Is stress a source of gravity?

[Q-reeus]

Q-reeus: "stability of a thin shell under static internal gas pressure"
Which is completely irrelevant to the scenario we are discussing; at least I thought it was.

Not really, but more below.

...Your scenario stipulates that the shell is "self-supporting"; that means there must be vacuum inside and outside the shell. (You also state, later on, that the radial pressure on the inner surface of the shell is zero; that will be true only if there is vacuum inside the shell.)

While I had not explicitly stated in #1 a fully evacuated environment, it was implied. So in essence, yes to the above. And further, surface radial pressure is zero at all instants at both inner and outer surfaces, more or less by definition of the model used.

A shell with internal gas pressure is not "self-supporting"

It can be. A balloon isn't, but a glass light bulb continues to be so (in that case negative relative internal-to-external pressure generally applies).

and both its static equilibrium configuration and the dynamics of its small oscillations about that equilibrium are different. So which case are we talking about?

Relevance of Ehlers model is this: replace static gas pressure with inertial forces of inward or outward radial acceleration. It represents a per unit area of shell radial acting 'pressure' of the same vectorial nature as gas pressure. One is a static thing, the other dynamic, but otherwise the same character. The balancing forces from elastic shell stress don't 'care' which it is. The Ehlers model shows tangent stresses do the balancing. It is impossible in that setting for radial elastic stresses to provide any balance. Spent several hours trawling for online material specifically stating the stress distributions for the breathing shell mode. Unfortunately the references were all oblique - overwhelmingly the focus is on mode patterns and frequencies. Hence the Elhers ref.

Now I probably got your jack up on this issue by using some emotive wording. Pardon please my personal failing that way - it's a habit hard to break. I want to keep this discussion, which imo is quite important, civil and pleasant as possible. So I will just venture a guess here (can't even say educated guess, as I have no professional qualifications or background of any kind) that the model you used was based on equilibrium conditions for a mathematically excised shell within a self-gravitating perfect fluid sphere. Then I can see how your findings would make good sense re force balances. Would that be about right?

 

[Q-reeus]

Then you have been misunderstanding my principal claim. From the begninning my principal claim is that your Komar-mass argument is invalid because any spacetime with GW's is non-stationary and the Komar mass is not defined on such spacetimes.

OK but this is a chicken-and-egg thing I have been trying to get across repeatedly. If as all you folks insist there will be no GW's for the shell scenario given, the Komar model should be valid to use! Can't have it both ways imo.

My argument is very general has nothing whatsoever to do with vibrating shells nor any of the other irrelevant details of the specific scenario.
Do you now agree with that or not?

Apart from my earlier comment, I have obviously to agree that a disturbance (GW's) means non-stationary spacetime. But that is not saying much apart from stating that motion = movement.

If you do not agree, then which part do you disagree with? Do you disagree that any spacetime with GW's is non-stationary,

How could I? As per above, by definition, GW's = non-stationary spacetime. Hardly the issue.

or do you disagree that the Komar mass is only defined on stationary spacetimes?

And this is where it get's to be slippery eel wrestling territory. Here's an excised piece from #9:

Honestly, there are truckloads of gedanken experiments accepted as valid that regularly fail to include every single possible factor and detail. How could Einstein get away with his use of trains and lights in SR setting when 'clearly' the masses involved are warping spacetime thus invalidating the flat spacetime postulated in SR. But of course we use reasonableness and accept such warping is of no real consequence.

Apply that to this case of a basketball sized shell vibrating in breathing mode. Just on the assumption pressure really does provide an uncompensated gravitating mass m_s as per #1, the fluctuation in m_s will be gravitationally minute. Any resulting monopole GW's so generated will be vastly smaller again in magnitude. Are we being remotely reasonable in saying such tiny perturbations (and again, recall they 'officially' can't exist anyhow) will seriously throw out the Komar expression. Yet once again, my appeal goes out to all, including you silent onlooker GR pros. Provide a sensible, qualitative and order of magnitude quantitative justification for the implied claim here that *any* non-stationary spacetime generated, no matter how exceedingly feeble, invalidates use of Komar expression.

If you do agree, then we can proceed to discuss details.

Up to you on that. I've said my piece above, for the umpteenth time really.

 

[PeterDonis]

No significant amount of energy or momentum (at least compared with the potential energy, which is what we are talking about) is required for example to knock out a pole sideways which has a clean frictionless surface at the ends.

Sorry, posting too early in the morning. I meant the part about "magically" replacing the pole with a slightly shorter one. The part about knocking the pole sideways is fine; in that case the pole will expand outward at both ends, with the "wave" of expansion propagating inward from both ends towards the center. As it does so, the internal stresses in the pole will gradually be relieved, starting at the ends and working in towards the center.

No, this is wrong by basic mechanics!

Yes, I realized that after I posted; that's why I deleted that part. You type fast.

 

[PeterDonis]

It can be. A balloon isn't, but a glass light bulb continues to be so (in that case negative relative internal-to-external pressure generally applies).

Relevance of Ehlers model is this: ...

Let's first get clear about just what scenario we are discussing. See below for the model I've been using; is that the scenario you want to discuss, or is it something else?

So I will just venture a guess here (can't even say educated guess, as I have no professional qualifications or background of any kind) that the model you used was based on equilibrium conditions for a mathematically excised shell within a self-gravitating perfect fluid sphere. Then I can see how your findings would make good sense re force balances. Would that be about right?

The model I have been basing my posts on is a thin spherical shell made of perfect fluid-type matter with vacuum inside and outside the shell. The only caveat to such a model is that a shell made of an actual perfect (or near-perfect) fluid, like air, would not be self-supporting; it would collapse under its own gravity. I'm assuming that a typical solid material (like metal or plastic or wood) which *can* support itself under its own gravity can still be described by a perfect fluid-style stress-energy tensor. That seems reasonable for the kind of scenario we're discussing.

 

[PeterDonis]

OK but this is a chicken-and-egg thing I have been trying to get across repeatedly. If as all you folks insist there will be no GW's for the shell scenario given, the Komar model should be valid to use! Can't have it both ways imo.

The Komar mass doesn't apply to your scenario, strictly speaking, because it's non-stationary; that's true whether or not GWs are emitted. What makes it non-stationary is the oscillation of the shell; the metric is time-dependent in the region in which the shell oscillates.

 

[Jonathan Scott]

I meant the part about "magically" replacing the pole with a slightly shorter one. The part about knocking the pole sideways is fine; in that case the pole will expand outward at both ends, with the "wave" of expansion propagating inward from both ends towards the center. As it does so, the internal stresses in the pole will gradually be relieved, starting at the ends and working in towards the center.

Shortly after that, one can push a replacement pole sideways into the gap (or even have it standing by the original so that when the original has been removed, the masses will fall together onto the shorter pole). I have specified the new pole as being slightly shorter to allow for the masses having fallen slightly closer together. The new pole will then take up the same stress at the original (apart from tiny corrections for being closer together), and will contain whatever "something" the original pole contained, as if it had transferred from the old pole by "magic". Certainly the Komar mass stress term would be approximately the same.

Is that clearer now?

 

[Dale]

How could I? As per above, by definition, GW's = non-stationary spacetime. Hardly the issue.

OK, so then the only possible point of disagreement is the issue of whether or not the Komar mass is defined in a non-stationary spacetime:

I can, of course, provide several references that state explicitly the the Komar mass is only defined on stationary spacetimes, including your OP. So I don't think that is actually the issue. I think you understand quite clearly that it is not defined in non-stationary spacetimes. From the above it seems that the issue is that you believe that, even though it is not defined, it is a good approximation:

And this is where it get's to be slippery eel wrestling territory. Here's an excised piece from #9:

Honestly, there are truckloads of gedanken experiments accepted as valid that regularly fail to include every single possible factor and detail. How could Einstein get away with his use of trains and lights in SR setting when 'clearly' the masses involved are warping spacetime thus invalidating the flat spacetime postulated in SR. But of course we use reasonableness and accept such warping is of no real consequence.

There are several ways that you can justify an approximation.

1) You can do a full non-approximated calculation of your quantity of interest and demonstrate that the approximated calculation is close.
2) You can expand the quantity of interest as an infinite series with terms of strictly decreasing magnitude and stop when the next term gets small enough.
3) You can expand the quantity of interest as the approximated plus some error term and determine some upper bound on the error term or expand the error term as in 2. (Btw, this approach is very common in the analysis of GWs, called linearized EFE or perturbative analysis. If you want to pursue your analysis this is the approach I would recommend.)
4) You can parameterize your degree of approximation and establish a maximum value for the parameter based on your measurement errors.

The gedanken experiments that I am aware of can be justified by one or more of those above methods. You have not justified your approximation in any of those ways nor provided any other justification besides your unsubstantiated assertion that it is small, such as:

Apply that to this case of a basketball sized shell vibrating in breathing mode. Just on the assumption pressure really does provide an uncompensated gravitating mass m_s as per #1, the fluctuation in m_s will be gravitationally minute. Any resulting monopole GW's so generated will be vastly smaller again in magnitude.

You certainly haven't demonstrated that. Just as you challenged my claim that the magnitude of the errors were equal to the magnitude of the GWs, so I challenge your claim that the magnitude of the errors are small. And just as I had to drop my claim since I wouldn't justify it, so I expect you to drop your claim if you won't justify it. A repeated assertion that they are small is not a justification.

 

[Q-reeus]

Let's first get clear about just what scenario we are discussing. See below for the model I've been using; is that the scenario you want to discuss, or is it something else?

The one presented in #1. It has the nice advantage of being physically realizable as is.

The model I have been basing my posts on is a thin spherical shell made of perfect fluid-type matter with vacuum inside and outside the shell. The only caveat to such a model is that a shell made of an actual perfect (or near-perfect) fluid, like air, would not be self-supporting; it would collapse under its own gravity.

Yes and that's a big caveat. In order to prevent collapse a fluid shell must be enclosed within some other supporting structure, itself solid. Hence the total system is now more complex and extended.

I'm assuming that a typical solid material (like metal or plastic or wood) which *can* support itself under its own gravity can still be described by a perfect fluid-style stress-energy tensor. That seems reasonable for the kind of scenario we're discussing.

No it can't for the reason above. Solids can bear static shear, uniaxial, and biaxial stresses impossible for a fluid. As the Ehlers bit demonstrates, tangent stresses supply all the support against any radial acting forces, whether static (gas, gravity) or inertial (vibratory motion). They have to, because radial surface forces are nonexistent. Any small interior elastic radial forces are providing internal balancing, they can do no more than that.

And simple approximations, like assuming uniform tangent stress for a thin shell in this setting are fine. We want the essentials - mechanical oscillation that entails sinusoidal exchange between motion/momentum and stresses/elastic energy. But all that was laid out in #1. Only thing missing there was a specific model setting stresses against motion. And imo entirely superfluous. We know the shell is a mechanical oscillator exhibiting the usual dynamics. Scaling behaviour for various parameters are what matters, and they are readily enough determinable from basic mechanics. I see no need to go beyond #1 re any further modelling. One unimportant error there was to miss a factor of 2 relating to biaxial stress contribution, but that's it.

 

[Q-reeus]

The Komar mass doesn't apply to your scenario, strictly speaking, because it's non-stationary; that's true whether or not GWs are emitted. What makes it non-stationary is the oscillation of the shell; the metric is time-dependent in the region in which the shell oscillates.

First time that bolded bit has been presented - previously I was being told it was the presence of ultra-feeble GW's. There was mention of shell oscillation invalidating, but no real explanation how. And nobody bothered to explain exactly what region(s) non-stationary spacetime referred to.
So what does this translate at exactly? Is it referring specifically to there being motion in the non-zero SET region (shell wall)? If so, in what way does this specifically impact on Komar expression? And how does it get around the constancy of system total energy? Just saying it invalidates is no real answer here.

 

[Q-reeus]

You certainly haven't demonstrated that. Just as you challenged my claim that the magnitude of the errors were equal to the magnitude of the GWs, so I challenge your claim that the magnitude of the errors are small. And just as I had to drop my claim since I wouldn't justify it, so I expect you to drop your claim if you won't justify it. A repeated assertion that they are small is not a justification.

I'm running late as usual, but will repeat in essence what I just wrote in #59 - some expert needs to set out just what is it that invalidates and exactly how and to what degree. when that is presented in comprehensible form, sensible positions can be taken. Till then, further disputing over higher moral ground on this is counterproductive. :zzz:

 

[Dale]

I'm running late as usual, but will repeat in essence what I just wrote in #59 - some expert needs to set out just what is it that invalidates and exactly how and to what degree.

The fact that it is not defined in a non-stationary spacetime invalidates it completely. Any assertion that the error is small needs to be justified mathematically. Frankly, I don't think that it is even possible to do since the error from the actual quantity and an undefined quantity is undefined.

 

[PeterDonis]

Shortly after that, one can push a replacement pole sideways into the gap (or even have it standing by the original so that when the original has been removed, the masses will fall together onto the shorter pole). I have specified the new pole as being slightly shorter to allow for the masses having fallen slightly closer together. The new pole will then take up the same stress at the original (apart from tiny corrections for being closer together), and will contain whatever "something" the original pole contained, as if it had transferred from the old pole by "magic". Certainly the Komar mass stress term would be approximately the same.

Is that clearer now?

Ah, ok. As the masses fall onto the new pole, the new pole will compress. As it compresses, pressure will build up inside it, in an inverse process to the one by which the old pole expanded as pressure was relieved. During this process, there will be some motion of the new pole (no net motion of its center of mass, but motion of parts of the pole, which counts as well).

 

[PeterDonis]

The one presented in #1. It has the nice advantage of being physically realizable as is.

Which is? Obviously if I could have told from #1 whether you intended the shell to have vacuum inside and outside, or something else, I wouldn't have had to ask about it.

No it can't for the reason above. Solids can bear static shear, uniaxial, and biaxial stresses impossible for a fluid. As the Ehlers bit demonstrates, tangent stresses supply all the support against any radial acting forces, whether static (gas, gravity) or inertial (vibratory motion). They have to, because radial surface forces are nonexistent. Any small interior elastic radial forces are providing internal balancing, they can do no more than that.

I agree a proper model for a solid should not assume that the pressure is isotropic; so perhaps a better term would be "quasi-perfect fluid", where the SET is diagonal (no shear stresses--if the system is spherically symmetric that is certainly going to need to be the case), but the radial pressure can be different than the tangential pressure.

And simple approximations, like assuming uniform tangent stress for a thin shell in this setting are fine.

Yes, I would agree with this approximation.

We want the essentials - mechanical oscillation that entails sinusoidal exchange between motion/momentum and stresses/elastic energy.

But again we come back to the question: what drives the oscillation? But I'll defer that until I get a definite answer to which scenario you want to discuss, as I asked above.

 

[PeterDonis]

First time that bolded bit has been presented

Actually, pervect mentioned it way back in post #18. He gave the definition of a stationary metric there, and said that oscillating shells do not meet that definition. Why they do not should be obvious from what he said, but just to make sure, I stated it explicitly in my post.

As far as treating the Komar mass integral as an approximation for a spacetime that is "almost stationary", I personally don't have any problem with that in principle (though some others may), but DaleSpam is right that approximations need to be justified. Since you are the one who is claiming that GR is wrong, as he said and as I have said before, it is up to you to justify whatever approximations you are making, and to justify the claim that the Komar mass should be conserved to whatever level of approximation you are using.

 

[pervect]

I haven't seen anything on using the Komar formalism with quasi-stationary spacetimes, unless you count the approach that derives the Bondi mass as a consequence of asymptotic time translations (which is in Wald).

If you can find some \xi^b such that \nabla_a \xi_b "small" in in the vacuum regions of the space-time, you should have an approximately conserved flux integral.

Something else that MIGHT work, is to say that if the actual pressure is negligibly different from the equilibrium pressure anywhere, the approach using Komar mass could give sensible approximate results. It's at least clear from your arguments (or a similar case, thinking about what happens if a shell under tension containing a very high pressure cracks and the contents explode) is that if the actual pressure is considerably different from the equilibrium pressure, the approach does NOT give sensible results. That doesn't really prove that it WILL give snesible results if the conditions are met, but it seems intuitively promising. Which may or may not mean it actually works.

 

[PeterDonis]

In looking around for references about gravitational waves and how they are generated, I came across this thread from PF from 2005:

https://www.physicsforums.com/showthread.php?t=60805

In it pervect gives the simple reason why monopole GWs are prohibited: Birkhoff's Theorem, which states (at least this is one way of stating it) that the metric in an exterior vacuum region of any spherically symmetric spacetime must be the Schwarzschild metric. See here:

http://en.wikipedia.org/wiki/Birkhoff's_theorem_(relativity )

Since the Schwarzschild metric contains no GWs, there can be no spherically symmetric (monopole) GWs. Since this hasn't been explicitly mentioned in this thread, I thought I'd mention it.

 

[Jonathan Scott]

Ah, ok. As the masses fall onto the new pole, the new pole will compress. As it compresses, pressure will build up inside it, in an inverse process to the one by which the old pole expanded as pressure was relieved. During this process, there will be some motion of the new pole (no net motion of its center of mass, but motion of parts of the pole, which counts as well).

True, but that's not the point of the example, and as I mentioned before the amount of internal potential energy is minimized for a sufficiently light and stiff pole and can certainly be much smaller than the gravitational potential energy of the system.

My point is to show that in a dynamic situation the stress part of the Komar mass is something which can vanish from one object and reappear later in another, so it isn't even approximately like a conserved quantity.

 

[TrickyDicky]

My point is to show that in a dynamic situation the stress part of the Komar mass is something which can vanish from one object and reappear later in another, so it isn't even approximately like a conserved quantity.

I think this is pretty evident (I don't know why peter donis keeps saying otherwise) and yes it is strange but it seems to be consistent with other well known and "weird" facts of GR.
We all agree (I think) that first of all the komar mass is not a valid concep in non-stationary situations and therefore we don't necessarily expect it to be conserved in those situations.
Second, we all agree (I think) that in GR what is conserved strictly is energy-momentum, not necessarily energy or mass by themselves (only in spacetimes with timelike killing vector it is energy strictly conserved).
So what you are describing about komar mass is what is expected according to what we agree about.
Maybe what is more difficult to explain is that in the setting you describe energy seems to be approximately conserved in the dynamical case unlike the komar mass.

 

[Q-reeus]

I haven't seen anything on using the Komar formalism with quasi-stationary spacetimes, unless you count the approach that derives the Bondi mass as a consequence of asymptotic time translations (which is in Wald).

If you can find some \xi^b such that \nabla_a \xi_b "small" in in the vacuum regions of the space-time, you should have an approximately conserved flux integral.

Something else that MIGHT work, is to say that if the actual pressure is negligibly different from the equilibrium pressure anywhere, the approach using Komar mass could give sensible approximate results. It's at least clear from your arguments (or a similar case, thinking about what happens if a shell under tension containing a very high pressure cracks and the contents explode) is that if the actual pressure is considerably different from the equilibrium pressure, the approach does NOT give sensible results. That doesn't really prove that it WILL give snesible results if the conditions are met, but it seems intuitively promising. Which may or may not mean it actually works.

Pervect, thanks for stepping back in with some interesting observations. Unfortunately there are sufficient caveats there to make it essentially impossible for me to absolutely defend using Komar expression (or presumably any similar one like ADM or Bondi). This leaves me in an invidious position. As a layman I am being required by some to take on the role of supreme GR expert in order to prove that Komar expression doesn't fail badly enough to throw out the basic argument of #1. Of course that I have long acknowledged I cannot do, yet not doing just that will gaurantee my 'failure' on this issue. I have special respect for how you handle matters in general. so I invite you please to consider the following scaling argument.

Take the case in #1 - and specifically we make it gravitationally small - basketball sized, and all in vacuo. As a typical mechanical oscillator, we know from basic mechanics it will have a natural frequency scaling as (E/ρ)^1/2, with those quantities defined in #1. Let's suppose at some specific value of E/ρ, whatever it is that puportedly ensures pressure is exactly canceled out as contribution to time varying gravitating mass m is actualy so. Now change just one parameter. Say E is made n times higher. Frequency of oscillation f rises by a factor n^1/2, and specifying amplitude of pressure oscillation is kept the same, radial displacement amplitude drops in the ratio 1/n. So radial velocity amplitude is a factor n^-1/2 smaller. If Komar redshift were somehow ever important as factor here, it has now been reduced owing to the reduced displacement amplitude (fluctuations in gravitational potential, dependent on radius R). Similarly for anything relating to velocity of motion - reduced as a factor. We notice that pressure is solely unaffected here. In the limit as E goes very high, every other physically reasonable contributor tends to zero. The graphs can all intersect at one point at most. If cancellation is a general principle, those graphs must match at all points, an obvious absurdity.

Get my point here? And the same kind of thing comes up if mass density ρ is altered. Or size scale (radius R and shell thickness δ grow/shrink in same proportion). It was all related in #1, but keeps getting buried under recycled issues here. Can there be any way around the above? I think not. Making things overly complex won't change the basic scaling arguments one iota imo. Now please no-one else jump in here first, I'm asking for a response from Pervect. Other subsequent responses are then welcome - in principle.

 

[Dale]

This leaves me in an invidious position. As a layman I am being required by some to take on the role of supreme GR expert in order to prove that Komar somehow doesn't fail badly enough to throw out the basic argument of #1.

You are putting yourself in the role of the supreme GR expert by your claims that the recognized experts are wrong. If you don't have the wherewithal to back up your claims that you are smarter than they are then don't make the claims. You cannot have it both ways. If you are expert enough to find these subtle flaws that have been overlooked for decades then you are expert enough to produce a valid exposition of the errors you have found.

Did you really expect to make a major theoretical breakthrough without doing some math?

 

[Q-reeus]

You are putting yourself in the role of the supreme GR expert by your claims that the recognized experts are wrong. If you don't have the wherewithal to back up your claims that you are smarter than they are then don't make the claims. You cannot have it both ways. If you are expert enough to find these subtle flaws that have been overlooked for decades then you are expert enough to produce a valid exposition of the errors you have found.

Did you really expect to make a major theoretical breakthrough without doing some math?

So you, DaleSpam, deliberately ignored my request to butt out till Pervect said his piece. Figures - true to your form. Your tactic of continually recycling accusations already supposedly settled is one reason I have little respect for anything much you say. I still occasionally fume over your bloody minded decision to shut me down here: https://www.physicsforums.com/showthread.php?t=498821 And despite qualifier in my last line in #1 there, your charge of 'perpetual motion machine' is interesting in light of the fact that elsewhere, including this very thread, you openly espouse that conservation of energy fails in GR. Hypocrisy - born of a fanatical ideological/religious devotion to Holy GR (bettet not forget to add 'imo'). And I could go on and on.

But since you consider yourself pretty savvy on this issue (otherwise how could you so persistently accuse me of getting it all wrong), and 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.

 

[PeterDonis]

True, but that's not the point of the example, and as I mentioned before the amount of internal potential energy is minimized for a sufficiently light and stiff pole and can certainly be much smaller than the gravitational potential energy of the system.

My point is to show that in a dynamic situation the stress part of the Komar mass is something which can vanish from one object and reappear later in another, so it isn't even approximately like a conserved quantity.

But nobody is saying that the stress part of the Komar mass is supposed to be conserved; only the total mass integral is (in cases where the spacetime is stationary, which in your examples it isn't anyway). Stress can be exchanged for other types of "energy" in dynamic situations. As I have shown, the conservation law covariant divergence of SET = 0 is *always* satisfied, even in non-stationary cases; that conservation law is the only one that always has to apply.

 

[PeterDonis]

I think this is pretty evident (I don't know why peter donis keeps saying otherwise)

I'm not saying otherwise; I've simply been pointing out explicitly that in each case where stress "vanishes", it doesn't do so "instantaneously"; it is gradually, continuously "exchanged" for some other piece of the SET in accordance with the local conservation law, covariant divergence of SET = 0.

 

[Jonathan Scott]

But nobody is saying that the stress part of the Komar mass is supposed to be conserved; only the total mass integral is (in cases where the spacetime is stationary, which in your examples it isn't anyway). Stress can be exchanged for other types of "energy" in dynamic situations. As I have shown, the conservation law covariant divergence of SET = 0 is *always* satisfied, even in non-stationary cases; that conservation law is the only one that always has to apply.

Firstly, just to be clear: The normal stress terms represent the force per unit area perpendicular to the selected axis, which is equivalently the rate per area at which that component of momentum is flowing through that plane at that point. The conservation law says that if you consider a tiny cube of material and there is a gradient in this pressure between one side and another, then that will be matched by a rate of change of the relevant component of momentum density, so that overall that component of momentum is conserved. The divergence of each row of the tensor being zero expresses the conservation of energy and each of the three components of momentum.

There is no problem with a sudden change in the forces, for example if objects collide or break apart. Energy and momentum still flow continuously.

The Komar mass expression is based on internal stresses, which can appear or disappear almost instantly. The integral of this stress with respect to volume is not a particularly meaningful quantity except that in a static situation (with no acceleration) it happens to exactly match the gravitational potential energy. Note that if something causes the start of some acceleration, the Komar mass expression is already broken even before the acceleration has the chance to change the configuration measurably.

 

[Mentz114]

Q-reeus, can you write a Lagrangian for your scenario in terms of fields (scalar, vector or tensor) , so it's quadratic in the fields. You'll need a term for any waves in there. If so it's easy to calculate the EMT which would be a good place to start solving the EFE.

 

[Q-reeus]

Q-reeus, can you write a Lagrangian for your scenario in terms of fields (scalar, vector or tensor) , so it's quadratic in the fields. You'll need a term for any waves in there. If so it's easy to calculate the EMT which would be a good place to start solving the EFE.

Sorry but the answer is no. My only claim on all this is that via scaling arguments given in e.g. #1 and #69, there seems no way around pressure being in general an uncompensated gravitating mass term, given it's status as source in SET. Just a short while ago , a PM message suggested Birkhoff's theorem was solid on this. I had raised it myself earlier on, but only in the context of saying one should not use it unless it's conceptual basis did not entail a philosophical bind. In other words, defeating an argument by means of a theorem the argument is trying to show is suspect. What I'm getting at is, if cancellation is somehow there, we should be able to point to the terms in SET that physically do that. And I can;t see it, for the reasons given. So if what you are asking for is considered necessary here, someone with the math skills will have to do it. Again though, where is there an achiles heel in my scaling arguments?

I can think of just one conceivable factor. Assume pressure + time rate of change of radial momentum flow somehow exactly cancels. But I see the latter, if a legitimate source term at all, as self-cancelling owing to it's vector form in a spherical geometry. By contrast pressure terms as source of gravitating mass just add scalar-like in Komar expression. Additionally, shell geometry ensures tangent stresses will be highly 'levered' wrt radial momentum rate of change, as compared to a linear situation (say a bar in axial vibration mode). Not much else to add at this stage. [EDIT: this is really a moot argument, since afaik there is simply no room for time rate of change of momentum/energy flow density as part of the SET. So using it implies inventing a whole new SET, yes?]

 

[TrickyDicky]

So now not only gravitational energy is not a SET source, but we have (see JScott and Peterdonis discussion inspired by Q-reeus OP) pressure components that are explicitly in the SET of a stationary mass acting as if they weren't gravitational sources in a dynamical context.
Wish someone could clarify this a bit.

 

[PeterDonis]

Firstly, just to be clear: The normal stress terms represent the force per unit area perpendicular to the selected axis, which is equivalently the rate per area at which that component of momentum is flowing through that plane at that point.

...

The Komar mass expression is based on internal stresses, which can appear or disappear almost instantly.

The distinction you are making between "normal stress terms" and "internal stresses" is not correct, at least not when assigning physical meaning to the components of the stress-energy tensor. *All* stresses in the material are captured in the SET, regardless of whether you think of them as "internal stresses" in a small element of material or as "normal stresses" at a surface between two elements. From the point of view of the SET and GR, "internal stresses" and "normal stresses" are not two different things, but two different ways of looking at the same thing.

 

[Jonathan Scott]

The distinction you are making between "normal stress terms" and "internal stresses" is not correct, at least not when assigning physical meaning to the components of the stress-energy tensor. *All* stresses in the material are captured in the SET, regardless of whether you think of them as "internal stresses" in a small element of material or as "normal stresses" at a surface between two elements. From the point of view of the SET and GR, "internal stresses" and "normal stresses" are not two different things, but two different ways of looking at the same thing.

Sorry, I didn't intend any distinction between these terms. The same stress term is both normal (perpendicular to plane) and internal (only present within the materials of the system, not in the gaps between).

 

[Jonathan Scott]

So now not only gravitational energy is not a SET source, but we have (see JScott and Peterdonis discussion inspired by Q-reeus OP) pressure components that are explicitly in the SET of a stationary mass acting as if they weren't gravitational sources in a dynamical context.
Wish someone could clarify this a bit.

So do I!

I don't know what the actual geometric effect of the stress term is on the shape of space-time as described by the LHS of the Einstein Field Equations, and I don't have the patience to try to work it out at the moment, but it does seem odd that this stress can come and go very rapidly (far more rapidly than changes of energy or momentum).

Note that the Komar mass expression is a scalar "pseudo-energy" value formed by integrating terms of the tensor over a volume and adding the results together. It seems possible to me that stress could be a source term in the full tensor yet come and go suddenly if this meant that the shape of space on the other side of the equation changed in a way which only had a local effect.

What I find difficult to believe is that something relating to stresses could have any effect on the distant field, as the volume integral of the stress is not a conserved quantity in dynamic situations. There is probably an integral involving acceleration terms as well for which the total value is conserved in this situation, but as my examples with poles illustrate, it is difficult to see how this "something" could flow from one place to another continuously.

 

[Mentz114]

Does pressure gravitate ?

I think astrophysics says 'yes'. A large cloud of hydrogen could not collapse to sufficient pressure to ignite fusion unless the ever-increasing pressure worked with gravity, and not against it.

I'm trying to find some backing for this in Peebles' book and other sources, like Tolman-Oppenheimer-Volkov spacetimes.

 

[PAllen]

The gist of this thread seems to be:

- using some approximate (at best) arguments, and some general rules of thumb about 'sources of gravity', applied to a problem that is quite non-trivial to do in GR to high accuracy, we create a contradiction because these are claimed to lead to a result that contradicts a rigorous theorem with no qualifiers that was proved all the way back in 1923 stood up to all further analysis since (Birkhoff's theorem)?

The only logical conclusion is that our collection of adhoc arguments fails to accurately produce a cancellation which we know must happen. This situation is routine throughout physics and math. If I evaluate (1/7+1/7+1/7+1/7+1/7+1/7+1/7-1) on my machine the result is not zero! OMG - math is inconsistent. Gravitational waves are an extremely low energy phenomenon notoriously difficult to evaluate numerically.

 

[Jonathan Scott]

Does pressure gravitate ?

I think astrophysics says 'yes'. A large cloud of hydrogen could not collapse to sufficient pressure to ignite fusion unless the ever-increasing pressure worked with gravity, and not against it.

I'm trying to find some backing for this in Peebles' book and other sources, like Tolman-Oppenheimer-Volkov spacetimes.

I think that's irrelevant by several orders of magnitude.

The overall net pressure across a surface in any sort of near equilibrium is going to be the gravitational pressure. The energy corresponding to the volume integral of that is similar to the potential energy of the system. The additional gravitational force due to the gravity of the potential energy is a second-order effect which is extremely tiny.

 

[PAllen]

Does pressure gravitate ?

I think astrophysics says 'yes'. A large cloud of hydrogen could not collapse to sufficient pressure to ignite fusion unless the ever-increasing pressure worked with gravity, and not against it.

I'm trying to find some backing for this in Peebles' book and other sources, like Tolman-Oppenheimer-Volkov spacetimes.

I agree it must. A simple argument: Integrating pressure over volume is proportional to COM KE of constituents. Clearly, the latter must gravitate (it can be radiated away, reducing mass), thus obviously the former must gravitate. The stress energy tensor is written in terms of pressure, but the effect must be consistent with basic energy balance.

However, I am not sure about the astrophysics argument. I seem to recall that fusion ignition can be explained with Newtonian gravity. Two-Fish Quant would presumably know for sure as this was his field.

 

[Jonathan Scott]

The gist of this thread seems to be:

- using some approximate (at best) arguments, and some general rules of thumb about 'sources of gravity', applied to a problem that is quite non-trivial to do in GR to high accuracy, we create a contradiction because these are claimed to lead to a result that contradicts a rigorous theorem with no qualifiers that was proved all the way back in 1923 stood up to all further analysis since (Birkhoff's theorem)?

I agree that if Q-reeus accepts the basics of GR, then Birkhoff's theorem seems to rule out any effect on the external field due to radial pulsations of any sort.

However, I think that the question in the title of this thread is still interesting, as my "pole" examples demonstrate that stress can come and go suddenly, without apparently "flowing" anywhere new, yet one would expect something which was effectively supposed to act as a gravitational source term to be better behaved.

My primary point with these examples was more specifically to demonstrate that the Komar mass expression breaks down as soon as acceleration enters the picture, so it can't be used even as an approximation, but I'm still puzzled about how stress could act as a source and be able to vanish so rapidly.

 

[PAllen]

I agree that if Q-reeus accepts the basics of GR, then Birkhoff's theorem seems to rule out any effect on the external field due to radial pulsations of any sort.

However, I think that the question in the title of this thread is still interesting, as my "pole" examples demonstrate that stress can come and go suddenly, without apparently "flowing" anywhere new, yet one would expect something which was effectively supposed to act as a gravitational source term to be better behaved.

My primary point with these examples was more specifically to demonstrate that the Komar mass expression breaks down as soon as acceleration enters the picture, so it can't be used even as an approximation, but I'm still puzzled about how stress could act as a source and be able to vanish so rapidly.

Those are interesting questions. I think the next place to look would be ADM mass, which is (I think) the simplest form that applies rigorously to dynamic situations, with proper conservation properties - given the asymptotic assumptions (which don't appear to hold for our universe, but are typically assumed to be 'effectively true' at 'cosmologically short' time scales).

 

[Mentz114]

I agree it must. A simple argument: Integrating pressure over volume is proportional to COM KE of constituents. Clearly, the latter must gravitate (it can be radiated away, reducing mass), thus obviously the former must gravitate. The stress energy tensor is written in terms of pressure, but the effect must be consistent with basic energy balance.

However, I am not sure about the astrophysics argument. I seem to recall that fusion ignition can be explained with Newtonian gravity. Two-Fish Quant would presumably know for sure as this was his field.

Yes. I've found a very interesting recent paper by A. Mitra where he says

Thus the comoving (local) Active Gravitational Mass Density (AGMD) \rho_g = \rho + 3p indeed appears to increase due to the 'weight' of pressure, It is however important to note that this pressure contribution is actually due to the field energy contribution (when computed in quasi-Cartesian coordinates): 3p = t^0_0 and the field energy density is positive as long as p is positive.

The paper is "Einstein energy of FRW metric" http://uk.arxiv.org/abs/0911.2340v2

 

[Q-reeus]

The gist of this thread seems to be: - using some approximate (at best) arguments, and some general rules of thumb about 'sources of gravity', applied to a problem that is quite non-trivial to do in GR to high accuracy, we create a contradiction because these are claimed to lead to a result that contradicts a rigorous theorem with no qualifiers that was proved all the way back in 1923 stood up to all further analysis since (Birkhoff's theorem)?

If it's as you say, mind pointing out just how the argument given in #69 falls flat? Can you identify just where and how compensation to pressure comes about independent of any parameter value there? Specifically.

Gravitational waves are an extremely low energy problem notoriously difficult to evaluate numerically.

But the issue is not exclusively about GW's, even though I used them in both examples [1] and [2] in #1. The G-clamps example [2] can be made a static problem - just stress up the setup once. A gravitating field that is now completely static ensues. Apply my scaling argument, and please give some reasoned counterargument against my claim there can be no parameter (e.g. Young's modulus E) independent match between work in stressing, and field energy resulting. Unless one denies there is such a thing as field energy I suppose.

 

[Q-reeus]

I agree it must. A simple argument: Integrating pressure over volume is proportional to COM KE of constituents. Clearly, the latter must gravitate (it can be radiated away, reducing mass), thus obviously the former must gravitate. The stress energy tensor is written in terms of pressure, but the effect must be consistent with basic energy balance.

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.

 

[Q-reeus]

I agree that if Q-reeus accepts the basics of GR, then Birkhoff's theorem seems to rule out any effect on the external field due to radial pulsations of any sort.

Wel I'm not agreeing at all unless someone shows me how #1 and #69 can logically fail. A chain is as strong as it's weakest link, and I need showing that my arguments are not pointing to the existence of such in BT. Must go :zzz:

 

[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.