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We consider the vector and the Lichnerowicz wave equations on theSchwarzschild spacetime, which correspond to the Maxwell and linearizedEinstein equations in harmonic gauges (or, respectively, in Lorenz and deDonder gauges). After a complete separation of variables, the radial modeequations form complicated systems of coupled linear ODEs. We outline a preciseabstract strategy to decouple these systems into sparse triangular form, wherethe diagonal blocks consist of spin-$s$ scalar Regge-Wheeler equations (forspins $s=0,1,2$). Building on the example of the vector wave equation, which wehave treated previously, we complete a successful implementation of ourstrategy for the Lichnerowicz wave equation. Our results go a step further thanprevious more ad-hoc attempts in the literature by presenting a full andmaximally simplified final triangular form. These results have importantapplications to the quantum field theory of and the classical stabilityanalysis of electromagnetic and gravitational perturbations of theSchwarzschild black hole in harmonic gauges.

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