The linearised conformal Einstein field equations: Petrov-type D backgrounds
Abstract
Although gravitational wave effects can be studied using the full non-linear Einstein field equations, linear perturbation theory remains an active area of research, e.g. black hole perturbation theory, quasinormal modes, post-Newtonian and post-Minkowskian expansions. With this motivation in mind, in this talk we derive a conformal Teukolsky equation obtained by linearising the CEFEs on a Petrov type D background. This calculation is timely given the growing relevance of the hyperboloidal approach in black-hole perturbation theory, where conformal compactification is introduced at the level of an already linearised effective wave equation. However, here, instead of starting from a reduced linear wave equation, we derive the conformal Teukolsky equation directly from the (non-linear) CEFEs, a formulation in which the conformal factor is, in general, a dynamical variable. We discuss how, in the non-linear equations, there is a coupling between the conformal factor and the curvature; however, when linearised around a Petrov-type D background, the conformal factor decouples from the equations governing the dynamics of the $\phi_0$ and $\phi_4$ components of the rescaled Weyl tensor. Although the conformal Teukolsky equation is not tied to any particular coordinate system, it is shown that, when using hyperboloidal coordinates, it reduces to the equation used in the literature obtained through a coordinate-based, effective approach. Besides exploring the use of hyperboloidal coordinates, we also particularise the conformal Teukolsky equation to the case of a conformal representation of the Kerr spacetime adapted to the geometry near spatial and null infinity.
