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

Abstract

In this short note we shall demonstrate that given a smooth solution γ to the linearised Einstein equations on Schwarzschild which is supported on the l ≥ 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 γ 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.

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

“…Following [5], this gauge (there mostly called Lorenz ) has found important applications in the gravitational self-force literature [7]. More recently, the work [26] has used harmonic gauge in the proof of global non-linear stability of Kerr-de Sitter black holes, while in [29,30] 2 and [27,28] the vector field method was used, directly in real-space, to study stability and decay of perturbations of Schwarzschild, with [31] an addendum that implicitly considers an upper triangular decoupling of the Lichnerowicz wave equation under the transverse-traceless conditions, ∇ ν p µν and p λ λ = 0, without referring to or comparing with the fully diagonal decoupling of the same system by [9].…”