Metric Reconstruction and Second Order Perturbation for Generic Spherically Symmetric spacetime

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.