At the 1957 Chapel Hill conference, Feynman (as 'Mr Smith') was credited with a stroke of brilliance in settling an argument over whether GW's (gravitational waves) were real, via his simple 'sticky bead' argument: https://en.wikipedia.org/wiki/Sticky_bead_argument#Feynman.27s_argument
But was it really brilliance or basic conceptual blunder that incredibly has held sway in the GR community ever since? Let's see, by stepping back from the trees, and viewing the forest as a whole. That is, let's see if the ubiquitous deforming ellipsoid '+' and 'x' polarizations, as per standard illustrations e.g.
https://en.wikipedia.org/wiki/Gravitational_wave#Effects_of_passing
which are supposedly locally experienced, actually makes geometric sense when viewed on a global basis.
The basic thing about such GW's is their purely transverse traceless nature - purely spatial component shear deformations entirely transverse to the propagation vector k.
[Leaving aside pedantic arguments where the observer may hypothetically have some huge transverse Lorentz boost relative to the distant GW source.]
To pin it down as simply as possible, take the case of a single linear quadrupole oscillator. The quintessential two masses vibrating via a coupling spring. Which by analogy to an electric dipole oscillator in EM, is the primitive from which the far-field radiation of any other source distribution can be built via combination of such primitives.
Well at least for the monochromatic GW field, which is all that need be considered to come to a decisive conclusion.
We note that the intensity pattern has a maximum in the transverse plane (equator), diminishing to zero at the poles. See e.g.
http://www.roe.ac.uk/~lmb/animations.html
Also, far-field '+' GW polarization is aligned with projection of the oscillation axis z.
Consider then the accompanying png illustration. A central mass-quadrupole oscillates along axis z, and we consider the far-field effect at an imaginary spherical shell centred about the oscillator. The local or 'tree level' view is illustrated by the two smaller ellipses, centred about and orthogonal to the arbitrary radius vector k. Which here lies in the equatorial plane, thus normal to the oscillation axis z. A passing GR style GW supposedly will deform a circular array of beads as shown - stretching/squashing in a plane normal to k and with ellipse major axes aligned along alternately z, and a tangent to the equator at that radius. The generalization to arbitrarily oriented radius vector k is hopefully obvious.
But now we go global i.e. 'forest level' view. Instead of Feynman's short stick-and-2-beads though, here it's been extended to form a continuous circular hoop lying on the equator (solid line narrow ellipse), centred about axis z. With a uniformly distributed array of beads (black dots) free to slide along the hoop. Thus an array of Feynman's stcks'n'beads joined end-to-end. it matters not whether one allows gaps or not between such individual sticks - as will become very clear. Also shown (solid line) is a meridian line (line of longitude), and one can imagine a similar array of beads strung out uniformly on a hoop so oriented. The broken line ellipses will be explained later.
The arrangement just described corresponds to an unpertubed notionally flat Minkowski spacetime. What will - no - what could really happen - if a GR variety pure tensor GW now arrives to perturb things? Only pure transverse shear deformations being allowed?
Take firstly the case of the equatorial hoop. Supposing at a certain moment, the '+' polarization gives maximal circumferential dilation in the equatorial plane, thus simultaneously contraction along the meridians! Hmm... Which is simply the local 'deforming ellipse' case everyone is familiar with (vertically squashed small ellipse) BUT - extended globally. Does it take a huge IQ to figure out that the beads in the equatorial hoop, purely by symmetry, cannot have any motions along the hoop? The 'sticky beads' are stuck, even if perfectly frictionless! Dilation/contraction along lines of latitude makes zero sense. Something obvious when viewed globally ('forest view'), only seemingly sensible if viewed as a local perturbation ('tree level' view i.e. the small ellipses). Feynman got it badly wrong.
Supposing one relaxes the pure transverse stipulation within GR and allows the equatorial hoop to 'breathe' radially. WE are necessarily introducing longitudinal GW's here! First observation is this runs counter to the usual argument that a 'rigid' hoop does not appreciably respond to the (impossible) shear deformations - only the free-to-slide beads should. Snookered there. But anyway, just suppose the hoop did freely expand. By how much would it need to radially expand to yield relative motion between the beads (but still with zero sliding along the hoop), consistent with a 1/r drop in GW induced transverse motion amplitudes? Which is supposedly what a GR GW will induce.
The reader should quickly figure out that radial expansion/contraction amplitude would have to be, in the far zone, a constant entirely independent of distance r from source. In such a scenario, radial displacements (i.e. longitudinal, along propagation vector k direction) become overwhelmingly larger (proportionately) than any purported transverse motions. Increasingly so as r increases. Does that sound like a physically reasonable proposition?! Reverting to an array of Feynman sticks with gaps between them, which then admits free 'hoop' expansion, will not rescue things in the least. A constant longitudinal oscillation amplitude is simply unphysical.
For the case of beads on a meridian hoop (or stick array if you wish), the situation is not quite so utterly obvious, since the symmetry constraint is not as totally restrictive. One could imagine the beads somehow alternately spreading apart along the hoop then contracting, with maximal motion amplitudes nearest the equator, diminishing to zero at the poles. However, we note that at the same time instant, contraction along meridians implies such an oriented hoop will want to deform inwards - conflicting basically with the 'need' of expansion for latitudinal hoops. Moreover the amplitude of such inward (or outward at the other half-cycle) motion would again have to be independent of r - for the same reason as for latitudinal hoops.
The relevance to above of those dashed-line ellipses should now be evident - the hypothetical 'breathing' motions of latitudinal and meridian hoops are intrinsically out of phase. And of course such conflicting motions are a desperate concession running counter to standard GR GW lore. That only admits appreciable motion of the beads, not supporting 'rigid' hoops. Casting aside the hoops and beads, it should now be evident the supposed pure transverse pure shear metric deformations are self-contradictory. Pure shear deformations - torsional ones - are allowed on a spherical shell. But most definitely not ones consistent with GR's TT-gauge quadrupole mode GW's.
Clearly, there is no way out of this mess for the standard paradigm. GR's and similar tensor gravity theories brand of pure tensor GW's are logical absurdities. Just why that wasn't realized at the very inception of GR is a matter for historians and maybe sociologists or even psychiatrists to ponder.
So how then to explain the positive results now evidently confirmed by aLIGO?
The reader is encouraged to take a good look at the following:
http://arxiv.org/abs/1503.04866
https://inspirehep.net/record/1353103/plots
http://arxiv.org/abs/1502.00333
Unlike the logically impossible GW's of GR and similar, Carver Mead's G4v vector GW's are logically self-consistent, having a close correspondence with those of an EM quadrupole oscillator.
But was it really brilliance or basic conceptual blunder that incredibly has held sway in the GR community ever since? Let's see, by stepping back from the trees, and viewing the forest as a whole. That is, let's see if the ubiquitous deforming ellipsoid '+' and 'x' polarizations, as per standard illustrations e.g.
https://en.wikipedia.org/wiki/Gravitational_wave#Effects_of_passing
which are supposedly locally experienced, actually makes geometric sense when viewed on a global basis.
The basic thing about such GW's is their purely transverse traceless nature - purely spatial component shear deformations entirely transverse to the propagation vector k.
[Leaving aside pedantic arguments where the observer may hypothetically have some huge transverse Lorentz boost relative to the distant GW source.]
To pin it down as simply as possible, take the case of a single linear quadrupole oscillator. The quintessential two masses vibrating via a coupling spring. Which by analogy to an electric dipole oscillator in EM, is the primitive from which the far-field radiation of any other source distribution can be built via combination of such primitives.
Well at least for the monochromatic GW field, which is all that need be considered to come to a decisive conclusion.
We note that the intensity pattern has a maximum in the transverse plane (equator), diminishing to zero at the poles. See e.g.
http://www.roe.ac.uk/~lmb/animations.html
Also, far-field '+' GW polarization is aligned with projection of the oscillation axis z.
Consider then the accompanying png illustration. A central mass-quadrupole oscillates along axis z, and we consider the far-field effect at an imaginary spherical shell centred about the oscillator. The local or 'tree level' view is illustrated by the two smaller ellipses, centred about and orthogonal to the arbitrary radius vector k. Which here lies in the equatorial plane, thus normal to the oscillation axis z. A passing GR style GW supposedly will deform a circular array of beads as shown - stretching/squashing in a plane normal to k and with ellipse major axes aligned along alternately z, and a tangent to the equator at that radius. The generalization to arbitrarily oriented radius vector k is hopefully obvious.
But now we go global i.e. 'forest level' view. Instead of Feynman's short stick-and-2-beads though, here it's been extended to form a continuous circular hoop lying on the equator (solid line narrow ellipse), centred about axis z. With a uniformly distributed array of beads (black dots) free to slide along the hoop. Thus an array of Feynman's stcks'n'beads joined end-to-end. it matters not whether one allows gaps or not between such individual sticks - as will become very clear. Also shown (solid line) is a meridian line (line of longitude), and one can imagine a similar array of beads strung out uniformly on a hoop so oriented. The broken line ellipses will be explained later.
The arrangement just described corresponds to an unpertubed notionally flat Minkowski spacetime. What will - no - what could really happen - if a GR variety pure tensor GW now arrives to perturb things? Only pure transverse shear deformations being allowed?
Take firstly the case of the equatorial hoop. Supposing at a certain moment, the '+' polarization gives maximal circumferential dilation in the equatorial plane, thus simultaneously contraction along the meridians! Hmm... Which is simply the local 'deforming ellipse' case everyone is familiar with (vertically squashed small ellipse) BUT - extended globally. Does it take a huge IQ to figure out that the beads in the equatorial hoop, purely by symmetry, cannot have any motions along the hoop? The 'sticky beads' are stuck, even if perfectly frictionless! Dilation/contraction along lines of latitude makes zero sense. Something obvious when viewed globally ('forest view'), only seemingly sensible if viewed as a local perturbation ('tree level' view i.e. the small ellipses). Feynman got it badly wrong.
Supposing one relaxes the pure transverse stipulation within GR and allows the equatorial hoop to 'breathe' radially. WE are necessarily introducing longitudinal GW's here! First observation is this runs counter to the usual argument that a 'rigid' hoop does not appreciably respond to the (impossible) shear deformations - only the free-to-slide beads should. Snookered there. But anyway, just suppose the hoop did freely expand. By how much would it need to radially expand to yield relative motion between the beads (but still with zero sliding along the hoop), consistent with a 1/r drop in GW induced transverse motion amplitudes? Which is supposedly what a GR GW will induce.
The reader should quickly figure out that radial expansion/contraction amplitude would have to be, in the far zone, a constant entirely independent of distance r from source. In such a scenario, radial displacements (i.e. longitudinal, along propagation vector k direction) become overwhelmingly larger (proportionately) than any purported transverse motions. Increasingly so as r increases. Does that sound like a physically reasonable proposition?! Reverting to an array of Feynman sticks with gaps between them, which then admits free 'hoop' expansion, will not rescue things in the least. A constant longitudinal oscillation amplitude is simply unphysical.
For the case of beads on a meridian hoop (or stick array if you wish), the situation is not quite so utterly obvious, since the symmetry constraint is not as totally restrictive. One could imagine the beads somehow alternately spreading apart along the hoop then contracting, with maximal motion amplitudes nearest the equator, diminishing to zero at the poles. However, we note that at the same time instant, contraction along meridians implies such an oriented hoop will want to deform inwards - conflicting basically with the 'need' of expansion for latitudinal hoops. Moreover the amplitude of such inward (or outward at the other half-cycle) motion would again have to be independent of r - for the same reason as for latitudinal hoops.
The relevance to above of those dashed-line ellipses should now be evident - the hypothetical 'breathing' motions of latitudinal and meridian hoops are intrinsically out of phase. And of course such conflicting motions are a desperate concession running counter to standard GR GW lore. That only admits appreciable motion of the beads, not supporting 'rigid' hoops. Casting aside the hoops and beads, it should now be evident the supposed pure transverse pure shear metric deformations are self-contradictory. Pure shear deformations - torsional ones - are allowed on a spherical shell. But most definitely not ones consistent with GR's TT-gauge quadrupole mode GW's.
Clearly, there is no way out of this mess for the standard paradigm. GR's and similar tensor gravity theories brand of pure tensor GW's are logical absurdities. Just why that wasn't realized at the very inception of GR is a matter for historians and maybe sociologists or even psychiatrists to ponder.
So how then to explain the positive results now evidently confirmed by aLIGO?
The reader is encouraged to take a good look at the following:
http://arxiv.org/abs/1503.04866
https://inspirehep.net/record/1353103/plots
http://arxiv.org/abs/1502.00333
Unlike the logically impossible GW's of GR and similar, Carver Mead's G4v vector GW's are logically self-consistent, having a close correspondence with those of an EM quadrupole oscillator.
