GR: What is the polarization vector of a gravitational wave? I am learning about gravitational waves. I would like to understand what we mean by a polarization vector of gravitational waves. How are they defined?
 A: The Einstein field equation for a metric $g_{mu\nu}~=~\eta_{\mu\nu}$ $+~h_{mu\nu}$, for very small $h_{mu\nu}$ become linearized. For a gravity wave we take the traceless metric $\tilde h_{\mu\nu}~=~h_{\mu\nu}~-~Tr(h)\eta_{\mu\nu}$ obeys the equation
$$
\square\tilde h_{\mu\nu}~=~0.
$$
We take the traceless components in the matrix
$$
\tilde h_{\mu\nu}~=~\left(\begin{matrix}0&0&0&0\\0&h_{++}&h_{\times\times}&0\\
0 & h_{\times\times} & -h_{++} & 0\\0&0&0&0\end{matrix}\right).
$$
where the components obey the Laplace equation with $\square h_{++}~=~0$ and $\square h_{\times\times}~=~0$. These are equations for the two polarization directions of the gravitational wave. 
This is a helicity $2$ wave corresponding to two polarizations. A weak gravity wave is in some ways similar to an entangled state of two photons, or diphoton. This is one reason the graviton is thought to have spin $=~2$.
This carries over for strong gravitational waves as well. The solution set for gravtational radiation are type N solutions, which have no longitudinal component and so the gravitational wave is massless, or equivalently the graviton is massless. Other solutions such as type II and III physically correspond to gravitational radiation close to the source and partially bound to itself. In these more complicated situations there is a sort of self-mass. However, for the far field configuration of type N solutions the wave has two transverse degrees of freedom and no longitudinal degree of freedom.
A: You typically talk about a polarization tensor, which is the perturbation
$$H_{ij}=\left(\begin{matrix} h_+ & h_\times & 0 \\
h_\times & -h_+ & 0 \\
0 & 0 & 1\end{matrix}\right)$$
for a wave traveling in the $\hat{z}$-direction.
You can use perturbation theory to assume this is a small perturbation to the background metric $g_{\mu\nu}$ which is often taken to be flat Minkowski space.
