# Calculation of interaction of gravitational waves with a black hole

We have a well understood interacting electromagnetic system, an electromagnetic wave interacting with an atom. We use perturbation theory to calculate what happens in such a system. The result is of course the phenomenon of stimulated emission, which has an important application in LASERs.

My question: Is there any way to do the gravitaional counter-part of the problem, that is interaction of a black hole with a gravitational wave(well, I agree that the correspondense is approximate of course), may be using AdS/CFT ?

• Are you asking about "scattering" of a gravitational wave off a BH? May 26, 2022 at 10:34
• that is why i described the "corresponding" scenario in electromagnetism. When we say an e.m. wave scatter off an atom, (do?) we get the stimulated emission from the atom May 26, 2022 at 10:39
• May 26, 2022 at 14:19

You can calculate the propagation of gravitational waves on a background metric using pertubation theory in first order. You start with a pertubation of the metric tensor given by: $$\begin{equation} \widetilde{g}_{\mu\nu} =g_{\mu\nu}+\delta g_{\mu\nu} \qquad\text{with}\qquad |\delta g_{\mu\nu}|<1 \end{equation}$$ $$\begin{equation} \widetilde{g}^{\mu\nu} =g^{\mu\nu}-\delta g^{\mu\nu} \qquad\text{with}\qquad |\delta g_{\mu\nu}|>1. \end{equation}$$ The different signs were explained by me here already. You can now calculate the Christoffel symbols, the Riemann curvature tensor and the Einstein tensor only in first order of the pertubation.
In a vacuum, we get the equation $$G_{\mu\nu}[g]=0$$ (equivalent to $$R_{\mu\nu}[g]=0$$) for the background metric and $$\delta G_{\mu\nu}=0$$ for the gravitational waves out of the field equations. Using a corresponding pertubation tensor (similar to the correspondence between the Ricci and Einstein tensor): $$\begin{equation} \delta g_{\mu\nu}' =\delta g_{\mu\nu}-\frac{1}{2}(g^{\kappa\lambda}\delta g_{\kappa\lambda})g_{\mu\nu}, \end{equation}$$ $$\begin{equation} \delta g_{\mu\nu} =\delta g_{\mu\nu}'-\frac{1}{2}(g^{\kappa\lambda}\delta g_{\kappa\lambda}')g_{\mu\nu}, \end{equation}$$ the equation describing the scattering of gravitational waves is given by: $$\begin{equation} \square\delta g_{\mu\nu}' +2R_{\kappa\mu\lambda\nu}\delta{g^{\kappa\lambda}}'=0. \end{equation}$$ For the Schwarzschild metric in particular using this equation results in the Regge-Wheeler equation.