Do gravitational waves cause time dilation or not? There are a lot of questions on this site about gravitational waves and time dilation, and some of the answers are contradictory.
I have read this question:
Do gravitational waves cause time dilatation?
where Tom Andersen says:

In other words, if there was a beam of gravity waves, and one person was in the waves, the other not, the person who experienced the waves would have a small difference in their watch as compared to the person who was not in the wave zone.

Can a gravitational wave produce oscillating time dilation?
where peterh - Reinstate Monica says:

As you can see, it changes only the space coordinates. And only the transversal ones. If there is a change also in the time coordinate, it is not a gravitational wave any more. So, the short answer in the literal sense is a clear no.

Do gravitational waves affect the flow rate of time?
where G. Smith says:

I am reasonably sure that they do cause time to slow down and speed up in an oscillatory way for nearby observers.

So for the sake of argument, let's say there is a non-planar GW and there are two photon-clocks, one of them is in the way of the GW, the other is not affected by the GW.
As the GW passes through one of the clocks, the mirrors will come closer and farther in an oscillatory way, because of the GWs effect of stretching and squeezing spacetime itself. Thus, the clock that is affected by the GW, will seem to relatively (compared to the other clock) tick slower and faster. 
Question:


*

*Do gravitational waves cause time dilation or not?

 A: At least the first order expansion for the covariant metric
$$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$$
with the Minkowski tensor
$$\eta_{\mu \nu} = \left(
\begin{array}{cccc}
 1 & 0 & 0 & 0 \\
 0 & -1 & 0 & 0 \\
 0 & 0 & -1 & 0 \\
 0 & 0 & 0 & -1 \\
\end{array}
\right)$$
and the perturbation
$$h_{\mu \nu} = \left(
\begin{array}{cccc}
 0 & 0 & 0 & 0 \\
 0 & h_{+} & h_{\times} & 0 \\
 0 & h_{\times} & -h_{+} & 0 \\
 0 & 0 & 0 & 0 \\
\end{array}
\right)$$
which give the contravariant metric
$$ \text{g}^{\mu \nu }\to \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \frac{h_{+}+1}{h_{+}^2+h_{\times}^2-1} & \frac{h_{\times}}{h_{+}^2+h_{\times}^2-1} & 0 \\ 0 & \frac{h_{\times}}{h_{+}^2+h_{\times}^2-1} & \frac{1-h_{+}}{h_{+}^2+h_{\times}^2-1} & 0 \\ 0 & 0 & 0 & -1 \\ \end{array} \right) $$
does not show any time dilation, since the $g_{\rm t t}$ and $g^{\rm t t}$ components are $1$.
A: Using a classic result from R. Isaacson (1968), we know that gravitational waves are transverse and follow null geodesics.
Putting these two statements together results in the GWs having no time dilation effects.
An interferometer like LIGO is really measuring the changing proper distance between two test masses as a GW passes.
One way to think about that distance measurement is to measure the proper travel time of something with known speed.
You can make a practical measurement of the time interval between the emission and re-detection a photon that traveled from one mass to the other and back.
Changes in that time interval correspond to changes in the proper distance traveled.
An atomic clock fixed at a single location would not tick faster or slower in the presence of GWs.
If you are very near to a source of GWs, there is some ambiguity.
the math
Isaacson's result was derived in the "high frequency limit", so it holds for arbitrary amplitude GWs provided the wavelength of the GW is small compared to the radius of curvature of the background space.  This is an appropriate limit for any GW in Minkowski space ($R\rightarrow\infty$) and many other scenarios too.
These two facts are consequences of the gauge fixing and lead to the typical cartesian metric construction for a GW propagating in the $\hat{z}$ direction:
$$ \mathbb{h} = \mathbb{A} e^{-i \vec{k}\cdot \vec{x}}, $$
$$ \mathbb{A}\rightarrow A_{\mu\nu} = \left(
\begin{array}{cccc}
 0 & 0 & 0 & 0 \\
 0 & h_{+} & h_{\times} & 0 \\
 0 & h_{\times} & -h_{+} & 0 \\
 0 & 0 & 0 & 0 \\
\end{array}
\right),\quad\quad
\vec{k} \rightarrow k^\alpha = (\omega, 0, 0, k), $$
where the tensor amplitude $\mathbb{A}$ is a function of the two polarization states predicted by GR $h_{+/\times}$, $\omega$ is the wave frequency, and $k$ is the wavenumber.
To satisfy the Einstein field equations $\vec{k}\cdot\vec{k} = 0$.
That's the null geodesic statement.
For example in Minkowski space $(\omega = k)$.
In cartesian coordinates the wavevector $\vec{k}$ points in the $\hat{t}$ and $\hat{z}$ directions.
To satisfy the transverse condition the GW must have zero amplitude in its $t$ and $z$ components.
If the total metric is $\mathbb{g}_\mathrm{tot} = \mathbb{g} + \mathbb{h}$.
We can calculate the proper time between two events A and B:
$$\Delta \tau = \sqrt{-\Delta \vec{x} \cdot \Delta \vec{x}} = \sqrt{- \Delta x^\mu \Delta x^\nu (g_{\mu\nu} + h_{\mu\nu}) }$$
$$\Delta \vec{x} = \vec{x}_B - \vec{x}_A$$
Since the interval $\Delta \vec{x}$ is time-like, we can boost to a reference frame where $\vec{x}_A$ and $\vec{x}_B$ are colocated, and compute $\Delta\tau$ there.  Now
$$ \Delta \tau = \sqrt{-\Delta x^\mu \Delta x^\nu (g_{\mu\nu} + h_{\mu\nu}) } = \Delta x^t \sqrt{-g_{tt} + 0} $$
The proper time measured between two events is not altered by a GW.
other polarizations?
Some extensions to GR predict additional polarization states beyond the two transverse ones in GR.
Some of these other polarization states are longitudinal and would cause time dilation.
The addition of non-GR polarizations changes the rate of energy loss to GWs in binary systems, so there are tight experimental constraints on these theories from binary pulsar measurements and from LIGO's direct detections of GWs.
near field
All of the above is about "far field radiation", where the GWs are far from their source.
The transverse radiation decays in amplitude as $1/r$.
If you are near a source of GWs there are additional longitudinal-like, non-linear terms that decay as $1/r^2$.
In calculations about generating GWs we typically match near field solutions about the source motion to far field solutions about radiation.
These near field longitudinal modes don't matter in the far field since they decay so much faster, so they are typically ignored.
But technically there is a tiny, non-zero time dilation effect.
For astrophysical sources like LIGO has detected, this time dilation is very much suppressed compared to the already tiny spacial stretching.
I personally think of the near field longitudinal modes as "time varying gravitational potential" not "gravitational radiation".
To get technical, the near field longitudinal modes are Petrov Type III regions of spacetime while the transverse radiation is Petrov Type N.  
A: I agree with G.Smith. A gravitational wave passing by causes distortions of space-time which means distortions of space and time. This distortion can be imagined as a gravitational potential well which inevitably involves time dilation. - Remember the Shapiro time delay. The only difference regarding curvature is that here we have Ricci curvature whereas gravitational waves are due to Weyl curvature.
