Living Inside a Hedgehog: Higher-dimensional Solutions that Localize Gravity

TL;DR

This work analyzes gravity localization on a 3-brane in D = n+4 dimensions with a bulk cosmological constant, exploring strictly local defects, global hedgehog defects, and bulk p-form fields. It shows that strictly local defects cannot localize gravity for $n \ge 3$, motivating global defects or bulk gauge fields as mechanisms to trap gravity on the brane. Global hedgehog defects can produce localized gravity with an exponential warp factor, but may introduce singularities at infinity; the associated Newtonian corrections scale as $1/(cr)^{n+1}$. In contrast, certain bulk p-form fields yield regular, gravity-localizing solutions with constant γ and corrections to Newton's law that resemble the standard RS2-like behavior, independent of $n$ for $n \ge 3$. Overall, the paper identifies viable higher-dimensional pathways to 3-brane gravity localization and clarifies the role of bulk matter content in shaping the brane-world Newtonian potential.

Abstract

We consider spherically symmetric higher-dimensional solutions of Einstein's 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 $n\geq 3$, for strictly local topological defects. We discuss global topological defects that lead to the localization of gravity and estimate the corrections to Newton's 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 Newton's law on the 3-brane are parametrically the same as for the case of one transverse dimension.

Figures (1)

• Figure 1: The $n=3$ solution for $Q^2/\gamma$ (dashed line) and $Q^2 c^2$ (solid line), as a function of $Q^2 \Lambda_7/M_7^5$. Only the branches with $c^2 >0$ and $\gamma>0$ lead to solutions that localize gravity.