Living inside a hedgehog: higher-dimensional solutions that localize gravity

We consider spherically symmetric higher-dimensional solutions of Einsteinʹs equations with a bulk cosmological constant and n transverse dimensions. In contrast to the case of one or two extra dimensions we find no solutions that localize gravity when n⩾3, for strictly local topological defects. We discuss global topological defects that lead to the localization of gravity and estimate the corrections to Newtonʹs law. We show that the introduction of a bulk “hedgehog” magnetic field leads to a regular geometry and localizes gravity on the 3-brane with either a positive, zero or negative bulk cosmological constant. The corrections to Newtonʹs law on the 3-brane are parametrically the same as for the case of one transverse dimension.