The finite-distance gravitational deflection of massive particles in stationary spacetime: a Jacobi metric approach
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Regular Article - Theoretical Physics
The finite-distance gravitational deflection of massive particles in stationary spacetime: a Jacobi metric approach Zonghai Li1, Junji Jia2,a 1 2
MOE Key Laboratory of Artificial Micro- and Nano-structures, School of Physics and Technology, Wuhan University, Wuhan 430072, China Center for Astrophysics and MOE Key Laboratory of Artificial Micro- and Nano-structures, School of Physics and Technology, Wuhan University, Wuhan 430072, China
Received: 13 December 2019 / Accepted: 18 January 2020 © The Author(s) 2020
Abstract In this paper, we study the weak gravitational deflection of relativistic massive particles for a receiver and source at finite distance from the lens in stationary, axisymmetric and asymptotically flat spacetimes. For this purpose, we extend the generalized optical metric method to the generalized Jacobi metric method by using the Jacobi– Maupertuis Randers–Finsler metric. More specifically, we apply the Gauss–Bonnet theorem to the generalized Jacobi metric space and then obtain an expression for calculating the deflection angle, which is related to Gaussian curvature of generalized optical metric and geodesic curvature of particles orbit. In particular, the finite-distance correction to the deflection angle of signal with general velocity in the the Kerr black hole and Teo wormhole spacetimes are considered. Our results cover the previous work of the deflection angle of light, as well as the deflection angle of massive particles in the limit for the receiver and source at infinite distance from the lens object. In Kerr black hole spacetime, we compared the effects due to the black hole spin, the finitedistance of source or receiver, and the relativistic velocity in microlensings and lensing by galaxies. It is found in these cases, the effect of black hole spin is usually a few orders larger than that of the finite-distance and relativistic velocity, while the relative size of the latter two could vary according to the particle velocity, source or observer distance and other lensing parameters.
1 Introduction 100 years ago, Eddington et al. [1–3] firstly verified the general relativity through the deflection of light in the solar gravitational field. Nowadays gravitational lensing (GL) becomes a powerful tool in astrophysics and cosmology. For examples, a e-mail:
[email protected] (corresponding author)
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it is used to measure the mass of galaxies and clusters [4–6] and to to detect dark matter and dark energy [7–12]. One of the main quantities in the study of GL is the deflection angle. Various approaches relying on the geodesics were built to calculate it. In 2008, Gibbons and Werner [13] proposed a geometrical and topological method of studying the gravitational deflection of light in a static and spherically symmetric spacetime using the Gauss–Bonnet (GB) theorem. Later, Werner [14] extended this method to the rotating and stationary spacetimes by using the Randers– Finsler geometry. In Gibbons–Werner method, the deflection angl
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