Is stress a source of gravity?

[TrickyDicky]

According to wikipedia "mass in GR" it is simply impossible to define mass(energy) in GR in the general case, precisely because the gravitational energy not being a source issue, so I guess that even though it seems common sense to consider pressure by itself a source of gravity, there is no rigorous way to show it in GR unless we use some simplifying assumption like no time dependency or asymptotic flatness that are not found in reality.

 

[Mentz114]

Another must-read from Mitra -

"Does Pressure Increase or Decrease Active Gravitational Mass Density?", arXiv:gr-qc/0607087v4 27 Oct 2006

 

[TrickyDicky]

Another must-read from Mitra -

"Does Pressure Increase or Decrease Active Gravitational Mass Density?", arXiv:gr-qc/0607087v4 27 Oct 2006

Again, he seems to be talking about the static case only.

 

[pervect]

My take on the issue is this:

It's already known that one can't find a general expression for "mass" or a "source term" that is a tensor quantity

So, in general, I think it's hopeless to look for a truly general simple, scalar "source term". It just doesn't exist - at least not as a tensor.

I think one will also find that most discusssions of mass involve studying the metric near infinity - very few can be reduced to an actual integral involving components of the stress-energy tensor.

 

[PAllen]

This bit is imo a conflation of pressure as contributor to stress/strain energy, and that due to pressure all by itself. Read my comments in #1 on that.

I am thinking purely physically. Imagine a shell with pressurized gas inside. Increase pressure of gas. Gravitational mass increases. How one factors this into increase of mass due to internal energy versus 'pressure itself' I don't care. But physically, other things being equal, increasing pressure must increase gravitational mass. [edit: in such a scenario, to increase pressure you would normally have to add energy. Is the mass increase due to increased energy or increased pressure? It all depends on how you add things up. Mass+KE or mass plus pressure term should work in some form. Mass + KE + pressure term probably double counts and is not right. Mentz's reference seems to amount to support this intuition].

 

[Jonathan Scott]

I am thinking purely physically. Imagine a shell with pressurized gas inside. Increase pressure of gas. Gravitational mass increases. How one factors this into increase of mass due to internal energy versus 'pressure itself' I don't care. But physically, other things being equal, increasing pressure must increase gravitational mass.

For a gas, the potential energy of the pressure is effectively in the kinetic energy of the molecules, so that extra energy must increase the mass trivially. In this case, the stored energy is like the energy in a spring, and in the ideal case the total energy stored is a half of the pressure times the volume (in a similar way to the energy in a compressed spring being a half of the final force times the distance compressed). You can also similarly store energy by squeezing an elastic material, and again it will be physically present in the compressed material.

The sort of pressure in the "Komar mass" case is very different. In this case the energy equivalent is calculated by integrating the pressure in each plane through an object, which then gives the total force through that plane, which in the static case must exactly balance the gravitational force perpendicular to that plane, and when those elements are integrated over the direction perpendicular to the plane to complete the volume, the result simply multiplies the force by the distance between the sources, giving the potential energy. This value is determined entirely by the gravitational potential of the configuration and is completely unrelated to the type of material, including its elasticity and density. There could be some energy due to compression in the material itself, for example in the form of increased electric fields within squeezed materials, but this does not get included in the Komar mass expression. If the object is sufficiently rigid and light, there could be a negligible amount of energy actually stored in it.

 

[PeterDonis]

as my examples with poles illustrate, it is difficult to see how this "something" could flow from one place to another continuously.

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:

(1) Initial state: Two masses at rest, held apart by pole #1. Pole #2, slightly shorter than #1, sitting beside pole #1. "Source" is rest mass of two masses, plus rest mass of two poles (these stay the same throughout), plus pressure in pole #1, plus stored energy in pole #1 due to compression (because compression makes the pole's energy density, SET component T_00, slightly larger on average than it would be if the pole were unstressed). Entire "source" is also multiplied by the average "redshift factor" across the system (more precisely, the "redshift factor" is inside the integrand). This can also be thought of as adding a "gravitational binding energy" term (which will be negative since the "redshift factor" is less than 1), but that assumes that the "binding energy" can somehow be separated out, when it really can't; it's really a multiplier.

(2) Pole #1 removed (slid to the side to allow the masses to fall towards pole #2). "Source" is all rest masses, plus stored energy (from increased density) and pressure in pole #1 is gradually being "exchanged" for kinetic energy of pole #1 as it expands (however, this part will "drop out", see next item), and for kinetic energy of two masses as they fall (this is the key part that stays). Average "redshift factor" will get slightly smaller as the masses fall.

(3) Pole #1 completely expanded, zero stress. Masses just about to hit pole #2 (we assume things are set up so they work out this way, to keep it simple). "Source" is all rest masses, plus kinetic energy of two falling masses. Average "redshift factor" continues to get slightly smaller as the masses slow down and come to rest after they hit pole #2 (see next item).

(4) Pole #2 compressed, masses again at rest. "Source" now is all rest masses, plus stored energy and pressure in pole #2. Also, "source" is now multiplied by a somewhat smaller "redshift factor" than it was in (1) above, since the system is now more compact.

So the overall "conversion" of "source" (to the degree that the Komar mass is approximately conserved in this scenario) is from pressure (and stored energy due to density increase) to KE and back to pressure (and stored energy) again, plus the correction for the change in "redshift factor".

 

[PeterDonis]

pressure in pole #1 is gradually being "exchanged" for ... kinetic energy of two masses as they fall (this is the key part that stays).

I should add that "exchange" is not really the right word here, since we can increase the KE of the two masses when they hit pole #2 by making pole #2 shorter, regardless of the initial pressure and density increase in pole #1. But that is accounted for by the change in "redshift factor", which will be larger if we make pole #2 shorter.

 

[PeterDonis]

There could be some energy due to compression in the material itself, for example in the form of increased electric fields within squeezed materials, but this does not get included in the Komar mass expression.

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.

 

[Dale]

Your tactic of continually recycling accusations already supposedly settled is one reason I have little respect for anything much you say.

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.

now that you have unrespectingly broken my request in #69, answer my scaling argument given there. And I mean something that makes sense. Yes, that's right genius - your turn to put up or shut up.

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.

 

[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.