Scalar field in a minimally coupled brane world: no-hair and other no-go theorems

TL;DR

This paper investigates scalar fields in a simplified brane-world model, extending no-hair theorems from general relativity and identifying possible global structures and exotic objects like wormholes under specific conditions.

Contribution

It extends no-hair and no-go theorems to minimally coupled brane-world scenarios with scalar fields, clarifying the possible causal structures and exotic solutions.

Findings

Global causal structures match those in GR with a cosmological constant

Proved no-hair theorem for scalar fields with non-negative potential

Allowed traversable wormholes with high matter densities in strong fields

Abstract

In the brane-world framework, we consider static, spherically symmetric configurations of a scalar field with the Lagrangian $(\d\phi)^2/2 - V(\phi)$, confined on the brane. We use the 4D Einstein equations on the brane obtained by Shiromizu et al., containing the usual stress tensor $T\mN$, the tensor $\Pi\mN$, quadratic in $T\mN$, and $E\mN$ describing interaction with the bulk. For models under study, the tensor $\Pi\mN$ has zero divergence, so we can consider a "minimally coupled" brane with $E\mN = 0$, whose 4D gravity is decoupled from the bulk geometry. Assuming $E\mN =0$, we try to extend to brane worlds some theorems valid for scalar fields in general relativity (GR). Thus, the list of possible global causal structures in all models under consideration is shown to be the same as is known for vacuum with a $Lambda$ term in GR: Minkowski, Schwarzschild, (A)dS and Schwarzschild-(A)dS. A no-hair theorem, saying that, given a potential $V\geq 0$, asymptotically flat black holes cannot have nontrivial external scalar fields, is proved under certain restrictions. Some objects, forbidden in GR, are allowed on the brane, e.g, traversable wormholes supported by a scalar field, but only at the expense of enormous matter densities in the strong field region.