Geodesic motion of test particles around the scalar hairy black holes with asymmetric vacua

Abstract An asymptotically flat hairy black hole (HBH) can exhibit distinct characteristics when compared to the Schwarzschild black hole, due to the evasion of no-hair theorem by minimally coupling the Einstein gravity with a scalar potential which possesses asymmetric vacua, i.e, a false vacuum $$(\phi =0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and a true vacuum $$(\phi =\phi _1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ϕ</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . In this paper, we investigate the geodesic motion of both massive test particles and photons in the vicinity of HBH with $$\phi _1=0.5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0.5</mml:mn> </mml:mrow> </mml:math> and $$\phi _1=1.0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1.0</mml:mn> </mml:mrow> </mml:math> by analyzing their effective potentials derived from the geodesic equation. By fixing $$\phi _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> , the effective potential of a massive test particle increases monotonically when its angular momentum L is very small. When L increases to a critical value, the effective potential possesses an inflection point which is known as the innermost stable circular orbit (ISCO), where the test particle can still remain stable in a circular orbit with a minimal radius without being absorbed by the HBH or fleeing to infinity. Beyond the critical value of L , the effective potential possesses a local minimum and a local maximum, indicating the existence of unstable and stable circular orbits, respectively. Moreover, the HBH possesses an unstable photon sphere but its location slightly deviates from the Schwarzschild black hole. The trajectories of null geodesics in the vicinity of HBH can also be classified into three types, which are the direct, lensing, and photon sphere, based on the deflection angle of light, but the values of impact parameters can vary significantly than the Schwarzschild black hole.