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{WaveEq2} \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.

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

Substitution of the gauge conditions into the Einstein tensor yields the following. The $00$ equation is \[G_{00}=-2\triangle \Psi =0,\] thus, assuming vanishing boundary condition (at infinity or elsewhere), $\Psi=0$. Then the $0\alpha$ equations take the form \[G_{0\alpha}=\tfrac12 \triangle w_\alpha =0,\] and therefore $w_\alpha =0$ as well. Then the $\alpha\beta$ equations are reduced to \[G_{\alpha\beta}=(\delta_{\alpha\beta}\triangle -\partial_\alpha \partial_\beta)\Phi -\square s_{\alpha\beta}=0.\] Taking the trace, we see that \[\triangle \Phi=0,\] which means that $\Phi=0$ and the remaining non-trivial equation is \[\square s_{\alpha\beta}=0.\] Note, that such representation was only possible in vacuum, i.e. for gravitational wave solutions.