What is meant by the transverse nature of gravitational waves? Gravitational waves, like the electromagnetic waves, are also transverse. By transversality of the EM waves, we mean that ${\vec E}\cdot\vec{k}={\vec B}\cdot\vec{k}=0$ i.e., the accompanying electric and the magnetic field (which are two $3$-vectors) vibrate in a plane perpendicular to the direction of propagation specified by the unit vector along ${\vec k}$. A gravitational wave $h_{\mu\nu}(z,t)=h^{(0)}_{\mu\nu}\sin(kz-\omega t)$ is a rank-$2$ tensor.

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*How can I understand what is meant by the transverse nature of gravitational waves $h_{\mu\nu}$, mathematically? Physically/geometrically, which quantities vibrate perpendicular to the spatial direction $\hat{k}$?

Note The comment points out that transversality of gravitational wave means $\partial^\mu h_{\mu\nu}=0\leftrightarrow k^\mu h_{\mu\nu}=0$. But I find it hand to interpret geometrically because of two main reasons. First, $k_\mu$ is not a $3$-vector but a $4$-vector and second, all the $16$ components of $h_{\mu\nu}$ are not physical.
 A: Mathematically the transverse condition for gravitational waves (GW) is $k^\mu h_{\mu\nu} = 0$, where $k^\mu$ are the components of the wave vector and $h_{\mu\nu}$ are the components of the GW perturbation to the metric tensor.
the wave vector in GR
As you point out, the wave vector is a 4D object in GR.  It still points in the propagation direction of the wave, just like the waves we may be more familiar with.  In cartesian coordinates the wave 4-vector can be written:
$$\vec{k}\rightarrow(k^t, k^x, k^y, k^z).$$
More generally we can break the 4-vector into timelike and spacelike components and write it as (in geometric units, where $G=c=1$):
$$\vec{k}\rightarrow(\omega, \mathbf{k}).$$
The time component, $k^t = \omega$, is the angular frequency of the wave, and $\mathbf{k}$ is the usual wave 3-vector.  All waves propagate through both space and time, which is represented by the direction of $\mathbf{k}$ and the non-zero $k^t$.  A GW traveling in the $\hat{z}$ direction would have a wave 4-vector $$\vec{k}\rightarrow(\omega, 0, 0, k).$$
GWs propagate at the speed of light, along null geodesics.  Mathematically, this can be represented as $\vec{k}\cdot\vec{k} = 0$.  For GWs propagating through flat Minkowski space this becomes
$$\vec{k}\cdot\vec{k} = k^\mu \eta_{\mu\nu} k^\nu = 0,$$
where $\eta_{\mu\nu}$ are the components of the Minkowski metric.
For our $\hat{z}$ propagating GW,
$$\vec{k}\cdot\vec{k} = -\omega^2 + k^2 = 0.$$
The timelike and spacelike parts of the wave vector have equal magnitude, so the wave propagates through both time and space at the same rate.  This is the definition of a lightlike or null path.  All null 4-vectors have zero magnitude.
transverse GWs
The transverse condition $k^\mu h_{\mu\nu} = 0$, tells us the directions that the GW perturbations affect space-time.  For $\hat{z}$ propagating GWs, all of the $h_{t\mu}$ and $h_{z\mu}$ components must vanish to satisfy the condition.  Only four of the possible 16 components can be non-zero: $h_{xx}, h_{xy}, h_{yx}, h_{yy}$.  Symmetry of the metric requires $h_{xy} = h_{yx}$.  Further, a gauge condition sets the trace of the perturbation to zero, so $h_{xx} = -h_{yy}$. There are two independent components in the GW metric, which represent the two  polarizations of GWs.
The transverse condition says GWs only perturb the spacetime in directions perpendicular to their propagation.  The "$\hat{z}$ propagating" wave really moves in both the $\hat{t}$ and $\hat{z}$ directions, as we saw above. The GW does not affect spacetime in those directions.
The GW squishes and stretches spacetime only in the $\hat{x}$ and $\hat{y}$ directions.  The physical property that oscillates is the proper separation between two points in spacetime.
$$\Delta s_{a\rightarrow b} = \int_\vec{a}^\vec{b} \sqrt{g_{\mu\nu}dx^\mu dx^\nu}$$
For a GW propagating through Minkowski space in the $\hat{z}$ direction, the metric is:
$$g_{\mu\nu} \rightarrow \begin{pmatrix}
-1 & 0 & 0 & 0 \\
 0 & 1+h_+ & h_\times & 0 \\
 0 & h_\times & 1-h_+ & 0 \\
 0 & 0 & 0 & 1 \\
\end{pmatrix},
$$
where $h_+$ and $h_\times$ represent the two polarizations.  We can define them like $h_{+/\times} = A_{+/\times}\sin(\omega t - k z)$ for a simple wave, or however you like.
The most straightforward way to see the change in proper separation is to calculate the integral for different separations, e.g. let $a$ be the origin $\vec{a}\rightarrow(0,0,0,0)$ and $\vec{b}\rightarrow(0,\ell,0,0)$.
Or try $\vec{b}\rightarrow(0,0,0,\ell)$ or $\vec{b}\rightarrow\frac{1}{\sqrt{2}}(0,\ell,\ell,0)$.  It helps to make simplifying assumptions which can reduce the integral to
$$\begin{align*}
\Delta s &\approx \sqrt{ g_{\mu\nu} \Delta x^\mu \Delta x^\nu} \\
\Delta x^\mu &= b^\mu - a^\mu
\end{align*}
$$
A: The typical image of transverse gravitational waves is that as the wave travels thru spacetime, space itself stretches perpendicular to the direction of travel ($\hat{k}$).  So space starts out "square", then get stretched vertically, goes back to square, gets stretched horizontally, goes back to square, etc.
You can Google "transverse" "gravitational waves" (with the quotes) and go to their Images section to see many examples of this.  I don't know if I can copy-and-paste an image from a website here due to copyright issues.
