1+1+2 electromagnetic perturbations on general LRS spacetimes: Regge–Wheeler and Bardeen–Press equations

Abstract

We use the (covariant and gauge-invariant) 1+1+2 formalism developed by Clarkson and Barrett (2003 Class. Quanum Grav. 20 3855–84), and develop new techniques, to decouple electromagnetic (EM) perturbations on arbitrary locally rotationally symmetric (LRS) spacetimes. Ultimately, we derive three decoupled complex equations governing three complex scalars. One of these is a new Regge–Wheeler (RW) equation generalized for LRS spacetimes, whereas the remaining two are new generalizations of the Bardeen–Press (BP) equations. This is achieved by first using linear algebra techniques to rewrite the first-order Maxwell equations in a new complex 1+1+2 form which is conducive to decoupling. This new complex system immediately yields the generalized RW equation, and furthermore, we also derive a decoupled equation governing a newly defined complex EM 2-vector. Subsequently, a further decomposition of the 1+1+2 formalism into a 1+1+1+1 formalism is developed, allowing us to decompose the complex EM 2-vector, and its governing equations, into spin-weighted scalars, giving rise to the generalized BP equations.