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Is There Really a Strictly Conserved StressEnergy Tensor in GR? 
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#73
Nov512, 01:43 PM

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PF Gold
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I don't know how far one could go with this analogy; I think it is more useful to get at the issue via nonlinearity and the observation that SET being the only source of Ricci curvature does not mean (even close) that SET can be directly related to effective gravity at a distance (except in special cases, e.g. where Komar volume integral is valid). 


#74
Nov512, 01:50 PM

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[Edit: To forestall a potential question from Qreeus, who remembers me talking in previous threads about the field at a given event being due to nonzero SET somewhere in the past light cone of that event: the words "directly related" in the above are key. Ultimately, wherever there is Weyl curvature, there must have been a nonzero SET "source" somewhere in the past light cone. The Weyl curvature in the Schwarzschild exterior vacuum region around a gravitating body is ultimately due to the nonzero SET inside the body. (And if the "body" is a black hole, the Weyl curvature of the hole is ultimately due to the nonzero SET inside the body that collapsed to form the hole.) But that's only "ultimately"; the connection is indirect, since there may be a lot of "empty" spacetime in between, so to speak, and so the properties of "empty" spacetimei.e., of solutions of the vacuum EFEplay a role in determining what the Weyl curvature is.] 


#75
Nov612, 08:54 AM

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http://www.physicsforums.com/blog.php?b=4293 


#76
Nov712, 10:40 AM

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In respect of the first one. Find in difficult to avoid concluding that ADM mass as you have derived it, starting with EFE's and culling out an expression that corresponds to gravitating mass M, there is not here a de facto recognition that curvature explicitly contributes to that M  as directly part of the source and not just modifier of T_{ab} (which √(g_{tt}) is). So my impression is GR is made 'consistent' by way of a rather cunning and circuitous route, to put it diplomatically. In respect of the second one. The vexed issue of nonlocalizability has it seems a majority consensus 'yes' (localization of gravitational field energy is impossible). But there are those who say 'no'  that this is not a consistent or coherent position. That article also brings in Feynman's sticky bead argument which you also refer to in that 3rd and final blog in the series. Quite frankly the more I try and make sense of the sticky bead argument, the less sense it seems to make. This is probably an issue for a separate thread, but since it has been used here as justification for energy in GW's, and thus sensibility of ADM mass, shall here briefly outline the problem as I see it. From that Wiki article: The only way one could posit relative motion imo is to interpret the metric stretching as giving rise to tidal 'g' accelerations everywhere in the transverse plane. That seems like a geometrical impossibility for plane wave situation  to me only for something like spherically symmetric Schwarzschild geometry would everywhere transverse tidal 'g' make physical sense. But that is always there accompanied by comparably sized radial component too, and diminishes rapidly at large r no matter how strong the proper acceleration of a stationary observer is there (say for supermassive BH). One cannot have in a plane wave (strictly spherical but we are dealing with GW's at very, very large r from source) the necessary diverging radial vectors that apply in SG case. I'm wondering whether HulseTaylor binarypulsar results might actually indicate a nonconservative process  orbital decay purely owing to field retardation effects. Yet another way conservation of energy can fail in GR? Just when you thought it was all done. 


#77
Nov712, 11:15 AM

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Also, don't confuse curvature with the metric. The ADM mass depends on the metric; curvature does not appear in it. Of course if the metric is something other than Minkowski, curvature is present, but that's not the same as curvature explicitly appearing in the ADM integral. It doesn't. [Edit: I see the Wiki article describes this wrong; it says "parallel to the wave velocity". AFAIK Feynman proposed the thought experiment as I have described it just above. But I'll check some sources to confirm.] 


#78
Nov712, 07:29 PM

P: 1,115

However let's ignore for now the matter of radial motions. Consider as example where two nonspinning neutron stars collide head on. Resulting in predominantly axial quadrupolar ringdown. This should give, in equatorial plane, harmonic GW stretch/compression along polar and azimuthal directions  NS and EW. But is this logically consistent with stickybead argument? Consider at large r from source we have a circumferential hoop (stick joined onto itself) centred about polar axis and lying in equatorial plane. With a uniformly dispersed array of beads strung out along the hoop. So there are ringdown GW's passing through. Clearly we need consider only the azimuthal EW GW component. At the point of maximal azimuthal metric change  halfway between maximum dilation and compression, let's suppose your pov is correct and rigidity of hoop prevents any appreciable azimuthal stretch and thus any accompanying radial motion of hoop. Now please explain how each and every sticky bead decides which way to move in this situation  east or west. I'll save you the trouble  by symmetry there can be no such motions. But this is unfair you may say  a continuous hoop is different to a straight stick. OK then, let's fix that by cutting up the hoop into equal pieces, which at large r, each such 'stick' well approximates to a straight stick. Further, our cutting up introduces a small gap between each 'stick' to allow interferencefree radial 'breathing' in and out. Does this make a whit of difference to whether the beads, now strung out on an azimuthal array of sticks, will know to move left or right? Seems clear the answer is no different to before; not at all! Stretch/compression along lines of longitude (polar axis here) logically should follow essentially the same  but not quite. Maximum deformation amplitude along lines of longitude at equator, goes to zero at the poles. And this weak second order stretch deformation gradient implies a translational force on beads  so then motion along a hoop so placed and oriented. But this weak gradient will die off as 1/r^{2} with distance and thus cannot be considered a true wave property. Also, for a short stick there is essentially the same motion induced as for beads thus no relative motion. For me this illustrates there is something nonsensical with the stickybead argument  it has become unstuck. And with it a famous traditional argument for physically real GW's. Notice we have stuck to the original assumption that only transverse relative beadstick motions can in principle exist. Two topics now I guess but best to have this thrashed out here as it all ties together. 


#79
Nov712, 08:03 PM

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In the case of pure plane waves, there are no tidal changes in the longitudinal direction at all. Put another way, if I have two thin, flat objects both placed transverse to the waves and very close together, and initially at rest relative to each other, there will be no relative motion between them longitudinally; they will simply undergo the same transverse oscillations, but slightly out of phase. (A note: if there is a single flat object but its thickness is significant relative to the wavelength, there will be shear stress induced in the object because the transverse vibrations at the front surface will be slightly out of phase with those at the back surface. I suppose this could lead to radial relative motion because of internal forces within the object, as long as the net radial momentum was zero. I was not intending to talk about that case since it's more complicated, and we're only trying to answer the question of whether GWs can heat up an object at all, not investigate the details of various ways it could do so.) (I should also note that I haven't been able to confirm what model Feynman actually had in mind; it's possible that he *was* thinking of a more complicated case than pure plane waves. More to come on that if I can find a reference.) 


#80
Nov712, 08:18 PM

P: 1,115

Peter  I have just time for short comment. Any real GW source produces at large r a spherical wavefront  by spherical it is only implied the wave phase is a function of r and not of θ or phi (spherical polar coord's). My arguments are correct re need for radial motions  just try imagine stretching a balloon without it's radius growing! A nonsense. And btw no matter how great the radius from source (so it all looks like planewave situation), easy to find that relative phase differential between stretch and compression components is constant. Please give this more thought. Must go.



#81
Nov712, 08:28 PM

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#82
Nov812, 03:02 AM

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#83
Nov812, 09:24 AM

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#84
Nov812, 10:39 AM

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#85
Nov812, 03:22 PM

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Of course, you can do local approximations, but then there is no advantage to expanding as local spherical waves rather than local plane waves. The symmetry argument doesn't apply, and you add needless complication to your approximation terms without adding accuracy. 


#86
Nov812, 03:39 PM

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However, I can see doing a geometric optics approximation where we model the GWs, globally, as expanding spherical wavefronts of "graviton pulses", similar to the way "photon" wavefronts are modeled as spherical in SR as an approximation, even though they're really not (the lowestorder EM radiation is dipole so the spherical term is 0 for that as well). Of course the GW wavelength has to be much, much smaller than the size of the spheres for this to work, i.e., the GWs have to be high frequency. I suspect that the paper Qreeus linked to about HFGWs was doing something along those lines. But that is still only an approximation. Furthermore, it's a useless approximation for trying to decide if GWs carry energy, because a "yes" answer to that question is built into the geometric optics approximation in the first place. That approximation assumes that the "gravitons" are massless particles carrying some finite amount of energy and momentum (if they carried zero energy and momentum they would have infinite wavelength, which obviously violates the smallwavelength assumption). 


#87
Nov812, 09:37 PM

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See also (perhaps you should inform author of gross ignorance in using the term 'spherical wave' in respect of antenna radiation! What an ignoramus!) Also try Get used to it folks  spherical wave simply refers to phase of wavefront, and need have no bearing on angular dependence of field strength or direction  savvy?! I never once used the term spherically uniform field or monopole field or monopole moment  that all came from inside your heads. Now, assuming your bonfire for the straw man has burnt out, listen up. Have been feeling my way on this issue  beginning with #76. Some statements made in #78 I now see are wrong (phase differential bit and what flowed from that), but stand by the overall thrust. It needs considerably more refinement and better presentation, and that I intend to do, but hands are tied up at the moment elsewhere. Sufficient to say I'm now sure GW's are a phantom. One more thing. Since you and DaleSpam have not heeded my request to leave it all for now  you might as well make good on that undertaking to provide reference material for Feynman's 'strange' stickybead argument that had stick pointing along propagation axis. Have you found one yet? 


#88
Nov812, 10:54 PM

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Of course, since you didn't intend to imply anything about anything other than the phase then your argument becomes a non sequiter. A GW has more than just phase, so these other components need not be spherically symmetric as your argument requires: 


#89
Nov812, 11:28 PM

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