would have been compressed to being of no thickness at all.

This "Komar stress-energy" is definitely NOT conserved. Momentum is conserved during the changes, but the integral of the stress-energy over a pole goes from the potential energy to zero when it is moved out of the way.

 

[Jonathan Scott]

Yes, it does. It's in T_00, the time-time component of the SET. If the material is compressed, its density increases; that is reflected as an increase in T_00. If other (non-gravitational) field energies also increase, those increases will also show up as an increase in T_00.

I meant that any additional energy due to compression does not appear in the Komar stress-energy term. It does of course appear within the ordinary energy.

 

[Dale]

It seems you didn't get the message from my comments in #71. I deny not only the objective validity of every point(scoring) you make above, I despise the attitude behind them.

That much is certainly clear.

You make it a habit not just in this thread but on numbers of others of continually raising false representations of what I both have said and mean

Whenever I have actually done that it has only been because you fail to present what you mean in a clear and unambiguous manner. This is also a common impediment to Peter Donis' efforts to communicate with you. When we try to make things clear and unambiguous by bringing in math, you reject all such attempts in preference for vague statements in English that inevitably leads to misunderstandings. One of the reasons for the math you avoid is precisely to eliminate this issue that you are complaining of here. I am willing to fix it, are you?

- over and over in a deliberate campaign of psychological warfare by attrition. I'm thoroughly sick of having to trawl back through previous entries, just in order to show this or that statement of yours is bogus. And the longer the thread becomes, the more emotionally and physically enervating that becomes. Which I believe is your deliberate intent - get me to give up out of sheer exasperation.

My intent is actually to get you to stop trying to dodge the issue at hand. I do, in fact, hope that you find it psychologically uncomfortable, not to motivate you to leave, but to motivate you to actually confront the problem in your logic.

Unless you arrange for my permanent ban here at PF - and I wouldn't put that past you. So here's my message to you DaleSpam: Draw up a list of persons you vow never to respond to - and make sure my name is at the top of the list. OK! (and I won't, out of reciprocity on that arrangement, bother to answer your #101).

I am not attempting to ban you, and have never done so. However, if you continue to post unsubstantiated nonsense claiming to debunk GR then I will continue to respond. If you continue to duck the issue then I will continue to point out that you are doing so.

So, are you either ready to post a proof justifying the approximation of using the Komar mass in a non-stationary spacetime, or do you conceed that the Komar mass is indeed undefined in a non-stationary spacetime? (of course, there is always the third option: to dodge the question and get angry at me personally).

 

[TrickyDicky]

Looking back over some recent and not so recent entries here, a pattern emerges. Beginning with myself then Jonathan Scott (and TrickyDicky with acute observations), Specific arguments of principle are raised via some simple gedanken experiments. The response in general (not by all) is to refer to the inapplicability of say Komar expression, without offering a viable alternative expression that is applicable. That or just saying that mass or energy or whatever is ill defined in GR in general - meaning the specific points raised are unresolvable in principle. An amazing stance from my pov. If it's the case that mass/energy-momentum etc is so ill-defined, then pray tell how is it that Birkhoff's theorem is not by that outlook also subject to uncertainty?. Strikes me as faintly rediculous to argue that Komar falls over because some ultra-tiny perturbation of spacetime is present somewhere. No qualitative or quantitative justification for showing any such tiny pertubations should be treated no differently than in other disciplines (EM, mechanics), as something rightly ignored in context. Not adding up imo. And of course I've had it continually thrown in my face that it's up to me to provide a rigorously mathematical proof.Will someone do what I asked back in #1 - point to which SET terms, for either example [1] or [2], can be shown to offer parameter independent cancellation of pressure, so as to specifically justify Birkhoff's theorem? And please take note of a point raised time and again - I for one do not accept as valid overthrowing a counterexample by the very theorem that counterexample is calling into question. Please, someone out there in PF land - deal with the specifics of the two examples given back in #1 + #69. Show just specifically how it all comes out right for GR - or not! I don't enjoy continually repeating on this - tackle the specifics. And if it's felt that ADM or whatever is a better model to use, go ahead and work from that - justifying it's use. Failure to show how any other terms can reasonably act to cancel pressure should be a sign something is wrong, not with my offering, but GR.

Let's recall exactly what Birkhoff's theorem says wrt what we are discussing here. The theorem which has been proved in many different ways, says in lay terms that a spherically symmetric vibrating shell (monopole radial pulsations which would be the only possible ones) in vacuum cannot propagate any disturbance into the surrounding space. This amounts to saying that the very spherical symmetry of the system cancels any spherically symmetric disturbance, so there is no such thing as a monopole GW if we want to keep the system spherically symmetric, this also guarantees any exterior metric to the shell must be static.
This is usually understood in the sense that in the exterior of a spherically symmetric shell there is no notion of the interior radial magnitude of the shell and therefore there's no way for the metric to propagate a perturbation of it.
Note that even in not-vacuum solutions of the EFE with spherical symmetry like FRW metric there is no propagation of GWs (only for perturbed forms there are).
All these are purely geometric results independent of GR as a physical theory.

 

[Dale]

Jonathan Scott, sorry, I was going to respond to your post of long ago to me, but I got wrapped up in the struggle to get Q-reeus' to actually confront the issue of the validity of the Komar mass.

I have found that the best way to think of the divergence of the stress energy tensor is to think of a 4D box around a region of spacetime (not necessarily small). The 4 divergence being 0 says that any energy or momentum which enters one side of the box will leave another side of the box. It is important to recognize that a stress is the same as a momentum flux.

So, suppose that you have a stress which suddenly increases. Thinking in 4D, that means that one side of our box has two regions, the region of low momentum flux and the region of high flux. By the 4-divergence, this increased momentum flux in the side of the box must correspond to increased flux out of some other side of the box. There are two possibilities, either it can go out one of the other spatial sides of the box, i.e. a corresponding stress change on that side, or it can go out the time side of the box, i.e. a change in the momentum of the material leaving the box.

You can make that box as small or as large as you like, and the principle will hold. Any change on one side must be balanced by a corresponding change on another side.

 

[Jonathan Scott]

Jonathan Scott, sorry, I was going to respond to your post of long ago to me, but I got wrapped up in the struggle to get Q-reeus' to actually confront the issue of the validity of the Komar mass.

I have found that the best way to think of the divergence of the stress energy tensor is to think of a 4D box around a region of spacetime (not necessarily small). The 4 divergence being 0 says that any energy or momentum which enters one side of the box will leave another side of the box. It is important to recognize that a stress is the same as a momentum flux.

So, suppose that you have a stress which suddenly increases. Thinking in 4D, that means that one side of our box has two regions, the region of low momentum flux and the region of high flux. By the 4-divergence, this increased momentum flux in the side of the box must correspond to increased flux out of some other side of the box. There are two possibilities, either it can go out one of the other spatial sides of the box, i.e. a corresponding stress change on that side, or it can go out the time side of the box, i.e. a change in the momentum of the material leaving the box.

You can make that box as small or as large as you like, and the principle will hold. Any change on one side must be balanced by a corresponding change on another side.

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.

 

[PAllen]

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.

And? It is well known that Komar mass is not conserved in non-stationary spacetime. That's why it shouldn't be used for such a scenario.

 

[Q-reeus]

Not giving up on me it seems. OK DaleSpam, I'm touched enough to break my own vow and give this another shot. Doubtless will regret it. Just don't expect me to go trawling like I have - it's just not worth it personally. Some comments on your #108:

Whenever I have actually done that it has only been because you fail to present what you mean in a clear and unambiguous manner. This is also a common impediment to Peter Donis' efforts to communicate with you. When we try to make things clear and unambiguous by bringing in math, you reject all such attempts in preference for vague statements in English that inevitably leads to misunderstandings. One of the reasons for the math you avoid is precisely to eliminate this issue that you are complaining of here. I am willing to fix it, are you?

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. Hence the demand for a rigorous high level maths proof from me, knowing that will not be forthcoming. You and Peter and others here at times freely use simple non-rigorous arguments where it suits, so yes I'm more than annoyed when there is carte blanche refusal to meet me at that level. Emotive words, combined with obstinate rejection of a straight forward request - 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. And Peter is well aware of the trouble I had just trying to get acceptance of the correct basic stress distribution in a shell. It was painfully circuitous and I have no patience left for the 'yes-it-is, no-it-isn't', 'yes-you-did', no-I-didn't' situations that developed there.

My intent is actually to get you to stop trying to dodge the issue at hand. I do, in fact, hope that you find it psychologically uncomfortable, not to motivate you to leave, but to motivate you to actually confront the problem in your logic.

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. What I have done is set out simple but I maintain logically rigorous counterexamples that try and break out of that circular bind. And no-one it seems, certainly not yourself, is prepared to specifically point to any failing in that logic. It is you and others that pointedly refuse to address the logic there. And how many times have I appealed by now? Don't ask. The 'logical' reaction that comes back to me is assertions that I'm stupid or bad or unreasonable or terribly incoherent, so why would anyone bother to deal straight with the specifics I present. Because that would be sensible and fair, that's why.

I am not attempting to ban you, and have never done so.

Quite a relief.

However, if you continue to post unsubstantiated nonsense claiming to debunk GR then I will continue to respond. If you continue to duck the issue then I will continue to point out that you are doing so.

There you go again - pejorative words gauranteeing an angry reaction. When will you learn to be more circumspect? You know I'll just say you are ducking my challenge. And you know there is nothing I am ducking, no matter how many times you say otherwise.

So, are you either ready to post a proof justifying the approximation of using the Komar mass in a non-stationary spacetime, or do you conceed that the Komar mass is indeed undefined in a non-stationary spacetime? (of course, there is always the third option: to dodge the question and get angry at me personally).

Just read my above comments. And when you personally address the specifics raised in #1,69, as I asked in #71, there can be a proper basis for further discussion. If you are not prepared to do so, I'm entitled to conclude you cannot find a flaw, and certain conclusions follow. There is no doubt in my mind there has been a flurry of PM correspondence on that involving yourself. Yet no one steps up to address what is a simple argument. Be the hero and break this hoodoo DaleSpam, even if you fall bravely. Be the hero.
[Late edit: I see you're in good sniping form with your #110. Sigh.]

 

[Jonathan Scott]

And? It is well known that Komar mass is not conserved in non-stationary spacetime. That's why it shouldn't be used for such a scenario.

Exactly. Q-reeus was clearly hoping it was at least "approximately" conserved, which is not the case. As I've previously mentioned in this thread, using the "poles" illustration, it isn't conserved when any motion or even acceleration is involved, even when the acceleration hasn't yet got anywhere.

There is a related puzzle that in the original stress-energy tensor this term appears to be part of the gravitational source term on the RHS of the Einstein equations, and I for one find it difficult to understand how something apparently non-conserved can be involved there. However, I know that's a very tricky area to understand, so I'm not expecting it to be solved in a PF thread.

 

[Q-reeus]

Let's recall exactly what Birkhoff's theorem says wrt what we are discussing here. The theorem which has been proved in many different ways, says in lay terms that a spherically symmetric vibrating shell (monopole radial pulsations which would be the only possible ones) in vacuum cannot propagate any disturbance into the surrounding space. This amounts to saying that the very spherical symmetry of the system cancels any spherically symmetric disturbance, so there is no such thing as a monopole GW if we want to keep the system spherically symmetric, this also guarantees any exterior metric to the shell must be static.
This is usually understood in the sense that in the exterior of a spherically symmetric shell there is no notion of the interior radial magnitude of the shell and therefore there's no way for the metric to propagate a perturbation of it.
Note that even in not-vacuum solutions of the EFE with spherical symmetry like FRW metric there is no propagation of GWs (only for perturbed forms there are).
All these are purely geometric results independent of GR as a physical theory.

Yes I understand that IF there is zero fluctuation in gravitating mass m going on, everything you say makes perfect sense and I would never have used the oscillating shell model in #1. But the whole point of using it is as a nice test bed to check on BT via a consistent application of how SET terms are supposed to contribute there. As for the proofs of BT, there would need to be an explanation of just how that SET balancing act is incorporated for me to consider taking it as gospel. As I have said earlier in #76 and #105, if one can't find any reasonable way to balance SET terms yet maintain Birkhoff's theorem holds rigorously, it logically implies new, de facto SET terms are being invoked.

 

[PAllen]

Yes I understand that IF there is zero fluctuation in gravitating mass m going on,

Birkhoff's theorem does not assume this. It proves that the assumptions of spherical symmetry forces this to be true. You are interchanging conclusion with assumption.

I think the core of your error is reasoning from false premises: x appears in stress energy tensor, therefore contributes directly to gravitating mass; Komar mass formula at least approximately describes mass for non-stationary situations. These are both simply false, while BT is a rigorous mathematical theorem. Also a pure math theorem is that ADM mass is conserved in asymptotically flat spacetime. Therefore, if you used ADM mass, you would find you whole argument about varying gravitational mass collapses. The ADM theorems help explain why BT works.

 

[Dale]

I thought I already explained that earlier, with a similar description,

You may have, I was too focused on the other discussion.

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.

I cannot speak for PeterDonis. It sounds like you and I agree then, that the stress energy tensor is conserved. Specifically, it sounds like we agree that in the case of an instantaneous change in pressure the zero divergence of the stress-energy tensor still holds at each event without any sort of delay. Is that a correct representation of your opinion?

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 and would go further. The Komar mass is only defined in a static spacetime, so not only is it not conserved in other spacetimes it doesn't even exist in them.

 

[Jonathan Scott]

I cannot speak for PeterDonis. It sounds like you and I agree then, that the stress energy tensor is conserved. Specifically, it sounds like we agree that in the case of an instantaneous change in pressure the zero divergence of the stress-energy tensor still holds at each event without any sort of delay. Is that a correct representation of your opinion?

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.

 

[Q-reeus]

Q-reeus: "Yes I understand that IF there is zero fluctuation in gravitating mass m going on,"

Birkhoff's theorem does not assume this. It proves that the assumptions of spherical symmetry forces this to be true. You are interchanging conclusion with assumption.
I think the core of your error is reasoning from false premises: x appears in stress energy tensor, therefore contributes directly to gravitating mass; Komar mass formula at least approximately describes mass for non-stationary situations. These are both simply false, while BT is a rigorous mathematical theorem. Also a pure math theorem is that ADM mass is conserved in asymptotically flat spacetime. Therefore, if you used ADM mass, you would find you whole argument about varying gravitational mass collapses. The ADM theorems help explain why BT works.

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?

 

[Jonathan Scott]

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?

The Komar mass expression is mathematically equal to the conventionally expected value for the effective total mass-energy of a system, equal to the sum of the local mass-energy for each component minus the potential energy which would need to be extracted to form the system from components initially at infinity. This does not mean it is a true description of the arrangement of mass-energy within the system.

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.

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. In GR, this effect cancels even more powerfully than in Newtonian theory, as the field due to a particular component particle effectively points to its anticipated position at the current time taking into account both velocity and acceleration, so the field is effectively that of a consistent "snapshot" of the whole sphere at a particular time, rather than seeing near and distant motions being out of phase.

 

[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 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".