Hierarchical formulation of the self-gravitating, n-dimensional, charged scalar field in spherical symmetry in affine null formalism

Abstract

We develop an affine-null characteristic formulation of the Einstein-Maxwell system coupled to a charged complex scalar field in $n$-dimensional spherical symmetry. By introducing suitable auxiliary variables, the main field equations are cast into a hierarchical system of radial hypersurface equations, supplemented by a transport equation for the scalar field. We discuss the associated characteristic initial-boundary value problem for asymptotic, vertex and null-boundary configurations, and derive the corresponding asymptotic quantities and balance laws. As a consistency check, we recover the scalar-free Reissner-Nordström-Tangherlini family, including both the non-extremal and extremal branches, directly from the hierarchy. The resulting framework provides a systematic setting for the study of charged scalar dynamics and exact black-hole solutions in higher-dimensional affine-null coordinates.