Phantom hairy black holes and wormholes in Einstein-bumblebee gravity

Abstract

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 $$\ell &gt;-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , the phantom field is admissible and the conventional scalar field is forbidden; if $$\ell &lt;-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , the phantom field is forbidden and the conventional scalar field is admissible. By defining the Killing potential $$\omega ^{ab}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ω</mml:mi> <mml:mrow> <mml:mi>ab</mml:mi> </mml:mrow> </mml:msup> </mml:math> , 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 96:251101), the first law cannot be constructed at all. We also show there still exists a stable circular orbit around the BP black hole. When the bumblebee potential is linear, we find that the phantom potential and the Lagrange-multiplier $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> behave as a cosmological constant $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> .