Is stress a source of gravity?

[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: