Gravitational wave solutions to the Einstein field equations It is well known that general relativity predicts gravitational waves, but I would like to know how. What solution(s) to the Einstein field equations yield something which can be interpreted as a wave(-like) phenomenon?
 A: Typically solving the full Einstein equations is rather difficult, so to calculate stuff about gravitational waves people typically use the following approximation
$$ g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} $$
That is, they approximate the full metric $g_{\mu\nu}$ as some perturbation of flat Minkowski spacetime. This approximation is called 'linearized gravity', as one uses a linear approximation of the full Einstein equations to calculate the dynamics of these small perturbations $h_{\mu\nu}$. See e.g. chapter 7 in Carroll for this. Working out the Einstein equations in this regime one typically finds as a solution for $h_{\mu\nu}$ of the following form for a `gravitational' wave propagating in the $z$ direction:
$$ h_{\mu\nu} = \begin{pmatrix}
0 & 0 & 0 & 0 \\
0 & h_+ & h_\times & 0 \\
0 & h_\times & h_+ & 0 \\
0 & 0 & 0 & 0 \\
\end{pmatrix}$$
Since the metric gives us the distance between two points $ds^2 = g_{\mu\nu}dx^\mu dx^\nu$ we see that such a gravitational wave really causes behavior that in any experiment would be identical to stretching the $x$ and $y$ directions of our space-time.
Just like an electromagnetic wave has two polarization directions perpendicular to its momentum, a gravitational wave moving in the $z$ direction also has two polarizations, here indicated as $h_+$ and $h_\times$. The $h_+$ polarization of the gravitational wave (moving in the $z$ direction) stretches and squeezes the spacetime in the $x$ and $y$ directions. The $h_\times$ polarization stretches and squeezes the spacetime diagonally in the $x$ and $y$ directions

This is a linear approximation to the Einstein equations which means that we are neglecting the energy carried by the gravitational wave that would itself distort the space-time curvature and cause more gravitational waves. When we neglect that gravitational waves actually carry energy themselves, react with each other and create more gravitational waves. This is a legitimate approximation as gravitational waves with a small amplitude (and of relatively large wavelengths) carry very little energy indeed. We find that General Relativity in this 'linear regime' looks very much like the Maxwell equations (i.e. electromagnetic waves also pass right through one another without interacting, we say their equations are 'linear').
Gravitational waves with large amplitudes and very short wavelengths (high frequencies) carry a lot of energy, and we can no longer neglect the interactions between gravitational waves of this sort (or the interactions between them and the additional gravitational waves they could emit themselves). This self coupling of gravitational waves, that becomes more and more important at higher energies, makes the Einstein equations so hard to solve and is what gives rise to all sorts of complications, both in classical GR as the quantization of this theory. However, studying one gravitational wave on its own in the limit that it has a long wavelength (and therefore carries relatively little energy) is a very legitimate approximation of the theory.
