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(all discussions here are in the extreme weak field approximation about Minkowski space)

For the last couple of years I've been looking into the production and reception of radio frequency gravitational waves. It's kind of a retirement project the main goal of which is to get a better understanding of GW through futile calculations. I've gotten to the point where one of the approaches I've been looking at might actually may lead to a feasible measurement. Feasible here means likely beyond my resources but well within the existing technology.

As singularly hopeless as this may sound, it may be possible to produce and subsequently detect evanescent gravitational waves. Evanescent waves are non-propagating near fields. They occur in all wave phenomena in one guise or another. For an isolated time harmonic GW source these fields die off as ##(kr)^{-5}## whereas the radiation fields die off as ##(kr)^{-1}##.

The particular example I've been considering uses garden variety quartz crystal resonators in the HF radio band. A typical 4MHz fundamental mode crystal is a quartz disk 8mm in diameter and about 0.42mm thick. There are two circular electrodes plated about 4mm in diameter plated on the front and back faces of the disk. When a sinusoidal voltage is applied a volume shear mode is excited in the plane of the disk along the "x" crystal axis (and no I don't know which way this axis points or even if it's direction is controlled by the manufacturer. Generation of gravitational waves isn't in the spec sheet).

Back to evanescent GW. The experiment would be to place two 4MHz crystals near one another. Near in this case is can-to-can which puts the crystal center to center distance at 3.45mm. A quick calculation yields,

##(kr)^{-5} = (\frac{\lambda}{2\pi r})^5 = 5\times 10^{17}##

The complete back of the envelop is,

##16 \pi G c^{-4} = 4.15\times 10^{-43} \frac{s^2}{kg\;m}##

##T_{x y} = 3.35\times 10^{6} \times Q## Pa

This is for a drive current of 0.5 A (exceeds the spec but these things are cheep). Typical Q values run 50,000 to 100,000 so ##T_{x y}## is also a "big" number. Also I'd like to mention that ##T_{x y}## depends on the mechanical stress induced in the material and is not really a direct function of the crystal mass. The metric strain is the integral of the green function times the ##T_{x y}## so a factor on the crystal volume appears,

##V = 2\pi R^2 = 2.11\times 10^{-8} m^3##

The metric strain is approximately,

## h_{x y} = G V T k (kr)^{-5} \approx \times 10^{-22}## [edit: I've removed Q which is in T. Q is included in T]

Working a similar estimate for the received voltage gives about 50 pico volts. This prompted me to ask people who might know good approaches to measuring at these levels. I posted this yesterday on the Electrical Engineering forum,

Signal measurement ~50 pico volts

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# I Are Evanescent Gravitational Waves Measurable

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