Algebraically special perturbations of the Schwarzschild solution in higher dimensions

TL;DR

This work analyzes algebraically special perturbations of the generalized Schwarzschild solution in arbitrary dimensions using the Kodama–Ishibashi master framework. A central finding is that, for d>4, regular perturbations compatible with algebraic speciality are strictly limited to parameter variations of the background (mass and K^{d-2} moduli) or simple angular/linear momentum additions, with no time-dependent branches; in contrast, four dimensions admit infinite time-dependent families generalizing Couch–Newman. The analysis hinges on the gauge-invariant condition δΩ_{ij}=0 and shows how this constraint interacts with tensor, vector, and scalar master equations to forbid new solutions in higher dimensions. The results have implications for the reconstructability of metric perturbations from δΩ_{ij} and underscore the relative scarcity of higher-dimensional algebraically special perturbations, while highlighting a loop-hole tied to smoothness assumptions on K^{d-2}.

Abstract

We study algebraically special perturbations of a generalized Schwarzschild solution in any number of dimensions. There are two motivations. First, to learn whether there exist interesting higher-dimensional algebraically special solutions beyond the known ones. Second, algebraically special perturbations present an obstruction to the unique reconstruction of general metric perturbations from gauge-invariant variables analogous to the Teukolsky scalars and it is desirable to know the extent of this non-uniqueness. In four dimensions, our results generalize those of Couch and Newman, who found infinite families of time-dependent algebraically special perturbations. In higher dimensions, we find that the only regular algebraically special perturbations are those corresponding to deformations within the Myers-Perry family. Our results are relevant for several inequivalent definitions of "algebraically special".