I need some little help here

I'm reading about gravitational waves and particularly about gravitational waves described by the linearized non-homogeneus Einstein's equations

\begin{equation} \left( -\frac{\partial^2}{\partial t^2} + \nabla^2 \right)\bar h_{\mu\nu}=-16\pi T_{\mu\nu} \end{equation}

These three classic books says the solution to ths equation is know as "retarded solution", given by

From Schutz \begin{equation} h_{\mu\nu}(t,x^{i})=4\int\frac{T_{\mu\nu}\left( t-R, y^{i} \right)}{R}d^{3}y \end{equation}

From Hartle

\begin{equation} h^{\alpha\beta}(t,\vec x)=4\int d^{3}x \frac{\left[T^{\alpha\beta}\left( t', x' \right)\right]_{ret}}{\left| \vec x - \vec x' \right|} \end{equation}

From Schneider

\begin{equation} h^{\alpha\beta}(t,\vec x)=-\frac{4G}{c^4}\int \frac{ T^{\alpha\beta}\left( t-\frac{\left|y \right |}{c},\vec x + \vec y \right)}{\left | \vec y \right |} d^{3}y \end{equation}

well, in this point i guess the three last equations representing the same solution to the wave equation, so my questions are

  1. This retarded solution, does where it come from? None of those books says where come from

  2. what is the physical situation that describe it? Until I know $T$ is the Tensor Stress-Energy related to the mass that deform the spacetime, I guess that mass is the source of the gravitational waves but if that the case. The gravitational waves eventually will be so far from the source that the mass or T in this case will be zero, or T is related to another mass?

  • 4
    $\begingroup$ Any book on EM should explain how to solve the wave equation with sources. See Jackson, for example. $\endgroup$ – Javier Oct 17 '16 at 18:49

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