# Detection of gravitational waves with rotating resonator

Attempts at detecting gravitational waves started with resonant-mass detectors.

One gets some high Q mass and watches it's vibrations. When gravitational wave passes through the mass the latter gets stretched/shrunk a little. The electrostatic connections between the body atoms return them to the original position but this creates some oscillations. Vibrations can accumulate thanks to high mechanical quality factor.

Unfortunately, even with all improvements like using cryogenic temperatures to decrease thermal noise and increase Q and improvements of amplifiers, detectors of this type have no actual detections under their metaphorical belt.

But what if resonator was tuned to frequency f and rotated exactly at f / 2 and there is incoming gravitational pulse with duration N / f (or N cycles of the resonator) and amplitude A?

Then instead of a single pulse of amplitude A and duration N / F resonator will "feel" N stretches and N shrinkages of the amplitude A / N and duration 1 / f.

Acelerated motion equation: S = a * t^2 / 2. That means that if pulse is n times shorter and has 1/n amplitude then it is equivalent to the n times force. 1/n distance and n times force mean same level of energy.

If N < Q then we will effectively accumulate energy and we will get N times more energy in the resonator - we can add several orders of magnitude if we manage to rotate resonator really fast or if we look for very low frequency waves.

As resonator stretches, it's rotational speed will change - this may be easier to detect than the vibrations themselves.

We may use space based resonator or maybe try to use rotational motions and resonances of some molecules.

Was this idea of using rotating resonator for detection of gravitational waves ever considered?

• How do you propose rotating the resonator without imparting any unwanted wobble? Commented Dec 16, 2023 at 11:05
• I can't follow this argument. In the standard setup, the resonator is "strained" $N$ times, each of amplitude $A$. In your setup, the resonator is stretched $2N$ times, each of amplitude $A/2$. Commented Dec 29, 2023 at 9:29
• @ProfBob S = a * t^2 / 2 . Moving object half the distance in half the time requires double acceleration. Acceleration == force, so half distance on double force means same energy per cycle. But there are twice as many cycles, so double total energy. And we can do not only 2 rotations per wave cycle, but as many as we can manage. Commented Dec 31, 2023 at 6:48
• The impulse response of the Explorer bar to a short duration gravitational wave $τ \approx 1/f_{OG}$ approximated by a pulse is written: $\xi(t) \approx \frac{2L}{\pi ^{2}}e^{\frac{−ω_{0}t}{2Q} }Re[h^{+}(ω_{0})]ω_{0}τ \sin(ω_{0}t)\;\;\;(20)$, the problem is to involve this quantity in a rotating system? (*) ens.psl.eu/sites/default/files/2017_PSI_sujet_phyU.pdf Commented Jan 2 at 18:25

I doubt anyone has seriously considered a macroscopic rotating resonator gravitational wave detector such as you describe, since it would blow itself apart even before it could be swamped by noise. There is, however, at least one related idea that I discuss in an update at the end.

Mechanical Failure

The strain at the centre of a cylinder rotating about its central diameter is $$\sigma_0=\frac{1}{2}\rho\omega^2 L^2$$ where $$\rho$$ and $$L$$ are the cylinder's density and length, and $$\omega$$ is its angular frequency. The resonant frequency of the cylinder is $$f=\frac{v}{2L}$$ where $$v=\sqrt{E/\rho}$$ is the speed of sound calculated from the Young's modulus $$E$$. Your proposal rotates the bar at half the resonant frequency, which then leads to a central strain $$\sigma_0=\frac{\pi^2}{8}E\,\sim E$$ Most materials have a yield strength about three orders of magnitude less than their Young's modulus, so your bar will disintegrate long before it reaches its desired rotational speed. For example, the Young's modulus of aluminium is about 72000 MPa, but its tensile strength is only 84 MPa. Even perfect single crystals have yield strengths $$\sim E/10$$, so even a giant diamond or silicon crystal would explode.

Coupled Noise

Aside from this inevitable disintegration, rotational energy leaking into the acoustic modes would swamp any gravitational wave signals. The Allegro and AURIGA resonant gravitational wave detectors were 2300 kg aluminium cylinders 3 m long and 0.6 m in diameter. Such a cylinder rotating about its central diameter at half its resonant frequency would have about 6 GJ of rotational kinetic energy (equivalent to more than a tonne of TNT explosive energy). Gravitational wave excitations would be about $$2\times 10^{-30}$$ J. Even the slightest imperfection in the system or the uniformity of the local static gravitational field would couple the rotational and acoustic modes. It is extremely unlikely that this coupling could be suppressed by 40 orders of magnitude.

Other Ideas

Great experiments often start with kicking around vague ideas, but it is not at all clear what you are suggesting with your alternate idea that one could "maybe try to use rotational motions and resonances of some molecules." Without more details, this is similar to saying "maybe we could try to use magic". I can't think of anything plausible, but you may be more creative than me. You'd need to specify a system, including what molecules, how you'd align the molecules, how you'd rotate them, what the signal would be, and how you'd detect the signal.

Frequency Upconversion and Downconversion (Added 4 Jan 2024)

Rotating at half the resonant frequency is very problematic, but it turns out that rotating at lower frequencies is potentially useful.

Braginsky proposed to measure beats in the angular velocity of two crossed dumbells rotating at a frequency slightly offset from half the gravitational wave frequency. (As noted in the question, measuring changes in rotational speed may be easier.) The rotation down-converts the signal of one circular polarization component of a gravitational wave with frequency $$\omega_g$$ and up-converts the other, i.e. $$\omega_g \rightarrow \omega_g \pm 2\omega_{rot}$$.

Braginsky used the down-converted signal to improve sensitivity, but more recently it has been proposed to use the up-converted signal to move low frequency signals into higher frequencies which have lower noise:

• I think we can imagine a system without rotation but which plays the same role: a quartz oscillator, the gravitational wave will modulate the resonator circuit in frequency, we suppose that at the moment when the crystals starts to dilate, the gravitational wave compresses it and at the moment when it is going to compress the gravitational wave dilates it ???? Commented Jan 3 at 20:10
• I don't think disintegration is quite the issue you think it is. There is plenty of interest in detecting gravitational waves at the Hz scale, it's just that current gravitational wave detectors can't do it yet. I also don't think vacuum is such a big issue. What about a rotating interferometer? Problem solved - the vacuum should feasible; it is for LIGO! Commented Jan 4 at 9:46
• Thanks @AXensen. The question is specifically about resonant detectors - not interferometers - rotated at half their resonant frequency. I agree there is no problem rotating a resonant detector at 1 Hz, but that is far outside their narrow sensitive range (typically ~kHz). The strain in the supports for a 4 km LIGO-like interferometer rotating at ~1 Hz would be tens of GPa (from $1/2 \rho \omega^2 L^2$), so it would also disintegrate. My vacuum comment was a bit of an afterthought, so I'll need to think about your interesting LIGO vacuum comparison. Commented Jan 4 at 14:21
• Thanks @The-Tiler. Yes, it is interesting to think about alternatives without rotation, but quartz oscillators are typically pretty small. Wouldn't one large enough to be a gravitational wave detector be essentially a resonant bar - just made out of quartz instead of typical aluminium? If you drive it at the same frequency you want to detect, how do you extract the gravitational wave signal from the much stronger driving force? Commented Jan 4 at 14:38
• +1 for the nice answer; recently the heterodyne approach has been revived, but using oscillating electromagnetic fields instead of rotation. Commented Jan 4 at 21:48