If you take Einstein's field equations, \begin{equation} R_{\mu\nu}-\tfrac{1}{2}g_{\mu\nu}R = -\kappa T_{\mu\nu}, \end{equation} and you insert the metric \begin{equation} g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}, \end{equation} then you get a field theory in flat spacetime for the propagation of the perturbation $h_{\mu\nu}$. This is 'linearised GR' and it's a first place to start for a description of gravitational waves.
The equations result from the linearisation process are a set of wave equations of the form \begin{equation} \square^{2}\bar{h}_{\mu\nu} = - 2\kappa T_{\mu\nu}. \end{equation} Here, $\bar{h}_{\mu\nu} = h_{\mu\nu} - \tfrac{1}{2}\eta_{\mu\nu}h$ is called the 'trace-reversed' perturbation, and $h=\eta^{\mu\nu}h_{\mu\nu}$ is the trace.
If you want to solve this animal, then you can follow the analogous problem in electromagnetism and use a retarded Green's function. You can guess the form of the Green's function and with a little refinement; this is how I have solved it.
However, I recently read that one can solve this equation without guesswork. Apparently, you have to impose the boundary conditions, that \begin{equation} \displaystyle\lim_{t\to-\infty}\left[\frac{\partial}{\partial r} + \frac{\partial}{c\partial t}\right]\,r\bar{h}_{\mu\nu}=0 \end{equation} where the limit is taken along any surface $ct + r = constant$, together with the condition that $r\,\bar{h}_{\mu\nu}$ and $r \partial_{\rho}\bar{h}_{\mu\nu}$ be bounded by this limit. The book I saw this in said that the physical meaning of this is that there's no incoming radiation from past null infinity.
What am I looking for?
- What exactly are these three conditions telling us,
- Why these conditions have the form that they do, and
- What is means that a surface has constant $ct + r$.
