Scalar fields in multidimensional gravity. No-hair and other no-go theorems

TL;DR

This paper extends no-hair and no-go theorems for scalar fields in multidimensional gravity, showing constraints on black holes, particlelike solutions, and wormholes in higher-dimensional models with various internal spaces.

Contribution

It generalizes known four-dimensional results to higher dimensions, including multiple internal Einstein spaces and arbitrary potentials, establishing new nonexistence theorems.

Findings

Asymptotically flat black holes cannot have scalar hair if V≥0

No particlelike solutions exist for V≥0

Wormholes are non-existent under broad conditions

Abstract

Global properties of static, spherically symmetric configurations of scalar fields of sigma-model type with arbitrary potentials are studied in $D$ dimensions, including space-times containing multiple internal factor spaces. The latter are assumed to be Einstein spaces, not necessarily Ricci-flat, and the potential $V$ includes contributions from their curvatures. The following results generalize those known in four dimensions: (A) a no-hair theorem: in case $V\geq 0$, an asymptotically flat black hole cannot have varying scalar fields or moduli fields outside the event horizon; (B) nonexistence of particlelike solutions in models with $V\geq 0$; (C) nonexistence of wormholes under very general conditions; (D) a restriction on possible global causal structures (represented by Carter-Penrose diagrams). The list of structures in all models under consideration is the same as is known for vacuum with a cosmological constant in general relativity: Minkowski (or AdS), Schwarzschild, de Sitter and Schwarzschild--de Sitter, and horizons which bound a static region are always simple. The results are applicable to a wide range of Kaluza-Klein, supergravity and stringy models with multiple dilaton and moduli fields.