Peter - great effort in producing those two well-argued follow-up blogs Does Gravity Gravitate: The Sequel , Does Gravity Gravitate: The WaveAnd now the third (and final) post in the series is up:
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 Tab (which √(gtt) 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 non-localizability 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:
Two basic issues. First, as I understand it a GW involves purely transverse shear deformations of just spatial components of metric (zero dilational component). How can that even in principle allow induced motion of a bead along the propagation axis? Makes no sense imo, even if there is an unstated assumption stick length is long wrt, or at least appreciable fraction of, GW wavelength. Second, even when orienting stick orthogonal to propagation axis, induced motion of bead on stick seems nonsensical. Do not these shear deformations have as analogy the orthogonal stretching and un-stretching of a rubber sheet? Then the stick and bead and anything else gravitationally small existing in this 'rubber sheet' act as just figures drawn on it, hence must co-deform with the rubber. Thus would be undergoing motions (or rather deformations) only relative to an undetectable background flat metric. Hence no detectable relative motion of bead wrt stick, making any kind of local detection or energy absorption impossible in principle. Evidently Eddington adopted the lifelong view that along this or similar line of argument, GW's were merely coordinate artifacts - 'ripples in the coordinates' and thus unphysical. Considered now antiquated thinking, was he wrong?The thought experiment was first described by Feynman (under the pseudonym "Mr. Smith") in 1957, at a conference at Chapel Hill, North Carolina. His insight was that a passing gravitational wave should in principle cause a bead on a stick (with the stick parallel to the wave velocity) to slide back and forth, thus heating the bead and the stick by friction. A gravitational wave pulse will stretch spacetime behind the bead, pushing the bead forward; after the wave passes through the bead the stretching will occur in front of the bead, accelerating the bead in the opposite direction. This heating, said Feynman, showed that the wave did indeed impart energy to the bead and stick system, so it must indeed transport energy.
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 super-massive 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 Hulse-Taylor binary-pulsar results might actually indicate a non-conservative 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.
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