Is stress a source of gravity?

[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) [itex] \rho_g = \rho + 3p[/itex] 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): [itex]3p = t^0_0[/itex] 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.

 

[Dale]

A chain is as strong as it's weakest link

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

 

[PAllen]

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

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

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

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

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

 

[Q-reeus]

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

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

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

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

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

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

 

[Q-reeus]

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

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

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

 

[Q-reeus]

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

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

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

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

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

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

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