# Boundary conditions of relativistic wave solutions?

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 $ct + r$.

Given boundary conditions allow for a unique solution of the wave equation. If you do it your way and immediately guess the retarded Green's function you are fine, but in principle there is an infinite amount of solutions which have to be fixed by boundary conditions. By imposing Sommerfeld conditions, you make sure that only the retarded solution survives, which is the only physical solution: there is no incoming radiation from infinity. $ct+r=constant$ assures that you are actually treating past null infinity (which is not clear from the first condition alone). Furthermore, $r\bar{h}_{\mu\nu}$ and $r\partial_\rho\bar{h}_{\mu\nu}$ have to be bounded in order to assure that the limit at $-\infty$ is well defined.