Phantom hairy black holes and wormholes in Einstein-bumblebee gravity

TL;DR

This paper explores how Lorentz violation in Einstein-bumblebee gravity allows for new hairy black hole and wormhole solutions with scalar fields, modifying the traditional no-hair theorem and thermodynamic laws.

Contribution

It introduces novel hairy black hole and wormhole solutions in Einstein-bumblebee gravity, analyzing the effects of Lorentz violation on black hole hair and thermodynamics.

Findings

Lorentz violation alters the no-hair theorem for black holes.

Phantom and conventional scalar fields are allowed depending on LV coupling.

The Smarr formula and first law are preserved with LV, but not without it.

Abstract

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 $\omega^{ab}$, 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 $\lambda$ behave as a cosmological constant $\Lambda$.