# Gravitational waves detection using lunar retroreflectors

According to Wikipedia, the wavelength of a gravitational wave is about 600,000 km. The Earth moon distance is about 384,400 km. Considering that the latter is roughly half the former, it seems to me that there would be an instant during which the space between Earth and moon were almost entirely squeezed or stretched as a gravitational wave passes through the Earth moon system. I'm curious why the lunar reflectors cannot be used to detect those waves more cheaply and easily than LIGO, the kilometer long dedicated interferometer that actually succeeded in detecting them. To do so, it needed to detect space-time variations on the order of a hundredth of the diameter of the nucleus of a hydrogen atom (2.40×10−15 m). Considering the ratio of 600,000:1, wouldn't a retroreflector detection require about 600,000 times less resolution/accuracy than LIGO (1.44x10-9 m), achievable without a super-sensitive interferometer?

• Well, $10^{-9}$ m is about $10^{10}$ times smaller than the daily change in surface-to-surface distance caused by tidal forces in the solid earth and moon, so that is still going to need some pretty exotic instrumentation to measure it! Commented Nov 20, 2019 at 22:51
• Could one not make use of an Earth orbiting satellite and take advantage of the fact that the tidal change is very slow compared to the passage of a gravity wave? Commented Nov 20, 2019 at 22:55
• As I understand it, gravity waves are detectable because the universe shortens in the north-south direction at the same time that it lengthens in the east-west direction (this having to do with the fact that space-time is described by a tensor, or because gravitons are spin 2, or whatever your choice of reasons is). So you need two sensors, ideally at right angles to each other. So your idea would work, if only we had two or more moons. It could be that if we put retro-reflectors at the moon's Trojan points that we could do this. I don't know if it's been considered. Commented Nov 20, 2019 at 22:56
• Or a single additional micro satellite to position an additional retroreflector at a suitable position in space, perhaps in lunar orbit. Commented Nov 20, 2019 at 23:16
• The LISA concept has a constellation of three spacecraft arranged in an equilateral triangle with sides 2.5 million km long. Commented Nov 20, 2019 at 23:28

There are a few issues with what you've described. In no particular order:

1. The length of the arms determines the frequency of wave that can be detected. So, for example, LIGO (ground-based) and LISA (proposed space-based) have vastly different length of arms. That doesn't change the resolution, it changes which frequencies each detects / would detect. This is analogous to changing the length of your antenna for electromagnetic waves. You need a different size antennas for different frequencies of radiation. That does not by itself say how good your amplifier has to be to detect small signals in any particular frequency.

2. Related, you'd have to check to see if there are plausible astrophysical sources of gravitational waves at the frequencies to which your moon-reflector system would be sensitive. (I don't know the answer off-hand because it's not a length that I've thought about much.)

3. You still need extreme resolution. I'm not exactly clear what you meant with the ratio you stated, but you seemed to think you could have a less sensitive instrument because of different length arms. That's not the case - you still need very sensitive measurements of change in distance. Ala what's above, you've changed the frequency to which you're sensitive but not necessarily the tolerances on the arm-length change measurement.

4. The moon is seismically active. I'm not expert on moonquakes, but I think it's likely that it's active enough to blow out your tolerance on the measurement, keeping in mind the extreme measures that LIGO takes to isolate itself from seismic noise and the fact that moon reflectors were put there in the 1960s with no thought for this type of precision.

5. You need a differential measurement in different directions. LIGO uses two arms. LISA has three legs, although interferometry is done pairwise on them. You'd need to put something in orbit at a comparable length in order to have your second leg. That's its own engineering feat. (Not necessarily impossible, but not as ready to go as you might hope.)

6. Assuming you are sending the laser beams from earth, you need simultaneous line of sight from the ground station to both the moon and the end of your new leg. That's going to be a problem if you want continuous coverage.

7. You'd need an extremely good model of atmospheric effects on the propagation of the laser beam as these are probably bigger than the change in distance that you're trying to measure. (LIGO pumps the arms to vacuum. LISA would operate in open space.)

I could probably go on, but this may give the idea that what you're suggesting is much more complicated than perhaps it seemed at first.

• An important one you missed: light intensity. The LIGO arms store lots of laser power, but the lunar laser ranging only became feasible about a decade ago when, after decades of effort, the team's started to get more than one photon on average reflected from each laser pulse.
– rob
Commented Nov 21, 2019 at 6:50
• Yes, that's a good one too, @rob. I'll give it a little time but I may add to the list if there are other suggestions. I definitely did not intend for this to be an all-inclusive list. Commented Nov 21, 2019 at 15:20

Gravitational waves are transverse, not longitudinal. You're talking about comparing the earth-moon distance to the wavelength, but if the direction of propagation is parallel to the earth-moon line, then there is no effect.