A response to AlphaNumeric re closed thread

Evidently you are just not willing to answer my #18 wrt your claim in #16 that stress induces a change in mass (whether active, passive, or inertial) ONLY if elastic deformation accompanies that stress. Which restated means - elastic energy is the only source of mass change owing to an applied stress. And your #20 continues to evade answering on that matter. Why?

As for your recapitulation of those relativistic conservation laws you gave in #16, well my response is rather simple. I believe it is right and proper to test the internal consistency of such laws - by way of an at least potential counterexample. Do you deny that sort of thing is a legitimate scientific procedure and is in fact done regularly? And that the 'disk-brake' setup scenario I have described quite clearly enough is such a one? The issue is fundamentally qualitative - what is the character of stress as source term, and then what are the implications. If you are so completely confident, what is the problem with testing for internal consistency of those conservation laws against the disk-brake setup? Something terribly complex there? I don't think that is the explanation. As i said - it's all down to the character of stress as source term(s). And in case you want to spin things as making me out in favor of violation of energy-momentum conservation here - just check my #5 as one of many instances I have made my own opinion clear!

Now, would you do me this much courtesy and actually answer what I posed to you in #18?! A straight answer.

 

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I only have a few minutes before I have to go out for the day but I'll comment on this bit for the time being....

Of course models must be tested and evaluated regularly. But the question is how your example is going about testing GR. The only thing which matters is whether \(T^{ab}_{;b}=0\). Does adding your additional terms change this? If not then further discussion is not required. If it does then you have violated the field equations. If your scenario gives \(T^{ab}_{;b}=0\) then by definition local energy and momentum conservation is satisfied. Then the only way it could contradict relativity is if \(G_{ab} \neq 8\pi T_{ab}\) but you cannot disprove that with a thought experiment, only assert it or provide experimental evidence it is mistaken.

Sorry, I have to go out for the day. I'm such a heartless bastard that I volunteer for a charity on Sundays and that's more important than walking you through a procedure you should have already done.

 

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Not so much a heartless bastard as a monumentally evasive and disingenuous one, and also quite boastful of your 'charity work' which I imagine is a fun jaunt if truth be known. And btw I've done my share of charity work too, but don't boast about it. Anyway, when and if you can summon up enough courage to face the music over your monumental gaffe re character of stress as source term, perhaps this thread can get down to real business. Or perhaps you really have seen the writing on the wall and will opt to close it down like last one. And yes, I'm sure it was only by my making a song and a dance along those lines that this thread was not closed early on. Regardless of your denials to the contrary.

 

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You seem to be ignoring how I have repeatedly explained how to address the question of energy conservation in general relativity. Each and every time I'd gone through details, at length, you've ignored them. I also asked you if you understood the mathematical methods I used, you ignored that question too. Is a simple yes or no too much?

You also 'forget' how I offered to help you submit your work to a journal. You utterly ignored that yet continue to try to frame the discussion as me somehow being afraid. You say I'm afraid yet I offer to help. You imply Awal's banning was my doing when it wasn't. You repeatedly insinuate I am 'itching' to close and delete this thread when I haven't. You imply I wish to shut down the discussion when I closed the previous thread for being massively off topic yet I clearly stated you could start a new one. Seriously, you project so hard we could point you at a wall and call you a cinema.

'Quite boastful'? Saying it once? Tell me, do you really view the world with such disdain and suspicion that you immediately warp everything without even realising it or are you just trying to be deliberately dishonest and generally an unpleasant person? No doubt if I'd said nothing other than "I'm only going to post something short" you'd have spin it as "You're running away!". I say nothing and you complain. I respond with lengthy quantitative details and you complain. I repeatedly address the issue of energy conservation and you ignore it all. Essentially you just want to whine.

So the repeated lengthy details I've given about how to address your issue of whether or not energy conservation exists in your scenario doesn't count as 'real business'? I've shown how your scenario is not a problem, that the only way you can not have local energy conservation is to add in something which violates \(T^{ab}_{;b}=0\). But if you do that then you have \(G^{ab}_{;b} \neq 0\), which cannot be so since that's an algebraic identity. You've failed to respond to any of that. Instead you refuse to get down to 'real business' unless I address the point in a particular way. Ignoring alternative ways to address the problem doesn't make them any less valid.

I can only conclude you are now purely trolling. Again and again and again with this insinuation when you have nothing but your own rampant paranoia to base it on.

I'll repeat my offer : If you write up your work I'll format it into the appropriate LaTeX style necessary to submit it to review at reputable journals so that you can then submit it to a journal for review and it'll be evaluated on its scientific merits and not stylistic presentation. Clearly you don't want to get down to 'real business' with any of the 'heavy hitters' here, you just want to continue this imaginary persecution complex you have. Why did you ignore this offer previously? If you think I don't know enough relativity to discuss this subject properly then why are you still here? Why haven't you put in some time to write up your work? You clearly have a chip on your shoulder in regards to the 'heavy hitters' here so why do you even bother?

Wow, you're like Alex Jones, the paranoid wacko from Info Wars. He makes paranoid predictions "X will happen, there's a secret government task force working on it!" and when X doesn't happen he claims "Well I was still right because X was changed to Y after I exposed X!". Essentially there is nothing I can say which can alleviate your paranoia and persecution complex because if I say "Yes, I was going to close the thread" then you say "Aha! I knew it!" while if I say "I have no intentions to close the thread" then you say, as you just did, "Only because I called you on it!". Essentially your paranoia becomes a no-lose scenario, your distorted view of the world becomes self reinforcing.

Of course if you continue with the various misrepresentations I'll be well within my moderator rights to give you an infraction warning. As I've previously said, you've done more than enough to warrant one now (last week even!) but despite your paranoid misrepresentations I don't use my moderator powers much.

And so to your question. Does the passive mass increase? I don't know with any real confident, the concept is one which I have not come across myself at any point within relativity and what masses in GR I do know about make me unwilling to just guess without further investigation. Could I find out the specifics and answer if I put in some time and effort? Probably. Do I think you're worth that? No, I've got other books to read. Do I think the question is relevant to your claim about an energy conservation issue? Nope, as my posts have repeatedly addressed that without needing to consider the masses.

See, I have no issue putting my hands up when I don't know something, despite your delusional claims otherwise. This thread isn't going to disappear, it isn't going to mysteriously edited, you aren't going to be banned (at least not for asking me a question I cannot immediately answer). The qualitative concept is straight forward, particularly in Newtonian gravity, but in general relativity I am apprehensive to just give an off the cuff guess because the knowledge I do have of GR informs me that the notion of 'mass' can be a slippery thing when considering the Einstein field equations, as illustrated by the difficulty and complexity in defining mass in general relativity. The definitions such as the Komar mass and Bondi mass I am familiar with, using them to show that the mass of a Schwarzchild black hole with metric \(ds^{2} = \left(1 - \frac{2GM}{c^{2}r}\right) c^{2} dt^{2} + \ldots\) is indeed M so something I remember doing as a homework problem years ago. This is why I asked you to give a formal definition.

But what does my lack of an answer get you? Are you any closer to demonstrating your claim? No. In fact you're further away because I've explained at length, numerous times, how everything boils down to \(T^{ab}_{;b}=0\) and if your additional terms do not violate that then the discussion is over. I said this several posts ago but you've failed to respond to any such things, despite them being the 'real business' you supposedly wish to talk about. if you were talking about the Komar mass the details are important. Notice how the notion of a Killing vector comes up often in that link, almost like what I've been talking about in regards to Killing vectors, metrics, contractions with tensors like \(T_{ab}\) were all in the right general area to address the main point you try to make.

Despite the self delusions hacks on forums like to tell themselves I don't claim to know everything and have said so many times and don't mind being wrong. Being mistaken about something and being told you're mistaken is the best way to learn. Unfortunately I do not think this is something you're familiar with, since you clearly have issues with people like myself, who have put in the time and effort to grasp a wide range of topics to a working degree. As I've demonstrated I have a working familiarity with numerous bits of GR, though not exhaustive, enough to get by in various domains related to differential geometry, relativity and quantum field theory. Would I like to know more? Sure and I'm fortunate enough to be paid to learn more and do more. Do I have some kind of paralysing fear people might find out I'm not omniscient? Nope.

As I've said, the sorts of things you insinuate about me or my (possible) actions in regards to this thread say more about you than me. If you really believe that people like myself are always afraid of having gaps in our knowledge exposed then, quite frankly, I pity you. This "Oh I bet you're dying to close the thread!" repetitiveness from you illustrates how you cannot grasp that someone might be comfortable with their level of knowledge and with themselves enough to say "I don't know" or "I was wrong". Your constant paranoid insinuations suggests that if our roles were reversed you'd not hesitate to remove the thread. Like I said, if that is the case I feel sorry for you. Too many internet hacks see someone who has some working knowledge and project their insecurities onto said person. "You are confident about something in physics? Why surely you must believe yourself all knowing!". Farsight is functionally innumerate, cannot do any mainstream physics and has zero working models in his 'work' yet claims its worth 4 Nobel Prizes. Once when challenged about a claim of his he said "I bet I could beat you at arm wrestling!". Arm wrestling, now that is a sign of insecurity.

You denounced my mentioning that I spent today working for a charity, calling it boastful. If I were a boastful person, of the kind Farsight or Sylwester are, then I'd be posting my work all the bloody time, even when no one asked about it and no one cared. I don't. I don't start threads much either, which many a hack does in a thinly disguised attempt to talk about their own claims. You started this whining thread rather than getting down to 'real business' directly, in contravention to the forum rules which say complaints about moderator actions in regards to thread closures should be handled via PM.

I don't need to come onto a forum to post work to get existential meaning, I am more than satisfied with my day to day work within the company I work for. This means I am not desperate to spew my work all over the place and be seen as 'knowing it all', unlike people like Farsight. Your inability to grasp that someone like myself could hold such a view is really quite sad. Anyway, enough of my elaboration on why you have misplaced grievances...

Now perhaps you'd like to address the multiple posts I've made where I go through, in detail, how to check energy and momentum (and anything else) conservation within general relativity. Thanks for you unwillingness to reply to said posts you've moved exactly zero distance closer to justifying your claim there's an issue with GR in regards to energy conservation. To justify your claim please show that your scenario necessarily leads to a violation of \(T^{ab}_{;b}=0\). Anything else, namely discussions of passive mass or Komar mass or what paranoid conspiracies you think are against you on this forum, is irrelevant. If you cannot do so then I indeed addressed, several times, the point you've been trying to make.

 

Not true - I've told you my position on that in #21. And it was elsewhere that someone pointed out, or rather admitted, that the vanishing of the divergence of the stress-energy tensor, the conservation of energy-momentum law in GR, strictly holds good ONLY FOR A POINT VOLUME! Which imo makes it about as useful as a bad tooth. Should I laugh or cry?

The answer is no. I never claimed to understand all the mathematical machinery of GR, but you disingenuously pitched your responses as though I was somehow sufficiently expert - and went on AFTER such presentation to say that I probably didn't understand it! What a hypocrite. Then why do that in the first place? A genuine person would naturally ASK first about my level of expertise and then tailor a response to suit - but not you!

See above. Disingenuous - since you had shrewdly figured already that I'm not a GR expert and have never so much as suggested otherwise. So a worthless 'offer' and you knew it. My claim is simply to have an insight on the implications of stress as source term. As manifest in that simple gedanken experiment you utterly refuse to deal with! The idea was real experts could look at it and either accept there is a real issue with stress in GR, or calmly and honestly provide a detailed explanation of how and why it all works out 'normally' in the given scenario. Something you have utterly refused to do! Given your now confessed abyssmal ignorance of nature of stress in GR, clearly you are no expert and that explains in part your continued evasion.
And I repeat, it is valid and proper to test any theory/theorem/principle by way of specific example. Einstein was famous for doing that - using gedanken experiments to arrive at SR and beginnings of GR. And that process is now just so much old hat - is it?

And yes you are afraid since you have continually evaded my question on nature of stress as source term. Beginning in #1 and culminating in my call for clarification in #18 of your pitiful 'iffy' and 'sort of' one-line response that finally came in #16. And since you have continued to evade, with finally in #24 the sole exception of an embarrassing admission that I shall come to.

Two lies in one! Firstly, I never claimed A-wal was banned - by you or anyone. Either back up that lie by citing me saying such, or retract. A futile request since you response is always just to invent some 'issue' that supposedly calls for me to 'retract' in turn.
Second - it was YOU that claimed A-wal was banned! This verbatim from your #12:

Likely a Freudian slip on your part. I did some easy checking. He is still listed as member to this day!:
http://www.sciforums.com/memberlist.php?ltr=A&pp=50&sort=username&order=asc
Join date shown as 03-03-13. Now check the ban list, which is in date descending order:
http://www.sciforums.com/banlist.php
A-wal NEVER appears there over the entire span of his still existent membership.
So you lied twice. Or would you rather portray this as merely gross incompetence coupled with recklessness on your part - an administrator who is supposed to know the ins-and-outs of this place like the back of one's hand?! And to have minimally checked before making such a false assertion. Feeling good about yourself?
And lastly on the A-wal matter. His threads have not vanished - another lie. But you or presumably some good buddy had managed to remove all means to find them through any publicly available method at SF. By luck perhaps I had saved the URL's:
http://www.sciforums.com/showthread.php?133852-Can-Anyone-Answer-These-Black-Hole-Problems
http://www.sciforums.com/showthread.php?133851-Relativity-Made-Simple
Take note - in future you or whoever will need to be more thorough and actually physically remove the threads themselves!

Blah blah blah....but the bizarre antics, Freudian slips, and lying I have covered above is sufficient to cover that bit of diversionary hype. Let's continue with another piece from #24:

'Whining thread'?! What gall! That phrase you have used often in a trolling way, and I have said virtually nothing in response, because I recognize your provocative style and intent. But let's recall how you shut down that previous thread. In #44:
http://www.sciforums.com/showthread...tation-in-GR&p=3062075&viewfull=1#post3062075

So, instead of specifically censuring the two irresponsible thread wreckers and simply removing those posts - the proper thing to do, you instead used that in part as excuse to close it. And then had the audacity to declare that if I at all protested this heavy-handed autocratic response of thread lock, I was to be labeled 'a paranoid conspiracy hack and doesn't deserve any discussion anyway'. Ultra-trolling behavior - specifically designed to incite an angry response at such utter autocratic unreasonableness. Which would give you the intended excuse to then ban. But naturally you won't admit to that calculated intent - ever. As you made such slurs in public, it was fitting and proper to address them in public - not by way of some invisible PM! And the embarrassing contrast is now back for all to see - those two A-wal threads that, against all your expectations, I had saved URL's of, are back to haunt you. I can freely cite attitudes and responses in them, and contrast to how you have dealt with me. An amazing contrast indeed. You close threads because of 'terrible presentation' and 'bad attitudes'? Just check out those lengthy A-wal threads! Which although arranged to be unfindable, are still physically there and were never closed. Ace hypocrite!!

So the great string theorist and GR 'expert' AlphaNumeric frankly admits, finally, that he doesn't know the answer to my simple question - 'does passive mass increase'? I had put it far more precisely than just that, but we all know the full context to that question, and this admission is telling!
You continually berate me for my ignorance of basics of GR, yet yourself now admitting to being stupefyingly ignorant of the most basic properties of stress in stress-energy tensor!! Having in your single-line response in #16 misidentified it as contributor to energy density term. Which it is not at all. Unbelievable!
Well tell you what AN, may I suggest you do some careful reading of just one Q&A passage here:
https://en.wikipedia.org/wiki/Mass_...simple_examples_of_mass_in_general_relativity

Specifically, the fourth example there, beginning with "Imagine that we have a solid pressure vessel enclosing an ideal gas."
It's merely a brief coverage of one topic dealt with in that article by Ehlers et. al that I cited much earlier in that closed thread. Once you have digested it - maybe the penny will drop. Just maybe you will then acknowledge that your single line answer in #16 I cited in #18 is way off the mark. That answer was, in so many words, claiming pressure contributes only indirectly, via the energy-density term - the first diagonal in the stress-energy tensor. Dead wrong! The 9 stress terms in tensor are completely separate to the energy-density term. Contribution to mass - active, passive, inertial, from stress-induced elastic/hydraulic/pneumatic energy is fully subsumed in the one energy-density term. But you have shown total ignorance of that fact.

Made my own position on contrasting character briefly but clearly enough in e.g. 2nd para in #9 here:
http://www.sciforums.com/showthread...losed-thread&p=3062384&viewfull=1#post3062384
Your foolish comments about 'pressure magically coming from somewhere' made earlier re my 'disk-brake' scenario was disingenuous tripe. Nevertheless, that little gedanken experiment does imo expose something truly 'magical' about character of stress as it is portrayed in GR. If and as you realize that, admit it, then maybe something useful can happen. But I doubt it, given your continual penchant of responding with a wall of trolling accusations. And occasionally recapitulated standard 'proofs' that imo 'disk-brake' scenario strongly calls into question.

See my above comments!!
In the meantime, anyone else want to chime in on this issue. Markus - are you good to tackle the core issue here, and not just dance around the edges as happened in #2 and #4? No obligation naturally and would understand reasons for remaining silent. A pity though if AN just manages to totally shut out any rational discussion of the real core issue.

 

Markus, you never took up my invite in last para here:
http://www.sciforums.com/showthread...losed-thread&p=3067728&viewfull=1#post3067728
I note your display here of a good knowledge of GR, including #10 "9. ENERGY IN GENERAL RELATIVITY". But try applying then to the apparently naive derivation here:
http://www.sciforums.com/showthread...tation-in-GR&p=3061964&viewfull=1#post3061964
If the circular geometry seems somehow a bit awkward, well just choose something even simpler - say a uniformly moving flat belt subject to a patch of uniform frictionless pressure. How can one escape the need to either ditch conservation of energy-momentum, or as I strongly prefer, role of pressure as source? Well, are you willing to attempt to apply your GR math know-how to that case and show there is no actual paradox? Do we agree it's not much use having impressive math if it can't then be applied to a relatively simple situation such as I have presented?

 

You don't have to do any "impressive maths" here, because, since the covariant derivative of the metric tensor vanishes

\(\displaystyle{g{^{\mu \nu }}_{||\nu }=0}\)

then so does the Einstein tensor, since it is just a linear combination of the metric tensor and its derivatives :

\(\displaystyle{\left ( R_{\mu \nu }- \frac{1}{2}g_{\mu \nu }R\right )_{||\nu }=0}\)

And because the Einstein field equations tell us that the Einstein tensor is exactly equal to the SEM tensor multiplied by a constant, then that means of course that the covariant derivative of the SEM tensor must vanish as well :

\(\displaystyle{T{^{\mu \nu }}_{||\nu }=0}\)

In other words, total energy and momentum is always conserved globally. Note that the object T is the full stress-energy-momentum tensor, including the pressure terms. Therefore you can have pressure as a source term in the field equations while at the same time not violating conservation of energy-momentum, or the field equations. There is no paradox here. In fact I don't know why you would think that there is, in the first place.

The above is a general way to show this without having to go through any tedious calculations for specific scenarios; however, if you wish to propose a specific stress-energy-momentum tensor and a metric here, then I will be happy to check for you that the covariant derivative really does vanish, i.e. that energy is conserved.

Perhaps you are not aware of this, but the definition of the Stress-Energy-Momentum tensor is not arbitrary; it is the conserved Noether current associated with the fact that we find certain symmetries on our space-times, specifically translation invariance. The meanings of the tensor's elements follow directly from there. No one just decided one day that there needs to be a "pressure term" in it; rather this is the direct consequence of symmetry considerations through Noether's theorem.

Now that this has been established and answered for you, can I respectfully ask you to please keep your issue confined to your own thread, and don't hijack this presentation. Thank you.

 

From: http://www.sciforums.com/showthread...ivity-Primer&p=3070997&viewfull=1#post3070997

OK Markus - back here in the other thread. Now you say my query has been answered in the rest of your #27 above linked. Not really, since I asked specifically how it works out in the given example, or any variation thereof. I agree entirely with your words in #27 "You don't have to do any "impressive maths" here...", but not with your words "There is no paradox here. In fact I don't know why you would think that there is, in the first place." But actually you should be very well aware, given our discourse between #2 and #5 here. Read again my response elsewhere - last main para in #21. What exactly is the problem with my position there, and with you or whoever doing as suggested? As I asked again in that other thread.

Well anyhow you asked above for a specific metric. Consider then a scenario where the setup is floating in deep space. Lab frame then is locally flat Minkowski metric to any degree required. Employ artificial g by mounting db ('disk-brake') on a large carousel set rotating at some fixed angular velocity ω wrt lab frame. Carousel and db spin axes orientations coincide, db mounted on carousel periphery at some mean radius r very large wrt db dimensions, hence subject to a centripetal mean acceleration g = -rω^2. Relative rotation speed of db wrt carousel is ω'. Just then substitute in expression I gave in #11:
http://www.sciforums.com/showthread...tation-in-GR&p=3061964&viewfull=1#post3061964

In present case, just substitute ω' for ω, and recall from #11 above that d is the moment arm vector from db spin axis to center of pressure applied by pads.
Yes I agree there is no "impressive math" required. A matter of dealing with the basic character of stress in GR setting. And see where it leads in the simple scenario given. Just why no-one so-far has been prepared to do that is a matter of speculation. Now it's your choice whether to attempt a specific resolution as requested, or just keep falling back on the generalized proof, as was AlphaNumeric's position. Or to just ignore this post entirely.

 

Can you explain to us why, having already been shown in the general case that energy and momentum is conserved in a very elegant way, one would need to waste time doing a very tedious specific calculation ?
Just wondering, because this really is beyond me. Is there anything mathematically wrong with the proof I have given ? I don't believe there is.

Anyways, you want to base it on a metric which is locally Minkowskian, and describes the space-time around an object which is basically a uniformly rotating disk. If we assume that the thickness of the disk is very much smaller than its radius, and if we further assume that the whole setup is stationary, static and electrically neutral, then we can use the ansatz proposed by Hartle and Sharp ( 1967 ) for just such a problem :

\(\displaystyle{ds^2=e^{2\mu }(d\rho ^2+dz^2)+\rho ^2e^{-2\nu }(d\phi -\omega dt)^2-e^{2\nu }c^2dt^2}\)

The solution of the Einstein vacuum equations ( i.e. the field equations for the space-time outside the spinning disk ) can be obtained by solving

\(\displaystyle{R_{\kappa \lambda }=0}\)

for the two unknown coefficient functions \(\mu, \nu\). I haven't done the maths, but just by looking at it I see no reason for such a solution not to exist, but it would certainly be very complicated. However, that is not what you are interested in; you want to know if energy as a whole is conserved if a brake is applied to the spinning disk. For this you'd have to examine the full interior field equations, being

\(\displaystyle{R_{\kappa \lambda }-\frac{1}{2}g_{\kappa \lambda }R=\frac{8\pi G}{c^4}T_{\kappa \lambda }}\)

with some unknown energy momentum tensor on the right, which includes terms for pressure, stress etc etc. All of these form sources of the gravitational field in the interior of the disk. There are now two ways to show that energy is conserved in this setup :

1. The hard way - find an explicit expression for the stress-energy-momentum tensor in our coordinate system, and calculate its covariant derivative to show that it vanishes
2. The smart way - show that the Einstein tensor on the left is divergence free. By the EFEs this tensor is equal to the stress energy momentum tensor, up to a constant. Almost no maths required at all.

(1) is totally impractical, because we would need to take into account all dynamics of a disk made of some material, its surface, and its interaction with the braking mechanism. The resulting expression would be so complicated as to be impossible to handle. But as I said already, you are welcome to try. If you give me an explicit expression for the SEM tensor I shall be happy to calculate its covariant derivative for you. I can already tell you though that the result will be exactly zero, hence it would just be a colossal waste of time to attempt such a calculation.

(2) on the other hand is so simple as to be almost trivial; the divergence of the Einstein tensor is

\(\displaystyle{\left ( R_{\kappa \lambda }-\frac{1}{2}g_{\kappa \lambda }R\right )_{||\lambda }=0}\)

because all the terms are linear combinations of the metric tensor and its derivatives. Since this covariant derivative always vanishes, we do not need to know the elements of the SEM tensor to realize that it must vanish too. I invite you to check for yourself.

So no matter how you approach this, energy conservation is never violated, regardless of whether there is a "pressure" term in the SEM tensor or not.

We are working on a smooth, differentiable manifold, and with an isolated system. If energy conservation holds in any infinitesimal volume, then, by Stoke's theorem, there is no net flux through a hypersurface enclosing an arbitrary number of such volume elements; in other words

\(\displaystyle{\oint_{\partial V}\sqrt{-det g}\xi ^{\mu }T_{\mu }^{\nu }d^3S_{\nu }=0}\)

with a Killing vector field \(\xi\). So energy momentum is conserved globally as well. The conservation of these quantities is a geometric result, and quite independent of any physical details of the scenario.

If you want to know more about the relativistic Hartle-Sharp disk, have a read here :

http://articles.adsabs.harvard.edu//full/1969ApJ...158L..65B/L000066.000.html

 

Thanks for that one. Only a few years after Roy Kerr's solution. This 'slide show' mentions some of the interesting physics associated with the Kerr metric analysis of spinning objects. Just for fun.

http://amalfi.colorado.edu/~rosalba/astro2030/KerrBH.pdf

 

The relevant physics on the exterior of the disc will be governed by solutions to the Ernst equation. On the plus side, this equation is integrable so we can read off all the conserved quantities (take the trace of the powers of the relevant Lax operator). However, to do real physics, we'll have to solve an initial-boundary value problem associated with the Ernst equation. This is monumentally complicated. There are a class of such problems that can be solved pseudo-analytically, by solving a matrix Riemann-Hilbert problem on a specific Riemann surface, but even then the problem is hard (Fokas & Lenells, 2009). In the general case you might be able to attack the problem in the large t-limit using the nonlinear steepest descent method of Deift & Zhou (Annals of Math. 1993), but this method in itself is very complicated.

If it's not clear, establishing analytical results in GR (or any theory governed by a set of nonlinear PDEs) is extremely hard. If it was easy, I'd be out of a job.

 

[edit]
I suggest reading the final section of this post first before typing a reply. Only replies of a particular kind will be entertained.
[/edit]

Actually that is something I regularly do with people but from experience when I ask people such as yourself about their level of knowledge they don't like it or assert it isn't relevant.

Actually my responses were pitched at the level of someone who has taken an introductory course in GR.

Now that you have clearly stated you do not understand it I can therefore conclude you are unfamiliar with the more subtle issues of how to define mass and energy precisely within general relativity, thus showing why you failed to appreciate the motivation behind my request you define specifically what you meant and why the Newtonian perspective given in the link you then provided was insufficient, despite you saying it was enough.

No, not at all. If you have a valid position it doesn't matter whether you're an expert or not, competent or not. All that matters is whether you can make your case properly. Given you are now being open about the shortcomings of what you understand and what you're been presenting it isn't my fault you haven't got a good enough case to present to a journal. Similarly it means you haven't presented a good enough case for your position and you know it.

You have insight into the specifics of the stress energy tensor but you don't know how to do much, if anything, with tensors? Quite....

If you find the responses from people here not to your liking then please find another place, no one is preventing you asking other people.

Please link to a post of mine where I said I am an 'expert' in general relativity. I know particular things about general relativity and I'd say I'm in the top 1% of the population for it but then so would anyone who has ever done a GR course since such a group is less than 1% of the population anyway. Within said group I'd say I'm pretty decent. I have passed various courses in GR and GR related topics, up to and including distinctions at graduate level, as well as having a PhD in the specifics of deformations to particular curved space-times but the domain of GR is vast and no one knows all of it. Am I an expert in even the domain my thesis was written in? Nope, having met the sorts of people who are experts in the domain I know I do not come close to most of them. Am I the worst? Nope.

And your use of the word 'abyssmal' is what Wikipedia would term a 'weasel word', an unnecessary and often inappropriate adjective or description based on personal bias. Do I have some experience working directly with the stress-energy tensor? Yes. Has it any relevance to the stuff I did during my PhD? Not directly. As such that puts the last time I worked directly with the field equations at 8 years ago. Considering I haven't recently thumbed through a GR textbook I'd say I'm not too rusty and despite your many predictions implying likely behaviour to the contrary, I have no issue saying so. Instead I have used what knowledge I can immediately recall and implement to address your query in alternative ways. There is more than one way to skin a cat, the fact your point can be addressed in a way not directly aligned with your preconceptions doesn't bother me at all.

Where did I say otherwise? In fact I specifically agreed with you. Funny how you keep calling me disingenuous but you have no problem trying to make it seem like I'm disagreeing with points you've made despite explicitly saying I agree with them!

You demand to be treated in a particular way and yet time and again show you aren't worthy of the necessary level of respect.

I responded in multiple ways. I asked you for clarification and you provided what you don't realise is itself a very 'iffy' definition. The concept of passive mass is simple enough within Newtonian gravity but you have yet to provide a sound definition within general relativity.

The notion of different types of mass, such as gravitational and inertial, is something I've seen discussed in a GR context but not the notion of passive mass. The notion, IMO, seems somewhat unnecessary or ad hoc within GR as the field equations take care of everything. The influence one object has on another is not give some kind of direct interaction as might be seen from Newtonian gravity but rather they both create a warped space-time and each moves according to that warping, such as along trajectories defined by the geodesic equation. Each object has no need to know what else contributed to the warping, only what the warping is. If you wish to try to shoe horn in concepts whose original definitions are heavily based on Newtonian concepts then fine but there's plenty of examples in GR where that can become a nightmare, as illustrated by the plethora of different mass definitions within GR.

My comments about A-wal were in response to your implication/accusation/whatever that something dodgy had occurred in regards to him and his interactions with myself, such as posts no longer being there. This occurs when someone is banned in a particular way, thus I concluded he had been banned. Did I check? No. Do I care he can still post? No.

You imply a conspiracy and that I and/or others on the moderating team have done something to cover up his posts. Ask yourself this, if I'd been involved in such an activity why did I mistakenly conclude from your comments he was banned? Wouldn't I know if he was or not? If I wished to cover up his posts why not just ban him and have the forum auto-delete all of his posts? That'd ensure complete and utter removal of any interactions between him and myself from the forum. And do you think if I wished to cover up something involving his posts I'd forget to remove his posts? The reason none of that occurred is because the conspiracy you're implying didn't/doesn't exist. This is typical of paranoid conspiracy theorising, ignoring how the most basic of facts contradict the proposed conspiracy.

And where did I say I know 'the ins and outs of this place like the back of my hand'. I am not an administrator, I am a moderator, I only have the power to modify this sub-forum and the astronomy one. And recently, as anyone who spends much time here will have noticed, I've not been around much due to massive work commitments. As such I've missed a considerable amount of the comings and goings on the site.

So all of this 'feeling good about yourself?' stuff is implying deception where none exists. You really do seem to struggle with the notion that someone like myself is able to have disagreements with people and just not go into some ego driven spiral of vengeance and conspiracy. I have no problem getting things wrong, I do not pretend to be all knowing or an expect in anything. Hacks often mistake my higher than average level of knowledge for particular domains of science and mathematics as delusions of omniscience. Compared to most I know a lot but then I've spent more than a decade doing maths and physics beyond the legally required level here in the UK so one would certainly hope that to be the case.

Oh and by the way, the ban list doesn't show if someone has been banned and then reinstated, it shows only those who are currently banned. If A-wal were to have been banned for say a week for some reason and that week had passed his name is automatically removed from the ban list when the account is unlocked.

Wow, there's some hypocrisy if ever I saw it.

Closed with a suggestion to start afresh with a post which addresses some of the requests people had made.

Heavy handed would have been a deletion. Closing a thread, after more than a day and a page of off topic posts, and suggesting a new one start up if the original post returns and wishes to continue is not a 'heavy-handed autocratic response'. You're not the first person to have such a thing happen but you are the first to make such a massive issue of it and imply an attempt to shut down discussion.

Like I said, if I'm so 'itching' to implement some kind of cover up then why didn't I do just that? Why not delete/ban all relevant posts/parties? Why allow this thread, which is now more about moderating actions than physics and thus should be in the 'open government' section? You have repeatedly said/insinuated I'm itching/desperate to do all these kinds of moderator power abusing and yet I haven't. Of the options open to me in regards to your original thread both "remove all off topic posts" and "close due to too high a fraction of off topic posts and a large delay in the OP returning" options are valid. Given you felt this to be a 'heavy-handed autocratic response' then the forum rules state you should take it up with admin over in the 'open government' forum. That is what it is for. But I've not sliced this thread up into pieces and moved (or deleted) the resultant new thread because I'm fine talking about it here if that is what you wished.

Nice quote mining. Let's look at the full sentence : "Of course if he interprets this closer as somehow a "OMG, you're a heretic!! We must shut him up!" then he's a paranoid conspiracy hack and doesn't deserve any discussion anyway. ". The first half of the sentence qualifies the second half, yet you only quoted the second half. Once again you show you are not above behaviour you complain about in others, ie misrepresentation.

The first half of the sentence (and the post as a whole) clearly states why the thread is being closed and makes it very clear further discussion is fine, even suggesting how to help such discussion move more smoothly. The final sentence was to pre-empt any insinuations from yourself that you are receiving here the same treatment you claim you experienced elsewhere. The post makes it clear the subject is not off limits. Rather it makes it clear that if the original poster, ie you, were to take a common type of moderator action as somehow an attempt to shut down discussion then said poster, ie you, is very much mistaken and would have to be of a particular mentality which makes discussion pointless as said poster, ie you, would show himself to be more about causing a scene than causing a discussion. And here we are, seeing that despite this pre-emptive comment you do imply some kind of 'shut them up!' conspiracy is going on.

You could have raised an issue about the thread closer if you wished, the post doesn't make it automatic such actions would label you a 'paranoid conspiracy hack'. For example, you could have PM'd me or started a thread in open forum asking something along the lines of "My thread was closed before I had a chance to response on the grounds of too much off topic discussion and a lack of response from me. I'm unhappy with this action and would prefer it is a moderator could remove the off topic posts and then reopen it, rather than I start a new thread". That's a civil way to query moderator actions and would have almost certainly been done upon request. Since such an action on your part wouldn't be insinuating there's some kind of "OMG, you're a heretic!! We must shut him up!" motivation behind the thread closure the first half of the sentence, the qualifier, would mean the second half of the sentence, the labelling, wouldn't apply.

However, since you have shown you do think there are ulterior motives going on, including in regards to A-wal, you have met the conditions of the first half of the sentence, the qualifier. You didn't have to but you did.

It wouldn't incite an angry response from someone who didn't think there's something dubious going on. It was pointing out that if someone views the action, an action which has been done many times before elsewhere without complaint, as anything other than 'forum tidying' then their ability to evaluate situations and actions would be severely compromised, as it would imply an overly paranoid point of view. There were clear grounds for closing the thread (though, as I've explained, this was not the only course of action) and so viewing the action as anything other than that was questionable, to say the least.

This is exactly what I'm talking about. You see intent where none exists. You assert desires others supposedly have without justification.

I'm now wondering if you are trying to be deliberately provocative. You have repeatedly asserted things about my supposed intentions for which you have no evidence and which are not consistent with actions I have taken or actions I could take but haven't. That in itself is enough to give you an infraction warning. There's enough in this thread to give you several. The catch 22 I have is that if I don't give you one you can continue to escalate this unfounded rhetoric and if I give you one you'll just say "Ah ha! I knew it!".

Then you could have used the open government forum or contacted an administrator.

Yet more misrepresentations from you. Where did I claim to be 'great' or an 'expert' in either? I'm competent in parts of them. I'm more familiar with them than most but that isn't 'great' or 'expert'. In fact I've said in the past the opposite, that I was closer to the bottom than the top of people within my research community for string theory.

This, yet again, is a comment from you worthy of an infraction.

Yet more 'weasel words'. And you again distort things. I showed clearly I could do a number of things within GR which pertained to the issue at hand. Compare that to yourself, who cannot do even the tensor calculus necessary to understand the specifics of the field equations. Using what knowledge I do have to address situations outside of previous experience is hardly something I'm ashamed of.

Notice how your link explicitly mentions the Komar mass, something I've previously brought up in regards to the problem of defining mass within general relativity. Notice how the section makes it clear that the answer is dependent upon the definition of mass. Specifically it says One might also ask about the answers to this question if one assumed that one were asking about the mass as it is defined in special relativity rather than the Komar mass". It explicitly says that if we defined mass differently then further complications occur. It explicitly highlights the problem with a non-isolated system, which is precisely what the question of what is causing the pressure brought up by myself and others was about. When you were asked to give your definition of passive mass you didn't provide anything of that kind of definition, instead reaching for a Newtonian based concept.

And where did I say anything about the energy-density term? I talked about the components of T but in regards to the field equations implying local energy-momentum conservation \(T^{ab}_{\quad;b}=0\).

Other than the fact the question of whether a system is isolated or not (ie something external influencing it) is precisely the issue brought up in regards to the differing definitions of mass given in that link you just gave to Wikipedia. Other than how the pressure vessel example in your link explicitly highlights the importance of distinguishing between considering the gas and considering the gas and the thing holding it under pressure, ie the vessel, which again pertains to points raised by Rpenner and myself.
Yeah, other than that

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In your opinion. And if your example calls into question GR's point of view why did you link to that wiki page, thus trying to use the GR point of view to make your case?

This harkens back to something previously raised, the need to test and retest models. We test models by doing experiments. Since we're not doing an experiment here we're purely within the realms of the model of gravity that is general relativity. What precisely are you considering 'magical' and what is being brought into question? Your wiki link points at some GR result, it doesn't illustrate anything 'magical', it illustrates GR can model such a scenario. Thought experiments are used in at least 2 ways to call into question a model : a sequence of logical properties of a model which lead to a conclusion qualitatively inconsistent with reality and two alternative ways to describe the same hypothetical scenario to see if different results are obtained. We're talking about the latter.

Can you show that within general relativity your disk brake example behaves in a way contradictory to general relativity? That's what is required. Nothing about that wiki link shows anything 'magical', it shows GR has a way of addressing such systems. You have not shown the standard proofs do not apply to the disk brake case. As your own link shows, pressure terms contribute to the Komar mass of pressured systems and by how much depends on the definition of what the system in question is. Seriously, what is magical? That the pressured vessel example includes pressure terms? That more than a single component of the stress-energy tensor contributes to mass?

Notice nowhere in your wiki link does it mention 'passive mass'. The notion is not relevant to the question of what the masses of the gas or pressure vessel are. All that mattered was a precise formal definition of the quantity in question, in this case the Komar mass. I repeatedly mentioned the Komar mass in previous posts and asked you for a quantitative definition, highlighting how mass is a slippery concept in GR. If you had said "I want the Komar mass of ...." then it would have been much more straight forward as you'd have given a proper definition, one valid within general relativity, and discussion about its relationship to the stress-energy tensor could have followed. Instead you use a Newtonian concept. I was honest enough to say I'm unfamiliar with 'passive mass' within general relativity. Your link illustrates why that is not unsurprising, GR handles things differently, instead dealing with other, rigorously defined, notions of mass. As my repeated mentioning of it illustrates I'd heard of the Komar mass. I also knew it depends on the stress-energy tensor in a non-trivial manner. But given your admitted lack of knowledge about this sort of stuff in GR you couldn't properly define what was of interest. Now I don't claim to know all about the Komar mass but I know enough to be able to do (and to have done) basic calculations with it, including showing that \(K_{ab} \equiv 2T_{ab}-Tg_{ab}\) is such that \(K_{00} = \rho + 3P\) when \(T_{ab} = (P+\rho)u_{a}u_{b} + P g_{ab}\) in an orthonormal frame (ie a convenient one). This then feeds into the Komar mass in a nice way. In fact I just checked the lecture notes I made years ago and indeed exactly that calculation is done within them, showing how the Komar mass indeed gives the Schwarzchild metric black hole mass as the M in the \(1-\frac{2GM}{r}\) metric coefficients. It involves a Killing vector in the 0 component, ie \(X = \partial_{t}\), another thing I mentioned previously but which you didn't understand.

By your own links there are multiple definitions of mass in GR. By your own links different definitions require different amounts of information. By your own links the definitions do not refer to such things always as 'passive mass'. By your own link your definition of 'passive mass' was Newtonian, not relativistic. By your own links the quantities you were interested in pertained to the Komar mass, something I'd repeatedly mentioned and am familiar with. I doubt any of this will cause you to stop with your 'weasel words' and repeated "The great string theorist and GR 'expect' AlphaNumeric" misrepresentations but at least others can see the situation. For them I'll be a bit more specific, to show the precise sequence of steps in regards to the mass definition and how it isn't surprising in the least how the stress-energy momentum contributes more than just \(T_{00}\) :

The Komar mass is defined \(M_{K} = \frac{1}{4\pi}\int_{S^{2}|_{r=\infty}} \nabla_{a}X_{b}dS^{ab}\). The integral is defined at asymptotic infinity over a sphere enclosing 'the entire universe'. Using Stoke's theorem \(M_{K} = -\frac{1}{4\pi}\int_{\Sigma} \nabla^{a}\nabla_{a}X_{b}d\Sigma^{b}\). Since X is a Killing vector \(\nabla_{a}X^{a} = 0\) since by definition \(\nabla_{a}X_{b} = -\nabla_{b}X_{a}\) and by contraction with \(g^{ab}\) we get \(\nabla_{a}X^{a} = 0\). Killing vectors also satisfy \(\nabla^{a}\nabla_{a}X_{b} = -R_{bc}X^{c}\) by virtue of the identity \(\nabla_{a}\nabla^{a}X_{b} + R^{a}_{bac}X^{c} + \nabla_{b}\nabla^{a}X_{a} = 0\) which then becomes \(\nabla_{b}\nabla^{b} X_{a} + R_{ab}X^{b} = 0\). We now consider the field equations \(R_{ab} - \frac{1}{2}R g_{ab} = 8\pi T_{ab}\). Some rearranging gives \(R_{ab} = 8\pi \left( T_{ab} - \frac{1}{2}T g_{ab} \right)\) and so we can combine all of these to give \(M_{K} = \frac{1}{4\pi} \int_{\Sigma} \left( 2T_{bc} - T g_{bc} \right) X^{b} d\Sigma^{c}\). That's the origin of the term \(K_{ab} = 2T_{ab} - Tg_{ab}\) I mentioned previously. \(T_{00} = \rho\) is energy density and \(T_{ii} = P\) is pressure, giving \(T_{ab} = (P+\rho)u_{a}u_{b} + P g_{ab}\). If we're in the aforementioned convenient orthonormal frame then \(T_{ab} = \textrm{diag}(\rho,P,P,P)\) so \(T = 3P-\rho\) and giving \(K_{00} = 2T_{00}-Tg_{00} = \rho + 3P\). All of this is standard bookwork but the label 'passive mass' is not used. Apparently 'the great string theorist and GR expert AlphaNumeric' has abysmal knowledge because Q doesn't know enough GR to be able to even refer to things in their standard ways or give good enough definitions, all the while thinking there's something 'magical' going on, as GR has a much more complicated notion of mass (or rather, notions) than Newtonian gravity.

Remember, you started a thread complaining about me, after explicitly being asked to start a new thread with details so discussion could be made. If you'd been able to answer the questions I asked of you properly we could have spent the last page talking about the Komar mass and how it depends on the energy-momentum (stress-energy) tensor. Instead we get to hear all about the chip on your shoulder. Even if someone claims to be an 'expert' in a domain that isn't the same as claiming to know everything. Even very precise niche domains of science can be so fast no one can know it all and anyone who has done any science to a reasonable level will know and accept this.

You really cannot see how some people can be comfortable in their 'skin' enough to have no issue admitting mistakes or lack of knowledge, can you? Seriously, whatever issue you have with 'physics heavy hitters' you need to deal with it. If you really believe the nonsense you come out with, no seeing the dishonesty, misrepresentation and paranoia of your posts, I feel sorry for you.

You get this one last long post, as I had today off work so was willing to spend the time replying. Given the ever growing length of these back and fore posts further discussion about "I said... you said.... I said.... you said...." I'm not going to bother engaging you with. If you wish to 'talk shop' then Marcus looks like he's willing to indulge you, I've covered why the various insinuations about the dependency of certain mass definitions on stress-energy components is fully expected to be more elaborate than just the energy density term, why your 'passive mass' definitions were insufficient and how your wiki link is completely consistent and supportive of many things I've said, including my repeated mentioning of the Komar mass and various definitions of mass in GR.

If you wish to complain about the moderator action I undertook when I closed the previous thread then post a new thread in the open government forum. If you wish further engagement from me in regards to this 'magical' issue you mention then a new thread can be made, provided it includes a precise definition of the mass you're using and a demonstration using said algebraic definition that something is wrong with GR, ie it is inconsistent. As the Komar mass description in your own wiki link illustrates, precision is everything, consider that an illustration of the quantitative detail required for further discussion.

Further insinuation of there being an attempt to remove/silence/whatever A-wal will be given an infraction warning. As explained, I am not an admin so I cannot remove posts outside my designated forums. Furthermore there are automatic ways to remove all posts by a user, banned or not and A-wal is not banned. These all illustrate no attempt to scrub the forum has been made, contradicting your implications.

I'll leave this thread open, though if you or Marcus want your discussion to be moved to a new thread, free of the clutter here, then PM me.

Beyond that I'm going to call it a day. If you wish to go around The Internets declaring me an idiot or a liar or someone involved in elaborate forum conspiracies to remove discussions so be it. I'm happy for anyone else to read this post and see that I am familiar with the relevant concepts once it becomes clear what you're talking about, no thanks to your inability to provide definitions/terminology expected of the subject. The original moderator action, closing the thread after a day and more than a page of off topic replies while giving the option for a new thread, which prompted your complaints I stand by.

 

Firstly I should say thanks for taking some considerable time and effort in answering. But to your questions. I would basically just repeat that quote from #21 and apply specifically to the case in point. If the proof you gave is indeed valid, internally consistent, there should be no evident issue presented in any given scenario. But I see that further on you claim the actual scenario I gave is terribly complicated, something an earlier respondent claimed also. The opposite is so. Dead simple. It really gets down to one thing which will come to later.

This is going the wrong way - steering it to basically to an Ehrenfest disk type situation. No. Forget rotation, forget disks for now. It's all about basic character of stress in GR.

I said above and have often elsewhere, the core issue is dead simple, and imo considerably simpler than your preferred approach (2). Does stress induce a change in mass, quite separate to any associated and material dependent elastic energy? How many times have I asked for a straight answer now? We all know the answer - gave a Wiki reference in #25 for instance. Why is it so hard to answer that one simple question? Once you can answer that simple as it gets question, the next one is equally simple. We just have to ask this: Take any axially symmetric arrangement - say a horizontal bar pivoted about a central horizontal shaft. Apply 'squeezing' pressure to part of one arm such that the clamping forces exactly balance hence induce no moment of themselves. If 'just stress' is a source of mass, is there a moment induced owing to said mass differential between those two arms? Only one stab at this one.

You no doubt figured it out right. There is inevitably a net moment about the central axis and a tendency (whether or not realized owing to friction etc.) for that bar to tilt under gravity. Is that something difficult or in any way doubtful? How many equations need be solved to arrive at that conclusion? None of course. Far from being the super complex problem you and one or two others have made it out to be, nothing could be simpler. I repeat to the point of going blue in the face - IF stress is a source of mass, it must be equally a source of active, passive and inertial mass - otherwise equivalence principle fails dramatically. The rest is dead simple and follows as per above. There exists a differential that upsets an otherwise symmertic balance - easily seen to be just that owing to the stress induced in one arm. Complicated? Really?

What seems to be hard is to accept the consequences. Instead of a bar, replace with a circular disk. We now have full axial symmetry - well not quite. Stress (compressive) applied to one side has made it heavier. Oh - we have a 'disk-brake'! As you very well know, I idealized to frictionless pressure, so nothing prevents that differential in mass = moment arm from inducing rotation. Which in principle keeps going indefinitely. IF stress is a legitimate source term that is. It should be obvious that arguments such as e.g. 'hour-glass' elastic deformation of disk under stress might somehow prevent motion are spurious. Need I point out why? Disk stiffness can be arbitrarily high, just as friction can be arbitrarily low. Valid idealizations.

Whats more though, you can have all the real world friction, hysteresis etc. you want, but the basic issue remains - and is dead simple. Induce rotation by some external means, and the stress-induced mass differential -> moment arm is always there, on top of any heating etc. from all those 'terribly complex' real-world factors folks are so anxious to include. Strangely these same folks are perfectly comfortable discussing idealized 'eternal Schwarzschild black holes' for instance, even though such never exist in the real universe. Oh well.

If so, I invite you to answer above (which is only restating my position from the start), and offer a resolution to your satisfaction. One where stress acts according to GR. Start by answering that simple question - stress, all by itself, decoupled from elastic energy part, is or isn't a source of mass. Nothing complex at all.

At another site the matter of gravitational wave loss of say a binary system was discussed. It was there that it was admitted that even though the system loss was due entirely to non SEM component GW's, the net mass shrunk. And such that the SEM divergence was always zero. When pressed about the obvious contradiction, that the SEM divergence law holding only for a point came up - and that by an expert in GR.

It appears to be an Ehrenfest type scenario dealt with, as mentioned earlier. Not in the least relevant. Can we just at least settle that simple question - the one on stress as source of mass?

 

AlphaNumeric - I have just quickly glanced through your wall of text response in #30. You have more stamina for this kind of thing than I care for. It's pointless to continue as there is no common ground. If you wish to focus on my points raised in last response to Markus in #31 and answer them in the context of which they are given, well and good. Otherwise, why go on like this?

 

This is the bit where being a layman is actually fun - because occasionally you stumble across new things. I had never heard of the Ernst equation before, and I am certainly not too proud to admit that.
Learned something new today - thanks Guest254

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Precisely my point. And the proof is valid and internally consistent. If you disagree, then please point out exactly where in the ( very short ) proof you think I went wrong.

The basic character of stress in GR is simply that it is a form of energy. As such, it forms a source of the gravitational field, just like all forms of energy.

I don't get you. As mentioned above, stress is a form of energy. So is mass ( density ). Both of them are sources of the gravitational field, but that does not mean that they are neccesarily interdependent. In the SEM tensor you will find separate terms for both, in fact you can decompose the tensor like so :

\(\displaystyle{T^{\mu \nu }=T_{mass}^{\mu \nu}+T_{stress}^{\mu \nu }}\)

Your desire to interpret everything in terms of "mass" is misleading at best, and erroneous at worst. Instead you should think only in terms of energy, because that is what the SEM tensor really represents. It is a measure of the total energy of a system, which is the sum of individual contributions from mass density, stress, momentum etc etc. Going the other way, i.e. thinking of energy as mass, will only lead to "massive" confusion, as this thread clearly shows.

So, to sum up - the source of the gravitational field is energy. Mass is a form of energy. Stress is a form of energy. If you put mass under stress, then the total energy of the system increases, but not the original mass itself. This is as direct an answer as I can give to a question which misses the mark from the outset. Of course you can induce a moment in a rotating bar, which will change the form of the SEM tensor, and thus the geometry of space-time. There is no problem with this, and in particular no conservation laws are violated.

I should reiterate also that the source term in the SEM tensor is mass density, not mass. There's an important difference to be taken note of here.

No one said that - stress is not a source of mass. Stress is a form of energy, and therefore a source of the gravitational field.

So you have taken a disk and made the energy density across its volume non-uniform by applying stress to some area within it ? So what ? I fail to see the issue. All that has changed is how the energy is distributed within the system. Where is the paradox, or the violation of GR which you seem to imply ?

Resolution to what, exactly ? I still don't see the problem which needs resolving.

Again, I don't see the contradiction...?
Look again at the flux integral I gave in my previous post - for a non-stationary system such a binary star, the hypersurface through which flux is calculated is obviously also non-stationary in this case; the net flux remains zero still. Total energy ( binary + waves ) remains always conserved.
What is not conserved is total energy within a stationary volume anywhere within that space-time, because, due to the presence of gravitational waves, you'd have a non-vanishing net flux through its surface.

This is all perfectly analogous to classical mechanics, btw.

Stress is not a source of mass. Stress is a source of gravity, because it is a form of energy. All forms of energy are sources of gravity.
I refer you again to the actual meanings of the SEM components, either on my GR primer or on Wikipedia. You will not find "mass" in there, only energy density; if you put stress on a mass, the mass does not change. All that changes is the total energy, and how it is distributed throughout the system in question. Consequently the SEM tensor would change, and thus the geometry of space-time.

This all boils down to a very simple thing - GR does not concern itself with mass, it only concerns itself with energy in its various forms.

 

I have this answered for you already. Mass is energy. If you put that mass under stress, you increase the total energy of the system, but not the amount of original mass. It is that total energy and how it is distributed which is the source of the gravitational field, not just mass.

So, applying stress has an effect on the gravitational field, but not on the original amount of mass.

I can only reiterate again - stress is not a source of mass, it is just another form of energy like aforementioned mass itself also is. It is energy that is the source of the gravitational field, therefore stress effects the geometry of space-time without varying the amount of mass in the system.

Just discard your concepts of mass, inertial mass, relativistic mass etc etc as sources of the field. This just leads to terrible confusion. All the SEM tensor measures is total energy at every point within the system. If this tensor varies in any way, then so does the resultant gravitational field. It is as simple as that. However, calculating the precise dependence of the gravitational field on stress or pressure in a specific scenario like the one you gave is exceedingly difficult and tedious; it is easy to give qualitative results ( this is the true power of GR ), but very hard to quantify them.

Note also that, given individual contributions by various forms of energy, you can construct a unique SEM tensor, but the reverse is not true. Given just a metric tensor, i.e. a gravitational field, the individual elements of the SEM tensor are not uniquely determined. Not surprisingly, since it measures only the total energy of the system. You can deconstruct such a total in infinitely many ways.

 

Yet again - check it by way of a gedanken experiment! And you know which one.

You are off the rails here. Of course according to GR it is a source of gravitational field. But it is most definitely NOT a form of energy. On a formal level it's true stress and energy density have the same dimensional units:
Stress has dimensions force/area = Ma/L^2 = (ML/T^2)/L^2 = M/(LT^2) (a = acceleration)
Energy density has dimensions (force x L)/L^3 = force/area = above.
However dimensional equivalence does not make them the same beast and functionally the two are chalk and cheese. Again my 2nd para in #9:

GR makes the extraordinary assumption that stress contributes an 'energy density equivalent' based on, for uniaxial stress, of sweeping one cube face through the entire length of a stressed unit cube, at a constant force given by stress x area of that sweeping face. It being taken the stress is acting normal to that face. And this contribution can be positive OR negative, depending purely on sign of stress. This is 'magic' - there is no such sweeping occurring - no such energy change is really happening.

Compare to genuine elastic energy contribution, where the actual swept volume of unit cube is just strain = σ/E, always in elastic solids a tiny fraction of unstressed volume. And since force grows linearly, average value is just half the maximum. Hence we have W = (1/2)σ^2/E. Entirely material dependent, parametric wrt stress, and genuinely an energy input, entirely unlike the pretend 'pure stress' linear contribution.

In short, as stiffness rises, genuine elastic energy density shrinks in inverse proportion, whereas stress 'contribution' is virtually unaffected - fractional volume change -σ/E tends to 0 as E rises. Taken to the limit, elastic energy density vanishes, but stress 'magically' contributes to the max - for zero energy input. Which imo is an absurdity. But there is no need to go to that limit to see that the non-linkage between stress as source and true energy input allows wonderful things, e.g. that disk-brake 'overbalancing wheel'.

Someone is being misled yes, but who? Stress is NOT a form of energy - as per above! And SEM means stress-energy-momentum. No - they are NOT 'all energy'. ONLY the very first diagonal term is energy (density). And it includes any and all energy density owing to stress. As I have clearly stated umpteen times before - see yet again my second para in #9!
Hence there IS always some interdependence (except for shear stress terms - which for solids contribute zero to field - except for elastic energy density contribution). But it is indirect and entirely material dependent in the case especially of solids. As per above!

This is all utterly wrong! The source of the field - according to GR - is energy+stress+momentum, NOT just energy. And the confusion re mass is yours not mine. It is commonly understood that in context of use here, the term mass means gravitating mass = sum of all terms that contribute, i.e. SEM tensor. You are confusing matter = rest mass with mass as in gravitating mass. It is the term matter = rest energy = rest mass (at absolute zero and zero pressure) + thermal energy + stress-induced elastic/hydraulic/pneumatic energy that is understood to be solely the energy contribution. Mass as in gravitating mass in GR includes all of SEM. And you should have known I clearly meant the latter. Or stress as a component of that gravitating mass where used.

Then recognize what that means when disk replaces bar! I made it clear enough - 'overbalancing wheel'.

Only if one ignores the obvious - a stress-induced moment arm is continually there and induces rotation - 'for free' - unavoidable IF stress is a source term. Or non-conservatively contributes to power if rotation is induced externally. Repeat repeat repeat....

Oh please Markus, an utter triviality - we all understand the difference!! I certainly do.

Wrong. As before gone over. Stress - in GR - IS a source of gravitating mass, but is NOT of itself a form of energy. And whatever contributes to the gravitational field constitutes gravitating mass.
Again from the Wiki article: https://en.wikipedia.org/wiki/Mass_in_general_relativity

Unambiguous - pressure (triaxial stress) is separate to total energy, and both contribute to the mass = gravitating mass. Going blue in the face again - repeat ad nauseum until it gets through! And btw, that Komar mass is defined here is an irrelevancy in context of something so gravitationally tiny as a 'disk-brake'. Correction factor owing to gravitational potential is infinitesimal, and anyway only has distant not local meaning. This answers someone else who's recent post harped on this irrelevant distinction.

Because owing to that non-uniformity you acknowledge exists, we have a turning moment and the freeking thing wants to turn - and keep turning in the case of disk!!! And it is supposedly stress - NOT energy density, that is major contributor to this imbalance! Again - as material stiffness rises, elastic energy density shrinks in direct proportion, but stress contribution is virtually unaffected. A continuous moment directly implies here 'overbalancing wheel' = perpetuum mobile. But you cannot connect the dots, after so long explaining?

Well I'm about to give up trying. I suspect you simply are turning a blind eye.

No it's not, but let's drop this one as a distracting side issue.

Wrong again on both counts. See my earlier comments.

Wrong - just read the actual text from that article I reproduce above. And no more of this straw-man definitional confusion over the use of 'mass'. I believe it is clear enough now as to what means what and where.

As per above. A straw man issue (gravitating mass is not equated to matter = rest mass), and as I have explained above, it is wrong to lump all terms under energy. As you yourself claimed - at times!!
Your #37 brings in nothing new and just repeats the same erroneous claims and conclusions, so will not bother responding with tedious repetition of correctives.

Quick summary:
Stress is NOT energy density - they are totally different beasts with quite different functional form and material dependence/independence. Which is at the heart of my argument.

Mass as I have used it always unless specifically said otherwise, referred to gravitating mass, which is a catch-all and includes all and/or any component of contributing SEM terms - including but not limited to rest mass/energy. Context alone made that clear and there was no excuse for running with a straw-man here.

 

The words "evident issue" and "scenario" are prone to interpretation. Q-reeus means "any appearance of counter-intuitive results arising from any set of my assumptions glued onto General Relativity would demonstrate the proof is wrong" while Markus Hanke reads "Since the proof was proven from certain assumptions, there will be no logical contradiction arising from any set of initial conditions that are consistent with those assumptions, i.e. there will be no logical issue presented in any admissible scenario".

Here Q-reeus is wrong, since a proof is only as good as its assumptions and one does not discredit a purported proof by substituting new assumptions that logically contradict the first assumptions. One may, if one chooses point out that the purported proof had gaps in its logic where intuitive guesses need to be replaced with logic and calculation if one expects to convince others. Alternatively, one may examine the assumptions themselves and declare them unsuitable, leaving the proof untouched but demonstrably irrelevant. As Q-reeus has done neither of those things, the proof looks both relevant and convincing.

The problem here is that Q-reeus trusts his Newton-trained intuition -- and rejects geometrical theorems in the language of tensor calculus -- even though General Relativity is entirely a geometrical statement in the language of tensor calculus. Without logic and calculation is no gedanke in Q-reeus' gedanken experiment -- it is merely a collection of assertions from both inside and outside of General Relativity that the poster doesn't have the first idea what to do with, the arrogance of the incompetent to try and dictate to those that studied General Relativity formally what is or isn't a difficult problem, and the illogic of trying to argue with the conclusion of a theorem rather than its premises.

Example: If I assume \(a^2 + 1 = 0\) this is incompatible with the assumption that \(x^2 \geq 0\) for all numbers, but not of itself an issue for I may be working in the complex numbers where both \(a = \sqrt{-1}\) and \(a = -\sqrt{-1}\) are solutions. Complex numbers are very natural in the theory of polynomials but not the numbers Newton and Euclid grew up with.

I have encountered similar in numerous Special Relativity discussions where posters trust their Euclidean Geometry and their Newtonian preconceptions to the point that they commit errors and blame Special Relativity for their woes even when they themselves have made assumptions contradictory to Special Relativity which lead to their discovery of contradictions. They trust only their intuition, as it's good enough to get them to work and back each day, but fail utterly in comprehending the physical theories the see fit to declare as "self-contradictory" when they are only airing personal misconceptions.

Example: Of all time-like paths between two events in space-time, a time-like geodesic is the one with the longest proper time. In curved space-time there may be more than one, however. This is a simple statement of geometry, with awkward phrasing in the language of Newtonian preconceptions.

I rely on my intuition daily, but I trust in what I can prove and calculate. As a result of which my intuitive grasp of four-dimensional geometry has improved. But if I wish to communicate something, I will write it in the language of algebra, logic, mathematics because my geometric intuition is not easily shared. Here, not only is Q-reeus apparently hostile to the idea of meeting people half-way, insisting rather that they do all the hard work so that he may throw stones at it, but Q-reeus lacks the willingness to train his intuition to serve the cause of learning physics better. As a result, Q-reeus mistakenly thinks he is doing physics better than anyone in the twentieth century, but actually is not contributing to his own welfare or that of Mankind.

 

Another useful tool to model spacetime. This is a basic example of how it's used. Neat site.

http://www.encyclopediaofmath.org/index.php/Ernst_equation