Metric Reconstruction and Second Order Perturbation for Generic Spherically Symmetric spacetime

TL;DR

This work develops a perturbation framework for general asymptotically flat, spherically symmetric spacetimes to enable beyond-GR analyses of ringdown signals, including quadratic (second-order) effects. It extends the Teukolsky formalism to a modified, decoupled equation with generalized source terms and demonstrates a metric reconstruction method in t r-symmetric spacetimes that avoids the Hertz potential. By fixing gauge degrees of freedom order-by-order, the authors show that second-order perturbations retain a Teukolsky-like structure with GR-like and beyond-GR source terms, enabling QQNM studies. The approach provides a practical path to test gravity theories with future gravitational-wave observations, while acknowledging limitations for certain modified gravity models and outlining directions such as quantum-inspired black hole models.

Abstract

Higher-order perturbations during the ringdown phase are essential for testing gravitational theories. This requires a perturbation framework that extends beyond General Relativity, as well as an appropriate method for reconstructing the spacetime metric. In this work, we address these challenges within the context of general spherically symmetric spacetimes. We introduce a modified Teukolsky equation for perturbative calculations in asymptotically flat, spherically symmetric spacetimes. The metric reconstruction method, which does not rely on the Hertz potential, is extended to $tr$-symmetric spacetime, allowing for the calculation of metric components under specific gauge conditions. Additionally, we present a second-order perturbation theory applicable to generic spherically symmetric spacetimes.