On Existence of Self-Tuning Solutions in Static Braneworlds without Singularities

TL;DR

The paper proves a strong no-go theorem: for static braneworlds with broken bulk Lorentz invariance, a non-singular background that satisfies the NEC on both the bulk and the brane and has positive brane energy density cannot exist with a flat 3D brane. The analysis derives explicit NEC inequalities in terms of warp-factor derivatives and shows that, unless the Lorentz symmetry is effectively restored ($a=b$), these conditions drive the solution toward singularities. It also exhibits the Lorentz-invariant RS2 limit as a viable exception, and discusses Lifshitz-like backgrounds and a potential evasion via positive 3D curvature, though the latter is phenomenologically problematic. Overall, the work clarifies the constraints on self-tuning static braneworlds and delineates the boundaries between NEC-satisfying, non-singular configurations and inevitable singularities under a broad class of assumptions.

Abstract

A static self-tuning SO(3)xZ_2 symmetric and translation invariant braneworld setup with flat brane is considered. We discuss the null energy conditions (NEC) for matter on the brane and in the bulk and prove that for the static regular background with broken Lorentz invariance the NEC and positiveness of the total energy density on the brane and NEC in the bulk cannot be satisfied simultaneously. Then we give some examples and elaborate some special cases. For instance, we provide a macroscopic solution for a background with Lifshitz scaling.