Is There Really a Strictly Conserved Stress-Energy Tensor in GR?

[PeterDonis]

Looking back over previous posts:

there is an enormous number of random HFGW emitters - colliding ions etc., churning out a random but time-averaged highly smooth and spherically symmetric flux of GW's

...

To repeat a previous statement directed elsewhere -try and get out of your head this image of orbiting massive bodies as coherent GW emitter here. Previously explained just why HFGW scenario was chosen.

The flux may look spherically symmetric (when averaged) far away from the source. But if you looked closely enough, in the small regions where the GWs are being generated, you would see systems of "orbiting massive bodies" with nonzero quadrupole moments. (Yes, the "massive bodies" would be ions, not neutron stars. So what? The EFE doesn't care; it applies equally well in both cases.) If that isn't present on *some* scale, GWs will not be generated.

 

[Q-reeus]

Well I missed this aspect of your scenario due to the deleted link. However, with the clues there I was able to find what I suspect is the right paper. Irrespective of whether the paper is right in detail [it didn't seem to address what seems like an obvious concern - the gravitational analog of destructive interference; I am concerned that the formula they started with is not valid for huge numbers of dense particles; and that in that scenario, the net GW escaping would be many orders magnitude smaller than even the small number they compute],

Not so - not if you are drawing some analogy with incoherent EM radiation. For the latter it is well known that destructive and constructive interference exactly cancel on average and net power is simply the sum over all emitters [in optically transparent region - i.e. photosphere], each such treated as an independent and isolated source. Various good textbooks on classical optics will confirm that. I see no reason why the same would not hold in HFGW case - authors of article I cited certainly took that position, and there was no follow-up refutation article I'm aware of. Also remember that in EM case, there is continual scattering, absorption/re-emission going on, so that it takes perhaps tens of thousands of years for energy of a typical x-ray core genetated photon to finally escape, but by then that energy has degraded to optical frequency photons. Not so in HFGW case - once generated, further scattering probability is miniscule and it simply sails right out of interior region.
There is however the matter of whether to treat this emission process classically or quantum mechanically. If the former, the HFGW flux is overall exceedingly smooth and homogeneous on all but the smallest scale of interest. But assuming quantization - gravitons - then evidently E = hf applies equally to EM or gravitational quanta. In which case HFGW emission becomes a statistically extremely improbable event but nonetheless necessary on a time-averaged basis. Then one might more closely model stellar HFGW emission to that of a chunk of radioactive material weakly outputting gamma rays, rather than say a frosted light bulb picture. Not that any of this now matters much.

Birkhoff's theorem is irrelevant. Precisely to the extent that there is a GW contribution to ADM mass, the space-time is not static and Birkhoff does not apply. Remember, even using the paper's figures we are talking about microscopic energy compared to the radiation. The radiation alone means Birkhoff applies only approximately. The microscopic GW would add a microscopic further deviation from SC static geometry.

The idea there was to highlight that if, by your own proposed scenario below, or mine in last main para of #8 (enclosing multiple heat shields), we have only appreciable net GW emission, then there is a dilemma. Gravitating mass of spherically symmetric body is in GR deemed owing entirely to SET contribution, and SET is strictly conserved. Yet energy is streaming out in form of exclusively non-SET GW's. So on the one hand the spacetime should remain perfectly static (SET = source of gravitating mass conserved), on the other it cannot because energy is being continually lost, and for this geometry conservation of energy should strictly hold true. Anyway I resign to it having lost import given what you present below as standard GR position on that sort of scenario.

I think a much sharper form of your idea would be to imagine a bunch of massive massive balls at absolute zero (well, as close as possible) inside a trapping shell (also at absolute zero) in an empty, asymptotically flat universe. Assume all collisions are perfectly elastic (this might not be possible in an SET that satisfies plausible energy conditions; but let's ignore that for now). However, infinitesimal the rate, the motion of the balls would eventually cease, with KE having been radiated as GW. Given the decay of binary pulsar orbits as a model, it would seem necessary that this would happen in principle.

An interesting alternate arrangement to mine of #8, given we are talking in-principle only situations.

How to make sense of this? SET zero divergence is an infinitesimal conservation, evaluated in in the context of local curvature (here varying at the scale under consideration). The divergence is covariant divergence with incorporates the metric which incorporates the fluctuating geometry. Counter-intuitive as it may be, it is possible for the SET covariant divergence to be everywhere zero at all times, while integrating at null infinity (to get Bondi mass, which excludes the radiated energy) shows a declining mass due to radiated GW and reduced KE of the balls.

Thanks for at least providing a clear enough position statement on how GR community handles this. One thing though I brought up in #21 aught to be repeated here. Scenario of spherically symmetric body emitting steady isotropic flux of HFGW's in an asymptotically flat spacetime. I trust we agree that despite steady decline in mass of central object, there is here an overall conservation of energy - inclusive of GW emission to infinity - at least on ADM definition. Well this means energy is here a more robust entity (conserved) than gravitating mass (not conserved assuming energy conserving conversion of SET contributing material to GW's). An interesting situation. Energy strictly conserved. SET strictly conserved. Gravitating mass - strictly owing to strictly conserved system SET, not conserved. This btw also covers as a response to your #35. I won't and haven't been disputing the problem of overall energy 'balance' in GR. Just that in a situation where energy is well defined and deemed conserved, gravitating mass is not conserved if GW generation occurs. Not in GR. This is naturally talking about when integrating everything out to infinity - not just central mass.

In no way do I dispute how counter-intuitive this is. Many scientists have expressed dissatisfaction with state of energy conservation in GR. It is perfectly reasonable to take this as evidence that GR is not a final theory. However, it has no weight at all in showing that GR is internally inconsistent. The only way to do that is to show, mathematically, that one chain of derivation leads to answer x, and another to answer y, and both are without error. Good luck with that.

I sympathise with said many scientists. It's evidently futile to continue this line of attack given the position statement you have kindly provided (and PeterDonis has evidently been saying the same in a slightly different way). My own personal view of this standard GR position has 'unintuitive' as too mild a word - illogical, absurd, preposterous comes closer. The title of this thread in retrospect should have been "Is there a 1:1 correspondence between a strictly conserved SET and gravitating system mass?" to which the answer is evidently no. As to answer x or y disagreeing, that will probably only show up in more refined observational data in cases where a system's dynamics may appreciably depend on whether gravity is SET source or not. Or maybe that has already been found!

 

[Q-reeus]

The flux may look spherically symmetric (when averaged) far away from the source. But if you looked closely enough, in the small regions where the GWs are being generated, you would see systems of "orbiting massive bodies" with nonzero quadrupole moments. (Yes, the "massive bodies" would be ions, not neutron stars. So what? The EFE doesn't care; it applies equally well in both cases.) If that isn't present on *some* scale, GWs will not be generated.

I see you have completely shifted ground from position in #32 - good. Please read #37 which sums up my position on your (standard GR) position re conserved SET/gravitating mass. There is still the matter of null Nordtvedt results to consider, but first - is there anything further on 'conserved SET vs non-conserved gravitating mass' issue to discuss? I think not. For you it is a non-issue yawn, for me a sad shake of head, and that it seems is that.

 

[Dale]

My own personal view of this standard GR position has 'unintuitive' as too mild a word - illogical, absurd, preposterous comes closer.

There is only one standard for "illogical" and it is the one PAllen mentioned. Until you can provide that (which you haven't), all you can claim is "unintuitive".

As far as other possible pejoratives go, if GR is absurd, preposterous, etc., then insofar as GR is experimentally validated it seems that the universe itself is absurd, preposterous, etc. That shouldn't be too surprising given the experimentally confirmed weirdness of the universe; an accurate theory must accurately reflect that weirdness. Don't blame the theory for the universe.

 

[Q-reeus]

As far as other possible pejoratives go, if GR is absurd, preposterous, etc., then insofar as GR is experimentally validated it seems that the universe itself is absurd, preposterous, etc. That shouldn't be too surprising given the experimentally confirmed weirdness of the universe; an accurate theory must accurately reflect that weirdness. Don't blame the theory for the universe.

It's a matter of just how weird it really is in the final wash-up, and we are far from having observationally established that yet. Although I suspect there is already data providing a differing viewpoint. As per my last two sentences in that #37 post you quoted. I plan on having more to say on that anon.

 

[Dale]

Sure, just back it up with some solid evidence. Otherwise all you have is the usual "unintuitive" complaint, which is a subjective matter of experience and opinion.

 

[PeterDonis]

Gravitating mass of spherically symmetric body is in GR deemed owing entirely to SET contribution

No, it isn't. It also depends on the metric. If the metric changes, the "gravitating mass" changes, even if the SET is conserved. If GWs are present, the metric is changing.

 

[PeterDonis]

I see you have completely shifted ground from position in #32

Not really. Read the last couple of sentences of that post again; then read my post just before this one, in response to a sentence of your post #37. The "gravitating mass" does *not* just depend on the SET; it also depends on the metric. I've pointed that out a couple of times now, but you still do not seem to realize the implications.

There is still the matter of null Nordtvedt results to consider

Not really; that's the same issue as with GWs. The answer is what I stated (again) just above.

 

[Q-reeus]

No, it isn't. It also depends on the metric. If the metric changes, the "gravitating mass" changes, even if the SET is conserved. If GWs are present, the metric is changing.

For static spherically symmetric mass, by metric changing this amounts to redshift factor changing, therefore total of gravitating mass, right? Basically Komar mass definition. But if GW's make no contribution to SET, why should gravitating mass change, and therefore associated metric? There is no tautology here?

 

[Q-reeus]

"There is still the matter of null Nordtvedt results to consider"

Not really; that's the same issue as with GWs. The answer is what I stated (again) just above.

The more I have looked at articles on defining or trying to define gravitational mass, the more of a quagmire it seems to become - e.g. http://relativity.livingreviews.org/open?pubNo=lrr-2009-4&page=articlese1.html
Given how it is, there is likely no prospect of getting agreed upon notion of how to split gravitational binding energy (if even that can be agreed on itself) into a matter and field part. So am abandoning any further discussion re Nordtvedt results. :zzz:

 

[PAllen]

For static spherically symmetric mass, by metric changing this amounts to redshift factor changing, therefore total of gravitating mass, right? Basically Komar mass definition. But if GW's make no contribution to SET, why should gravitating mass change, and therefore associated metric? There is no tautology here?

If you have a mostly spherical body radiating:

1) It is not static. By definition - the metric is changing, both inside and outside the body, even for pure GW. (For EM radiation, the SET is undergoing first order change as well; for pure GW, the SET is changing as well, but in a way that preserves zero covariant divergence at each point).
2) For both EM radiation and GW, the greater the radiation, the less the exterior matches the SC metric.
3) Therefore, for a radiating body, Komar mass is inapplicable, period. You need to use Bondi mass (if you want a model of mass excluding the radiation).

This has been explained numerous times.

 

[PeterDonis]

For static spherically symmetric mass, by metric changing this amounts to redshift factor changing, therefore total of gravitating mass, right Basically Komar mass definition.

As PAllen said, if there is radiation present (any kind), the metric is not static. However, if the radiated power is very small, the metric might be usefully viewed as "quasi-static", meaning it is approximated reasonably well by a succession of "static" states with slowly decreasing Komar mass integrals. However, this is an approximation, and it doesn't mean that the system is actually static.

But if GW's make no contribution to SET, why should gravitating mass change, and therefore associated metric?

Because GWs *are* changes in the metric. That's what they're *made of*. If the metric isn't changing, there are no GWs. Once again, I've said this several times, but apparently you haven't grasped the implications.

 

[PeterDonis]

Therefore, for a radiating body, Komar mass is inapplicable, period. You need to use Bondi mass (if you want a model of mass excluding the radiation).

A side comment: the Bondi mass isn't useful if one is trying to find a connection between externally measured mass and the SET, because the Bondi mass (like the ADM mass) is found by integrating the metric coefficients, not the SET components. The Komar mass is an integral over SET components, so it intuitively seems like a better choice for seeing how "mass" corresponds to "amount of stuff"; but of course, since the Komar mass also depends on the metric, it doesn't "fix" the issues involved with physically interpreting integrals over curved spacetimes (I put "fix" in scare-quotes because there is *no way to fix those issues; they're there, and we just have to deal with it).

 

[Q-reeus]

If you have a mostly spherical body radiating:

1) It is not static. By definition - the metric is changing, both inside and outside the body, even for pure GW. (...for pure GW, the SET is changing as well, but in a way that preserves zero covariant divergence at each point).

Presence of GW's, even of the coherent variety owing to a single emitter, cannot of themselves be altering metric locally (i.e. where they are passing through) in any time-averaged way, according to GR. Because by definition they are non-SET source thus not a source of Ricci curvature period, or of Weyl curvature on time-averaged basis. So all you can legitimately mean by above quote is that changed metric owing to GW emission is indirect, via reduction in gravitating mass of body. Therefore conversion (read loss) of SET contributing media to non-contributing GW's. Therefore zero covariant divergence of SET is imo without proper sense when clearly there is no equivalent conserved integral form. Gravitating mass is disappearing from body with no balancing/conserving flux of gravitating media out of system (you know - GW's don't gravitate).

Hence gravitating mass just disappears from universe - down the cosmic sink-hole. Whilst simultaneously 'SET is everywhere preserved'. No necessary connection to net energy content either which may be entirely conserved (ADM basis). Why bother having having such a SET divergence law given in the real world extended spacetime arena it's a proper, accurately predictive integral form that is needed? A lot of GR folks must see something useful to it I can't. Perhaps the sheer mathematical beauty and symmetry is so appealing. No chance Nature might have a different view. But we have been over this and restating position yet again is just wasting words. I maintain standard GR position on this is a grand oxymoron, you folks clearly don't. So in summary, thanks (nearly) all for contributing but time to close the chapter, hopefully amicably agreeing to disagree, and move on.
[PS - Just caught your latest entries Peter. Enough has been argued - please take above as my response to your posts also.]

 

[PAllen]

Presence of GW's, even of the coherent variety owing to a single emitter, cannot of themselves be altering metric locally (i.e. where they are passing through)

Ignoring rest for now. This is false. GW is nothing but changing metric. GW without changing metric is like saying EM with no changing E or B field.

 

[PAllen]

A side comment: the Bondi mass isn't useful if one is trying to find a connection between externally measured mass and the SET, because the Bondi mass (like the ADM mass) is found by integrating the metric coefficients, not the SET components. The Komar mass is an integral over SET components, so it intuitively seems like a better choice for seeing how "mass" corresponds to "amount of stuff"; but of course, since the Komar mass also depends on the metric, it doesn't "fix" the issues involved with physically interpreting integrals over curved spacetimes (I put "fix" in scare-quotes because there is *no way to fix those issues; they're there, and we just have to deal with it).

Agreed, but I'm not trying to relate externally measured mass to SET. I'm trying to relate decline in externally measured mass to emission of radiation. For that, Bondi mass is appropriate. And I don't dispute your earlier post that, for practical (approximate) purposes, you can use Komar mass for slowly changing bodies. But, obviously, if you are trying to address issues of principle in conservation of energy in GR, you can't be approximate.

 

[PeterDonis]

Presence of GW's, even of the coherent variety owing to a single emitter, cannot of themselves be altering metric locally

Apparently you still don't understand what "GWs *are* changes in the metric" means.

in any time-averaged way

Time-averaged or not time-averaged, makes no difference. Either way the metric *does* change.

Because by definition they are non-SET source thus not a source of Ricci curvature period

This part is true, yes. GWs are "waves of changing Weyl curvature". Which means that this...

or of Weyl curvature on time-averaged basis.

...is wrong. You should really look at the actual theory of GWs before making these claims.

So all you can legitimately mean by above quote is that changed metric owing to GW emission is indirect

No, it isn't. It's as direct as can be: the GWs *are* the changes in the metric. They are identical.

via reduction in gravitating mass of body

This is backwards. The reduction in the gravitating mass of the body is due to the changes in the metric, not the other way around.

Gravitating mass is disappearing from body with no balancing/conserving flux of gravitating media out of system (you know - GW's don't gravitate).

This is not correct. Remember the second part of my blog post, where I said that gravity as a quantum field is self-interacting? And how the classical limit of that quantum field theory, the Einstein-Hilbert action, leads to a field equation, the EFE, which is *also* nonlinear, i.e., self-interacting? That is equivalent to saying that GWs *do* "self-gravitate"--GWs do interact with each other.

The reason this self-interaction doesn't show up in any practical sense is that any GWs that we have any hope of detecting here on Earth in the foreseeable future are so weak that even if we detect the GWs themselves, we have no hope of detecting the much smaller self-interactions between them. But you are talking "in principle", and in principle, GWs *do* interact with each other, and *do* "gravitate" in the sense you are using the term here.

Why bother having having such a SET divergence law given in the real world extended spacetime arena it's a proper, accurately predictive integral form that is needed?

Because the integral form is *not* needed to make predictions. For example, in the binary pulsar case, AFAIK, the calculations are done using the EFE (they are numerical calculations since nobody knows any closed-form analytical solution for two bodies orbiting each other in GR), not using any integral forms. Again, you should really learn more about the actual theory, and how actual predictions are made, before making these claims.

A lot of GR folks must see something useful to it I can't.

Yes, they do.

Perhaps the sheer mathematical beauty and symmetry is so appealing.

It is, but that's not the reason the theory has survived.

No chance Nature might have a different view.

In all experiments to date that I'm aware of, Nature's view matches GR's. Do you have any examples where that's not true?

I maintain standard GR position on this is a grand oxymoron

And I maintain that this is because you don't understand what the "standard GR position" actually says; your criticisms are not of GR, but of your own straw-man version of GR that doesn't match the basic theory and doesn't match how the theory is actually used.

 

[Mentz114]

The Vaidya metric is a simple example where the gravitating mass is reducing because radiation is carrying away gravitating 'stuff'. The exterior is not a vacuum, but has the geometrical optics type SET.

http://en.wikipedia.org/wiki/Vaidya_metric

 

[PeterDonis]

Agreed, but I'm not trying to relate externally measured mass to SET. I'm trying to relate decline in externally measured mass to emission of radiation. For that, Bondi mass is appropriate.

Yes, agreed.

 

[PeterDonis]

The Vaidya metric is a simple example where the gravitating mass is reducing because radiation is carrying away gravitating 'stuff'.

Yes, but in this case the radiation is EM, and so is associated with a non-zero SET. So it's easier intuitively to see the connection between radiated energy and reduced gravitating mass in the body.

 

[Dale]

I maintain standard GR position on this is a grand oxymoron

And your position is without evidence.

 

[Q-reeus]

Q-reeus: "Presence of GW's, even of the coherent variety owing to a single emitter, cannot of themselves be altering metric locally (i.e. where they are passing through)"
Ignoring rest for now. This is false. GW is nothing but changing metric. GW without changing metric is like saying EM with no changing E or B field.

Hopefully just a misunderstanding here. Your quote has selectively omitted the part that modifies and that imo crucially matters: "...in any time-averaged way, according to GR." This is in contrast to equivalent energy flux of EM radiation - which does alter metric locally on time-averaged basis precisely because EM radiation is treated as a SET contributor. The momentary fluctuations in Weyl curvature (as stated zero when averaged over a complete cycle or otherwise on a stochastic averaging measure) add nothing to gravitating energy density (hence Weyl curvature from GW's does not induce an added Ricci curvature component). I was talking about contribution to system gravitating mass - and by definition GW's cannot be doing so - GW energy density is strictly by GR a non-gravitating energy density. Agreed? That's the point being made, or rather summarized in part in #49. An outgoing energy flux that contains no SET contribution. Hence, to labor the point yet again - the books do not balance gravitating mass-wise, despite what a covariantly divergence-free SET would suggest!

 

[Q-reeus]

Q-reeus: "Gravitating mass is disappearing from body with no balancing/conserving flux of gravitating media out of system (you know - GW's don't gravitate)."
This is not correct. Remember the second part of my blog post, where I said that gravity as a quantum field is self-interacting? And how the classical limit of that quantum field theory, the Einstein-Hilbert action, leads to a field equation, the EFE, which is *also* nonlinear, i.e., self-interacting? That is equivalent to saying that GWs *do* "self-gravitate"--GWs do interact with each other.

The reason this self-interaction doesn't show up in any practical sense is that any GWs that we have any hope of detecting here on Earth in the foreseeable future are so weak that even if we detect the GWs themselves, we have no hope of detecting the much smaller self-interactions between them. But you are talking "in principle", and in principle, GWs *do* interact with each other, and *do* "gravitate" in the sense you are using the term here.

Not accepting this at all. Your blog contrasted the 'yes' self-interaction/self-gravitation of quantum gravity theories with 'no' classical GR position that holds gravity does not gravitate - period. Not a contributor to SET in any form - period. I quote you verbatim from https://www.physicsforums.com/blog.php?b=4287:

It's important to note that there is no contradiction between the two answers we have just described. "Gravity" in the two answers means two different things: gravity as a quantum field does gravitate (the field interacts with itself), but gravity as the classical tensor satisfying the Bianchi identity doesn't gravitate, because there is nothing "left over", once the Bianchi identity is satisfied, to contribute to the source on the RHS of the EFE.

To sum up what we've said so far: we've talked about two possible ways to answer our title question, and they lead to opposite answers:

(1) In order to ensure conservation of the source, the complete Einstein tensor, including *all* contributions from gravity, must appear on the LHS of the EFE; there is nothing left over to contribute to the "source" on the RHS of the EFE. So in this sense, gravity does *not* gravitate.

You have left yourself wide-open on this. A pity because I wanted to finish it up in #49 on a good note, but can't tolerate the blatantly conflicting statements you have made here. I will not bother with the rest of your post - above is key issue. Will the real PeterDonis please stand up.

 

[PAllen]

Hopefully just a misunderstanding here. Your quote has selectively omitted the part that modifies and that imo crucially matters: "...in any time-averaged way, according to GR." This is in contrast to equivalent energy flux of EM radiation - which does alter metric locally on time-averaged basis precisely because EM radiation is treated as a SET contributor. The momentary fluctuations in Weyl curvature (as stated zero when averaged over a complete cycle or otherwise on a stochastic averaging measure) add nothing to gravitating energy density (hence Weyl curvature from GW's does not induce an added Ricci curvature component). I was talking about contribution to system gravitating mass - and by definition GW's cannot be doing so - GW energy density is strictly by GR a non-gravitating energy density. Agreed? That's the point being made, or rather summarized in part in #49. An outgoing energy flux that contains no SET contribution. Hence, to labor the point yet again - the books do not balance gravitating mass-wise, despite what a covariantly divergence-free SET would suggest!

Weyl curvature contributes to gravitational mass. This is easily seen in the SC geometry, for which SET=0 and Ricci curvatrue=0, everywhere. Komar mass volume integral is zero (or undefined, perhaps, because of the singularity). Meanwhile, ADM mass = Bondi mass (in this geometry they are equal) = M parameter of metric.

In the case of GW flowing out of some region, in a spacetime asymptotically flat at infinty, the ADM mass stays constant, the Bondi mass decrease. Each is computed using Weyl curvature in the case where there is only vacuum outside said region (because both are defined in terms of limit of metric integration as you go to infinity).

The books balance at infinity for spacetimes meeting certain boundary conditions. Otherwise, they don't balance at all. For our universe, it appears they don't balance at all. There are actually many lines of evidence for the proposition the conservation of total energy cannot be achieved in an expanding universe, and it may be considered a plus that GR predicts this.

 

[PeterDonis]

Your blog contrasted the 'yes' self-interaction/self-gravitation of quantum gravity theories with 'no' classical GR position that holds gravity does not gravitate - period. Not a contributor to SET in any form - period.

You apparently failed to read this in what you quoted:

"Gravity" in the two answers means two different things

The contrast I was drawing was *not* between a classical GR view of gravity and a quantum view of gravity. It was between two different meanings of the word "gravity": "gravity" as "the LHS of the EFE, vs. the RHS of the EFE" vs. "gravity" as "a massless, spin-two field". Gravity in the first sense does not gravitate; gravity in the second sense does.

*Both* of these senses of the word "gravity" are part of GR, and *both* answers to the question are part of GR. I was most emphatically *not* trying to contrast a "GR answer" to the question with some other theory's answer. To briefly recap what I said in the blog post:

(1) The EFE has "gravity" on the LHS, and "stress-energy" on the RHS. The RHS is the "source" that produces the gravity on the LHS, and there is no "stress-energy due to gravity" on the RHS of the EFE. So in this sense, gravity does not gravitate.

(2) The EFE is nonlinear, because the action it is derived from (the Einstein-Hilbert action) is nonlinear, because that action is the classical limit of the quantum field theory of a massless, spin-two field, which is nonlinear. "Nonlinear" means "self-interacting". So in this sense, gravity does gravitate.

Notice that *both* answers refer to the EFE; *both* answers are therefore "GR" answers. They just refer to different properties of the EFE, which is why they are different answers.

can't tolerate the blatantly conflicting statements you have made here.

There is no conflict. You need to read more carefully. What's more, you need (IMO) to read with a real intent to understand, instead of just looking for things you can contradict.

This material is not easy; I understand that. I have been reading about GR, thinking about it, working problems in it, and discussing it with others, for about 25 years now. Many people here on PF have been doing it longer than that. We know this is not stuff you can grasp overnight. But coming into it with the attitude that "if I see an apparent contradiction and nobody can explain it to my satisfaction, GR must be wrong, inconsistent, flawed, etc." is not likely to get you anywhere. The fact that the theory *is* consistent and experimentally verified (to 14 decimal places) within its domain of applicability does *not* guarantee that there will be an explanation for it that you can intuitively grasp.

Your response to this is basically "I trust my intuition more than I trust your assertions that the theory is correct even though you can't explain it to my satisfaction." I understand that that seems like a reasonable response to you. That doesn't change the fact that it's wrong. Nature doesn't care about your intuitions. It doesn't care about *my* intuitions. It also doesn't care about whether I can explain to you why the things I am saying are correct.

Also, to be clear, I am *not* saying that you should just abandon your intuitions and blindly accept what I am telling you, or what anyone else here on PF is telling you. Feynman, who has been quoted several times now in this discussion, once said: "What I cannot create, I do not understand". I'm the same way, and I suspect you are too. The things I am saying in these threads, and that I put into my blog posts, are things I have created; that's the only way I can understand them. Of course my "creations" aren't original; I'm just rediscovering for myself paths of reasoning that many, many people have followed before me. But I only understand the paths that *I* have followed myself.

The reason I'm responding to your questions is that I hope that, at some point, one of those paths of reasoning will open up for you. I have been assuming that that's why you are posing the questions in the first place: here's this theory that everybody says is correct, but you can't see any path of reasoning that gets you to where everybody says they are, and you would like some help in finding it. I understand that it's frustrating when people keep on insisting there's a path, and pointing in various directions, and all you see is underbrush. Unfortunately, that's just an indication of how hard the paths are to find in this neck of the woods.

 

[Q-reeus]

Weyl curvature contributes to gravitational mass. This is easily seen in the SC geometry, for which SET=0 and Ricci curvatrue=0, everywhere. Komar mass volume integral is zero (or undefined, perhaps, because of the singularity). Meanwhile, ADM mass = Bondi mass (in this geometry they are equal) = M parameter of metric.

You haven't filled in here an identifying detail but I can pretty well assume this is talking about an exterior BH spacetime. That way your words makes sense. Clearly you are here saying curvature (Weyl) of the external field acts as it's own source - gravity gravitates of necessity. But only if it's Weyl curvature giving rise to further Weyl curvature. Yet oddly (on an intuitive level) the EFE's forbids any similar situation of Ricci curvature acting as it's own further source. Interesting. At last that issue now appears cleared up. Got a very different slant on Weyl curvature - as non-source some time back: https://www.physicsforums.com/showpost.php?p=3781365&postcount=26 https://www.physicsforums.com/showpost.php?p=3781823&postcount=28 etc.
[further perusing there, and this a better one: https://www.physicsforums.com/showpost.php?p=3786286&postcount=46]

Possibly explainable as completely complimentary but if it is representative of what may be termed 'GR logic', I fear never being able to quite get the hang of it. No need to question where the severe bouts of headache have been coming from.

In the case of GW flowing out of some region, in a spacetime asymptotically flat at infinty, the ADM mass stays constant, the Bondi mass decrease. Each is computed using Weyl curvature in the case where there is only vacuum outside said region (because both are defined in terms of limit of metric integration as you go to infinity).

And at last is adding up for me in a far more rational way.

The books balance at infinity for spacetimes meeting certain boundary conditions. Otherwise, they don't balance at all. For our universe, it appears they don't balance at all. There are actually many lines of evidence for the proposition the conservation of total energy cannot be achieved in an expanding universe, and it may be considered a plus that GR predicts this.

This bit I for now at least have no basic argument with.

 

[Q-reeus]

They just refer to different properties of the EFE, which is why they are different answers.

Not from my reading of that blog. A further quote:

In other words, on the "gravity as just another quantum field" view, classical GR is just a low-energy effective field theory; it is what you get when gravity is too weak for its quantum nature to show up. (Don't be misled by that "too weak", btw; in the sense of the term used here, gravity is "too weak" at, and well inside, the horizon of a stellar-mass black hole.)

Not different properties on my reading - just hugely differing levels of significance. So huge there is no effective non-linearity owing to field self-contribution in classical GR - where it is in fact taken as precisely zero. Hence structure of EFE's - with only field curvature on LHS and only non-gravitational field contributing SET source on RHS. A consistent position throughout that blog, as per my previous quote. It has struck me now as per last entry in response to PAllen's last post that there is a gravity gravitates thing in GR - just completely divorced from EFE's. But as per those [STRIKE]two[/STRIKE] three links there - your own position on role of Weyl curvature as or not as source could do with some clarification. In that thread my distinct impression was you denied Weyl curvature (wasn't specifically referred to as Weyl curvature there, but that's what was meant) could act as source of gravitation.

If just once in your blog post there was a statement saying there is a sizable contribution to mass from GW Weyl curvature, likely none of this would be happening. It would have clicked for me - EFE relationships are just one part of the scene to consider. You were sort of saying something along that line in #52 - GW's do self-gravatate, but tied it to quantum model and as per above quote, levels of non-linearity are thence ridiculously tiny in classical setting even inside of BH! In other words, GW's by that logic of no account as source of gravitating mass flux in scenario discussed.

Hope we can finish this up on a high note - I'm sure you have been meaning well, as the rest of your post is conveying.

 

[PeterDonis]

Not different properties on my reading - just hugely differing levels of significance. So huge there is no effective non-linearity owing to field self-contribution in classical GR - where it is in fact taken as precisely zero.

This is wrong. There *is* "effective nonlinearity" in classical GR. That's the point. Once again, please read more carefully: I said a number of times that *the EFE* is nonlinear.

Hence structure of EFE's - with only field curvature on LHS and only non-gravitational field contributing SET source on RHS.

That's *not* the same as saying the EFE is linear. Where did I say that writing the EFE that way makes it linear? (Just as a general point, you can't make a nonlinear equation linear by rearranging terms.)

For example, the Schwarzschild solution itself is a manifestation of the classical nonlinearity of the EFE. Consider: we have a *vacuum* solution (no nonzero SET anywhere) which is *curved*, and the curvature is all Weyl curvature (as it must be since there is no nonzero SET anywhere, and only nonzero SET can produce Ricci curvature). Only a nonlinear equation can produce this kind of solution with zero "source" on the RHS.

(Mathematically, you can see that the EFE is nonlinear by looking "under the hood" of the Einstein tensor on the LHS; you will see that it contains products of derivatives of the metric, i.e., it is quadratic in derivatives of the metric. By contrast, Maxwell's Equations are linear in derivatives of the EM field.)

It has struck me now as per last entry in response to PAllen's last post that there is a gravity gravitates thing in GR - just completely divorced from EFE's.

As you can see from my example above, the presence of Weyl curvature is not "completely divorced" from the EFE. It is still constrained by having to solve the *vacuum* EFE, i.e., the EFE with zero source (i.e., zero SET). This probably deserves a separate thread if you want more discussion of it, since it's a general point about the relationship between the EFE and the various tensors that describe aspects of gravity.

In that thread my distinct impression was you denied Weyl curvature (wasn't specifically referred to as Weyl curvature there, but that's what was meant) could act as source of gravitation.

If "source" means "what's on the RHS of the EFE", then yes, Weyl curvature (or *any* curvature) does not act as a "source" of gravitation. But that doesn't mean Weyl curvature can't "propagate", i.e., that Weyl curvature at one event can't lead to Weyl curvature at another event, without any nonzero SET appearing in between. See above.

[Edit: I realize that "propagate" is a bad word; unfortunately, we don't have a word for the way that *tidal* Weyl curvature (the kind that's present in the Schwarzschild solution) can be present in a source-free solution of a nonlinear field equation, as opposed to propagating waves of Weyl curvature (which could be present even if the EFE were linear, just as EM waves can be present in a solution of the source-free Maxwell Equations, which are linear).]

If just once in your blog post there was a statement saying there is a sizable contribution to mass from GW Weyl curvature, likely none of this would be happening. It would have clicked for me - EFE relationships are just one part of the scene to consider. You were sort of saying something along that line in #52 - GW's do self-gravatate, but tied it to quantum model and as per above quote, levels of non-linearity are thence ridiculously tiny in classical setting even inside of BH!

I can see how some of the language in the blog post could be confusing on this point. I'll try to fix it up to make clear that the nonlinearity appears at the classical level, not just the quantum level. Bear in mind that the nonlinearity does not just show up in GWs, and just because it's too weak to detect in GWs (or at least any GWs we are likely to detect in the foreseeable future) doesn't mean it's too weak to detect *anywhere*. See comments above about the Schwarzschild solution.

[Edit: As a further note, you say "GW Weyl curvature". GWs are *not* the only kind of Weyl curvature. There are no GWs in Schwarzschild spacetime, but there is Weyl curvature.]

 

[PeterDonis]

I can see how some of the language in the blog post could be confusing on this point. I'll try to fix it up to make clear that the nonlinearity appears at the classical level, not just the quantum level.

I have made some changes to the blog post (the original "does gravity gravitate?" post, not the follow-up, which is still in draft) to try to make the above issue clearer. Please feel free to comment.

 

[Q-reeus]

That's *not* the same as saying the EFE is linear. Where did I say that writing the EFE that way makes it linear? (Just as a general point, you can't make a nonlinear equation linear by rearranging terms.)

I certainly acknowledge that you have never claimed EFE's are linear equations.

For example, the Schwarzschild solution itself is a manifestation of the classical nonlinearity of the EFE. Consider: we have a *vacuum* solution (no nonzero SET anywhere) which is *curved*, and the curvature is all Weyl curvature (as it must be since there is no nonzero SET anywhere, and only nonzero SET can produce Ricci curvature). Only a nonlinear equation can produce this kind of solution with zero "source" on the RHS.

Agreed. But afaik the unavoidable non-linearity here is inherent in that metric spatial and temporal components are the base entities that vary on LHS. It's the next level of non-linearity that's in question - whether curvature is in part it's own source. Put very crudely, C = T can be non-linear eq'n whether or not T is itself inclusive of terms that are functions of C. Say C = T with T = T0+f(C) then one rewrites as C = T/(1-f(C)) which manifestly guarantees C is partially it's own source and in general also non-linear, or introduces added non-linearity over what otherwise might be - an explicit 'gravity gravitates' type relation.

(Mathematically, you can see that the EFE is nonlinear by looking "under the hood" of the Einstein tensor on the LHS; you will see that it contains products of derivatives of the metric, i.e., it is quadratic in derivatives of the metric. By contrast, Maxwell's Equations are linear in derivatives of the EM field.)

OK but as per above remarks, such non-linearity need have no bearing on whether LHS terms self-couple. Length and time scales can vary with length and time (inherent non-linearity) purely owing to coupling to RHS source matter, without there needing to be any input of curvature inducing more curvature.

As you can see from my example above, the presence of Weyl curvature is not "completely divorced" from the EFE. It is still constrained by having to solve the *vacuum* EFE, i.e., the EFE with zero source (i.e., zero SET). This probably deserves a separate thread if you want more discussion of it, since it's a general point about the relationship between the EFE and the various tensors that describe aspects of gravity.

Yes I agree to having overstated on that one. 'Propagation' must involve a non-zero SET in some manner.

[Edit: As a further note, you say "GW Weyl curvature". GWs are *not* the only kind of Weyl curvature. There are no GWs in Schwarzschild spacetime, but there is Weyl curvature.]

Understood that much. I can appreciate that there has been here differing interpretations on curvature and or past light-cone SET, as source in certain situations. Maybe nothing more fundamental than that. Anyway sorry for getting a little hot under the collar - apologies for evidently misinterpreting some of your earlier statements as contradictory when evidently really a combination of my limited understanding, and here and there some less than optimal terminology. Well thankfully I consider to have gained a little increased insight through it all. Just a little too much turmoil getting there for my taste. Must go. :zzz:

 

[PeterDonis]

the unavoidable non-linearity here is inherent in that metric spatial and temporal components are the base entities that vary on LHS.

No, that by itself is not enough. The spatial and temporal components of the EM field are the "base entities that vary" on the LHS of Maxwell's Equations, but that doesn't make them nonlinear. See below.

It's the next level of non-linearity that's in question - whether curvature is in part it's own source.

And that *is* present in the EFE, because it's *quadratic* in the derivatives of the metric components, whereas it's *not* present in Maxwell's Equations, because those are linear in the derivatives of the EM components.

 

[PeterDonis]

Anyway sorry for getting a little hot under the collar - apologies for evidently misinterpreting some of your earlier statements as contradictory when evidently really a combination of my limited understanding, and here and there some less than optimal terminology.

No worries. I certainly agree that some of the terminology is less than optimal.

Well thankfully I consider to have gained a little increased insight through it all.

I'm glad!

 

[PeterDonis]

Quick note: I have posted a follow-up, "Does Gravity Gravitate: The Sequel", on my PF blog:

https://www.physicsforums.com/blog.php?b=4288

Unfortunately, I ran up against the PF post length limit and still had more to cover, so there will be a second follow-up post, hopefully soon!

 

[PAllen]

One point to add is that plausible SET is derived by variation along with R (Ricci scalar) from matter Lagrangian. The variation of the Lagrangian incorporates metric terms in the SET, so you have metric on both sides. Even for a pure EM SET, you have the metric included in the SET. Thus, the metric is on both sides of the equation.

 

[PeterDonis]

One point to add is that plausible SET is derived by variation along with R (Ricci scalar) from matter Lagrangian. The variation of the Lagrangian incorporates metric terms in the SET, so you have metric on both sides. Even for a pure EM SET, you have the metric included in the SET. Thus, the metric is on both sides of the equation.

Just to be clear, the complete action in question is:

$$S = \int \left[ \frac{R}{16 \pi} + L_{M} \right] \sqrt{-g} d^4 x$$

where $R$ is the Ricci scalar, $L_{M}$ is the Lagrangian due to matter fields, and $g$ is the determinant of the metric tensor. The variation of the first term with respect to the metric gives the Einstein tensor, and the variation of the second term gives (minus) the SET. The total variation must be zero, which yields the Einstein Field Equation.

It's true that the variation of the second term with respect to the metric will include the metric. But it includes no *derivatives* of the metric, so it contains no information about the curvature (or even about the connection, which is first derivatives of the metric--curvature is second derivatives).