General relativity {GR} consist in the generalization of the indefinite Minkowski theory of special relativity {SR}, where the metric coefficients are slowly variations of space but independent of time. These metric coefficients are determined by the Einstein equations:
G=kT,
where G is the Einstein tensor, a slight modification of the Ricci tensor in order that its covariant divergence vanishes, and T is the stress-tensor whose covariant divergence must also vanish. The term k is a constant of proportionality, proportional to the ratio G/c^4, a small coefficient.
Gravity is said to be embedded on the left-hand side which is entirely geometrical, while the dynamics is described by the right-hand side, the energy and stress present in the system. That is, except for the gravitational energy which is already accounted for on the left-hand side. Consequently, the "gravitational force" disappears, and is replaced by free-falling bodies that follow geodesic equations:
total acceleration normal to the surface=acceleration on the surface + curvature terms=0
Flanagan & Hughes, in "The basics of gravitational wave theory", then ask:
"Does this mean that the gravitational wave (GW) has no effect? Certainly not! It just tells us that in the TT (traceless-transverse) gauge the coordinate location of a slowly moving free particle is unaffected by the GW. In essence, the coordinates move with the wave."
I wouldn't call thus "unaffected" at all! They then go on to claim that the GWs "cause the proper separation of two bodies to oscillate, even if the coordinate separation is constant." This is reminiscent of a longitudinal oscillation of all particles on a line are displaced to the left or right so that their spacing remain the same. However, we are told that GWs are transverse like their EMW counterparts.
Where then do gravitational waves come from? They come from a linearization of the Ricc tensor of the indefinite flat metric in which the linear terms satisfy a wave equation in vacuum (when the stress-tensor T vanishes). More specifically, Einstein found only the TT pieces of the perturbed metric coefficients carried energy, and satisfied the wave equation. The other metric pieces satisfied Poisson equations. According to Flanagan, "the non-TT parts of the metric do no exhibit radiative degrees of freedom." But don't tell me that because something obeys a wave equation means that it will necessarily radiate. And even if it did where in Einstein's equations is this possibility accounted for. All we have there are particles following geodesic paths which conserve, energy, momentum. angular momentum and anything else which would otherwise lead to energy loss---like radiation!
To make matters worse, all pieces of the linearly perturbed metric are non-local. That means that their calculation at a point requires "knowledge of the perturbation everywhere." However, observations that seeks to detect GWs are sensitive to the Riemann tensor only at one point in space. "For example the Riemann tensor components R_{itjt}, which are directly observable such as LIGO, are given in terms of the gauge invariant variables" some of which do not satisfy the wave equation and whose speeds are those of the "speed of thought" to use Eddington's terminology.
So now you tell us that the Riemann tensor components are linear combinations of perturbed metric components that do not obey the wave equation, just because they are "locally observable"?
Flanagan then goes on to claim: "...we described a splitting of metric perturbations into radiative, non-radiative, and gauge pieces. This splitting requires that the linearized Einstein equations be valid throughout spacetime. However, this assumption is not valid in the real Universe. Many sources of GWs are intrinsically strong field sources and cannot be described using linearized theory, and on cosmological scales the metric of our Universe is not close to the Minkowski metric. Furthermore, the splitting require knowledge of the metric throughout spacetime, whereas any measurements or observations can probe only finite regions of spacetime.
The conclusion Flanagan arrives at is absurd: "For these reasons it is useful to consider linearized perturbation theory in finite regions of spacetime, and to try to find gravitational radiation in this more general context." First, it is a more restricted context, and second why would anyone want to proceed with a theory---a linearized one at that---when the phenomenon it is try to describe is inherently nonlinear?!!
Through a contorted line of reasoning--if I may call it that--Flanagan arrives at an expression for "the proper separation of two particles oscillates with a fractional length change" h. I thought from the above discussion that the proper separation between free-falling bodies remains the same as a GW passes, only to learn now that it is a measurable quantity called by the name of "wave strain". "The proper distance we have calculated here is a particularly important quantity since it directly relates to the accumulated phase which is measured by laser interferometric GW observatories. The "extra" phase accumulated by a photon that travels down and back the arm of the laser interferometer in the presence of a GW is" the ratio that the distance the end mirror moves to the wavelength of the photon--meaning laser wavelength of course.
We obviously can say nothing about how much extra phase a single photon can accumulate. And if this isn't bad enough, the slight displacements of the mirror are well within the realm of Heisenberg's uncertainty principle, where no non-demolition observations can be performed no matter how advanced the technology has or will become. And if energy can't be localized in GR what are we doing with a theory from which we want to get infinitesimal displacements of a 40 kg mirror? Both the theory and apparatus are not adapted to the goal of measuring GWs--and we don't even know if they exist! As the Dutch team that was critical of the statical analysis of LIGO said: "If you want GWs, then GWs is what you will get."
This is
Notwithstanding all the hurdles they put for themselves, the authors then go on nonchalantly to analyze "geodesic deviations". To they think that these deviations from uniform motion, are really what will describe GWs? Just like in the cosmic wave background radiation, we have the cart before the mule. There, a sufficient amount of time has been reached after the big bang so as to allow thermal equilibrium to be achieved. And from the small fluctuations in the equilibrium thermal radiation one hopes to account for the formal of stars and galaxies. In a similar matter we are now looking from small deviations from geodesic motion to describe the cataclysmic events that created them! We are even told that the spectra obtained by LIGO carries the signature of the coalescence of black holes, and binary neutron stars. This is all the more surprising because GR can't even solve the two body problem, and there are no periodic solutions to the Einstein equations themselves. And even if there were periodic solutions to the full nonlinear Einstein equations, how would we know that they are reflected in the linearized solution. This would be like describing solitary waves as a superposition of monochromatic ones. It just can't be done!
And if all this is not as ludicrous as it may seem. The claim is made that "the stress-energy tensor must be conserved, which means [regular divergence=0] in linearized theory. Then Flanagan goes on to derive an expression for h that is NOT governed by a conservation equation. Obviously this is Einstein's quadrupole formula. But why pretend this has to do with the Einstein's equations? And why add on a "pseudo-tensor" to the stress tensor for "non-negligible self-gravity"--whatever that means. In the old days there was a raging controversy of whether or not GWs carried energy. If the pseudo-tensor describes gravitational energy it can be made equal to zero by a mere transform of coordinates. Who are we fooling here?
Comments