Abstract
In this short note we shall demonstrate that given a smooth solution to the linearised Einstein equations on Schwarzschild which is supported on the 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 Lorenz gauge. Here the Maxwell current is constructed from those gauge-invariant combinations of the components of 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. We in addition outline how one can apply this result to establish a decay statement for solutions to the linearised Einstein equations on Schwarzschild that are in a transverse and traceless gauge.
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