Reconstructing the Rippling Geometry around Spinning Black Holes*

The fabric of spacetime around a spinning black hole can ripple in intricate and surprising ways. These ripples are described by the (linearized) Einstein field equations, which are too difficult to tackle head on. In a key breakthrough, Teukolsky showed that the curvature of these ripples, which is encoded in a set of “Weyl scalars”, obeys a much simpler (separable) wave equation. For many purposes, rather than study the ripples themselves, it is sufficient to study their curvature (Weyl scalars). However, in this new era of gravitational-wave astronomy, it is becoming increasingly necessary to find the actual metric perturbation associated with these ripples, i.e., the precise way that they perturb the spacetime geometry.

The problem of determining a metric perturbation from its Weyl scalars is known as metric reconstruction. This problem was long ago solved in principle, but no one has explicitly carried out the complete procedure in full generality until now. We fill this gap by providing a direct connection between the physical degrees of freedom captured by the Weyl scalars and the perturbed spacetime geometry. Our method not only streamlines a complicated mathematical process but also offers a tool that can enhance numerical simulations and potentially aid in extending the perturbation theory of black holes beyond the linear regime.