Stress as supposedly conservative source term in GR

[kev01]

Given the complete absence of any meaningful feedback to my challenge to self-consistency of GR beginning at post #5 to evidently now permanently departed Yablon's
viewtopic.php?t=47
, and given the moribund state of SPF in general....
Here's an intended change of direction injection from stale and stalled 'Bell was an idiot' monotone theme at SPF.

Back to GR and imo foundational problems with it's mantra. One being the 'guaranteed fact' that 'stress-energy tensor' is 'divergenceless' - i.e. conservation of energy-momentum is foundational truth in GR - absent 'cosmic expansion' caveat.
Here's a very simple counterexample I came up with back in 2011 at PhysicsForums.com:
https://www.physicsforums.com/threads/c ... ke.498821/
As is evident, the then DaleSpam now Dale took a "Phineas zealot" position reminiscent of Numbers 25:6-13
Later, that same mod zealot enthusiastically participated in a thread explicitly claiming violation of conservation of energy, but took no thread-locking action. Go figure.

Hopefully, here, where ostensibly there is some true freedom to express 'heretical views', those conversant to some extent in GR will attempt to continue on from where that PF thread was savagely cut short. If you think you have found an easy reconciliation with 'divergenceless of stress-energy tensor' - good luck.
My claim is you will finish up with egg on face. Any takers here?

[FrediFizzx]

Given the complete absence of any meaningful feedback to my challenge to self-consistency of GR beginning at post #5 to evidently now permanently departed Yablon's
viewtopic.php?t=47
, and given the moribund state of SPF in general....
Here's an intended change of direction injection from stale and stalled 'Bell was an idiot' monotone theme at SPF.

Back to GR and imo foundational problems with it's mantra. One being the 'guaranteed fact' that 'stress-energy tensor' is 'divergenceless' - i.e. conservation of energy-momentum is foundational truth in GR - absent 'cosmic expansion' caveat.
Here's a very simple counterexample I came up with back in 2011 at PhysicsForums.com:
https://www.physicsforums.com/threads/c ... ke.498821/
As is evident, the then DaleSpam now Dale took a "Phineas zealot" position reminiscent of Numbers 25:6-13
Later, that same mod zealot enthusiastically participated in a thread explicitly claiming violation of conservation of energy, but took no thread-locking action. Go figure.

Hopefully, here, where ostensibly there is some true freedom to express 'heretical views', those conversant to some extent in GR will attempt to continue on from where that PF thread was savagely cut short. If you think you have found an easy reconciliation with 'divergenceless of stress-energy tensor' - good luck.
My claim is you will finish up with egg on face. Any takers here?

LOL! Hi Kev, yeah nobody wants to talk much on this forum anymore. Whatever. It's like a blog for me.

Happy Thanksgiving to all!

[kev01]

Given the complete absence....

LOL! Hi Kev, yeah nobody wants to talk much on this forum anymore. Whatever. It's like a blog for me.

Happy Thanksgiving to all!

Fair enough Fred. Still, I will take this opportunity to add a bit to OP. An alternate angle on the basic argument.
Take Birkhoff's Theorem. It states that the external gravitational field of a spherically symmetric mass distribution is entirely unaffected by any symmetric variations in e.g. density or radial velocity distributions. But is it fundamental Truth or merely a postulate? Let's test it via a specific example involving elastic stress, and see.
For instance, suppose a self-supporting thin elastic shell, say a perfectly spherical 'X-mas tree bauble' is placed in breathing mode radial oscillations.

The wall material will be under ~ sinusoidal wrt time, uniform transverse elastic stress/strain. Compression to tension back to compression...etc.
Clearly spherical symmetry precludes any contribution to external field variation from gross mass motion.
Further, there is a perfectly conservative interplay between elastic PE and KE - the TE sum is invariant (notional loss-free case).
Which means no contribution - there - to any gravitational wave source moments - monopole, dipole, quadrupole etc. Birkhoff's Theorem seems safe.
However...

At the time point of maximum compressive stress, there is a net positive stress contribution that is monopole moment in nature. And equally negative when under maximum tensile stress.
That there is no possible counteracting 'stress current' somehow analogous to the KE term in earlier mentioned elastic PE/KE -> invariant TE, simply vary the Young's modulus. The stiffer the bauble wall material, obviously the smaller the vibrational amplitude for a given maximal stress excursion.
That has no effect on the amplitude of stress contribution - which is simply linearly dependent on stress level. Unlike quadratic dependence for elastic energy density.

For an engineering solid e.g. steel, it's easy to show that stress as source is typically many orders of magnitude larger than corresponding elastic energy density as source.
From that it's clear that unlike all the other terms in RHS stress-energy-momentum tensor, stress as source cannot obey any conservative continuity relation! A cube of rubber compressed to the same final dimensions as a cube of diamond will require many orders of magnitude more energy to get to the same final stress state as the diamond. Yet both yield the same stress-only source contribution. Magic? A 'peculiarity' evidently not noticed in GR community.

We have a very straightforward counterexample to validity of Birkhoff's Theorem! Supposedly impossible monopole gravitational radiation! Provided of course stress is actually a source as per GR (and other rival theories for that matter).

Owing to the seeming 'magical' nature of stress as source, for quite some time I doubted stress was a bona fide source. Until one day, did a perturbative evaluation of a gas filled container subject to a uniform g field. Whereas a quadratic dependence of added gravitational passive mass on gas pressure was expected in accordance with pneumatic energy density, actually the first order contribution was linear in gas pressure. Hence stress/pressure is indeed a source of so-called passive gravitational mass at least.
Proving the same holds for active gravitational mass is more difficult to achieve but one assumes an equality will apply.

[FrediFizzx]

Given the complete absence....

LOL! Hi Kev, yeah nobody wants to talk much on this forum anymore. Whatever. It's like a blog for me.

Happy Thanksgiving to all!

Fair enough Fred. Still, I will take this opportunity to add a bit to OP. An alternate angle on the basic argument.
Take Birkhoff's Theorem. It states that the external gravitational field of a spherically symmetric mass distribution is entirely unaffected by any symmetric variations in e.g. density or radial velocity distributions. But is it fundamental Truth or merely a postulate? Let's test it via a specific example involving elastic stress, and see.
For instance, suppose a self-supporting thin elastic shell, say a perfectly spherical 'X-mas tree bauble' is placed in breathing mode radial oscillations.

The wall material will be under ~ sinusoidal wrt time, uniform transverse elastic stress/strain. Compression to tension back to compression...etc.
Clearly spherical symmetry precludes any contribution to external field variation from gross mass motion.
Further, there is a perfectly conservative interplay between elastic PE and KE - the TE sum is invariant (notional loss-free case).
Which means no contribution - there - to any gravitational wave source moments - monopole, dipole, quadrupole etc. Birkhoff's Theorem seems safe.
However...

At the time point of maximum compressive stress, there is a net positive stress contribution that is monopole moment in nature. And equally negative when under maximum tensile stress.
That there is no possible counteracting 'stress current' somehow analogous to the KE term in earlier mentioned elastic PE/KE -> invariant TE, simply vary the Young's modulus. The stiffer the bauble wall material, obviously the smaller the vibrational amplitude for a given maximal stress excursion.
That has no effect on the amplitude of stress contribution - which is simply linearly dependent on stress level. Unlike quadratic dependence for elastic energy density.

For an engineering solid e.g. steel, it's easy to show that stress as source is typically many orders of magnitude larger than corresponding elastic energy density as source.
From that it's clear that unlike all the other terms in RHS stress-energy-momentum tensor, stress as source cannot obey any conservative continuity relation! A cube of rubber compressed to the same final dimensions as a cube of diamond will require many orders of magnitude more energy to get to the same final stress state as the diamond. Yet both yield the same stress-only source contribution. Magic? A 'peculiarity' evidently not noticed in GR community.

We have a very straightforward counterexample to validity of Birkhoff's Theorem! Supposedly impossible monopole gravitational radiation! Provided of course stress is actually a source as per GR (and other rival theories for that matter).

Owing to the seeming 'magical' nature of stress as source, for quite some time I doubted stress was a bona fide source. Until one day, did a perturbative evaluation of a gas filled container subject to a uniform g field. Whereas a quadratic dependence of added gravitational passive mass on gas pressure was expected in accordance with pneumatic energy density, actually the first order contribution was linear in gas pressure. Hence stress/pressure is indeed a source of so-called passive gravitational mass at least.
Proving the same holds for active gravitational mass is more difficult to achieve but one assumes an equality will apply.

Well, do you have any math for the above? But you need to add in the consideration of gravitational torsion. General Relativity as usually stated is in fact not as general as it can be unless you allow torsion. Curvature-torsion is like electric field-magnetic field. A connected duality.
.

[kev01]

Well, do you have any math for the above? But you need to add in the consideration of gravitational torsion. General Relativity as usually stated is in fact not as general as it can be unless you allow torsion. Curvature-torsion is like electric field-magnetic field. A connected duality.
.

Torsion unless I'm mistaken is non-projecting. That is, it doesn't effect the gravitational field exterior to a source distribution. And even within a material body, is only relevant at a microscopic level i.e. interatomic spin-level interactions. Many GR buffs deny it exists at all - but for obvious reasons I won't rely on a vote of hands as authoritative rebuttal.

As for 'where is the math' - it's not needed per se. I gave the killer insight; 'just change the Young's modulus'. And the spherical symmetry of the particular scenario eliminates any expected potentially spurious counterclaims quite neatly. There is only the stress contribution to consider. And it does contribute! Birkhoff's Theorem bites the dust. And much more besides of course.

The 'magical' nature of stress as source comes out from examining the stress-energy-momentum tensor terms. The 00 energy density term will be elastic energy density ~ 0.5*stress*strain, plus the corresponding KE 0.5*density*(radial velocity)^2, and it's net time variation sums to zero for a loss-free oscillator.

Compare to ii stress terms. Contribution there is ~ stress. Just stress. It's greater than for elastic energy density by a factor of 2/(strain amplitude).
I have pointed out stress as source for stiff solids has that 'magical' quality of requiring negligible energy input to generate. Is that not bleeding obvious?

Go back to the spinning 'disk brake' with frictionless clamp pads. I never got to state the killer insight there, but it's the same; 'just change the Young's modulus'. Not obvious what the implications is in both scenarios? I spelt it out for oscillating bauble scenario. How can one recover conservation of energy-momentum in either situation?
You can't.

[kev01]

Looks like a deja vu case of blank stares = silent argument from incredulity = 'don't know where the error is but it must be there coz GR is just total Truth'. Sigh.

One more stab at getting some kind of useful feedback here at SPF. Below is reproduced Quote/response from post #3 at
https://www.physicsforums.com/threads/c ... ke.498821/

"Bill_K said: Your error consists in assuming that the forces at the compression point are horizontal. The motion of the disc at that point is downward. Above the point the disc is uncompressed, and compressed below it. To become compressed, it must be forcibly driven through the point. The brakes will have to exert a force with a small upward component, and this will tend to slow the rotation."

"It would be helpful to indicate to which part of my entry is being discussed, but I will take a reasonable guess and say it is the middle portion dealing with steady rotation of a clamped disk acted on by gravity. True there must be strain associated with stress, and so the disc under the pads takes on, very very slightly, an 'hour-glass' shape. But it is symmetric - repulsive forces one end is balanced by oppositely acting repulsive forces the other. The unstated assumption here was that the disk material experiences no hysteresis - no internal friction. Hence the flow of matter through the stressed region is conservative wrt purely elastic deformations. To assume otherwise then flow of a notionally inviscid fluid within a vertical tube section having a Venturi shaped portion would also experience a net upward force - violating the conservation of energy purely at the Newtonian level! Sorry, but off to bed."

Firstly, an acknowledgement that my above response was not entirely sufficient or entirely correct. I had properly answered Bill K at the level of his faulty argument, but....Owing to immersion in the applied g field, there is a vertical gradient in gravitational potential. Which means 'red-shifting' is operative. Not just of light, but applied forces. Two individuals in a tug-of-war at different elevations will result in the higher up one winning even though both are locally pulling equally on the connecting (negligible mass) rope.

That carries over to the slightly deformed disc with 'hour glass' deformation profile. In a zero g environment, the slight net vertical (in the disk plane) components of pinching forces will be symmetrically disposed and no net force in the plane of the disk results. Add gravity and 'red-shifting' kicks in. Result - there is a very slight excess of upward pinch force vs downward pinch force. Which will indeed oppose the excess downward gravitational force induced via disk 'brake pads' applied pressure.

The induced deformation will be directly proportional to applied pads pressure, thus also the tiny component of pinch forces acting in the disk plane.
Hence the net pinch force excess goes as the square of applied transverse compressive stress - once for stress level, once again for strain induced pinch angular component in the disk plane. Thus, without doing explicit calculations, one can say the quadratic in stress net pinch force functionally tracks with excess stored elastic energy density.

Which is precisely why pinch force excess cannot account for the much larger stress only contribution to gravitational excess force. Since it rises linearly not quadratically with stress amplitude! Add to that the earlier mentioned several times 'just vary Young's modulus'.
Not clear now what the strength of that argument is? Well - just ask how the tiny vertical components of pinch forces will vary as the disk material stiffness varies! Trends to zero as stiffness rises. Kind of obvious with a little thought - no?
So a double whammy for anyone trying to save 'divergenceless of stress-energy-momentum tensor'. See - no reams of complex math needed. Really.

What has been presented is qualitative for the most part but that's entirely sufficient since it hinges on logical factors covered quite adequately.
Anyone care to disagree? Yeah I know - no point trying to flog a dead horse. You all 'know' GR = Divine Truth even if the 'flaw' in my argument(s) is sort of 'not obvious'.

Cheers