Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations

Abstract

We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form complicated systems of coupled linear ODEs. We outline a precise abstract strategy to decouple these systems into a sparse triangular form, where the diagonal blocks consist of spin-𝑠 scalar Regge-Wheeler equations (for spins 𝑠 = 0,1,2). Building on the example of the vector wave equation, which we have treated previously, we complete a successful implementation of our strategy for the Lichnerowicz wave equation. Our results go a step further than previous, more ad-hoc attempts in the literature by presenting a full and maximally simplified final triangular form. These results have important applications to the quantum field theory of and the classical stability analysis of electromagnetic and gravitational perturbations of the Schwarzschild black hole in harmonic gauges.