Is stress a source of gravity?

[Q-reeus]

In GR normal stresses as per the three lower diagonal terms T₁₁, T₂₂, T₃₃ in the SET (stress-energy tensor) (e.g. http://en.wikipedia.org/wiki/Stress–energy_tensor) are source terms for gravitating mass. And afaik true for any similar theories of gravity. Owing to their being resolved into equal and oppositely signed normal stresses, the off-diagonal shear stress components cannot even formally be a source of gravitation in solids and so are of no relevance here. Elastic/hydrostatic energy with typically quadratic dependence on stress/pressure is formally part of the T₀₀ rest-energy term, and that role for stress is not in doubt.

My contention is that if normal stresses truly are a source for gravitating mass m, it implies the following:
[1] Existence of monopole GW's, which represents an internal inconsistency in GR.
[2] Generation of GW's violating the conservation of energy in general.

First, a situation illustrating [1]

Komar mass is considered a valid definition of gravitating mass m in a stationary metric setting. From Wikipedia article at http://en.wikipedia.org/wiki/Komar_mass#Komar_mass_as_volume_integral_-_general_stationary_metric
With these coordinate choices, we can write our Komar integral as

...Essentially, both energy and pressure contribute to the Komar mass. Furthermore, the contribution of local energy and mass to the system mass is multiplied by the local "red shift" factor

Evidently m above applies for any matter distribution assumed non-rotating and having a stationary center of energy. Apply that to the case of a perfectly elastic thin spherical shell vibrating at natural frequency f in the n = 0 'fundamental' membrane breathing mode (uniform sinusoidal oscillation in radius R). Spherical symmetry allows no mass dipole or quadrupole moments P, Q. Periodic exchange at frequency 2f between KE of radial motion and elastic energy in circumferential uniform biaxial stress/strain leaves total energy (integration over the T₀₀ term) time invariant. [Assuming for the moment a possible tiny monopole GW drain is entirely absent!] The off-diagonal SET shear terms play no role in determining m. For the momentum-energy flux terms T_i0 = -T_0i, having radial acting velocity vector character, spherical symmetry implies net cancellation. Even for other configurations such as a straight bar vibrating freely in fundamental axial mode, these 'magnetic' terms, although then non-zero, scale very differently as functions of say material elesticity and density to that for the stress terms considered below.

Which just leaves the T_ii stress terms, that are not time invariant. At minimum radial excursion there is positive (compressive) circumferential stress, and negative (tensile) stress at maximum excursion. For the biaxially stressed shell, let's say we have |T₁₁|max = |T₂₂|max = p₀ (c=1), with radial component |T₃₃| negligible. Choose time t = 0 when R is undergoing maximum inward motion. If the shell has a thickness δ << R there will be a harmonic monopole moment m_s = 4∏R²δp₀sin(2∏ft), owing solely to the almost purely biaxial stress. Implying radial acting monopole GW radiation owing to d/dt(m_s) = 8∏²R²δp₀fcos(2∏ft). (see e.g. http://www.tapir.caltech.edu/~teviet/Waves/gwave.html - with the g' monopole series there continued to 1/r radiative term). [Note that adding in a non-negligible radial T₃₃ contribution (thick shell case not considered) merely acts to redistribute the stress contributions in the Komar expression. All that matters is that pressure is the sole time varying net contribution to m] Contrary to the GR claim that the lowest possible GW mode is pure transverse quadrupolar. As to whether monopole GW generation is a conservative process here requires detailed calculations. It does seem to scale correctly wrt the relevant parameters. Not so for the next example involving forced vibrations.

Now, a situation illustrating [2], first introduced here: https://www.physicsforums.com/showpost.php?p=3790816&postcount=65 , necessarily cleaned up below:

Suppose two 'G'-clamps are welded back-to-back, and by means of say electric motors & batteries, the screws are periodically tightened and loosened. In this forced oscillation regime, frequency assumed well below mechanical self-resonance, inertial forces play no important role. By inspection periodic stresses in the assembly having a quadrupolar type distribution Q_s arise - compression in the screwed arms coinciding with tension in the opposite arms, and vice versa. If the screwed legs are taken as verticallly inclined, the stress moment Q_s would be linear and horizontal in orientation. Bending and shear stresses also present are self-cancelling wrt net pressure. This dominantly quadrupolar stress distribution acts as a source of quadrupolar GW's whose amplitude for a given driving frequency is directly proportional to the stresses (as T_ii source terms for m).

We have not so far included the usual contributions:

a) Gross matter motion under mechanical strain. This is expected to be overwhelmingly the dominant source of GW's. One also expects dominantly vertical strain motion, generating a net vertical linear quadrupole moment Q_m, orthogonal to that for stress generated Q_s. Hence little if any cross-coupling between the two. Further striking differences between Q_m and Q_s is the scaling wrt elastic constant E (Young's modulus), and material density ρ. Given a specified driving stress amplitude, strain is inversely proportional to E. Hence gross matter motion and thus Q_m scales accordingly. Additionally, Q_m is directly proportional to material density ρ. As Q_s is in this setting independent of both E and ρ, it is not possible for cross-coupling between these two GW sources to cancel anything in general.

b) Relativistic energy-momentum flux owing to redistributions of energy between the driving power source (battery etc.) and elastic strain energy in the clamps. Suppose this gives rise to a quadrupole moment Q_e. The same scaling feature wrt E mentioned above applies here also. As well, by careful arrangement of power sources one could eliminate any quadrupole moment term Q_e, leaving only insignificant higher-order terms.

It is this independence from E and ρ of stress contribution Q_s to GW amplitude, in this forced oscillation regime, that is critical. Plastic will flex far more than say steel. It follows back reaction from stress generated GW's must induce far greater power drain in the plastic clamps case than for the steel ones. Much longer 'stroke' for the same retarding 'force'. And given the E and ρ parameter dependence of all non-stress GW contributions, there is no way they can in general nullify the conclusion GW's owing to Q_s trend to 'for free' as E trends upward. There cannot be in general a conservative power balance. And importantly, this setting is in arbitrarily flat background metric - so Noether's theorem appears to be in serious trouble!

Of course as hinted in the title there is one possible ready cure for all this - pressure is in fact *not* a source of gravity. Assuming no fatal blunders in the foregoing, seems to me a stark choice has to be made. Has pressure as source ever been derived from first principles - as in direct calculation of motion contributed gravitating mass generated by a 'gas' of colliding particles? That might prove to be interesting. If anyone knows of such a study, please provide a reference to the literature. Why was stress inserted into the SET in the first place? Symmetry considerations perhaps - all the SET slots have to mean something physical? My suspicion as complete GR outsider is it was a carry over from SR, where pressure applied to a flowing fluid does exhibit inertial properties as a consequence of non-simultaneity. I believe it can be shown this inertial behaviour, implying a sort of 'mass' to pressure, is really a type of pseudo quantity that fails when stretched just a bit, but this is not the place to expand on that.

Don't expect all this to be taken lying down, so await breathlessly for sensible and constructive critiques.

 

[Dale]

Owing to their being resolved into equal and oppositely signed normal stresses, the off-diagonal shear stress components cannot even formally be a source of gravitation in solids and so are of no relevance here.

I am not sure what you mean by this. The 3D stress tensor at any single point can always be rotated into a coordinate system where the off-diagonal terms are 0 and the diagonal terms are called the principal stresses. I assume that the same is true of the 4D stress energy tensor at a point. But I am not at all sure that it can be done globally.

My contention is that if normal stresses truly are a source for gravitating mass m, it implies the following:
[1] Existence of monopole GW's, which represents an internal inconsistency in GR.
...
Komar mass is considered a valid definition of gravitating mass m in a stationary metric setting.

If you have GW's then the metric is, by definition, not stationary, so the Komar mass is not defined.

This claim here requires much more than a hand-waving argument like the one above. You need to actually derive some metric and show that it:
A) Is a monopole source
B) Is a solution to the EFE
C) Exhibits GW's

[2] Generation of GW's violating the conservation of energy in general.

Sure, in non static spacetimes energy is not generally globally conserved in GR.
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

 

[Q-reeus]

I am not sure what you mean by this. The 3D stress tensor at any single point can always be rotated into a coordinate system where the off-diagonal terms are 0 and the diagonal terms are called the principal stresses. I assume that the same is true of the 4D stress energy tensor at a point. But I am not at all sure that it can be done globally.

Sure rotation is possible as you say, but since in the SET off-diagonals represent shear stresses, we must simultaneously have T_ik + T_ki present as matching pair. Always then this resolves into orthogonal normal stresses of equal and opposite amplitude. A basic property of shear stress. Disagree?

If you have GW's then the metric is, by definition, not stationary, so the Komar mass is not defined.

Is this anything more than a trivial point? The GW's for given scenarios represent an extremely weak perturbation. Are you seriously suggesting that will throw out the validity of Komar mass used here? Surely non-stationary as significant factor re coe implies something like being in an FLRW setting or whatever where parallel transport issues etc. - 'counting difficulties' - lies at the heart of presumed failure of coe.

This claim here requires much more than a hand-waving argument like the one above. You need to actually derive some metric and show that it:
A) Is a monopole source
B) Is a solution to the EFE
C) Exhibits GW's

You are well aware I'm not some GR pro capable of doing that math. The 'hand-waving' is though sufficient imo to establish in principle what I have claimed. Let's see what others have to say. Meanwhile, what specific points do you find to be obviously in error?

 

[yoron]

As a possible first principle and definition of matter, pressure is a very nice idea. After all, that's what a star seems to use (gravity/pressure) to create elements, according to the main stream view. And matter is indeed coupled to gravity. Are you saying that pressure should be equivalent to gravity? And the 'stress' then would be the geometry?

 

[PeterDonis]

At minimum radial excursion there is positive (compressive) stress, and negative (tensile) stress at maximum excursion.

This is way too oversimplified. The radial components are not the same as the tangential ones, and the components vary radially (they are different on the inner surface of the shell than on the outer). This basically invalidates all of your reasoning about your case #1.

 

[Q-reeus]

Q-reeus: "At minimum radial excursion there is positive (compressive) stress, and negative (tensile) stress at maximum excursion."
This is way too oversimplified. The radial components are not the same as the tangential ones,...

And where did I suggest they were? Use of 'way too' suggests there are important factors completely overlooked. What are they exactly?

...and the components vary radially (they are different on the inner surface of the shell than on the outer). This basically invalidates all of your reasoning about your case #1...

No it doesn't. Notice I specified thin shell. For which circumferential biaxial shell stresses become arbitrarily close to uniform as thickness declines. Is it not clear to you radial stress in this thin shell situation is negligible? And anyway, what if one stupidly picked a thick shell scenario where stresses did vary significantly with radius, just to throw in unnecessary complication? In what way would that invalidate the essential argument? Would it invalidate in any way whatsoever the fact of a net sinusoidal (strictly - near sinusoidal) fluctuating pressure contribution - uncancelled by the other SET terms? Please, if there is some basic flaw, argue it on important principle, not by blowing up inessential details into major flaws. But despite that bit, glad to see you involved.

 

[Q-reeus]

As a possible first principle and definition of matter, pressure is a very nice idea. After all, that's what a star seems to use (gravity/pressure) to create elements, according to the main stream view. And matter is indeed coupled to gravity. Are you saying that pressure should be equivalent to gravity? And the 'stress' then would be the geometry?

Not sure your question is directed to me. If so, my argument goes like this: If pressure adds to gravity as assumed in GR, it causes problems listed. If it doesn't, that's a major problem for how GR is formulated.

 

[PeterDonis]

Use of 'way too' suggests there are important factors completely overlooked. What are they exactly?

See below.

Notice I specified thin shell. For which circumferential biaxial shell stresses become arbitrarily close to uniform as thickness declines. Is it not clear to you radial stress in this thin shell situation is negligible?

No, it isn't. It can't be, because without radial stress, the shell will collapse under its own gravity. And the radial stress has to change from inner to outer surface. That means that, for the EFE to hold at every event in the shell, the tangential stresses must vary too. For a thin shell, the variation becomes very sharp (i.e., very large spatial derivatives).

Please, if there is some basic flaw, argue it on important principle, not by blowing up inessential details into major flaws. But despite that bit, glad to see you involved.

You have already admitted that you can't write down an actual mathematical description of your scenario. That makes it very difficult to even understand what the scenario actually is, because you are using imprecise English to describe it. The only help I have in trying to make the imprecise English more precise is the details; hence, they are not "inessential". Without them I can't even analyze your scenario at all.

 

[Q-reeus]

Q-reeus: "...Is it not clear to you radial stress in this thin shell situation is negligible?"
No, it isn't. It can't be, because without radial stress, the shell will fall apart under its own gravity.

Huh? What? Did I say above the radial stress was exactly zero? Negligible wrt tangent stresses - that's what I said, and it's obviously the case, although you're welcome to argue otherwise. But I ask again - in what way would the addition of a significant radial stress (thick shell case) invalidate the general argument? But I'm thinking you probably will duck that question. Can sense where this is going below - and it's a pity.

Q-reeus: "Please, if there is some basic flaw, argue it on important principle, not by blowing up inessential details into major flaws..."

You have already admitted that you can't write down an actual mathematical description of your scenario. That makes it very difficult to even understand what the scenario actually is, because you are using imprecise English to describe it. The only help I have in trying to make the imprecise English more precise is the details; hence, they are not "inessential". Without them I can't even analyze your scenario at all.

What can I say to that. In essence you deny my arguments because it is not expressed in some full blown, complex mathematical model? I'm not seeing you deal with others on that basis, so why here? I have in fact used some math, just enough imo to help put the first scenario in clear enough terms. There really wasn't any need for that much - the idea was to apply a process of elimination there. And you couldn't follow it?! The second scenario is inherently beyond analytic solution if one demands 'the full maths'. In my opinion that would be a wholly unreasonable stance.

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. Anyway, better if you just come out and say plainly "I reject your arguments out of hand because they don't line up with established consensus opinion", if that is so and it seems to me to be so.
If it's not so and you genuinely can't fathom what #1 is all about, ask for help on any part therein and trust me I'll do my best to clarify any grey areas.
[EDIT: Have gone back and specifically added commentary re radial stress contribution, in light of your criticisms]

 

[Jonathan Scott]

I certainly find it hard to believe that Komar mass makes a real contribution to gravity.

If you have a static configuration of masses then the Komar mass stress contribution on the diagonal is positive and matches the internal (negative) potential energy of the system, and the mass of the set of particles making up the system is effectively decreased by twice the internal potential energy because of time dilation effects of all of the particles on one another. That seems to make a lot of sense and match the Newtonian model, as the overall energy of the system is simply decreased by the potential energy within the system.

However, if the configuration is not of minimal gravitational energy, for example held apart by poles, and one of the poles breaks or slips off its support, then the stress in that pole due to the gravitational force vanishes essentially instantly (probably at the speed of sound in the material of the pole). However, I would expect the overall mass-energy of the system and resulting gravitational field to remain constant at least initially, before anything starts moving.

I therefore think that the Komar mass adjustment is equal to the appropriate correction to make the potential energy come out right in the static case, but it is not the actual energy, as can easily be illustrated as follows.

If you consider the gravitational force between any two particles in the system on either side of a given surface and integrate that perpendicularly to the surface as the surface sweeps through the system, summing it for all pairs of particles you get the same (negative) totals as you get for the Komar (positive) mass diagonal integrals for each of the three axes, equal to the potential energy (assuming you only take each pair once). In Newtonian terms, this works as follows for each pair of particles:
$$
\int_0^r \frac{- G m_1 m_2}{r^2} dx = \frac{- G m_1 m_2}{r^2} r = \frac{-G m_1 m_2}{r}
$$
This means that if the system is static, the internal forces resisting the gravitational forces are equal to the gravitational forces, so the Komar mass integral is equal and opposite to the potential energy. However, even if the system is not static, the integral of the gravitational forces still gives the potential energy, so if that quantity is switched in sign and added to the time-dilated energy of the individual particles then that sum would still give the same overall energy without the requirement for being static.

This still leaves a question of how that potential energy contribution appears in the stress-energy tensor. My feeling is that it actually effectively adjusts the normal energy term, and has nothing to do with the other diagonal terms, but as usual for anything to do with gravitational energy, I don't really know.

 

[PeterDonis]

Huh? What? Did I say above the radial stress was exactly zero? Negligible wrt tangent stresses - that's what I said, and it's obviously the case

Why is it obviously the case? Have you done an equilibrium analysis that shows that the radial stress needed to support the shell against its own weight is negligible compared to the tangential stresses? I, for one, do not think it's at all obvious that that will be true.

But I ask again - in what way would the addition of a significant radial stress (thick shell case) invalidate the general argument?

I'm not even sure I understand your general argument yet. That's why I keep asking questions about details. See below.

What can I say to that. In essence you deny my arguments because it is not expressed in some full blown, complex mathematical model? I'm not seeing you deal with others on that basis, so why here?

I'm not necessarily asking for a full blown, complex mathematical model. What I *am* asking for is sufficient precision for me to be able to make some sort of estimate of what GR predicts for your scenario. With what you've given so far, I'm not sure I can do that. If I can't do that, I can't analyze your model; I can't say whether I think it's right or wrong at all. That's not denying your arguments; that's just saying I can't render a judgment on them one way or the other.

But yes, if I'm in that situation and you insist on a judgment from me, my judgment will be that GR is right, and if your argument is giving answers that are not consistent with GR, then there must be some subtle flaw in your argument that I'm not smart enough to see. That seems much more probable to me than the hypothesis that you actually have discovered a basic flaw in GR.

If it's not so and you genuinely can't fathom what #1 is all about, ask for help on any part therein and trust me I'll do my best to clarify any grey areas.
[EDIT: Have gone back and specifically added commentary re radial stress contribution, in light of your criticisms]

I'll go back and read through it again and see if anything else strikes me. But I think we're going to end up in the same place we've ended up in previous threads: your idea of what constitutes a sufficiently specified scenario for analysis is apparently much less stringent than mine, so I simply won't be able to say anything useful.

 

[PeterDonis]

This still leaves a question of how that potential energy contribution appears in the stress-energy tensor. My feeling is that it actually effectively adjusts the normal energy term, and has nothing to do with the other diagonal terms, but as usual for anything to do with gravitational energy, I don't really know.

Remember that the Komar mass integral has a "redshift factor" term in it; so for a non-static (or non-stationary) system, the contribution to the total mass of an infinitesimal bit of stress-energy changes as the curvature changes. In the example of two static masses held apart by a pole, if the pole breaks, the curvature change caused by the pole breaking will propagate, as you say, at the speed of sound in the pole; hence, so will the change in the Komar mass contribution of each little bit of stress-energy in the pole. (And in the two masses themselves, once the pole is no longer holding them up and they begin to fall--by that time, the curvature in their immediate vicinity has changed, and therefore so has the effective "redshift factor".)

So "potential energy" (or any other type of "gravitational energy") doesn't change the stress-energy tensor at all; what it changes is the "redshift factor" in the mass integral.

 

[Q-reeus]

Why is it obviously the case? Have you done an equilibrium analysis that shows that the radial stress needed to support the shell against its own weight is negligible compared to the tangential stresses? I, for one, do not think it's at all obvious that that will be true.

I could scrounge up a specific study on shell stresses, but let's apply KISS to this one. Ever blown up a balloon? How much air pressure can your lungs supply - maybe a fraction of a psi. That's the differential between inside and outside radial pressure. Keep blowing and the balloon will burst. Tensile strength of rubber several thousand psi. Point made?
BTW was in the process of editing my #9 as I had somehow overlooked this from your #8:

And the radial stress has to change from inner to outer surface. That means that, for the EFE to hold at every event in the shell, the tangential stresses must vary too. For a thin shell, the variation becomes very sharp (i.e., very large spatial derivatives).

In light of balloon example here, won't bother.

To get back to the essence of what #1 is all about, let's just concentrate on scenario [1] there. Simple really.
1) Do you agree that total energy is constant there (thus net T₀₀ contribution to m)?
2) Do you agree or not on off-diagonals being zero contributors to time-varying m? (DaleSpam hasn't responded to my points on that issue in #3, but no sweat)
3) Do you agree or not that T_i0, T_0i energy-momentum flux terms cancel to zero by reason of spherical symmetry?
4) Do you agree or not that all that's left is the diagonal pressure terms T_ii, and that these are clearly non-zero time varying?

As I said in #9, apply a process of elimination to above, and what does one find? Can't see the need of having to run complex numerical GR code through a supercomputer. And it would pay imo to avoid circular, self-referential arguments like demanding Birkhoff's theorem apply. The whole idea of a counterexample is to look for holes in such, imho. :zzz:

 

[Dale]

You are well aware I'm not some GR pro capable of doing that math.

I am just letting you know what is required in order to make a major theoretical breakthrough like the one you claim to have made. If you want to demonstrate that GR predicts monopolar GW's then you need to find a solution to the EFE's which both radiates GW's and is monopolar. Anything less will not accomplish the breakthrough you have claimed.

The fact that you cannot yet do the math does not change the requirements.

Is this anything more than a trivial point?
...
Meanwhile, what specific points do you find to be obviously in error?

It is hardly trivial to point out that the key equation used to make an argument doesn't even apply. Any argument which centers around a formula that doesn't even apply to the scenario being analyzed is a fundamentally flawed argument. Major theoretical advances shouldn't be based on obviously flawed arguments.

"Extraordinary claims require extraordinary evidence", and so far you haven't produced any.

 

[yoron]

I find it hard to see how you think. On the other hand I'm not familiar with the concept of Komar mass.The stress-energy tensor includes energy density, energy flux, momentum density, and momentum flux.

The pressure causing 'gravity' in the stress energy tensor I understand to be a result of the internal momentum flux?

You wrote "If pressure adds to gravity as assumed in GR, it causes problems listed. If it doesn't, that's a major problem for how GR is formulated."

Can you describe the problem in a simple way?

 

[Jonathan Scott]

Remember that the Komar mass integral has a "redshift factor" term in it; so for a non-static (or non-stationary) system, the contribution to the total mass of an infinitesimal bit of stress-energy changes as the curvature changes. In the example of two static masses held apart by a pole, if the pole breaks, the curvature change caused by the pole breaking will propagate, as you say, at the speed of sound in the pole; hence, so will the change in the Komar mass contribution of each little bit of stress-energy in the pole. (And in the two masses themselves, once the pole is no longer holding them up and they begin to fall--by that time, the curvature in their immediate vicinity has changed, and therefore so has the effective "redshift factor".)

So "potential energy" (or any other type of "gravitational energy") doesn't change the stress-energy tensor at all; what it changes is the "redshift factor" in the mass integral.

As far as I know, the "redshift" factor is exactly what I mentioned previously as the time-dilation effect on the particles of their own potential. This causes the effective energy of the system to be decreased relative to the local energy of the original components by twice the potential energy (as each interaction works both ways). I don't see why this should change in any way if, say, two parts of a supporting pole are pushed out of alignment with one another just enough to cause them to start to fall past each other. Any significant change in velocities and configurations is going to be far slower than the change in stress, and of course the fall could be stopped again a moment later.

Clearly in the Newtonian sense there has been no immediate change in the overall potential energy nor the kinetic energy of the system caused by the support being removed, yet for example if we consider two small masses held apart by a single light pole, the Komar mass term for the stress in that pole was previously equal in magnitude to the potential energy of the pair of masses relative to one another, but if the pole is disconnected it suddenly drops to zero. It does not seem plausible that this internal change could abruptly affect the overall energy of the system, or its strength as a gravitational source.

 

[Jonathan Scott]

I find it hard to see how you think. On the other hand I'm not familiar with the concept of Komar mass.The stress-energy tensor includes energy density, energy flux, momentum density, and momentum flux.

The pressure causing 'gravity' in the stress energy tensor I understand to be a result of the internal momentum flux?

You wrote "If pressure adds to gravity as assumed in GR, it causes problems listed. If it doesn't, that's a major problem for how GR is formulated."

Can you describe the problem in a simple way?

I don't know about the original poster, but I see the problem as the fact that pressure can come and go almost instantly, and it doesn't seem to make sense that the shape of space-time could be directly affected by a quantity which doesn't seem to be subject to any conservation law.

The total force perpendicular to any plane slicing through a static system is zero. The integral of the pressure over a plane gives the total force other than gravitational forces, so for a static system that force must be equal and opposite to the gravitational force. If this is integrated over three perpendicular planes which move through the system, the integral of the gravitational force is the potential energy for reasons mentioned in my previous post and for a static system this is opposite and equal to the stress part of the Komar mass. However, if the system is allowed to change, the pressure can immediately vanish, long before there is any visible change in the Newtonian potential energy or kinetic energy, and this appears to violate conservation of a form of gravitational source, regardless of whether it is actually "energy" or not.

 

[pervect]

Evidently m above applies for any matter distribution assumed non-rotating and having a stationary center of energy.

No. The requirement for having a stationary metric is that a coordinate system exists in which none of the metric coefficients are functions of time.

A coordinate-free description of the requirements is that there is a time-like Killing vector.

This explicitly rules out gravitational waves, and oscillating shells.

Basically the Komar mass approach takes advantage of these special symmetries, so it doesn't give a general solution to Einstein's field equations. It does make it easy to calculate gravitational fields in those systems that have the requisite symmetry. Unfortunately those systems that the approach applies to cannot include gravitational waves.

 

[yoron]

Interesting Johnathan. Haven't thought of pressure that way. Although it's a very theoretical description you give me here, with a lot of really difficult words in it :) I would expect gravity to obey 'c' myself?

You suspect it doesn't?
Your idea about pressure and the conservation laws?
That was a new angle to me, and interesting.
==

Aha, rereading Pervect "Basically the Komar mass approach takes advantage of these special symmetries, so it doesn't give a general solution to Einstein's field equations. It does make it easy to calculate gravitational fields in those systems that have the requisite symmetry. . . .

Unfortunately those systems that the approach applies to cannot include gravitational waves."

And that would be because? There is no arrow assumed for a Komar mass? I really need to look this up.

 

[PeterDonis]

I'll go back and read through it again and see if anything else strikes me. But I think we're going to end up in the same place we've ended up in previous threads: your idea of what constitutes a sufficiently specified scenario for analysis is apparently much less stringent than mine, so I simply won't be able to say anything useful.

Well, I went back and read through it again, and I was a bit pessimistic in the above quote; two things have struck me about the first of the two scenarios (the thin spherically symmetric shell). However, they don't change the verdict: GR is still right, and monpole GW radiation is still impossible.

Here they are:

(1) You are claiming that radial pressure is negligible and tangential pressure is not, *as a contribution to the Komar mass integral*.

(2) You are claiming that the pressures are the only things that change significantly as the system oscillates.

Let's take these in order. I feel like putting headings in this post, so here goes:

Radial vs. Tangential Pressure

The key thing you are missing here is simple: the radial pressure is positive throughout the shell (it has to be to keep the shell from collapsing under its own gravity). But the tangential pressure is *not*; it is positive (compressive) at the shell's inner surface, but *negative* (tensile) at the outer surface.

So in the Komar mass integral, which is taken over the entire shell, the contributions from radial pressure all add up; but the contributions from tangential pressure cancel each other out since the pressure changes sign. Therefore, radial pressure can contribute significantly to the final mass of the shell, but tangential pressure will not.

Interlude

As a segue from my first point to my second point, consider the question: *why* does the shell oscillate? For a system to oscillate, it has to have an equilibrium configuration, and if it is perturbed by a small amount away from that equilibrium, there has to be some restoring force that acts to bring it back.

Since we are talking about spherically symmetric oscillations, the above observation, all by itself, is enough to tell us something important: tangential forces are *irrelevant* to the dynamics of the oscillation. Only radial forces can play any role in a spherically symmetric oscillation. So again, there *must* be non-negligible radial pressure in this scenario, since the balance between radial pressure and gravity is what determines the equilibrium and the dynamics of the oscillation.

The restoring force is then obvious: if the shell is compressed slightly (i.e., its radius gets slightly smaller), radial pressure increases and pushes it back out again; if the shell is expanded slightly (i.e., its radius gets slightly larger), the opposite happens, radial pressure decreases and the shell collapses back inward again. *That* is why the shell oscillates.

What Varies with Time?

Armed with the above, we can now ask: what factors in the Komar mass integral vary with time? We know radial pressure does, as we just saw in the interlude. We don't care whether tangential pressure does or not (as I said in an earlier post, I would expect it to for the EFE to continue to hold), since its contributions integrated over the shell will cancel out. But is there anything else that does?

Yes, there is. The redshift factor also varies, because it depends on the radius of the shell. If the shell is compressed (and radial pressure rises), the redshift factor will get smaller, because the shell is more compact and so the potential within it is slightly more negative compared to "infinity". Thus, pressure gets larger but the redshift factor gets smaller, and the two effects cancel each other out to keep the Komar mass integral constant.

If the shell is expanded, the opposite happens: radial pressure gets smaller, but the redshift factor gets larger because the shell is less compact; so again, the two effects cancel each other out and the Komar mass integral remains constant.

Postscript

I said the Komar mass integral remains "constant", but actually that's an approximation. As pervect pointed out, a spacetime with an oscillating shell, even if the oscillation is spherically symmetric, is not stationary. I am basically assuming that the spacetime is "almost stationary", i.e., that the oscillations are small enough that the metric can be approximated by a stationary one. To that approximation, the Komar mass integral will be constant, and I have tried to give a physical picture of how that works. But strictly speaking, the mass integral will *not* be exactly constant because the spacetime is not stationary; however, that does not mean GWs will be emitted, because we've assumed spherical symmetry (i.e., zero dipole and higher moments). Strictly speaking, I should have written "no energy is lost to GWs" instead of "Komar mass integral remains constant" in the above. But the physical reasoning remains the same.

 

[PeterDonis]

I don't know about the original poster, but I see the problem as the fact that pressure can come and go almost instantly

Can it? In non-relativistic physics we often model systems as though it can, but in relativity, to be strictly correct, we can't do that. Pressure is part of the stress-energy tensor, and the stress-energy tensor obeys a conservation law at every event: its covariant divergence must be zero. This limits the possible changes in any component from event to event. For example:

However, if the system is allowed to change, the pressure can immediately vanish

No, it can't. Our Newtonian intuitions may make us think it can, but that's because our intuitions are inaccurate for this. For the pressure to actually "vanish" like this, the conservation law I just gave for the SET would have to be violated.

 

[Jonathan Scott]

Can it? In non-relativistic physics we often model systems as though it can, but in relativity, to be strictly correct, we can't do that. Pressure is part of the stress-energy tensor, and the stress-energy tensor obeys a conservation law at every event: its covariant divergence must be zero. This limits the possible changes in any component from event to event.

Again, as far as I know, the covariant divergence being zero expresses the conservation of energy and the momentum vector, which together form a four-vector quantity. Basically, for an infinitesimal "box" of space, the variation of one of these components with time is equal to the amount that passes through the walls of the box. This does not prevent a sudden (speed of sound) change in the pressure. A change in forces will eventually cause changes in the distribution of energy and momentum, but not instantaneously.

I agree that it's difficult to take everything into account correctly when in our experience gravitational energy is such a vanishingly small part of the total energy, but this particular point seems very strange to me.

Note that in contrast, if you add in the gravitational "tension" across space rather than the Komar "pressure" as the diagonal term, then that can be consistently treated like the energy of the field, giving a conserved flow of potential energy even in the non-static case (in a similar way to the energy of an electrostatic field). The gravitational "tension" is of course spread through space rather than being limited to following the internal structure, and if it is assumed to be proportional to the square of the field then that is easily matched to the requirement for the correct total force across a plane. However, this is not the way that GR describes things; this does not necessarily mean it is in conflict with GR, as there are multiple ways of looking at things (in that for example this method is assuming "forces"), but it does at least seem that way.

 

[Dale]

Basically, for an infinitesimal "box" of space, the variation of one of these components with time is equal to the amount that passes through the walls of the box. This does not prevent a sudden (speed of sound) change in the pressure. A change in forces will eventually cause changes in the distribution of energy and momentum, but not instantaneously.

Yes, instantaneously, that is what force is. Over every differential element any change in pressure requires an acceleration of the matter, changing the momentum flux and keeping the divergence 0.

 

[Q-reeus]

#14
If you want to demonstrate that GR predicts monopolar GW's then you need to find a solution to the EFE's which both radiates GW's and is monopolar. Anything less will not accomplish the breakthrough you have claimed.

There has been imo already and will in future be sufficient using my approach to prove the claims set out in #1 in principle, and hopefully someone with the GR math capabilities will take it upon themselves to model both situations [1] and [2] there, and prove it by that route beyond any doubt.

Q-reeus: "Is this anything more than a trivial point?...Meanwhile, what specific points do you find to be obviously in error?"
It is hardly trivial to point out that the key equation used to make an argument doesn't even apply. Any argument which centers around a formula that doesn't even apply to the scenario being analyzed is a fundamentally flawed argument. Major theoretical advances shouldn't be based on obviously flawed arguments.

First up here is to note what you judiciously excised in the above qouted passage from #3 - here it is:

The GW's for given scenarios represent an extremely weak perturbation. Are you seriously suggesting that will throw out the validity of Komar mass used here? Surely non-stationary as significant factor re coe implies something like being in an FLRW setting or whatever where parallel transport issues etc. - 'counting difficulties' - lies at the heart of presumed failure of coe.

You accuse me of resorting to vague hand-waving arguments. Let me throw that one back at you. How about this time responding to the above in some detail. Like exactly how and to what (in)significant level Komar expression is rendered useless. And if you argue it's not necessary 'coz I've got GR and a hundred thousand experts on my side', all I can say is, try your best to treat this as a level playing field, where straight logical argument counts for more than weight of opinion.

"Extraordinary claims require extraordinary evidence", and so far you haven't produced any.

None that meet your criteria, but let's see after your response to above point. And while we are at it, you also didn't respond to this:

Sure rotation is possible as you say, but since in the SET off-diagonals represent shear stresses, we must simultaneously have Tik + Tki present as matching pair. Always then this resolves into orthogonal normal stresses of equal and opposite amplitude. A basic property of shear stress. Disagree?

 

[Q-reeus]

The stress-energy tensor includes energy density, energy flux, momentum density, and momentum flux.

Last three are afaik different names for the one quantity (in normalized units).

The pressure causing 'gravity' in the stress energy tensor I understand to be a result of the internal momentum flux?

Some folks talk about it like that, but there is no net flux - equal and opposite flux yes but that's not a net flux. Pressure is pressure. I'm no expert in this, but that's my take.

You wrote "If pressure adds to gravity as assumed in GR, it causes problems listed. If it doesn't, that's a major problem for how GR is formulated."
Can you describe the problem in a simple way?

No simpler than shown in #1. Try several reads maybe. And look for a later entry where I try and straighten out some bad misconceptions that have arisen.

 

[Q-reeus]

No. The requirement for having a stationary metric is that a coordinate system exists in which none of the metric coefficients are functions of time.

A coordinate-free description of the requirements is that there is a time-like Killing vector.

This explicitly rules out gravitational waves, and oscillating shells.

Basically the Komar mass approach takes advantage of these special symmetries, so it doesn't give a general solution to Einstein's field equations. It does make it easy to calculate gravitational fields in those systems that have the requisite symmetry. Unfortunately those systems that the approach applies to cannot include gravitational waves.

Alright then, but to what significant extent is Komar definition invalidated for oscillating shell model (I've asked someone else on this, so this is a second opinion of sorts)? Bear in mind there isn't supposed to be any perturbation to metric either exterior or interior to the oscillating shell according to Birkhoff's theorem, correct? So I can't see the problem in principle, and given the extraordinary feebleness of any GW's for a small thin shell oscillating in any mode at all, can't imagine a valid problem in practice.

 

[Q-reeus]

(1) You are claiming that radial pressure is negligible and tangential pressure is not, *as a contribution to the Komar mass integral*.

For the scenario as actually given in #1 yes, but as stated there, it doesn't matter. Your whole conception of that model evidently is badly mistaken.

(2) You are claiming that the pressures are the only things that change significantly as the system oscillates.

Not quite, there's an important caveat. I claim pressure is the sole *net* contribution to a time changing m.

Let's take these in order. I feel like putting headings in this post, so here goes:

Radial vs. Tangential Pressure

The key thing you are missing here is simple: the radial pressure is positive throughout the shell (it has to be to keep the shell from collapsing under its own gravity). But the tangential pressure is *not*; it is positive (compressive) at the shell's inner surface, but *negative* (tensile) at the outer surface.

Wrong on every score. You have some completely different conception of what I gave in #1 - a thin elastic spherical shell (that means self-supporting). Note I said there vibrational energy exchange was between KE of shell radial motion, and elastic energy in shell (overwhelmingly) biaxial stress. Which clearly implies this is a gravitationally small system - say a basketball sized object. Contributions to dynamics from gravitational potential are minute, ca 10^-20 times the elastic/inertial ones. This is a system governed by the latter to all but an insignificant degree. If I had intended for gravitational potential and induced forces to be an important contributor, that would have been clearly spelt out in #1. A lot of barking up the wrong tree on this, and not just from you. Given that gravitational potential is insignificant here (and scaling argument shows it can be made arbitrarily small wrt pressure contributions), the Komar redshift factor is thus insignificant as 'nullifier' of fluctuating pressure contribution.

So it gets back to where else is there, other than straight SET terms in the small shell scenario? Nowhere imo. Consequently, all your remaining deductions are imo misplaced. And btw, you did not comment on my balloon argument re relative magnitudes (which itself is moot for a vibrating shell as per above, but still...)

On the matter of correct force balance fore a shell. For an oscillating shell in fundamental membrane mode, there is precisely zero radial pressure at either surface (but internal, relatively tiny radial pressure will generally exist). Overwhelmingly, restoring forces are owing to near uniform tangential stresses as claimed in #1. All stresses, tangent and radial, undergo periodic sign reversal at frequency f as set out in #1. It cannot be any other way. Try looking here maybe: hacks-galore.org/jao/thesis.pdf (sect. 3.4 and on) [Ehlers et. al. give more accessible info here: http://arxiv.org/abs/gr-qc/0510041 , http://arxiv.org/abs/gr-qc/0505040 (part 5 esp.)]. For thick shells, so called 'bending' contributions (radial gradient in tangent stresses) can be important, but none of that invalidates my key point. Which is there is always net positive shell stresses at inner excursion, sinusoidally changing to negative at outer excursion. As per #1!

 

[Dale]

The GW's for given scenarios represent an extremely weak perturbation. Are you seriously suggesting that will throw out the validity of Komar mass used here? Surely non-stationary as significant factor re coe implies something like being in an FLRW setting or whatever where parallel transport issues etc. - 'counting difficulties' - lies at the heart of presumed failure of coe.

You accuse me of resorting to vague hand-waving arguments. Let me throw that one back at you. How about this time responding to the above in some detail. Like exactly how and to what (in)significant level Komar expression is rendered useless.

It is exactly this extremely weak perturbation that you are interested in. You claim, without any evidence, that the errors due to using the Komar mass in the non-stationary spacetime are insignificant. I would expect that the magnitude of the errors in the Komar mass, though small, are exactly equal to the small magnitude of the GWs you claim.

Don't blame me for the fact that you made an obvious mistake in trying to analyze the non-stationary aspects of a non-stationary spacetime using an equation that is specifically and explicitly defined only in stationary spacetimes.

 

[Q-reeus]

It is exactly this extremely weak perturbation that you are interested in. You claim, without any evidence, that the errors due to using the Komar mass in the non-stationary spacetime are insignificant. I would expect that the magnitude of the errors in the Komar mass, though small, are exactly equal to the small magnitude of the GWs you claim.

That in turn is your claim without any evidence. Looks like a dead heat. So you can't specify either just why the spacetime will be non-stationary (Birkhoff's theorem does not say otherwise?), or to what level, and how a periodically varying non-stationary spacetime would not automatically imply existence of monopole GW's which conventionally shouldn't be. Well how about at least a link to the correct mass formula?

 

[Jonathan Scott]

Yes, instantaneously, that is what force is. Over every differential element any change in pressure requires an acceleration of the matter, changing the momentum flux and keeping the divergence 0.

I agree that a change in pressures and related forces causes an instantaneous change in acceleration and in the rate of change of energy density, but certainly in Newtonian theory this is very different from causing an instantaneous change in the energy or momentum (since acceleration only determines the rate of change of those), and normally such things apply also to GR at least as local approximations.

I don't see how we can get an instantaneous redistribution of something which is behaving (a) like energy for purposes of conservation of overall energy and (b) like energy for purposes of acting as a gravitational source. I know that the Komar mass only applies in the static case, but I would have expected some continuity for at least a short time as we for example remove some internal support.

 

[PeterDonis]

For the scenario as actually given in #1 yes, but as stated there, it doesn't matter. Your whole conception of that model evidently is badly mistaken.

I guess I was not pessimistic enough when I said I had been too pessimistic. :sigh: Apparently I was right the first time; I won't be able to say anything useful, because your idea of what it takes to actually specify a model is too different from mine. I have nothing more to add to what DaleSpam has already pointed out in response to your post.

 

[Dale]

That in turn is your claim without any evidence. Looks like a dead heat.

This is an excellent point, and I agree. I rescind my claim as to the amount of violation and stick only with the claim that the Komar mass is not defined for non-stationary spacetimes. That one I certainly can provide evidence for in the form of references if desired, but since you even mentioned it in your OP then I guess it is not a point of contention.

The fact that the key formula in your argument is not defined for the scenario contemplated completely invalidates your argument. I will just stop there because my claims about the amount of error are unnecessary and, as you point out, not backed up with any evidence.

So you can't specify either just why the spacetime will be non-stationary

Sure, I can, I just thought it was obvious.

As pervect mentioned, the technical definition of a stationary spacetime has to do with the existence of a timelike Killing vector, but the practical result is that in stationary spacetimes you can write the metric in a set of coordinates such that all of the components of the metric tensor are independent of the time coordinate. A metric with gravitational waves is, by definition, a function of time, and is therefore non-stationary.

Well how about at least a link to the correct mass formula?

I don't think that there is a single "correct" mass formula in GR which applies for all spacetimes. Other common mass formulas are the ADM mass, and the Bondi mass, but those also only apply for a subset of possible spacetimes.

 

[Q-reeus]

I guess I was not pessimistic enough when I said I had been too pessimistic. :sigh: Apparently I was right the first time; I won't be able to say anything useful, because your idea of what it takes to actually specify a model is too different from mine. I have nothing more to add to what DaleSpam has already pointed out in response to your post.

You prefer to finish it here as is? Very well. Just one final request though if you don't mind. Do you accept that given my clarification of what governs the dynamics of the shell in #1, elastic/inertial not gravitational, Komar redshift cannot be invoked to cancel out pressure as source?

 

[PeterDonis]

I don't see how we can get an instantaneous redistribution of something which is behaving (a) like energy for purposes of conservation of overall energy and (b) like energy for purposes of acting as a gravitational source.

You keep talking about "instantaneous" redistribution; it isn't. The conservation equation (covariant divergence of SET = 0) relates *rates of change* of the different SET components, such as pressure and momentum flux. If you are going to adopt a model coarse enough that one changes "instantaneously", then so must the other.

For example, consider your scenario of two masses held apart by a pole. You have stipulated that there is significant stress in the pole--i.e., that the pole's pressure makes a significant contribution to the Komar mass integral. That means that the pressure in any infinitesimal element of the pole *cannot* simply go to zero "instantaneously", unless that fluid element also "instantaneously" acquires a nonzero momentum flux that is "equivalent" to the pressure it had an instant before.

Here's a more "continuous" way to think about it: suppose at some instant of time we cut the supporting pole exactly in half and put the two halves slightly out of alignment. Consider the infinitesimal element of either half of the pole right at the location of the cut. What will be the immediate effect of the cut on its pressure? Answer: *none*. What will change "instantaneously" is the *rate of change* of its pressure--before the cut, that rate of change was zero; now it is negative. And the rate of change of the momentum of that infinitesimal element will also become nonzero, since it will start to fall.

Why is there still pressure on that element? And why will it start to fall? Because the pole as a whole was compressed, like a spring; and removing the constraint on the pole does not remove the compressive stress inside it. It just allows the pole to start re-expanding to its "normal" unstressed length. As it does so, the infinitesimal elements closest to the cut in the pole will start falling, then the ones further up, etc., etc. As each infinitesimal element starts to move, the pressure felt by that element starts to decrease. The *rates of change* of the momentum and the pressure are what are related by the conservation equation.

 

[Q-reeus]

This is an excellent point, and I agree. I rescind my claim as to the amount of violation and stick only with the claim that the Komar mass is not defined for non-stationary spacetimes. That one I certainly can provide evidence for in the form of references if desired, but since you even mentioned it in your OP then I guess it is not a point of contention.

I respect those comments, thanks.

The fact that the key formula in your argument is not defined for the scenario contemplated completely invalidates your argument. I will just stop there because my claims about the amount of error are unnecessary and, as you point out, not backed up with any evidence.

An odd mix of words there, but I guess it's a case of take it or leave it on that matter.

Sure, I can, I just thought it was obvious.
The technical definition of a stationary spacetime has to do with the existence of a timelike Killing vector, but the practical result is that in stationary spacetimes you can write the metric in a set of coordinates such that all of the components of the metric tensor are independent of the time coordinate. A metric with gravitational waves is, by definition, a function of time, and is therefore non-stationary.

There is not an obvious contradiction in that? Komar mass invalidated because of a non-stationary spacetime (monopole GW's), whilst simultaneously agreeing to claims there can be no such GW's, and hence no non-stationary spacetime to invalidate Komar expression! Food for thought maybe.

I don't think that there is a single "correct" mass formula in GR which applies for all spacetimes. Other common mass formulas are the ADM mass, and the Bondi mass, but those also only apply for a subset of possible spacetimes.

Accept that it's horses for courses in that respect, but I thought maybe a better model fitting the shell scenario. Anyway it has clicked for me when answering, and frequently editing my #27 - pressure will easily dominate any 'correcting' redshift factor, and it's easy to prove via a simple scaling argument. But nobody wants to know it seems so too bad.