Difference between revisions of "Wave equation"

Using the general expression for the linearized Einstein tensor, we see that it takes especially simple form \begin{equation} G_{\mu\nu}=\square\bar{h}_{\mu\nu}, \end{equation} thus the Einstein's equations are reduced to the wave equation \begin{equation} \label{WaveEq} \square \bar{h}_{\mu\nu}=\frac{16\pi G}{c^4}T_{\mu\nu} \end{equation} with the stress-energy tensor as the source (but note $16\pi$ in the numerator). This simple form makes the Lorenz gauge the preferred choice for studying most aspects of gravitational waves.

\begin{enumerate} \item Using the expression for variation of the metric perturbation under gauge transformations UNIQ-MathJax8-QINU, in the new frame we get \begin{align} 0&=\partialrtial_{\mu}\bar{h'}^{\mu}_{\nu} =\partialrtial_{\mu} \big[{h'}^{\mu}_{\nu} -\tfrac12 h' \delta^\mu_\nu\big]=\\ &=\partialrtial_{\mu}{h'}^{\mu}_{\nu} -\tfrac12 \partialrtial_{\nu}h' =\partialrtial_{\mu}({h}^{\mu}_{\nu} -\partialrtial^{\mu}\xi_{\nu}-\partialrtial_{\nu}\xi^{\mu}) -\tfrac12 \partialrtial_{\nu}(h-2\partialrtial_{\mu}\xi^{\mu})=\\ &=\partialrtial_{\mu}h^{\mu}_{\nu}-\tfrac12 \partialrtial_{\nu}h -\square \xi_\nu\\ &=\partialrtial_{\mu}\bar{h}^{\mu}_{\nu}-\square \xi_\nu, \end{align} thus \[\square \xi_{\mu}=\partialrtial_{\nu}\bar{h}^{\nu}_{\mu}.\] \item If the Lorenz gauge is already fixed, it will be preserved by any additional coordinate transformation with $\xi^\mu$ satisfying \[\square \xi^{\mu}(x)=0.\] \end{enumerate}