# 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?

• gravitational waves are really "weak" which justifies the approach of linearisation of Einstein's equations. From this you get something very similar to Maxwell's theory – Phoenix87 Jan 13 '15 at 11:40
• For a comprehensive book on solutions to the Einstein field equations, including gravitational waves, I would recommend 'Exact Solutions to the Einstein Field Equations' (Cambridge University Press). – JamalS Jan 13 '15 at 11:45
• The easiest way to understand this is through the "Weak Field Einstein Equations" (Google this exact phrase), which are treated well in "A First Course in General Relativity" by Schutz and also Sean Carroll's lecture that you can download from here – WetSavannaAnimal Jan 13 '15 at 11:51
• – Qmechanic Jan 13 '15 at 12:47

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
• Yes. You can think of $h_+$ and $h_\times$ as the two transverse polarization directions of a lightwave, but now for a gravitational wave. Both of these polarization directions have the effect of 'stretching' and 'squeezing' the spacetime in the directions perpendicular to its momentum. – JgL Mar 6 '17 at 10:11