Is stress a source of gravity?

[PeterDonis]

Below is my response to your #168:

There's no use in my responding point by point to most of this. First a general comment: I have already read through every post of yours of any length in this thread multiple times. Any information that I could possibly extract from those posts by repeated reading, I have already extracted; so if I'm still asking questions, it means no amount of additional reading of past posts is going to answer them. I'm not going to go back and read them any more; it won't do any good. If you can't figure out a short, succinct way of stating your arguments, then I simply don't have anything more that's useful to contribute to this discussion.

That said, there are a couple of quick specific items:

It was all about redistribution of energy - forming a mass-energy Q moment (or not, as I argued it could be avoided by careful placement)

If you do work on a system, you increase the overall integrated T_00 component of its SET (its energy). You can't "avoid" that by "careful placement" or "redistribution of energy". You can change how the increased T_00 is *distributed* within the material, but that doesn't matter for the Komar mass integral, which just sums all the individual pieces up anyway.

Firstly, screwed G-clamp legs provide arbitrarily high mechanical advantage.

Mechanical advantage reduces the force required from the "engine" (in your example, your thumb doesn't have to exert as much force). It does not change the amount of work that has to be done by the "engine" on the object being compressed; the reduced force simply gets exerted over a longer distance (in the case of a screw clamp, the distance over which the reduced force of the "engine", or your thumb, is exerted is not the linear distance traveled by the clamp end but the helical distance around the screw threads; that's what creates the large mechanical advantage). The total work done is what determines the amount of increase in the T_00 component of the Komar mass integral, what I was calling "energy stored in the material". Mechanical advantage is irrelevant.

 

[PeterDonis]

Of course it is the vacuum situation - there was never any case to reasonably conclude otherwise from #1 onward, and way back in #51 made it explicit.

Yes, I know you did, but then you keep making arguments which do not apply to that case. As in:

And would have hoped after so much discussion around this I have made it clear the role of referring to gas filled shell cases. To illustrate the general feature that a normal acting applied force, whether from gas acting on a surface, or inertial owing to the radial acceleration of the shell itself, is balanced by tangential (circumferential, azimuthal, whatever term you are comfortable with) hoop stresses, not radial stresses, within the shell.

Which shows that you completely missed the point of my last post: in a vacuum filled shell, the tangential stresses ADD to the "radial acceleration of the shell itself" due to the shell's self-gravity. They do NOT "balance" it. The "normal" component created from the tangential stresses by the curvature of the shell points INWARD, *not* outward--i.e., it points IN THE SAME DIRECTION as the "radial acceleration" caused by the shell's self-gravity. The *only* thing that can *balance* the shell's self-gravity is *outward radial pressure within the shell material itself*. If you can't comprehend this, then I don't see how we can have a useful discussion; you and I are simply talking past each other.

In static equilibrium there is just an unstressed shell sitting there doing nothing. That's it.

Yes. Which means that any small oscillations about this equilibrium will be small perturbations of an unstressed shell sitting there doing nothing--i.e., will average out to nothing. So what's the point?

The whole lesson from the analysis done in the article you refer to is that determining a proper force balance here necessarily involves curvilinear stresses acting over a finite area/volume

Yes, I explicitly said I wasn't disputing this.

Nevertheless, the balance is not some artifact of choosing just the right size of element as you seem to suggest - it holds for arbitrary size. Just check!

If you really want to harp on this, it should be a separate thread. I have already explicitly said that for this discussion I am fine with using the force balance as presented in the set of slides you linked to.

Huh!? As a reactive elastic structure, stress acts automatically either inward (internal gas pressure case) or outward (self-gravity case) to oppose whatever tends to deform away from equilibrium. The direction is allowed to change according to circumstance!

Yes, I know that.

[I should add here your use of 'inward' and 'outward' needs defining. From context it apparently references to the shell radius vector at the location considered. I would have used those terms as referring to the sign of tangent stresses acting across an elemental shell volume's edges - outward for tensile, inward for compressive. But I shall use your definition here.]

Look at the slides you linked to, where it shows how the tangential stresses across a curved shell element produce a normal component. That component points in the *opposite* direction to the pressure of the gas inside the shell. I was using the word "outward" to describe the direction of the gas pressure inside the shell, and "inward" to describe the (opposite) direction of the normal component of the shell stress on the curved element. It should be obvious that these terms are apt, since "inward" points towards the center of the sphere and "outward" points away from it. And by that same definition, the shell's self-gravity points inward--i.e., it points in the *same* direction as the normal component of the shell stress on the curved element. So those two things cannot possibly "balance" each other--they point in the *same* direction!

Result - radial component of outwardly acting biaxial tangent stresses

But those stresses act INWARD, *not* outward! Or if you are still hopelessly confused about the meanings of "inward" and "outward", the radial component of the shell stress points IN THE SAME DIRECTION as the shell's self-gravity. That's the critical point.

I think it's best to rest the vibrating shell in vacuum issue for now...

It's up to you; you're the one that originally proposed that scenario. If the discussion is going to keep going along the lines it has up to now, I agree it's pretty pointless; apparently we can't even use the words "inward" and "outward" and agree on their meaning.

the case of a self-gravitating stationary shell. Where it is agreed by all that Komar mass expression is supposed to hold. That was in fact just what got initially looked at here: https://www.physicsforums.com/showthread.php?t=541317

You may recall there was ongoing dispute over whether shell stresses could explain the metric transition from Schwarzschild shell exterior to Minkowski interior. I maintained extreme scepticism, but either way it never got to be solved. DaleSpam recently made a try but had to quit. So given the stationary nature of the problem, and that stress plays a crucial role there also, would you be agreeing to try and solve that case, for which a specific scenario was proposed in #17 there.

Sure, I'll go back and refresh my memory about where that discussion left off, and try to pick up from there.

Personally I would prefer a more general model that allows scaling effects to be checked.

If by "more general model" you mean "a model that includes small oscillations about the stationary equilibrium state", we would first have to agree on exactly what our measure of "energy conservation" is going to be. It can't be the Komar mass since that is only conserved for stationary systems. We could use the ADM mass or the Bondi mass, but those are insensitive to the details of the metric inside a bounded region if it looks asymptotically the same as you go to infinity. Which is kind of the point. If instead we are going to just look at the covariant divergence of the SET and check whether it is zero, that's fine, but you seem to think there's some kind of sleight of hand going on there, and I don't really understand why.

In short, I think the root questions at issue here are more general questions about what should "count" as a measure of "energy" or "mass" in GR, rather than specific questions about specific models.

Only thing is it would need to be specified as to what material model was used. A solid shell is assumed there, but if perfect fluid is needed for getting a solution, one must make the situation physically stable - 'ocean' on top of supporting thin shell 'earth' perhaps, with support shell of negligible mass.

I agree the material model needs to be better specified. However, I would also like to keep it simple enough that we can express things analytically. That basically means a "quasi-perfect fluid" SET, as I described earlier: only diagonal components (at least in the stationary equilibrium), but T_11 (radial pressure) does not have to equal T_22 = T_33 (tangential stress--the two components of that do have to be equal by spherical symmetry). I believe this can still produce a static equilibrium with appropriate stress distributions within the material.

I would also impose standard energy conditions: the strictest (and the one I would choose) would be to require that T_11 and T_22 = T_33 are both less than 1/3 of T_00.

 

[PeterDonis]

My suspicion remains that while Birkhoff's theorem applies in terms of outcome, it's for the wrong reasons - top down imposition that makes SET terms do strange acrobatics in e.g. vibrating shell case.

On re-reading I realized this deserves a separate response. You are misunderstand what Birkhoff's Theorem says. It doesn't say anything about the metric or the SET or anything else inside the interior non-vacuum region of the spacetime. It only says that as long as everything is spherically symmetric, the metric in the exterior *vacuum* region must be Schwarzschild. Therefore, since the Schwarzschild metric contains no GWs, the exterior of any spherically symmetric matter distribution, no matter how gnarly it gets inside, cannot contain GWs either.

Birkhoff's Theorem does not tell you *how* the gnarliness inside the spherically symmetric matter distribution gets contained so it doesn't produce GWs; it just tells you that it must get contained somehow *if* the distribution is spherically symmetric. As I've said before, what's really doing the hard work is the assumption of spherical symmetry, which is *very *restrictive on the matter distribution and the kinds of vibrations it can undergo.

 

[Q-reeus]

If you do work on a system, you increase the overall integrated T_00 component of its SET (its energy). You can't "avoid" that by "careful placement" or "redistribution of energy". You can change how the increased T_00 is *distributed* within the material, but that doesn't matter for the Komar mass integral, which just sums all the individual pieces up anyway.

But the power source (batteries) is counted as part of the system! Seems you figured it was external. Not so. And that's where a reread would help.

Mechanical advantage reduces the force required from the "engine" (in your example, your thumb doesn't have to exert as much force). It does not change the amount of work that has to be done by the "engine" on the object being compressed; the reduced force simply gets exerted over a longer distance (in the case of a screw clamp, the distance over which the reduced force of the "engine", or your thumb, is exerted is not the linear distance traveled by the clamp end but the helical distance around the screw threads; that's what creates the large mechanical advantage). The total work done is what determines the amount of increase in the T_00 component of the Komar mass integral, what I was calling "energy stored in the material". Mechanical advantage is irrelevant.

You missed the context here. Nothing to do with the T₀₀ shuffling. Everything to do with generating the T_ii stresses. Relevant passage again:

Second, as you point out, all the stress contributions to the integral have to cancel in static equilibrium; in Q-reeus' example, the stress contribution from the material has to be canceled by an equal and opposite stress contribution from the "engine" (whatever it is that is applying the force to the material). This is just the argument from the Ehlers paper that Q-reeus linked to, generalized...

You said there stress contributions (T_ii terms), - not elastic energy (part of T₀₀). So you actually meant the latter? Better to have been more specific if so. What made me certain you meant stress as stress, was equating it to the engine (motor) as force applier - *not* the batteries as energy source. It all added up to your arguing T_ii contributions in the lightly stressed motor somehow canceled much larger T_ii contributions in the clamps. So what were you meaning - T₀₀ balance, or T_ii balance?

 

[Q-reeus]

If by "more general model" you mean "a model that includes small oscillations about the stationary equilibrium state", we would first have to agree on exactly what our measure of "energy conservation" is going to be. It can't be the Komar mass since that is only conserved for stationary systems.

Keeping my commentary on #182 mercifully short, I'll just clarify this point. By more general I meant stick with the strictly static case (hence Komar fine), but make shell radius, thickness, and material density arbitrary parameters rather than fixed values as per #17; i.e. symbolic not numeric values. Would that be OK?

 

[Q-reeus]

On re-reading I realized this deserves a separate response. You are misunderstand what Birkhoff's Theorem says. It doesn't say anything about the metric or the SET or anything else inside the interior non-vacuum region of the spacetime. It only says that as long as everything is spherically symmetric, the metric in the exterior *vacuum* region must be Schwarzschild. Therefore, since the Schwarzschild metric contains no GWs, the exterior of any spherically symmetric matter distribution, no matter how gnarly it gets inside, cannot contain GWs either.

Birkhoff's Theorem does not tell you *how* the gnarliness inside the spherically symmetric matter distribution gets contained so it doesn't produce GWs; it just tells you that it must get contained somehow *if* the distribution is spherically symmetric. As I've said before, what's really doing the hard work is the assumption of spherical symmetry, which is *very *restrictive on the matter distribution and the kinds of vibrations it can undergo.

I get the drift, but it cannot just be 'If it's spherically symmetric - nothing else to think about '. There must be at minimum one extra assumption - the interior contents aren't allowed to magically change in net size over time. Conservation laws must internally hold - and that is an extra assumption. And that imo brings it back to role of stress ('conservation of stress'?). :zzz:

 

[PAllen]

I get the drift, but it cannot just be 'If it's spherically symmetric - nothing else to think about '. There must be at minimum one extra assumption - the interior contents aren't allowed to magically change in net size over time. Conservation laws must internally hold - and that is an extra assumption. And that imo brings it back to role of stress ('conservation of stress'?). :zzz:

Actually, there is no extra assumption. Spherical symmetry + vacuum outside a closed surface + Einstein field equatiorns + asymptotic flatness forces SC geomtry outside the closed surface. Put another way, conservation is consequence of EFE and the definition of the relevant tensors. GR is much more constraining than older theories. It is not possible, in principle, to suppose mass suddenly appears somewhere. This is why it is quite difficult in GR to describe propagation speed of gravitational force distinct from propagation of gravitational waves.

 

[PeterDonis]

But the power source (batteries) is counted as part of the system!

Yes, I know. Before the compression, the energy is in the T_00 of the batteries. After the compression, the same energy is stored in the material as additional T_00. The process of clamping, with respect to T_00, just transfers the energy from the batteries to the compressed material. So the T_00 contribution to the Komar mass integral is a wash; it's the same both before and after. That was my point.

It all added up to your arguing T_ii contributions in the lightly stressed motor somehow canceled much larger T_ii contributions in the clamps.

I was arguing for T_ii balance, but I wasn't trying to specifically argue "where" the T_ii is located at various stages. I was simply referring to Jonathan Stone's (valid) argument that, in static equilibrium (and both the "initial" and "final" states are states of static equilibrium--I was only referring to those states, not to what happens in between), the overall T_ii contributions *have* to cancel; if they didn't, there would be an unbalanced force somewhere in the system and there would not be a static equilibrium. So the T_ii contribution to the Komar mass integral is also a wash: it's zero before, and it cancels (and is therefore zero) after.

Keeping my commentary on #182 mercifully short, I'll just clarify this point. By more general I meant stick with the strictly static case (hence Komar fine), but make shell radius, thickness, and material density arbitrary parameters rather than fixed values as per #17; i.e. symbolic not numeric values. Would that be OK?

Yes, I agree, all parameters have to be variable (but they can only be functions of radius r, by spherical symmetry and the fact that the system is static).

 

[Dale]

First, I see from #175 you consider there to have been no personal issues.

It isn't that there are no personal issues, just that they are not relevant. I know you don't like me and don't respect me; you have made your dislike and disrespect abundantly clear. But as long as you post on PF then I will respond anyway and point out whatever I think is wrong with your claim du-jour.

My feeling is the SET zero divergence in GR is a top-down imposition that in the case of vibrating shell, means implicitly forcing what I would term the proper SET terms (T₀₀, T_i0, T_0i) into strange acrobatics in order to counter what I consider the improper SET terms (entire 9-member stress portion) physically implausable attributes.

OK, the fact that the SET seems strange to you is not terribly worrisome. Since you recognize that the SET has 0 divergence then I don't see that your scenario [2] is making any counter-GR claims. That you could produce GW's by carefully designed stresses or that the amplitude of those GW's would depend not only on the stress but also the stiffness of the material seem to be reasonable claims, IMO.

Throughout this thread it is only your scenario [1] that I have objected to.

 

[Dale]

I get the drift, but it cannot just be 'If it's spherically symmetric - nothing else to think about '. There must be at minimum one extra assumption - the interior contents aren't allowed to magically change in net size over time. Conservation laws must internally hold - and that is an extra assumption.

Well, the other assumption would clearly be that the EFE holds, otherwise you are not doing GR. If the EFE holds then the SET is automatically divergence-free, without requiring an additional assumption. So Birkhoff's theorem applies to any spherically symmetric exterior solution to the EFE. Magically induced spacetimes would not, in general, be solutions to the EFE.

 

[PAllen]

Actually, there is no extra assumption. Spherical symmetry + vacuum outside a closed surface + Einstein field equatiorns + asymptotic flatness forces SC geomtry outside the closed surface. Put another way, conservation is consequence of EFE and the definition of the relevant tensors. GR is much more constraining than older theories. It is not possible, in principle, to suppose mass suddenly appears somewhere. This is why it is quite difficult in GR to describe propagation speed of gravitational force distinct from propagation of gravitational waves.

Correcting this a bit: Only exterior vacuum (which excludes global EM fields), EFE, and spherical symmetry are assumed. That the result is asymptotically flat, static, and uniquely the SC geometry are all consequences.

Note also, that GW are vacuum (not part of SET), thus spherically symmetric GW are strictly prohibited.

 

[Q-reeus]

Yes, I know. Before the compression, the energy is in the T_00 of the batteries. After the compression, the same energy is stored in the material as additional T_00. The process of clamping, with respect to T_00, just transfers the energy from the batteries to the compressed material. So the T_00 contribution to the Komar mass integral is a wash; it's the same both before and after. That was my point.

But just in respect of internal energy flow from battery to elastic energy, it has always been my point too, so where was the argument then? I have though from the start included another factor here. That flow results, in general, in a changed quadrupole moment, therefore a changed field energy, which must be part of the overall energy balance. So not exactly 100% of battery energy goes to elastic. And further, when stress-only contribution to field is factored in, I maintain there is nominally no longer an overall energy balance. My cure, from the start, is to discount T_ii terms as field source.

I was arguing for T_ii balance, but I wasn't trying to specifically argue "where" the T_ii is located at various stages. I was simply referring to Jonathan Stone's (valid) argument that, in static equilibrium (and both the "initial" and "final" states are states of static equilibrium--I was only referring to those states, not to what happens in between), the overall T_ii contributions *have* to cancel; if they didn't, there would be an unbalanced force somewhere in the system and there would not be a static equilibrium. So the T_ii contribution to the Komar mass integral is also a wash: it's zero before, and it cancels (and is therefore zero) after.

Cancellation of stresses re force balance in a static body, equates to cancellation of Komar mass contribution from stress, only imo if one accepts that:

a) Field energy is discounted as source. There nominally exists in general a non-zero field and thus field energy owing to the T_ii quadrupolar distribution. Field energy is a parametric function of field strength - always positive regardless of sign of 'stress charge'.
[EDIT: As per my #168, this is only part of the equation. Interaction terms involving the much larger, always positively signed rest-energy T₀₀ field contributions will alter sign according to sign of T_ii terms. In the absense of local biasing effects (see b) below), such interaction terms cancel out exactly, leaving only that from the individual contributing terms. But biasing to some measure is an expected ubiquitous feature.]
The schizophrenic GR position is that such energy makes no contribution to the Komar mass, and yet GW's field energy *is* counted in the ADM mass - as per #131. There is some consistency here? Now when altered rest-energy from potential shift owing to T_ii's is factored in, sign of net change reverses. [Just got round to following up Mentz114's link in #92 to an arXiv article http://arxiv.org/abs/gr-qc/0607087 that discusses that aspect] But that brings us to the second reason.

b) Local depression/boosting of rest-energy owing to gravitational potential of T_ii terms are ignored. This was discussed in #173. Take again the example there of placing an unstressed sleeve around one G-clamp leg. Rest energy of that sleeve will be altered much more by the stress in enclosed leg than by the stress in the other legs. This is a local biasing effect not taken into account when applying the too simple rule 'stress contributions cancels out in a static body'. So even agreeing to field energy being omitted as source term, depression/boosting of rest-energy as per above is still there, and in fact would give twice the net change by discounting field energy as source.

 

[Q-reeus]

Actually, there is no extra assumption. Spherical symmetry + vacuum outside a closed surface + Einstein field equatiorns + asymptotic flatness forces SC geomtry outside the closed surface. Put another way, conservation is consequence of EFE and the definition of the relevant tensors. GR is much more constraining than older theories. It is not possible, in principle, to suppose mass suddenly appears somewhere. This is why it is quite difficult in GR to describe propagation speed of gravitational force distinct from propagation of gravitational waves.

Correcting this a bit: Only exterior vacuum (which excludes global EM fields), EFE, and spherical symmetry are assumed. That the result is asymptotically flat, static, and uniquely the SC geometry are all consequences.

Note also, that GW are vacuum (not part of SET), thus spherically symmetric GW are strictly prohibited.

I understand your position, but would be inclined to question the correctness of EFE's, in that above set of conditions. Because, shifting to the G-clamps case, it has to be agreed that as stiffness increases, the purported T_ii contributions generate a given amount of field energy for less and less energy input. Again, it doesn't matter if infinite stiffness, and hence zero input energy, is unobtainable even in principle. Graph the various acknowledged SET contributions. Totally different but, in the range of interest, monotonic functions of stiffness E. Claim everything balances energy-wise at some value of E. Just change E and 'balance' is lost. Cure - T_ii's do not act as source terms. All said umpteen times already. I'm quite aware no-one else agrees, but that's a blindingly obvious conclusion for me.

 

[TrickyDicky]

I understand your position, but would be inclined to question the correctness of EFE's, in that above set of conditions. Because, shifting to the G-clamps case, it has to be agreed that as stiffness increases, the purported T_ii contributions generate a given amount of field energy for less and less energy input. Again, it doesn't matter if infinite stiffness, and hence zero input energy, is unobtainable even in principle. Graph the various acknowledged SET contributions. Totally different but, in the range of interest, monotonic functions of stiffness E. Claim everything balances energy-wise at some value of E. Just change E and 'balance' is lost. Cure - T_ii's do not act as source terms. All said umpteen times already. I'm quite aware no-one else agrees, but that's a blindingly obvious conclusion for me.

The way I see it, there seems to be a very basic misunderstanding here, it is not possible to question the correctness of the EFE on the grounds of claiming the T_ii's do not act as source terms the way the whole stress-energy acts as a source. That is negating the mathematical meaning of the tensorial form of the equations.
It has been mentioned in this thread that what is strictly conserved under any circumstance according to the vanishing divergence of the stress-energy tensor is the momentum-energy as a whole, not being possible in general and for every circumstance (save the static or stationary cases already commented with timelike killing vectors that allow for the conservation of energy or mass or stress terms in themselves to be strictly defined) to single out specific components of the stress-energy tensor as sources in the same way the whole tensor as a mathematical object acts as the source of curvature. By their very nature the specific components of a tensor can't behave the same way the tensor itself. That doesn't necessarily mean that those components don't act as part of the source, simply that there's no meaningful way to express their conservation properties or their direct relation to curvature under every circumstance the way it is done with the stress-energy tensor's energy-momentum itself.

 

[PAllen]

I understand your position, but would be inclined to question the correctness of EFE's, in that above set of conditions. Because, shifting to the G-clamps case, it has to be agreed that as stiffness increases, the purported T_ii contributions generate a given amount of field energy for less and less energy input. Again, it doesn't matter if infinite stiffness, and hence zero input energy, is unobtainable even in principle. Graph the various acknowledged SET contributions. Totally different but, in the range of interest, monotonic functions of stiffness E. Claim everything balances energy-wise at some value of E. Just change E and 'balance' is lost. Cure - T_ii's do not act as source terms. All said umpteen times already. I'm quite aware no-one else agrees, but that's a blindingly obvious conclusion for me.

Well, the stress energy tensor has no meaning outside the EFE's. Further, the various ideas you have about how components of SET (which I usually just call T) contribute to gravity come from the EFE; more specifically, these simple heuristics are approximate conclusions of the EFE for special cases. Thus your methodology amounts to claiming a problem with EFE due to the fact that approximate conclusions of it, applied outside their domain of legitimacy, disagrees with exact conclusions of the EFE. Do you see that this is a logically absurd postion?

 

[Q-reeus]

The way I see it, there seems to be a very basic misunderstanding here, it is not possible to question the correctness of the EFE on the grounds of claiming the Tii's do not act as source terms the way the whole stress-energy acts as a source. That is negating the mathematical meaning of the tensorial form of the equations.
It has been mentioned in this thread that what is strictly conserved under any circumstance according to the vanishing divergence of the stress-energy tensor is the momentum-energy as a whole, not being possible in general and for every circumstance (save the static or stationary cases already commented with timelike killing vectors that allow for the conservation of energy or mass or stress terms in themselves to be strictly defined) to single out specific components of the stress-energy tensor as sources in the same way the whole tensor as a mathematical object acts as the source of curvature. By their very nature the specific components of a tensor can't behave the same way the tensor itself. That doesn't necessarily mean that those components don't act as part of the source, simply that there's no meaningful way to express their conservation properties or their direct relation to curvature under every circumstance the way it is done with the stress-energy tensor's energy-momentum itself.

Your point is made well and has been before, as also by PAllen in #195 and elsewhere, and by others. Accepting it means though that by definition there cannot be even in principle any independent check for self-consistency - ever. Probably time to retire this one-man crusade. But I will be interested to see what PeterDonis comes up with if the proposed resolution of static shell metric transition issue comes to pass. Stress distributions that are needed to patch from outer to inner - this I need to see. Must go :zzz:

 

[PeterDonis]

That flow results, in general, in a changed quadrupole moment, therefore a changed field energy, which must be part of the overall energy balance.

No, it isn't. The quadrupole moment affects how individual bits of energy (T_00) are distributed in the material, but it doesn't affect the overall "energy" (T_00 term) in the Komar mass integral, because the integral just sums up all the little bits anyway. Rearranging the bits doesn't change the sum. So:

So not exactly 100% of battery energy goes to elastic.

Incorrect; 100% of the battery energy does go to elastic energy *somewhere* in the material. Changing the quadrupole moment changes where, exactly, it goes, but it doesn't change the fact that it goes *somewhere*, which is all that's needed for it to show up in the T_00 term in the Komar mass integral.

And further, when stress-only contribution to field is factored in, I maintain there is nominally no longer an overall energy balance.

T_ii terms in the SET are not energy, they are pressure/stress. To determine whether they are the same "before" and "after", you need to look at pressure/stress balance, i.e., force balance, not energy balance. The energy balance is in T_00, and it is balanced, as I said above. The force balance must be there because the system is in static equilibrium, as Jonathan Scott argued (and as the Ehlers paper you linked to also argues, as I've pointed out before).

Cancellation of stresses re force balance in a static body, equates to cancellation of Komar mass contribution from stress, only imo if one accepts that:

a) Field energy is discounted as source.

If self-gravity is neglected, there is no "field energy" regardless of how you define that term. To include "field energy" at all, you have to look at a case where self-gravity is not negligible. Much of your further comments seem to be talking about self-gravity, which makes me confused because I thought you were neglecting it.

There nominally exists in general a non-zero field and thus field energy owing to the T_ii quadrupolar distribution.

The quadrupolar distribution doesn't affect the totals that go into the Komar mass integral; see above. The same logic applies to the T_ii terms here as applies to the T_00 term.

The schizophrenic GR position is that such energy makes no contribution to the Komar mass, and yet GW's field energy *is* counted in the ADM mass - as per #131.

That's because the Komar mass is only applicable when the system is stationary, and if it is stationary, there are no GWs emitted. In that case the Komar mass and the ADM mass are the same. In the non-stationary case, the Komar mass can't be applied, so there's no disconnect with the ADM mass, which can.

I realize this reply won't satisfy you, but that doesn't make it incorrect; it just means it doesn't satisfy you. As long as you keep pointing out things like the above, we are going to continue to use the GR framework to analyze the scenarios you pose, and to point out cases (like the Komar mass only applying in stationary spacetimes) where it doesn't match up well with your intuitions.

Your point is made well and has been before, as also by PAllen in #195 and elsewhere, and by others. Accepting it means though that by definition there cannot be even in principle any independent check for self-consistency - ever.

Why would you want an "independent" check for *self* consistency? Isn't that an oxymoron?

What we have is a mountain of evidence that GR makes correct predictions. That's why we feel justified in believing its predictions for phenomena within its known domain of validity.

But I will be interested to see what PeterDonis comes up with if the proposed resolution of static shell metric transition issue comes to pass. Stress distributions that are needed to patch from outer to inner - this I need to see.

Working on it.

 

[Dale]

I understand your position, but would be inclined to question the correctness of EFE's, in that above set of conditions.

What does this even mean? The EFE are obviously a well-formed set of equations, and we know that there are solutions to the EFE, so in terms of self-consistency they are clearly "correct".

By "correct" you could also mean that they do not agree with experimental evidence. This is, in fact, the only way to actually challenge GR, but you have not presented any such evidence here.

So what do you mean by "the correctness of EFE's"?

 

[PeterDonis]

Well, the stress energy tensor has no meaning outside the EFE's.

This isn't quite true as it's stated; I think you mean that the SET's meaning *within GR* is defined by its appearance in the EFE. The SET itself can be defined without the EFE; it's basically just the appropriate "wrapping up" into a single geometric object of the 3-D stress tensor and the relativistic energy-momentum 4-vector. (I believe there are also ways of deriving an SET as a variational derivative for any form of matter or energy that's describable by a Lagrangian.) But it's only in GR that the SET acts as the "source" on the RHS of a field equation, the EFE. The rest of your post is unaffected by this (probably rather pedantic) clarification.

 

[PAllen]

As one item of interest on this, in an earlier post Peter Donis explained at length how restrictive spherical symmetry is, and several have pointed out how the EFE require SET to have zero divergence (thus local conservation), everywhere. This may be well known to others, but I just came across a derivation in Synge's GR book that these restrictions result in the following:

You can choose the Ttt (T00) and the Trr (T11) components of SET as basically arbitrary functions of r and t. Then, all other components are completely determined. The only other ones that can be nonzero are: T22,T33, T01 and T10 (and, of course, these can only be functions of r and t). There is no freedom at all to adjust these other components while holding T00 and T11 to some value or expression. If you think you have, you have produced something that isn't a stress energy tensor.

 

[PeterDonis]

You can choose the Ttt (T00) and the Trr (T11) components of SET as basically arbitrary functions of r and t. Then, all other components are completely determined. The only other ones that can be nonzero are: T22,T33, T01 and T10 (and, of course, these can only be functions of r and t).

I don't have Synge's book, but this looks like what I've come up with as I work through the math of the static thin spherical shell. The only thing I'm not sure about is the constraint on T22 and T33; as far as I can tell these must be equal under spherical symmetry, but I'm not sure how they're constrained to a specific relationship with T00 and T11. (MTW talks about this some, but the main treatment there appears to be restricted to the perfect fluid case, where T22 = T33 = T11 is imposed as a condition of the model anyway, so I can't tell for sure how general their equations are supposed to be.) Can you give any more specifics about which particular components of either the EFE or the energy conservation condition (covariant divergence of SET = 0) Synge uses to derive a specific relationship between T22 and T33 and the other components?

 

[PAllen]

I don't have Synge's book, but this looks like what I've come up with as I work through the math of the static thin spherical shell. The only thing I'm not sure about is the constraint on T22 and T33; as far as I can tell these must be equal under spherical symmetry, but I'm not sure how they're constrained to a specific relationship with T00 and T11. (MTW talks about this some, but the main treatment there appears to be restricted to the perfect fluid case, where T22 = T33 = T11 is imposed as a condition of the model anyway, so I can't tell for sure how general their equations are supposed to be.) Can you give any more specifics about which particular components of either the EFE or the energy conservation condition (covariant divergence of SET = 0) Synge uses to derive a specific relationship between T22 and T33 and the other components?

I'm going out for the night, but I should be able to do this tomorrow.

 

[pervect]

If one is interested in the "hollow sphere" solution, there's a rather interesting way I thought of to model it, but I haven't taken the time to go through the math (and probably won' - but I thought I'd mention the idea).

You can imagine a hollow sphere as a thin, massless shell under compression (by massless I only mean that rho=0!), supporting a perfect fluid "ocean" above it - a fluid , that because it is perfect, has an isotropic pressure.

Note that if you assume the system is in equilibrium, the pressure at the surface of the fluid ocean must be zero, and this prevents one from just freely specifying the pressure at any given depth - the pressure vs depth is something you can calculate, not something you can just arbitrarily specify. If you want to model a case where the pressure at the surface isn't zero, you'd need another massless shell there to hold the pressure if you want the problem to be static.

If one wants to think about the non-static cases, it's easier to think about the non-relativistic case first, and I think the above picture helps, dividing it into an "ocean", with isotropic pressure, and a "support", some pressure vessel. But I'm not going to discuss it much more, except to say that there would obviously be sound / pressure waves traveling through the fluid at whatever the speed of sound is in the fluid is - and that it'd be messy to actually solve, but you could with enough effort right down the differential equations for it given the characteristics of the fluid.

The Schwarzschild solution for the perfect fluid case is well-documented in the literature, (and I'm too lazy to look it up! - or perhaps not motivated), and the boundary conditions for the massless sphere are pretty simple. As I argued much earlier, if you write the metric in the explicitly spherically symmetric Schwarzschild form, the coefficeint of dr^2 must be the same inside and outside the massless shell.

You can show this directly from Einstein's field equations, as I did in
https://www.physicsforums.com/showpost.php?p=3784270&postcount=202

To recap very quickly, one of the Einstein Field equations of the Schwarzschild metric involves only h(r) and rho, h(r) being the coefficient of dr^2. The pressure doesn't enter into the equation for h(r) at all.

This equation can be written as

<br /> 8 \, \pi \rho = \frac{ \left( dh/dr \right) }{r \, h^2} + \frac{1}{r^2} \left( 1 - \frac{1}{h} \right) \; = \; \frac{1}{r^2} \frac{d}{dr} \left[r \, \left(1 - \frac{1}{h} \right) \right]<br />

The page on Wald that this was originally taken from is in the original post, but you should be able to find a similar equation from whatever paper or text you use to look up the perfect fluid schwarzchild case.

If \rho=0, then r(1-1/h) is constant through the shell. We can take the limit of a shell approching zero thickness, say that the shell starts at r=r_0 and ends at r_1

Then r_0 (1 - 1/h(r_0) ) = r_1 (1 - 1/h(r_1) ) = constant.

As r_0 approaches r_1, h(r_0) must approach h(r_1).

Conceptually, you can solve for h(r) given \rho(r), because the pressure (isotropic or not) doesn't affect this solution for h(r).

This same equation is what gives rise to the Schwarzschild mass parameter M.

As far as the textbooks go, there's some disussion in MTW around pg 553 about "junction conditions", but it's rather a long read. It might be worthwhile as a "sanity check" if one really got into the problem, though,.

 

[Q-reeus]

What does this even mean? The EFE are obviously a well-formed set of equations, and we know that there are solutions to the EFE, so in terms of self-consistency they are clearly "correct".
By "correct" you could also mean that they do not agree with experimental evidence. This is, in fact, the only way to actually challenge GR, but you have not presented any such evidence here.
So what do you mean by "the correctness of EFE's"?

This is really all going in circles, but since you asked: I had no idea there was this requirement of a matchup between each term in the SET to each term in the EFE's until it was indirectly stated by you actually here: https://www.physicsforums.com/showpost.php?p=3563625&postcount=3 , and explicitly by Peter there in #20

My intuition on that, just looking at Komar expression, is that stress as contributor is as isotropic (scalar) source just like T₀₀. So if one considers some stressed volume element, in the weak gravity regime I would expect T_ii contribution to Weyl curvature (i.e. exterior to SET region) at a distant 'hovering' field point to act exactly the same as the rest-energy T₀₀ part as source. No 'vectorial' effects - orientation of stress in element has no effect at that distant field point. True or not? That a stressed element cannot exist in isolation has imo no bearing on the ability to analyze it's individual contribution. I raised this matter elsewhere but there was no feedback. Yet this matching thing seems to say there *is* a strong directionality, that apparently vanishes outside of the SET region itself. Cannot quite fathom the physicality of that.

Anyway, given this matching requirement between SET and EFE terms, then assuming e.g. my G-clamps scaling argument validly demonstrates non-physicality of stress as source (no-one else does but I'm not fazed), it follows there would be a reduced SET and reduced matching FE's. So it all gets down to being able to show that stress-as-source behaves as a proper physical quantity, or not. I note there is afaik no analogous quantity in any other classical field theory - elasticity/acoustics/EM. In those disciplines all source components individually and collectively obey the usual divergence/conitnuity relations, not just collectively. I'm getting the strong message here that in GR only collective need matter at all.

I will draw your attention to something raised in #173, but got no comment:

Last point here is the nature of any GW's resulting from periodically stressed G-clamps as per #1. Assuming a periodically time varying quadrupole-like distribution of stress leads to regular quadrupolar GW's is wrong. A true time-varying quadrupole source has mass currents flowing - hence both 'electric' and 'magnetic' components in accordance with the equipartition of energy rule surely applying for any periodic physically real wave. The absence of any 'stress current' rules out any 'magnetic' component for G-clamp scenario. Just the superposition of purely 'electric' monopole sources spatially displaced to look like a real quadrupole source. Is this consideration alone not fishy enough to rule out stress as genuine source? Or is 'electric' only GW's actually the case in GR?

So is it the case that GW's in GR have both 'electric' and 'magnetic' components obeying equipartition of energy? If so, recognize the odd behavour of stress as GW source.

Finally, I see Pervect in #203 mentioned as possible model that which I had suggested, last bit in #179 - 'ocean' above thin shell 'earth'. While positing negligible mass for the shell seems ok, stress in that shell, according to it's SET properties, couldn't be neglected. So if one wants a match all the way to flat interior, there is a two shell scenario. Maybe just good enough to look at the conditions within and at boundaries of 'ocean' part though. Of course a solid spherical shell has the advantage of a single shell scenario. (In https://www.physicsforums.com/showpost.php?p=3563851&postcount=6, Peter indicated the need for tensile hoop stresses in a self-gravitating shell. This should all prove to be interesting!)

 

[PeterDonis]

I had no idea there was this requirement of a matchup between each term in the SET to each term in the EFE's until it was indirectly stated by you actually here: https://www.physicsforums.com/showpost.php?p=3563625&postcount=3 , and explicitly by Peter there in #20

And even there I didn't really state it precisely. If you really weren't aware of this, then it *should* be stated precisely, because it's awfully tough to talk about GR if you don't know what it actually says. Approximations and heuristics and analogies and so forth are all very well, but as many of us have said many times before to you, if you are going to claim that GR is wrong about something, you need to actually look at the actual exact predictions GR makes. You can't prove a theory wrong by working with your own approximate, heuristic, hand-waving version of it.

So here's the more precise version: what is usually called the "Einstein Field Equation" is actually ten equations. There are ten because the two tensors that are related by the equation, the Einstein tensor and the SET, each have ten independent components. (In all of this I am assuming 4-dimensional spacetime; different numbers of dimensions mean different numbers of components. In 4-D spacetime, a symmetric tensor, which both of these tensors are, has ten independent components; this should be obvious if you think of each tensor as a 4 x 4 symmetric matrix.)

However, if the specific spacetime you are looking at has symmetries, the number of independent components is reduced, because the symmetries impose additional constraints. In a spherically symmetric spacetime, for example, there are actually only three independent components of the EFE; they are:

G_{00} = 8 \pi T_{00}

G_{11} = 8 \pi T_{11}

G_{22} = 8 \pi T_{22}

All seven other components of the EFE are either trivial (0 = 0) with spherical symmetry, or are determined by one of the above three equations (for example, since T_{33} = T_{22}, the "3-3" component of the EFE is identical to the "2-2" component written above).

Further, the above equations involve the Einstein tensor; what is that? It is "built" out of derivatives of the metric, by way of the Ricci tensor, which is worth noting: Weyl curvature does *not* contribute to the Einstein tensor, so Weyl curvature is not directly involved in the EFE at all. It is only determined indirectly, by solving the EFE and obtaining a metric, and then computing the curvature components from it.

Also, spherical symmetry reduces the number of independent components in the metric, just as it does with the SET; in a spherically symmetric spacetime, you can always find "Schwarzschild-type" coordinates in which the metric has only two independent components, g_{00} and g_{11}. It then turns out that the 0-0 component of the Einstein tensor only involves derivatives of g_{11}, while the 1-1 and 2-2 components involve derivatives of both g_{00} and g_{11}.

(I should note that in vacuum, when all the SET components are zero, the metric only has *one* independent component, since g_{11} = 1 / g_{00}. But that only holds in vacuum.)

Btw, the above also shows that, since we have three equations for five unknowns (three SET components and two metric components), we should be able to specify two arbitrary functions in the solution as it stands now, but only two. For example, we could specify T_{00} and T_{11}, and the three equations would then give us the two metric components *plus* T_{22}. This may answer the question I asked PAllen a couple of posts back, about what he referenced from the Synge book.

One final note: sometimes it works better to use one or more of the "conservation" equations (covariant divergence of SET = 0) instead of using some of the EFE components directly. There are four conservation equations (the divergence of a symmetric 4-D tensor is a 4-vector, so there is one equation per component of that vector), which in a general spacetime can be substituted for four of the EFE components (to keep the total number of equations at ten). However, in a spherically symmetric spacetime, only two of the conservation equations are non-trivial, the "0" component and the "1" component. In a static spacetime, the "0" component becomes trivial as well and only the "1" component is left, expressing hydrostatic equilibrium. In the textbook solutions for static spacetimes that I have seen, this equation is substituted for the "2-2" component of the EFE above to make the set of three equations easier to work with.

 

[Dale]

This is really all going in circles, but since you asked: I had no idea there was this requirement of a matchup between each term in the SET to each term in the EFE's until it was indirectly stated by you actually here: https://www.physicsforums.com/showpost.php?p=3563625&postcount=3 , and explicitly by Peter there in #20

My intuition on that, just looking at Komar expression, is that stress as contributor is as isotropic (scalar) source just like T₀₀. So if one considers some stressed volume element, in the weak gravity regime I would expect T_ii contribution to Weyl curvature (i.e. exterior to SET region) at a distant 'hovering' field point to act exactly the same as the rest-energy T₀₀ part as source. No 'vectorial' effects - orientation of stress in element has no effect at that distant field point. True or not? That a stressed element cannot exist in isolation has imo no bearing on the ability to analyze it's individual contribution. I raised this matter elsewhere but there was no feedback. Yet this matching thing seems to say there *is* a strong directionality, that apparently vanishes outside of the SET region itself. Cannot quite fathom the physicality of that.

Anyway, given this matching requirement between SET and EFE terms, then assuming e.g. my G-clamps scaling argument validly demonstrates non-physicality of stress as source (no-one else does but I'm not fazed), it follows there would be a reduced SET and reduced matching FE's. So it all gets down to being able to show that stress-as-source behaves as a proper physical quantity, or not. I note there is afaik no analogous quantity in any other classical field theory - elasticity/acoustics/EM. In those disciplines all source components individually and collectively obey the usual divergence/conitnuity relations, not just collectively. I'm getting the strong message here that in GR only collective need matter at all.

I will draw your attention to something raised in #173, but got no comment:

So is it the case that GW's in GR have both 'electric' and 'magnetic' components obeying equipartition of energy? If so, recognize the odd behavour of stress as GW source.

Finally, I see Pervect in #203 mentioned as possible model that which I had suggested, last bit in #179 - 'ocean' above thin shell 'earth'. While positing negligible mass for the shell seems ok, stress in that shell, according to it's SET properties, couldn't be neglected. So if one wants a match all the way to flat interior, there is a two shell scenario. Maybe just good enough to look at the conditions within and at boundaries of 'ocean' part though. Of course a solid spherical shell has the advantage of a single shell scenario. (In https://www.physicsforums.com/showpost.php?p=3563851&postcount=6, Peter indicated the need for tensile hoop stresses in a self-gravitating shell. This should all prove to be interesting!)

OK, so with all of that I still don't understand in what sense you mean "correct" when you question the "correctness of EFE's". Are you saying that you understand that it is a self-consistent set of equations but you believe that there is a simpler set of self-constent equations that makes all of the same experimental predictions? I.e. not so much questioning the "correctness" as the "minimalness".

Can you just be clear and concise? What do you mean when you "question the correctness of EFE's"? Specifically, in what sense do you consider that they might not be correct?

 

[Q-reeus]

OK, so with all of that I still don't understand in what sense you mean "correct" when you question the "correctness of EFE's". Are you saying that you understand that it is a self-consistent set of equations but you believe that there is a simpler set of self-constent equations that makes all of the same experimental predictions? I.e. not so much questioning the "correctness" as the "minimalness".

No, as said in #204 imo quite clearly enough, if stress is not legitimately a source term, both SET and EFE's are wrong, and the correct pair will be a reduced SET and FE's. As for correct predictions - can you cite any conclusive observational evidence for stress as source? In another thread I suggested the only possible one might be upper limit to size of neutron stars. But then observed that the EOS for NS's is still not well tied down, as is evident reading here: http://en.wikipedia.org/wiki/Neutron_star. So I would say there is no real observational evidence pressure acts as SET says it does (btw I doubt there will ever be experimental evidence).

Can you just be clear and concise? What do you mean when you "question the correctness of EFE's"? Specifically, in what sense do you consider that they might not be correct?

My intuitive explanation(s) above still not clear enough? Then I suggest repeat reads until it is.

 

[Q-reeus]

All seven other components of the EFE are either trivial (0 = 0) with spherical symmetry, or are determined by one of the above three equations (for example, since T33=T22, the "3-3" component of the EFE is identical to the "2-2" component written above).

Sure that symmetry reduction part presents no problems.

Further, the above equations involve the Einstein tensor; what is that? It is "built" out of derivatives of the metric, by way of the Ricci tensor, which is worth noting: Weyl curvature does *not* contribute to the Einstein tensor, so Weyl curvature is not directly involved in the EFE at all. It is only determined indirectly, by solving the EFE and obtaining a metric, and then computing the curvature components from it.

Was not saying otherwise - only asking whether and how rest-energy and stress as source terms qualitatively effect vacuum region differently. This has yet to be addressed btw! Or is it that nobody looks at it that way? Why not - something illegitimate with that question?

I would also like some feedback on the matter raised of GW's 'electric/magnetic' parts - is there an equipartition principle there or not? If so, how could stress contribution to GW's in G-clamp case yield any 'magnetic' part? Can't see it.

Also, spherical symmetry reduces the number of independent components in the metric, just as it does with the SET; in a spherically symmetric spacetime, you can always find "Schwarzschild-type" coordinates in which the metric has only two independent components, g00 and g11. It then turns out that the 0-0 component of the Einstein tensor only involves derivatives of g11, while the 1-1 and 2-2 components involve derivatives of both g00 and g11.

(I should note that in vacuum, when all the SET components are zero, the metric only has *one* independent component, since g11=1/g00. But that only holds in vacuum.)

OK thanks for patiently explaining some of these basics. I still come back to; even supposing the formal correctness of SET/EFE relationships as per above (assume for the moment no internal inconsistencies as per my examples), how does that gaurantee stress physically acts as per GR SET says? Have any of the stress generated curvature terms observational support? I doubt the sky would fall in if it turns out stress is not a real source - afaik even notional black holes don't require pressure as source in order to form. :zzz:

 

[PeterDonis]

Was not saying otherwise - only asking whether and how rest-energy and stress as source terms qualitatively effect vacuum region differently.

In the vacuum region, the RHS of all components of the EFE is zero. That means all the spacetime curvature in the vacuum region is Weyl curvature, *not* Ricci curvature. That's true generally, regardless of what symmetries the spacetime does or does not have. And as I said in my last post, Weyl curvature does not enter into the EFE directly at all; only Ricci curvature does.

So any relationship between the Weyl curvature in the vacuum region and individual SET components in the non-vacuum region is going to be very indirect: you first have to solve the EFE in the non-vacuum region, then determine from that what the boundary conditions are at the edge of the vacuum region, then solve the EFE *again* in the vacuum region subject to those boundary conditions. But in the case of spherical symmetry, even doing that is pointless, because we already know what the vacuum solution looks like: it is the Schwarzschild vacuum solution for any radius r > R, where R is the radius of the boundary between non-vacuum and vacuum. If the spacetime is spherically symmetric, this is true *regardless* of what's going on in the non-vacuum region; so there is *no* relationship between any particular SET component in the non-vacuum region and the metric in the vacuum region, except whatever contribution that SET component makes to the total mass M that appears in the Schwarzschild metric.

Note that in the general case, M can be a function of time; but if the spacetime is stationary or static, the metric is time-independent, so M is also. And in the stationary or static case, M is just the Komar mass integral (or the ADM mass, or the Bondi mass--they're all equal in this case). So SET components in the non-vacuum region affect the metric in the vacuum region only via their contribution to the Komar mass integral.

This has yet to be addressed btw! Or is it that nobody looks at it that way? Why not - something illegitimate with that question?

It's a legitimate question, and the above should address it.

I still come back to; even supposing the formal correctness of SET/EFE relationships as per above (assume for the moment no internal inconsistencies as per my examples), how does that gaurantee stress physically acts as per GR SET says? Have any of the stress generated curvature terms observational support?

First of all, as the above should make clear, there are no specific "stress generated curvature terms". There are specific components of the EFE in which stress appears as a "source", but those components involve more than one component of the metric, and more than one component of the curvature tensor (since any given component of the Einstein tensor, on the LHS of the EFE, mixes together derivatives of different components of the metric, or contractions of different components of the curvature). Once again, the relationship between stress and any specific curvature term, particularly Weyl curvature observed in an exterior vacuum region, is very indirect.

There are two regimes where stresses are significant compared to energy densities: neutron stars and cosmology. In both regimes including stress as a "source" on the RHS of the EFE is necessary to match observations.

In the case of neutron stars, without pressure acting as a "source", there would be no maximum mass for a neutron star; so given that we observe stars of 10 times or more the mass of the Sun, we would also expect to observe neutron stars of those masses. But we don't; all the neutron stars we have observed are no more than 1.5 times the mass of the Sun or thereabouts. That strongly suggests that there *is* a maximum mass for a neutron star, beyond which it will collapse to a black hole. That's the main observational consequence of pressure as a source that I know of that can be linked to a simple observation.

In the case of cosmology, the current "best-fit" cosmological model requires stresses as a source to match observations. More precisely, except for the "matter-dominated" phase of the universe's expansion (which lasted roughly from 100,000 years after the Big Bang until a few billion years ago when the expansion started accelerating), the presence of nonzero pressure in the SET is required to get the correct dynamics. In the "radiation dominated" phase, the pressure is the radiation pressure of light; in the current "vacuum dominated" phase (or in the early inflationary phase), the "pressure" is the cosmological constant. In fact, including the "pressure" due to the cosmological constant is the only way to get an accelerating expansion at all, so again we can link the consequence of pressure as a source to a simple observation.

Edit: Strictly speaking, I should say that "pressure" due to the cosmological constant is the only way to get accelerating expansion without using "exotic matter"--i.e., matter that violates one or more of the standard energy conditions. As far as I know nobody has seriously argued for trying to include exotic matter in cosmological models, since none has ever been observed.

 

[PAllen]

I don't have Synge's book, but this looks like what I've come up with as I work through the math of the static thin spherical shell. The only thing I'm not sure about is the constraint on T22 and T33; as far as I can tell these must be equal under spherical symmetry, but I'm not sure how they're constrained to a specific relationship with T00 and T11. (MTW talks about this some, but the main treatment there appears to be restricted to the perfect fluid case, where T22 = T33 = T11 is imposed as a condition of the model anyway, so I can't tell for sure how general their equations are supposed to be.) Can you give any more specifics about which particular components of either the EFE or the energy conservation condition (covariant divergence of SET = 0) Synge uses to derive a specific relationship between T22 and T33 and the other components?

The way Synge gets T22=T33 in terms of T00 and T11 is as follows:

The divergence zero condition Gij|j=0 for i=1, accounting for the many terms that are zero (including Christoffel symbols), ends up relating G22=G33 with G11,1 G01,0 G11 G01 G00. Of course, the same must be true for T. Synge has previously worked out which Christoffel symbols must be zero for spherical symmetry (in a metric expressed in coordinates which display that symmetry). Separately, relations between functions that determine the metric and T00 and T11 are obtained, and between T10 and T00. The end result is formulas involving only (regular) integration and differentiation for expressing the rest of T=G and the metric in terms of arbitrary T11 and T00. Even nonzero cosmological constant is allowed. No assumptions about nature of matter have been made. So far as I can tell, noting but spherical symmetry and divergence=0 have been assumed.

[EDIT: changes above for discrepancies in convention of time as coord. zero (as I and most here use) and time as coord 4 (as Synge uses).]