A Novel Method for Calculating Deflection Angle

In this paper, we introduce a method for calculating the deflection angle in the weak-field approximation, applicable to both null and timelike rays. By combining the trajectory equation $\mathcal{Z}(u)=(du/d\phi)^2$ and the `straight line' $u(\varphi)={\sin\varphi}/b$, we introduce a new function $\Phi(\varphi)$. The deflection angle can then be expressed as $\delta=\Phi(0)+\Phi(\pi)-\pi$, which directly depends on the impact parameter rather than the closest approach distance. This method offers a convenient and straightforward approach to calculations, avoiding the complexities of integration or iterative procedures. As an illustrative application, we compute the deflection angle for charged particle in the Kerr-Newman spacetime.Comment: 5 page

Deflection of charged signals in a dipole magnetic field in Kerr background

This paper investigates charged particle deflection in a Kerr spacetime background with a dipole magnetic field, focusing on the equatorial plane and employing the weak field approximation. We employ the Jacobi-Randers metric to unify the treatment of the gravitational and electromagnetic effects on charged particles. Furthermore, we utilize the Gauss-Bonnet theorem to calculate the deflection angle through curvature integrals. The difference between the prograde and retrograde deflection angles is linked to the non-reversibility of metrics and geodesics in Finsler geometry, revealing that this difference can be considered a Finslerian effect. We analyze the impact of both gravitomagnetic field and dipole magnetic field on particle motion and deflection using the Jacobi-Randers magnetic field. The model considered in this paper exhibits interesting features in the second-order approximation of ($M/b$). When $q\mu=2MaE$, the Jacobi-Randers metric possesses reversible geodesics, leading to equal prograde and retrograde deflection angles. In this case, the gravitomagnetic field and dipole magnetic field cancel each other out, distinguishing it from scenarios involving only the gravitomagnetic field or the dipole magnetic field. We also explore the magnetic field's impact on gravitational lensing of charged particles.Comment: 10 pages, 1 figur

The finite-distance gravitational deflection of massive particles in stationary spacetime: a Jacobi metric approach

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 receive 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 finite-distance of source or receiver, and the relativistic velocity in microlensings and lensing by galaxies. It is found in these cases, the effect of BH 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.Comment: 16 pages, 4 figure

Deflection of charged signals in a dipole magnetic field in Schwarzschild background using Gauss-Bonnet theorem

This paper studies the deflection of charged particles in a dipole magnetic field in Schwarzschild spacetime background in the weak field approximation. To calculate the deflection angle, we use Jacobi metric and Gauss-Bonnet theorem. Since the corresponding Jacobi metric is a Finsler metric of Randers type, we use both the osculating Riemannian metric method and generalized Jacobi metric method. The deflection angle up to fourth order is obtained and the effect of the magnetic field is discussed. It is found that the magnetic dipole will increase (or decrease) the deflection angle of a positively charged signal when its rotation angular momentum is parallel (or antiparallel) to the magnetic field. It is argued that the difference in the deflection angles of different rotation directions can be viewed as a Finslerian effect of the non-reversibility of the Finsler metric. The similarity of the deflection angle in this case with that for the Kerr spacetime allows us to directly use the gravitational lensing results in the latter case. The dependence of the apparent angles on the magnetic field suggests that by measuring these angles the magnetic dipole might be constrained.Comment: 13 pages, 3 figure

The deflection of charged massive particles by a 4-Dimensional charged Einstein-Gauss-Bonnet black hole

Based on the Jacobi metric method, this paper studies the deflection of a charged massive particle by a novel 4-dimensional charged Einstein-Gauss-Bonnet black hole. We focus on the weak-filed approximation and consider the deflection angle with finite-distance effects, i.e. the source and observer at finite distances from the black hole. To this end, we use a geometric and topological method, which is to apply the Gauss-Bonnet theorem to the Jacobi-metric surface to calculate the deflection angle. We find that the deflection angle contains a pure gravitational contribution $\delta_g$, a pure electrostatic one $\delta_c$ and a gravitational-electrostatic coupling term $\delta_{gc}$. We also show that the electrostatic contribution $\delta_c$ can also be computed by the Jacobi-metric method using the GB theorem to a charge in a Minkowski flat spacetime background. We find that the deflection angle increases(decreases) if the Gauss-Bonnet coupling constant $\alpha$ is negative(positive). Furthermore, the effects of the BH charge, the particle charge-to-mass ratio and the particle velocity on the deflection angle are analyzed.Comment: 11 pages, 5 Figures; conclusion part improved and reference adde