Massive vector field perturbations on extremal and near-extremal static black holes

摘要

We discuss a new perturbation method to study the dynamics of massive vector fields on extremal and near-extremal static black hole spacetimes. We start with, as our background, a rather generic class of warped-product metrics that consist of an m-dimensional spacetime and an n-dimensional Einstein space, and with respect to the behavior on the latter space, we classify the components of massive vector fields into the vector (axial)- and scalar (polar)-type components. On this generic background spacetime, we show that for the vector-type components the Proca equation reduces to a single homogeneous master equation, whereas the scalar-type components remain coupled. Then, focusing on the case of extremal and nearextremal static black holes in four dimensions, we consider the near-horizon expansion of both the background geometry and massive vector field by a scaling parameter λ with the leading-order geometry of λ → 0 being the so-called near-horizon geometry.We show that on the near-horizon geometry, thanks to its enhanced symmetry, the Proca equation for the scalar-type components also reduces to a set of two mutually decoupled homogeneous wave equations for two scalar variables, plus a coupled equation through which the remaining third variable is determined from one of the first two. Therefore, together with the vector-type master equation for a single variable, we obtain the set of three decoupled master wave equations, each of which governs each of the three independent dynamical degrees of freedom of the massive vector field in four dimensions. We further expand the geometry and massive vector field with respect to λ and show that at each order of λ the Proca equation for the scalar-type components can reduce to a set of two mutually decoupled inhomogeneous wave equations of which the source terms consist only of the lower-order variables, plus one coupled equation that determines the remaining third variable. Therefore, if one solves the master equation