Spherically symmetric scalar vacuum: no-go theorems, black holes and solitons

Summary: "We prove some theorems characterizing the global properties of static, spherically symmetric configurations of a self-gravitating real scalar field phi in general relativity (GR) in various dimensions, with an arbitrary potential V(phi), which is not necessarily positive-definite. The results are extended to sigma models, scalar-tensor and curvature-nonlinear theories of gravity. We show that the list of all possible types of space-time causal structure in the models under consideration is the same as the one for phi={rm const}, namely, Minkowski (or AdS), Schwarzschild, de Sitter and Schwarzschild-de Sitter, and all horizons are simple. In particular, these theories do not admit regular black holes with any asymptotics. Some special features of (2+1)-dimensional gravity are revealed. We give examples of two types of asymptotically flat configurations with positive mass in GR, still admitted by the above theorems: (i) a black hole with a nontrivial scalar field (`scalar hair') and (ii) a particlelike (solitonic) solution with a regular centre; in both cases, the potential V(phi) must be at least partly negative. We also discuss the global effects of conformal mappings that connect different theories and illustrate such effects for solutions with a conformal scalar field in GR."