<p>The <a href="https://astronomy.stackexchange.com/a/44977/23762">answer by @HDE 226868</a> 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, <a href="https://physics.stackexchange.com/questions/553487/do-gravitational-waves-cause-time-dilation-or-not">this</a> 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 <span class="math-container">$h_{\mu\nu}$</span> 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.</p>
<p>In the (mathematically rigorous) paper by <a href="https://journals.aps.org/prd/abstract/10.1103/PhysRevD.90.062002" rel="noreferrer">Koop and Finn (2014)</a>, they characterize the GW amplitude using the Riemann curvature tensor to &quot;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.&quot; See their Eq. (3.16) for that formula.</p>
<p>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 <a href="https://arxiv.org/abs/1501.00996" rel="noreferrer">Loeb and Moaz (2014)</a> about atomic clocks and gravitational waves.</p>
<p>The <a href="https://arxiv.org/abs/1501.00996" rel="noreferrer">Loeb and Moaz (2014)</a> 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 <a href="https://ui.adsabs.harvard.edu/abs/1994PhRvD..49.2658C/abstract" rel="noreferrer">Cutler and Flannagan (1994</a>). They cite a seminal paper by <a href="https://arxiv.org/abs/1307.4086" rel="noreferrer">A. Sesana (2013)</a>, 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:</p>
<blockquote>
<p>&quot;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 <a href="https://en.wikipedia.org/wiki/Hulse%E2%80%93Taylor_binary" rel="noreferrer">7</a>. In this gauge, an oscillating perturbation in the
time-time component of the metric, <span class="math-container">$h_{00}$</span>, would trigger periodic
variation in the Pound-Rebka time dilation and a mismatch be-
tween the ticking rate of clocks separated apart.&quot;</p>
</blockquote>
<p>Therefore, I think that <a href="https://arxiv.org/abs/1501.00996" rel="noreferrer">Loeb and Moaz (2014)</a> 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 <span class="math-container">$f$</span> as the redshifted frequency, not the intrinsic gravitational wave frequency.</p>
<blockquote>
<p>Why are there not yet any atomic clock instruments dedicated to registering time dilation caused by passing gravitational waves?</p>
</blockquote>
<p>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 <span class="math-container">$\sim 10^{-18}$</span>, which is precisely the numerical prefactor in the front of their Eq. (1).</p>
<blockquote>
<p>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)?</p>
</blockquote>
<p>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!</p>
<hr />
<p>EDIT: This was my first answer which is not very relevant for the OP. Although <a href="https://en.wikipedia.org/wiki/Pulsar_timing_array" rel="noreferrer">pulsar timing arrays</a> (PTAs) do not measure gravitational time dilation proper, as <a href="https://journals.aps.org/prd/abstract/10.1103/PhysRevD.90.062002" rel="noreferrer">pointed out by HDE 226868</a>, I'll keep it here for sake of clarity for my own progression in thinking about these questions.</p>
<p>The binary pulsar discovered by <a href="https://arxiv.org/abs/1501.00996" rel="noreferrer">Hulse and Taylor</a> 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 <a href="https://ui.adsabs.harvard.edu/abs/1994PhRvD..49.2658C/abstract" rel="noreferrer">LIGO</a> and <a href="https://arxiv.org/abs/1307.4086" rel="noreferrer">Virgo</a> via compact binary coalescences.</p>
<p>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 <a href="https://en.wikipedia.org/wiki/Hulse%E2%80%93Taylor_binary" rel="noreferrer">stochastic background</a> of gravitational waves. The various correlations that exist in the networks are handled in a myriad of ways.</p>
<p>The Nanograv consortium has been taking data for over a decade, and recently published <a href="https://en.wikipedia.org/wiki/Gravitational_wave_background" rel="noreferrer">this</a> 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.</p>
<p>There are other PTAs being designed/constructed so the future looks bright for this field!</p>

CC BY-SA 4.0
2021-07-23T01:30:15.763Z
It's a bit misleading to say that PTAs look for time dilation; it's just appropriately correlated changes in the time of arrival (which aren't caused by time dilation), not intrinsic time dilation in the usual sense.
HDE 226868
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2021-07-23T13:05:18.277Z
Ah I see this now. My bad. I wrote this answer at night half asleep - I've edited it to be more clear.
Daddy Kropotkin
CC BY-SA 4.0
2021-07-23T01:15:34.817Z
2021-07-23T17:06:24.757Z
2021-07-23T17:06:24.757Z
Daddy Kropotkin
<p>In Cartesian coordinates, the flat spacetime interval can be written in terms of invariant proper time <span class="math-container">$\tau$</span> as
<span class="math-container">$$c^2 d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2\ ,$$</span>
where <span class="math-container">$t$</span> is some universal time coordinate and the usual notation convention that <span class="math-container">$dt^2 = (dt)^2$</span> is used.</p>
<p>For all stationary observers, in the frame of reference for which <span class="math-container">$x, y, z$</span> are defined, then <span class="math-container">$dx= dy=dz=0$</span> and hence <span class="math-container">$d\tau = dt$</span> 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 <span class="math-container">$\tau$</span>, also measures <span class="math-container">$t$</span> and there is no time dilation between different stationary observers. Things change of course when observers start moving - that is Special Relativity.</p>
<p>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 <span class="math-container">$z$</span>-axis is:
<span class="math-container">$$c^2d\tau^2 = c^2dt^2 - (1+a_+\sin \omega t)dx^2 -2a_\times\sin(\omega t +\phi) dxdy - (1 - a_+\sin\omega t)dy^2 - dz^2\ , $$</span>
where <span class="math-container">$\omega$</span> is the GW frequency, <span class="math-container">$a_+$</span> and <span class="math-container">$a_\times$</span> are the amplitudes of the <em>tiny</em> GW perturbations, one for each of the possible &quot;plus&quot; and &quot;cross&quot; polarisations, and <span class="math-container">$\phi$</span> is an arbitrary phase difference between those polarisations.</p>
<p>If <span class="math-container">$dx=dy=dz=0$</span>, then you can see that it is <em>still</em> the case that <span class="math-container">$d\tau = dt$</span> and there is no time dilation between clocks at different locations.</p>
<p>This all assumes you are far from the source of gravitational waves, so that the waves can be considered transverse.</p>

CC BY-SA 4.0
2021-07-23T11:02:22.783Z
Repeating my comment to HDE 226868's answer: [Loeb and Maoz](https://arxiv.org/abs/1501.00996) explicitly say there are time-delay components to GWs (and give an approximate amplitude for the time-time component of the metric in the case of binary SMBHs -- although without a derivation -- and propose a scheme for measuring this using atomic clocks in separate orbits in space. Are they just wrong?
Peter Erwin
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2021-07-23T12:21:19.420Z
@PeterErwin that 2015 preprint hasn't been accepted to a journal. So yes, maybe they are wrong. They have a non-zero $h_{00}$ metric component by adopting the Newtonian gauge. It may come down to what you mean by "time dilation". I am not qualified to say that L+M have got it wrong. There are Doppler shifts and other effects associated with passing GWs that are predicted by the metric above that might be confused/conflated with "time dilation".
ProfRob
CC BY-SA 4.0
2021-07-23T23:06:22.087Z
@BenCrowell flat spacetime is introduced because the gravitational waves we observe on Earth are a tiny perturbation of that. Time dilation is commonly defined in terms of the ratio of $dt$ to $d\tau$ for clocks that are located at different, but stationary spatial coordinates.
ProfRob
CC BY-SA 4.0
2021-07-23T23:14:23.527Z
@BenCrowell correct. You work out $dt/d\tau$ for two observers and then take the ratio. If $dt/d\tau=1$, then that ratio is obviously 1.
ProfRob
CC BY-SA 4.0
2021-07-23T07:48:06.953Z
2021-07-24T13:30:55.067Z
2021-07-24T13:30:55.067Z
ProfRob
<p>General relativity predicts that there are only two possible polarizations of gravitational waves, the so-called &quot;tensor&quot; polarizations <span class="math-container">$+$</span> and <span class="math-container">$\times$</span>. It turns out you can show that <a href="https://physics.stackexchange.com/q/553487/56299">the tensor polarizations actually don't lead to time dilation</a>, making any attempted measurement of it pointless. The short answer, then, is that we don't expect to see any time dilation at all!</p>
<p>Now, you could argue that such an experiment would still be useful insofar as it could be used to search for alternative polarizations (the &quot;scalar&quot; and &quot;vector&quot; 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).</p>
<p>An individual transient signal would need five<span class="math-container">$^{\dagger}$</span> 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 <a href="https://arxiv.org/abs/1709.09660" rel="noreferrer">in the signal from GW170814</a> (more <a href="https://www.ligo.org/science/Publication-O1StochNonGR/" rel="noreferrer">here</a>) and at least found that purely tensor polarizations were strongly favored over purely scalar or purely vector polarizations. <a href="https://gwcenter.icrr.u-tokyo.ac.jp/en/" rel="noreferrer">KAGRA</a> has begun observations, and <a href="https://www.nature.com/articles/d41586-019-00184-z" rel="noreferrer">LIGO-India</a> should be completed by the middle of the decade, which will help break some of the degeneracies at work.</p>
<p>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 <a href="https://arxiv.org/abs/1802.10194" rel="noreferrer">with which to probe alternative polarizations</a>. 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 (<a href="https://arxiv.org/abs/1712.07132" rel="noreferrer">Cornish et al. 2017</a>), making some of this moot.</p>
<hr />
<p><span class="math-container">$^{\dagger}$</span>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 (<a href="https://arxiv.org/abs/2105.01521" rel="noreferrer">Chatziioannou et al. 2021</a>).</p>

CC BY-SA 4.0
2021-07-23T11:00:42.060Z
[Loeb and Maoz (2015)](https://arxiv.org/abs/1501.00996) explicitly say there are time-delay components to GWs (and give an approximate amplitude for the time-time component of the metric in the case of binary SMBHs -- although without a derivation) and propose a scheme for measuring this using atomic clocks in separate orbits in space. Are they just confused?
Peter Erwin
CC BY-SA 4.0
2021-07-23T13:17:07.497Z
The two tensor polarization states do not cause time dilation when derived in a flat background spacetime, but what about if a curved background is used instead, i.e., such as Kerr, which is more relevant for LIGO applications?
Daddy Kropotkin
CC BY-SA 4.0
2021-07-23T14:15:24.913Z
@PeterErwin I don't think I'm the person to adequately answer that. I've only ever seen the Newtonian gauge in the context of first-order scalar perturbations (and I've only ever seen GWs associated with first-order tensor perturbations), and since I can't track down the source for whatever derivation they're using for their first equation, I can't speculate on what their reasoning is. I also don't know whether it makes sense to associate SMBH binaries with scalar perturbations of any order. So I'm at a loss there.
HDE 226868
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2021-07-23T15:09:15.277Z
@HDE226868 I think quasinormal modes from tensor perturbations are used to study the ringdown after a compact binary merger, and are also used in extreme mass-ratio inspirals. In those cases, the background spacetime is curved, not Minkowski, and hence my question above. Also, I think they derived their Eq.(1) just by taking the leading term of the post-Newtonian approximation, which depends on the chirp mass to the 5/3
Daddy Kropotkin
CC BY-SA 4.0
2021-07-23T15:15:56.563Z
@PeterErwin I think Loeb and Maoz are using the post-Newtonian approximation since the SMBHs are are not merging in this case, rather their continuous g-wave emission is utilized. See their footnote 1 which cites Pound and Rebka 1956. Also, this paper https://journals.aps.org/prd/abstract/10.1103/PhysRevD.90.062002 that they cite when they discuss metric perturbations and time dilation explains in depth that there is a "background" curvature contribution that I think does not vanish when we are not in the limit of high frequency, as in the Loeb an Maoz paper.
Daddy Kropotkin
CC BY-SA 4.0
2021-07-23T16:40:53.593Z
Re *"will help break some of the degeneracies at work"*: In what way? Due to a higher total number of detectors/3D distribution of them? Or because the new detectors are qualitatively better? Or something else?
Peter Mortensen
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2021-07-23T19:09:14.283Z
@DaddyKropotkin surely what matters is where the GWs are measured, not what produced them. The GWs detect on Earth are essentially plane waves on a flat background. Sure, close to the generating source there will be longitudinal modes and non-zero $h_{00}$ terms.
ProfRob
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2021-07-23T19:32:18.957Z
@PeterMortensen Answer updated - I could have been clearer.
HDE 226868
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2021-07-23T01:06:47.890Z
2021-07-24T14:27:47.323Z
2021-07-24T14:27:47.323Z
HDE 226868
<p>Why are there not yet any instruments dedicated to registering time dilation caused by passing gravitational waves?</p>
<p>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)?</p>

CC BY-SA 4.0
2021-07-23T01:06:38.987Z
For uninformed folks like me could you add a link explaining "time dilation caused by passing gravitational wave(s)" showing that it has been predicted? *Thanks!*
uhoh
CC BY-SA 4.0
2021-07-24T00:42:39.980Z
A much better, and less self-assured question is, "Why isn't it necessary to adjust atomic clocks for time dilation caused by passing gravitational waves?"
RonJohn
<gravitational-waves><time-dilation><ligo>
Why are there not yet any instruments dedicated to registering time dilation caused by passing gravitational waves?
Alex
# Why are there not yet any instruments dedicated to registering time dilation caused by passing gravitational waves?

(Asked by: username_0 on 2021-07-23)

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)?

**Tags:** gravitational-waves, time-dilation, ligo
> **Comments:**
> **username_1**: For uninformed folks like me could you add a link explaining "time dilation caused by passing gravitational wave(s)" showing that it has been predicted? *Thanks!*  
> **username_2**: A much better, and less self-assured question is, "Why isn't it necessary to adjust atomic clocks for time dilation caused by passing gravitational waves?"  

---

## 3 answers

### Answer by username_4 on 2021-07-23. Score: 26 | Accepted answer

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](https://physics.stackexchange.com/q/553487/56299), 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](https://arxiv.org/abs/1709.09660) (more [here](https://www.ligo.org/science/Publication-O1StochNonGR/)) and at least found that purely tensor polarizations were strongly favored over purely scalar or purely vector polarizations. [KAGRA](https://gwcenter.icrr.u-tokyo.ac.jp/en/) has begun observations, and [LIGO-India](https://www.nature.com/articles/d41586-019-00184-z) 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](https://arxiv.org/abs/1802.10194). 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](https://arxiv.org/abs/1712.07132)), 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](https://arxiv.org/abs/2105.01521)).

> **Comments:**
> **username_6**: [Loeb and Maoz (2015)](https://arxiv.org/abs/1501.00996) explicitly say there are time-delay components to GWs (and give an approximate amplitude for the time-time component of the metric in the case of binary SMBHs -- although without a derivation) and propose a scheme for measuring this using atomic clocks in separate orbits in space. Are they just confused?  
> **username_3**: The two tensor polarization states do not cause time dilation when derived in a flat background spacetime, but what about if a curved background is used instead, i.e., such as Kerr, which is more relevant for LIGO applications?  
> **username_4**: @PeterErwin I don't think I'm the person to adequately answer that. I've only ever seen the Newtonian gauge in the context of first-order scalar perturbations (and I've only ever seen GWs associated with first-order tensor perturbations), and since I can't track down the source for whatever derivation they're using for their first equation, I can't speculate on what their reasoning is. I also don't know whether it makes sense to associate SMBH binaries with scalar perturbations of any order. So I'm at a loss there.  
> **username_3**: @HDE226868 I think quasinormal modes from tensor perturbations are used to study the ringdown after a compact binary merger, and are also used in extreme mass-ratio inspirals. In those cases, the background spacetime is curved, not Minkowski, and hence my question above. Also, I think they derived their Eq.(1) just by taking the leading term of the post-Newtonian approximation, which depends on the chirp mass to the 5/3  
> **username_3**: @PeterErwin I think Loeb and Maoz are using the post-Newtonian approximation since the SMBHs are are not merging in this case, rather their continuous g-wave emission is utilized. See their footnote 1 which cites Pound and Rebka 1956. Also, this paper https://journals.aps.org/prd/abstract/10.1103/PhysRevD.90.062002 that they cite when they discuss metric perturbations and time dilation explains in depth that there is a "background" curvature contribution that I think does not vanish when we are not in the limit of high frequency, as in the Loeb an Maoz paper.  
> **username_7**: Re *"will help break some of the degeneracies at work"*: In what way? Due to a higher total number of detectors/3D distribution of them? Or because the new detectors are qualitatively better? Or something else?  
> **username_5**: @DaddyKropotkin surely what matters is where the GWs are measured, not what produced them. The GWs detect on Earth are essentially plane waves on a flat background. Sure, close to the generating source there will be longitudinal modes and non-zero $h_{00}$ terms.  
> **username_4**: @PeterMortensen Answer updated - I could have been clearer.  

---

### Answer by username_3 on 2021-07-23. Score: 7

The [answer by @HDE 226868](https://astronomy.stackexchange.com/a/44977/23762) 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](https://physics.stackexchange.com/questions/553487/do-gravitational-waves-cause-time-dilation-or-not) 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)](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.90.062002), 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)](https://arxiv.org/abs/1501.00996) about atomic clocks and gravitational waves.

The [Loeb and Moaz (2014)](https://arxiv.org/abs/1501.00996) 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](https://ui.adsabs.harvard.edu/abs/1994PhRvD..49.2658C/abstract)). They cite a seminal paper by [A. Sesana (2013)](https://arxiv.org/abs/1307.4086), 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](https://en.wikipedia.org/wiki/Hulse%E2%80%93Taylor_binary). 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)](https://arxiv.org/abs/1501.00996) 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](https://en.wikipedia.org/wiki/Pulsar_timing_array) (PTAs) do not measure gravitational time dilation proper, as [pointed out by HDE 226868](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.90.062002), 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](https://arxiv.org/abs/1501.00996) 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](https://ui.adsabs.harvard.edu/abs/1994PhRvD..49.2658C/abstract) and [Virgo](https://arxiv.org/abs/1307.4086) 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](https://en.wikipedia.org/wiki/Hulse%E2%80%93Taylor_binary) 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](https://en.wikipedia.org/wiki/Gravitational_wave_background) 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!

> **Comments:**
> **username_4**: It's a bit misleading to say that PTAs look for time dilation; it's just appropriately correlated changes in the time of arrival (which aren't caused by time dilation), not intrinsic time dilation in the usual sense.  
> **username_3**: Ah I see this now. My bad. I wrote this answer at night half asleep - I've edited it to be more clear.  

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### Answer by username_5 on 2021-07-23. Score: 4

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.

> **Comments:**
> **username_6**: Repeating my comment to HDE 226868's answer: [Loeb and Maoz](https://arxiv.org/abs/1501.00996) explicitly say there are time-delay components to GWs (and give an approximate amplitude for the time-time component of the metric in the case of binary SMBHs -- although without a derivation -- and propose a scheme for measuring this using atomic clocks in separate orbits in space. Are they just wrong?  
> **username_5**: @PeterErwin that 2015 preprint hasn't been accepted to a journal. So yes, maybe they are wrong. They have a non-zero $h_{00}$ metric component by adopting the Newtonian gauge. It may come down to what you mean by "time dilation". I am not qualified to say that L+M have got it wrong. There are Doppler shifts and other effects associated with passing GWs that are predicted by the metric above that might be confused/conflated with "time dilation".  
> **username_5**: @BenCrowell flat spacetime is introduced because the gravitational waves we observe on Earth are a tiny perturbation of that. Time dilation is commonly defined in terms of the ratio of $dt$ to $d\tau$ for clocks that are located at different, but stationary spatial coordinates.  
> **username_5**: @BenCrowell correct. You work out $dt/d\tau$ for two observers and then take the ratio. If $dt/d\tau=1$, then that ratio is obviously 1.  

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