Second-order perturbations of the Schwarzschild spacetime: practical, covariant and gauge-invariant formalisms

Understand

High-accuracy gravitational-wave modeling demands going beyond linear, first-order perturbation theory.

• Particularly motivated by the need for second-order perturbative models of extreme-mass-ratio inspirals and black hole ringdowns, we present practical spherical-harmonic decompositions of the Einstein equation, Regge-Wheeler-Zerilli equations, and Teukolsky equation at second perturbative order in a Schwarzschild background.
• Our formulations are covariant on the $t$--$r$ plane and on the two-sphere, and we express the field equations in terms of gauge-invariant metric perturbations.
• In a companion Mathematica package, PerturbationEquations, we provide these invariant formulas as well as the analogous formulas in terms of raw, gauge-dependent metric perturbations.