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Why are there not yet any instruments dedicated to registering time dilation caused by passing gravitational waves?
Wouldn't it be interesting to augment LIGO/Virgo capturing of space distortion with simultaneous capturing time dilation (both caused by the same passing gravitational wave)?
General relativity predicts that there are only two possible polarizations of gravitational waves, the so-called "tensor" polarizations $+$ and $\times$. It turns out you can show that the tensor polarizations actually don't lead to time dilation, making any attempted measurement of it pointless. The short answer, then, is that we don't expect to see any time dilation at all!
Now, you could argue that such an experiment would still be useful insofar as it could be used to search for alternative polarizations (the "scalar" and "vector" polarizations) which would indicate that a different theory of gravity is warranted. On the other hand, this would be arguably be redundant, because there are other methods with which we can probe alternative polarizations in interferometric data, either by looking at individual sources or the hypothesized stochastic gravitational wave background (at the frequencies LIGO is sensitive to).
An individual transient signal would need five$^{\dagger}$ appropriately aligned detectors to fully characterize contributions of alternative polarizations, but the LIGO-Virgo collaboration was able to search for evidence of scalar and vector polarizations in the signal from GW170814 (more here) and at least found that purely tensor polarizations were strongly favored over purely scalar or purely vector polarizations. KAGRA has begun observations, and LIGO-India should be completed by the middle of the decade, which will help break some of the degeneracies at work.
A search of the stochastic background wouldn't require so many detectors because the signal is not coming from any one place in the sky, so it provides another strategy with which to probe alternative polarizations. The O1 observing run turned up no evidence of backgrounds with scalar or vector polarizations; that said, there was also no evidence of any background at all, tensor polarizations included. It's also possible that pulsar timing arrays may be able to shed light on the issue if a stochastic background is detected and there is substantial evidence for tensor polarizations but not alternative polarizations (Cornish et al. 2017), making some of this moot.
$^{\dagger}$A single interferometer's response to a gravitational wave is a sum of terms corresponding to individual polarizations. In more general theories of gravity, there are up to two tensor modes, two vector modes, and two scalar modes, but the class of interferometers LIGO and Virgo belong to can only measure a particular linear combination of the scalar modes, so we deal with five degrees of freedom. Therefore, five detectors are needed to determine how each mode (or combination or modes) contributes to the signal (Chatziioannou et al. 2021).
The answer by @HDE 226868 addresses the current attempts by LIGO/Virgo and PTAs to detect alternate gravitational wave (GW) polarization states, which have not been detected. In that answer, this SE question is cited, which shows that gravitational waves being interpreted as tensor perturbations of the flat (Minkowski) spacetime produces only two non-trivial polarization states which are not time-time components and thus do not cause time dilation. However, this does not mean that gravitational radiation cannot generally cause gravitational time dilation, since the components of the strain tensor $h_{\mu\nu}$ are not gauge-invariant quantities, so I think it might not be sufficient to just point at them and claim that there is no time dilation.
In the (mathematically rigorous) paper by Koop and Finn (2014), they characterize the GW amplitude using the Riemann curvature tensor to "provide a new, first-principles derivation of the response of modern, light-time gravitational wave detectors in terms of their interaction with spacetime curvature... Finally, the curvature-based response formula leads to a simpler calculation of light-time detector response than the corresponding calculations carried out using the metric perturbation approach." See their Eq. (3.16) for that formula.
Hence, they proved using pure differential geometry that gravitational waves can cause time dilation in a light-time detector, which provides fundamental justification for the ideas used in the paper by Loeb and Moaz (2014) about atomic clocks and gravitational waves.
The Loeb and Moaz (2014) paper outlines a proposed framework to detect the gravitational time dilation due to a gravitational wave that passes through a network of atomic clocks orbiting in space. They use the post-Newtonian approximation, specifically the leading-order mass quadrupole approximation, as seen in their Eq. (1) where the strain depends on the 5/3 power of the chirp mass, e.g. see Eq. (3.9) of Cutler and Flannagan (1994). They cite a seminal paper by A. Sesana (2013), whose Eq. (11) is equivalent to the Eq. (1) of Loeb and Moaz, and Sesana even derives it for us :). In the footnote 1 of Loeb and Moaz (2014), they state:
"In this paper, we adopt for pedagogical reasons a Newtonian gauge which is commonly used to describe the time-dilation ef- fect due to stationary gravity, as measured in the Pound-Rebka experiment 7. In this gauge, an oscillating perturbation in the time-time component of the metric, $h_{00}$, would trigger periodic variation in the Pound-Rebka time dilation and a mismatch be- tween the ticking rate of clocks separated apart."
Therefore, I think that Loeb and Moaz (2014) are just assuming that their Eq. (1) approximates the time-time component of the strain tensor, as means of having a crude approximation to work with for the sake of outlining the idea of the paper, by identifying $f$ as the redshifted frequency, not the intrinsic gravitational wave frequency.
Why are there not yet any atomic clock instruments dedicated to registering time dilation caused by passing gravitational waves?
Mostly because the sensitivity of atomic clock instruments has only recently reached the precision required to make gravitational time dilation measurements, and also because detecting gravitational waves is a rather recent accomplishment. As stated in the intro of Loeb and Moaz (2014), the precision of optical lattice atomic clocks has reached $\sim 10^{-18}$, which is precisely the numerical prefactor in the front of their Eq. (1).
Wouldn't it be interesting to augment LIGO/VIRGO capturing of space distortion with simultaneous capturing time dilation (both caused by the same passing gravitational wave)?
Yes indeed it would! But I think this would require using more sophisticated treatments of the background spacetime, which is dominated by the gravity of the solar system for LIGO/Virgo, rather than treating it as flat. Also, as @HDE 226868 points out, doing this with serious precision requires several ground-based interferometers, which will likely be reality in the future!
EDIT: This was my first answer which is not very relevant for the OP. Although pulsar timing arrays (PTAs) do not measure gravitational time dilation proper, as pointed out by HDE 226868, I'll keep it here for sake of clarity for my own progression in thinking about these questions.
The binary pulsar discovered by Hulse and Taylor in 1974 was the first binary pulsar to be discovered, and it was the first observational verification (later in 1975) of the existence of gravitational waves - however direct detection of gravitational waves did not occur until 2015 by LIGO and Virgo via compact binary coalescences.
Anyway, PTAs is a network of known pulsars whose delays of the time of arrival of pulses of light are correlated by a passing gravitational wave. Intuitively, such a gravitational wave would need to have a long wavelength, so a natural candidate has been the stochastic background of gravitational waves. The various correlations that exist in the networks are handled in a myriad of ways.
The Nanograv consortium has been taking data for over a decade, and recently published this paper announcing their progress. They are on the precipice of making a detection of the stochastic background, but there are some correlations that are still being worked out.
There are other PTAs being designed/constructed so the future looks bright for this field!
In Cartesian coordinates, the flat spacetime interval can be written in terms of invariant proper time $\tau$ as $$c^2 d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2\ ,$$ where $t$ is some universal time coordinate and the usual notation convention that $dt^2 = (dt)^2$ is used.
For all stationary observers, in the frame of reference for which $x, y, z$ are defined, then $dx= dy=dz=0$ and hence $d\tau = dt$ for all clocks that are stationary in that frame and the ratio of proper times is unity. This means the clock carried by the observer, which measures $\tau$, also measures $t$ and there is no time dilation between different stationary observers. Things change of course when observers start moving - that is Special Relativity.
The relevance of this, is that a gravitational wave (GW) applies a small perturbation to the metric, so the spacetime interval for a passing GW travelling along the $z$-axis is: $$c^2d\tau^2 = c^2dt^2 - (1+a_+\sin \omega t)dx^2 -2a_\times\sin(\omega t +\phi) dxdy - (1 - a_+\sin\omega t)dy^2 - dz^2\ , $$ where $\omega$ is the GW frequency, $a_+$ and $a_\times$ are the amplitudes of the tiny GW perturbations, one for each of the possible "plus" and "cross" polarisations, and $\phi$ is an arbitrary phase difference between those polarisations.
If $dx=dy=dz=0$, then you can see that it is still the case that $d\tau = dt$ and there is no time dilation between clocks at different locations.
This all assumes you are far from the source of gravitational waves, so that the waves can be considered transverse.
