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?
General Relativity – Gravitational Wave Solutions to the Einstein Field Equations
general-relativitygravitational-wavesgravitylinearized-theory
Related Solutions
In this answer, we take the point of view that the GEM equations are not a first principle by themselves but can only be justified via an appropriate limit (to be determined) of the linearized EFE$^1$ in 3+1D $$ \begin{align} \kappa T^{\mu\nu}~\stackrel{\text{EFE}}{=}~& G^{\mu\nu}\cr ~=~&-\frac{1}{2}\left(\Box \bar{h}^{\mu\nu} + \eta^{\mu\nu} \partial_{\rho}\partial_{\sigma} \bar{h}^{\rho\sigma} - \partial^{\mu}\partial_{\rho} \bar{h}^{\rho\nu} - \partial^{\nu}\partial_{\rho} \bar{h}^{\rho\mu} \right) ,\cr \kappa~\equiv~&\frac{8\pi G}{c^4}, \end{align}\tag{1}$$ where $$ \begin{align}g_{\mu\nu}~&=~\eta_{\mu\nu}+h_{\mu\nu}, \cr \bar{h}_{\mu\nu}~:=~h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h \qquad&\Leftrightarrow\qquad h_{\mu\nu}~=~\bar{h}_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}\bar{h}. \end{align} \tag{2}$$
There may be other approaches that we are unaware of, but reading Ref. 1, the pertinent GEM limit seems to be of E&M static nature, thereby seemingly excluding gravitational waves/radiation.
Concretely, the matter is assumed to be dust:$^2$ $$ T^{\mu 0}~=~cj^{\mu}, \qquad j^{\mu}~=~\begin{bmatrix} c\rho \cr {\bf J} \end{bmatrix}, \qquad T^{ij}~=~{\cal O}(c^0). \tag{3}$$
The only way to systematically implement a dominanting temporal sector/static limit seems to be by going to the Lorenz gauge$^3$ $$\partial_{\mu} \bar{h}^{\mu\nu} ~=~0. \tag{4}$$ Then the linearized EFE (1) simplifies to $$ G^{\mu\nu}~=~-\frac{1}{2}\Box \bar{h}^{\mu\nu}~=~\kappa T^{\mu\nu}. \tag{5}$$
In our convention, the GEM ansatz reads$^5$ $$\begin{align} A^{\mu}~=~&\begin{bmatrix} \phi/c \cr {\bf A} \end{bmatrix}, \qquad\bar{h}^{ij}~=~{\cal O}(c^{-4}),\cr -\frac{1}{4}\bar{h}^{\mu\nu} ~=~&\begin{bmatrix} \phi/c^2 & {\bf A}^T /c\cr {\bf A}/c & {\cal O}(c^{-4})\end{bmatrix}_{4\times 4}\cr ~\Updownarrow~& \cr -h^{\mu\nu} ~=~&\begin{bmatrix} 2\phi/c^2 & 4{\bf A}^T/c \cr 4{\bf A}/c & (2\phi/c^2){\bf 1}_{3\times 3}\end{bmatrix}_{4\times 4} \cr ~\Updownarrow~& \cr g_{\mu\nu} ~=~&\begin{bmatrix} -1-2\phi/c^2 & 4{\bf A}^T/c \cr 4{\bf A}/c & (1-2\phi/c^2){\bf 1}_{3\times 3}\end{bmatrix}_{4\times 4}. \end{align}\tag{6}$$
The gravitational Lorenz gauge (4) corresponds to the Lorenz gauge condition $$ c^{-2}\partial_t\phi + \nabla\cdot {\bf A}~\equiv~ \partial_{\mu}A^{\mu}~=~0 \tag{7}$$ and the "electrostatic limit"$^4$ $$ \partial_t {\bf A}~=~{\cal O}(c^{-2}).\tag{8}$$
Next define the field strength $$\begin{align} F_{\mu\nu}~:=~&\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}, \cr -{\bf E}~:=~&{\bf \nabla} \phi+\partial_t{\bf A}, \cr {\bf B}~:=~&{\bf \nabla}\times {\bf A}.\end{align} \tag{9} $$ Then the tempotemporal & the spatiotemporal sectors of the linearized EFE (1) become the gravitational Maxwell equations with sources $$ \partial_{\mu} F^{\mu\nu}~=~\frac{4\pi G}{c}j^{\mu}. \tag{10} $$ Note that the gravitational (electric) field ${\bf E}$ should be inwards (outwards) for a positive mass (charge), respectively. For this reason, in this answer/Wikipedia, the GEM equations (10) and the Maxwell equations have opposite$^5$ signs.
Interestingly, a gravitational gauge transformation of the form $$\begin{align}\delta h_{\mu\nu}~=~&\partial_{\mu}\varepsilon_{\nu}+(\mu\leftrightarrow\nu), \cr \varepsilon_{\nu}~:=~&c^{-1}\delta^0_{\nu}~\varepsilon, \end{align}\tag{11} $$ leads to $$\delta h~=~-2c^{-1}\partial_0\varepsilon \tag{12}$$ and thereby to the usual gauge transformations $$\delta A_{\mu}~=~\partial_{\mu}\varepsilon.\tag{13}$$ Such gauge transformations (13) preserve the GEM eqs. (10) but violate the GEM ansatz $\bar{h}^{ij}={\cal O}(c^{-4})$ unless $$\partial_t\varepsilon~=~{\cal O}(c^{-2}).\tag{14}$$ In conclusion, the Lorenz gauge condition (7) is not necessary, but we seem to be stuck with the "electrostatic limit" (8).
References:
- B. Mashhoon, Gravitoelectromagnetism: A Brief Review, arXiv:gr-qc/0311030.
--
$^1$ In this answer we use Minkowski sign convention $(-,+,+,+)$ and work in the SI-system. Space-indices $i,j,\ldots \in\{1,2,3\}$ are Roman letters, while spacetime indices $\mu,\nu,\ldots \in\{0,1,2,3\}$ are Greek letters.
$^2$ Warning: The $j^{\mu}$ current (3) does not transform covariantly under Lorentz boosts. The non-inertial frames that Wikipedia mentions are presumably because the $g_{\mu\nu}$-metric (2) is non-Minkowskian.
$^3$ The Lorenz gauge (4) is the linearized de Donder/harmonic gauge $$ \partial_{\mu}(\sqrt{|g|} g^{\mu\nu})~=~0.\tag{15}$$
$^4$ We unconventionally call eq. (8) the "electrostatic limit" since the term $\partial_t{\bf A}$ enters the definition (9) of ${\bf E}$.
$^5$ Warning: In Mashhoon (Ref. 1) the GEM equations (10) and the Maxwell equations have the same sign. For comparison, in this Phys.SE answer $$\phi~=~-\phi^{\text{Mashhoon}}, \qquad {\bf E}~=~-{\bf E}^{\text{Mashhoon}}, $$ $${\bf A}~=~-\frac{1}{2c}{\bf A}^{\text{Mashhoon}}, \qquad {\bf B}~=~-\frac{1}{2c}{\bf B}^{\text{Mashhoon}}.\tag{16}$$
If not, what are the other applications?
Calculating the relativistic precession of Mercury, for one. This post-diction was one of the key things that helped with the rapid acceptance of general relativity.
Modeling GPS, and calculating the orbits of LAGEOS and Gravity Probe B, for another. A full-blown general relativistic formulation works quite nicely on (and is absolutely essential for) black holes and neutron stars precisely because gravity about those extremely massive objects is simple. Earth's gravity field isn't so nice and simple. It's rather lumpy compared to a neutron star. One of the more recent models of the Earth's gravity field, Earth Gravity Model 2008 (EGM2008), is a 2159x2159 spherical harmonics model. How are you going to handle that with general relativity? The answer is to linearize the field equations.
Modeling the behavior of the solar system, for yet another. All three of the leading models of planetary ephemerides use a first order post-Newtonian approximation of gravity. (But apparently they're starting to wonder if they need to step beyond that. To second order.)
One last use: "weigh" the Earth. See my answer to the question "How is the mass of the Earth determined?" at the earth science stackexchange sister site.
Best Answer
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.