On the link between the Maxwell and linearised Einstein equations on Schwarzschild

In this short note we shall demonstrate that given a smooth solution $\gamma$ to the linearised Einstein equations on Schwarzschild which is supported on the $l\ge 2$ spherical harmonics and expressed relative to a transverse and traceless gauge then one can construct from it a smooth solution to the sourced Maxwell equations expressed relative to a generalised Lorentz gauge. Here the Maxwell current is constructed from those gauge-invariant combinations of the components of $\gamma$ which are determined by solutions to the Regge–Wheeler and Zerilli equations. The result thus provides an elegant link between the spin 1 and spin 2 equations on Schwarzschild.