Phantom hairy black holes and wormholes in Einstein-bumblebee gravity
In this paper we study Einstein-bumblebee gravity theory minimally coupled with external matter -- a phantom/non-phantom(conventional) scalar field, and derive a series of hairy solutions -- bumblebee-phantom(BP) and BP-dS/AdS black hole solutions, regular Ellis-bumblebee-phantom (EBP) and BP-AdS wormholes, etc. We first find that the Lorentz violation (LV) effect can change the so called black hole no-hair theorem and these scalar fields can give a hair to a black hole. If LV coupling constant $\ell>-1$, the phantom field is admissible and the conventional scalar field is forbidden; if $\ell<-1$, the phantom field is forbidden and the conventional scalar field is admissible. By defining the Killing potential $ω^{ab}$, we study the Smarr formula and the first law for the BP black hole, find that the appearance of LV can improve the structure of these phantom hairy black holes -- the conventional Smarr formula and the first law of black hole thermodynamics still hold; but for no LV case, i.e., the regular phantom black hole reported in [Phys. Rev. Lett. {\bf96}, 251101], the first law cannot be constructed at all. When the bumblebee potential is linear, we find that the phantom potential and the Lagrange-multiplier $λ$ behave as a cosmological constant $Λ$.