Finite-Distance Gravitational Deflection of Massive Particles by the Kerr-like Black Hole in the Bumblebee Gravity Model

Abstract

In this paper, we study the weak gravitational deflection angle of relativistic massive particles by the Kerr-like black hole in the bumblebee gravity model. In particular, we focus on weak field limits and calculate the deflection angle for a receiver and source at a finite distance from the lens. To this end, we use the Gauss-Bonnet theorem of a two-dimensional surface defined by a generalized Jacobi metric. The spacetime is asymptotically non-flat due to the existence of a bumblebee vector field. Thus the deflection angle is modified and can be divided into three parts: the surface integral of the Gaussian curvature, the path integral of a geodesic curvature of the particle ray and the change in the coordinate angle. In addition, we also obtain the same results by defining the deflection angle. The effects of the Lorentz breaking constant on the gravitational lensing are analyzed. We then consider the finite-distance correction for the deflection angle of massive particles.

Kerr black hole surrounded by a cloud of strings and its weak gravitational lensing in Rastall gravity

“…The spatial part of time-like geodesic in spacetime is the spatial geodesic in the corresponding Jacobi metric space and thus the Jacobi metric method has been widely used in the study of particle motion in curved spacetime. For example, it was used to study Kepler orbit [105,106], gravitational deflection of massive particles [75,89,90], the motion of charged particles [107], and Hawking radiation [108]. As mentioned above, the Jacobi-Maupertuis metric of a stationary spacetime is a Randers-Finsler metric described by ( ᾱ, β), say…”

“…[78], Ono et al proposed a generalized optical method to study the finite-distance deflection of light. Recently, Li et al [89,90] extended the study to massive particles case based on JMRF metric.…”

“…Haroon et al studied the finite-distance deflection of light by a rotating BH in perfect fluid dark matter with a cosmological constant [88]. Quite recently, by using Jacobi-Maupertuis Randers-Finsler (JMRF) metric and GB theorem, Li et al studied the finitedistance gravitational deflection of massive particles both in asymptotically flat spacetime [89] and asymptotically non-flat spacetime [90].…”