What are the differential equations that model a self-propagating gravitational wave in space-time? Light is a self-propagating wave, but it's very complicated.
Imagine, if you will, a wave in space-time that by assumption was self-propagating like light, except that it was a gravitational wave.
What are the differential equations and boundary conditions that would govern the transfer of a wave between two absorption points?
I'm familiar with differential equations, but not the specifics of differential geometry that might better address this.
 A: Wave equation for Gravitational wave(GW) comes from Einstein field equation in general relativity, with linearized approximation. Einstein equation is originally non-linear DE, but we can approximate it become linear. Set metric:
$$g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \; |h_{\mu\nu}| \ll 1$$
Calculate Christoffel symbol, Riemann curvature tensor, and so on. These things consist of many derivative terms of metric, but we only consider 1st-order $\mathcal{O}(h_{\mu\nu})$ terms. Then, Einstein equation will be reduced in linear DE. (I skip many details of process to derive wave equation from Einstein equation)
Small perturbation of metric $h_{\mu\nu}$ can be decomposed in each component $h_{00}, \; h_{0i}, \; h_{ij}$. For simplicity, we will use only spatial part with transverse gauge. Also, assume it is vacuum case. Then, DE is reduced like that:
$$\square h_{\mu\nu} = 0 $$
where $h_{\mu\nu}$ satisfies $h_{0 \nu} = 0$ (purely spatial), $ \eta^{\mu\nu} h_{\mu\nu} = 0$ (tracelss), $\partial_{\mu} h^{\mu\nu} = 0$ (transverse).
General solution of this DE is plane wave.
$$h_{\mu\nu} = C_{\mu\nu} e^{i k_{\sigma} x^{\sigma}}$$
and $C_{\mu \nu}$ will have such form. (assume propagating to $z$ direction)
$$C_{\mu\nu} = \begin{pmatrix} 
0 & 0 & 0 & 0 \\
0 & C_1 & C_2 & 0 \\
0 & C_2 & -C_1 & 0 \\
0 & 0 & 0 & 0
\end{pmatrix} $$
If we recover inhomogenous term in RHS,
$$\square h_{\mu\nu} \simeq 8 \pi G T_{\mu\nu} $$
then solution can be expressed with Green function and retarded time.
$$h_{\mu\nu}(t, \vec{x}) \simeq 8 \pi G \int \frac{1}{4\pi |\vec{x}-\vec{y}|} T_{\mu\nu}(t',\vec{y}) d^3 y $$
where $t = t' + |\vec{x}-\vec{y}|$
$\textbf{Edit}$
Here is a brief process of linearization:
Terms in Christoffel symbol are replaced to $h_{\mu\nu}$ instead of $g_{\mu\nu}$.
$$\Gamma^{\rho} _{\mu \nu} =  \frac{1}{2} \eta^{\rho \lambda} (\partial_{\mu} h_{\nu \lambda} + \partial_{\nu} h_{\mu \lambda} - \partial_{\lambda} h_{\mu\nu}  ) $$
$O((h_{\mu\nu})^2)$ order terms in Riemann curvature tensor are neglected. Also, Ricci tensor $R_{\mu\nu}$ and Ricci scalar $R$ have linear forms, too.
$$R_{\mu\nu\rho\sigma} = \eta_{\mu\lambda} \partial_{[\rho,} \Gamma^{\lambda}_{\sigma],\nu} + O((h_{\mu\nu})^2)  $$
Now, put them all to Einstein equation, then linearized form is yieleded.
$$ R_{\mu\nu} - \frac{1}{2} Rg_{\mu\nu}
= 8\pi G T_{\mu\nu} $$
$$\frac{1}{2} (\partial_{\sigma}\partial_{(\nu,} h^{\sigma} \; _{\mu)} -\partial_{\mu} \partial_{\nu} h - \square h_{\mu\nu} - \eta_{\mu\nu} \partial_{\rho}\partial_{\sigma} h^{\rho \lambda} + \eta_{\mu\nu} \square h   )  = 8\pi G T_{\mu\nu}  $$
Some redudant terms can be removed with transverse gauge assumption.
A: Light as a self-propagating wave in vacuum is governed by the free Maxwell equations:
$$\nabla^aF_{ab}=0$$The electric and magnetic fields which shows the oscillatory behavior are actually the components of the Maxwell tensor $F_{ab}$. One could ask - what are the analogues of $F_{ab}$ in GR? Note that one can express the vacuum Einstein's field equations $R_{ab}=0$ as divergence of Weyl tensor:
$$\nabla^aC_{abcd}=0$$ which looks quite similar to free Maxwell equations. There are other representations of these vacuum equations - for instance in 2-spinor formalism one can define a free zero rest mass (z.r.m.) field equation for a general spin n/2, for example see here. However, unlike free Maxwell equation, the vacuum Einstein's equations $\nabla^aC_{abcd}=0$ are not exactly linear in $C_{abcd}$, since both the connection $\nabla^a$ and $C^{abcd}$ depends on metric $g_{ab}$. Solutions of such non-linear differential equations are indeed hard to find. Some solutions for exact plane waves and spherical waves have been discussed here, here etc. If we are looking at small perturbations on background Minkowski-metric, then we can essentially linearize the vacuum equations as $$\partial^aK_{abcd}=0$$where $K_{abcd}$ is the linearized Riemann curvature and is traceless (like the Weyl curvature). $K_{abcd}$ is invariant under gauge transformation of linearized metric only when we consider perturbations about Minkowski-metric.
A: If you're interested in the full non-linear system, then the differential equations you need to solve are just Einstein's equations in vacuum ($R_{\mu \nu} = 0$).  You'll probably have to learn some differential geometry to go further than this.  In particular, writing this out in terms of a set of coordinates is a subtle issue, because in the context of differential geometry the very notion of coordinates is different than in most other fields of physics.
That said, there has been some work done on exact (non-linearized) plane-wave solutions.  A brief description of one such solution can be found in Sections 35.10–12 of Misner, Thorne, & Wheeler's Gravitation.  A more recent review of such solutions can also be found in Chapter 24 of Exact Solutions of Einstein's Field Equations by Stephani et al., though you'll definitely need some familiarity with differential geometry to absorb what's in there.
