arXiv subjects

Zonghai Li

Publications and source records attributed to Zonghai Li.

17 recordsLinked to original sources

arXiv2025

From Spray to Metric: The Geometric Construction of the Jacobi Metric

This paper develops a systematic approach to the geometrization of dynamics from the viewpoint of the geodesic equation. The method promotes a semispray to a spray through the imposition of suitable dynamical constraints, and the associated metric structure is extracted via reparameterization. When applied to static spacetimes, this spray-to-metric framework recovers the optical metric, the Jacobi metric for massive particles, and its generalization for charged particles in electromagnetic fields. We further show that a Randers-type Finsler metric arises naturally in the planar circular restricted three-body problem. By establishing a direct pathway from equations of motion to metric structures, this work offers a geometric perspective, independent of the traditional variational framework, may provide a basis for further studies on dynamical systems.

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arXiv2025

The bending of a straight line

In gravitational lensing under the weak-field approximation, the usual viewpoint is that light bending measures how a ray deviates from a straight line in Euclidean space. In this work, we take the opposite perspective: we ask how a straight line bends in a curved space, such as optical geometry--that is, how it deviates from geodesics. Using the Gauss-Bonnet theorem, we show that, at leading order, the deflection angle can be written as the integral of the geodesic curvature of a straight line in curved space. This reformulation emphasizes the global, coordinate-independent nature of the deflection angle and provides a complementary way of understanding the classical Gibbons-Werner method. To illustrate the idea, we apply it to three familiar spacetimes--Schwarzschild, Reissner-Nordstr\"om, and Kerr--and recover the well-known results. Furthermore, we extend the method to massive particles using the Jacobi metric, and illustrate it with the Reissner-Nordstr\"om spacetime.

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arXiv2024

Gravitational Lensing Using Werner's Method in Cartesian-like Coordinates

The Gibbons-Werner method for calculating deflection angles using the Gauss-Bonnet theorem and optical/Jacobi metric has become widely popular in recent years. Werner extended this method to stationary spacetimes, where the optical/Jacobi metric takes the form of a Finsler metric of Randers type, by adopting an osculating Riemannian metric. Werner's method is significant as it provides a concise expression for the deflection angle, retains applicability for gravitational lensing in Finsler geometry beyond the Randers type, and has the potential to stimulate widespread application of Finsler geometry across diverse fields. However, because of the cumbersome calculations required in Werner's method using conventional coordinates $(r,\phi)$, it has not been widely adopted. The aim of this paper is to alleviate the computational burden associated with Werner's method. To this end, we introduce Cartesian-like coordinates $(X,Y)$ to construct the osculating Riemannian metric and calculate the deflection angle using the Gauss-Bonnet theorem. We illustrate the current method with examples of the deflection of massive particles in Kerr spacetime, rotating Bardeen (Hayward) regular spacetime, and Teo rotating wormhole spacetime, respectively.

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arXiv2024

A Novel Method for Calculating Deflection Angle with Finite-Distance Correction

In recent study in Ref.[7] (arXiv: 2401.12525), we have introduced a method aimed at calculating the weak-field asymptotic deflection angle. This method offers an efficient computational approach that avoids the complexities of integration and cumbersome iterative procedures typically associated with deflection angle calculations. In the present paper, we expand upon this method to encompass finite-distance deflection scenarios, wherein it is postulated that both the distances from the source to the lens and from the observer to the lens are finite. Importantly, this extension naturally encompasses the case of a lens in asymptotically non-flat spacetime. As an illustrative example, we apply this method to compute the gravitational deflection angle of massive particles in Kerr spacetime while accounting for the effects of finite distance.

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arXiv2024

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.

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arXiv2024

Generating New Spacetimes through Zermelo Navigation

Zermelo navigation is not only a fundamental tool in Finsler geometry but also a fundamental approach to the geometrization of dynamics in physics. In this paper, we consider the Zermelo navigation problem on optical Riemannian space and, via Zermelo/Randers/spacetime triangle, explore the generation of new spacetimes from pre-existing ones. Whether the Randers metric has reversible geodesics corresponds to the presence of time-reversal symmetry in the generated spacetime. In cases where the Randers metric has reversible geodesics, we utilize a radial vector field to generate new static spacetimes from existing ones. For example, we can generate Schwarzschild, Rindler, de Sitter, and Schwarzschild-de Sitter spacetimes from flat spacetime. In fact, the Zermelo navigation method allows for the derivation of a variety of static spacetimes from flat spacetime. For multi-parameter spacetimes, they can be generated through various navigation paths. However, for some spacetimes, not all navigation paths may exist. In the second scenario, when the Randers metric does not have reversible geodesics, we employ a rotational vector field to transform non-flat static metrics into slowly rotating spacetimes. Alternatively, using a mixed vector field, we generate slowly rotating spacetimes starting from flat spacetime. We provide examples of generating Kerr spacetimes and Kerr-de Sitter spacetimes.

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arXiv2023

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.

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arXiv2022

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.

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arXiv2021

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

In this paper, an exact solution of Kerr black hole surrounded by a cloud of strings in Rastall gravity is obtained by Newman-Janis algorithm without complexification, and the influence of the string parameter $a_0$ on the black hole thermodynamics is studied. Furthermore, according to this black hole solution, we investigate the effect of a cloud of strings on gravitational deflection of massive particles. For a receiver and a source at infinite distance from the lens object, we use an osculating Riemannian manifold method. While the distance is finite, we apply the generalized Jacobi metric method. For the case between the two situations, the Jacobi-Maupertuis Randers-Finsler metric and Gauss-Bonnet theorem are employed. We find that the string parameter $a_0$ has obviously modification on the gravitational deflection of massive particles. Our result is reduced to the deflection angle of light by Kerr black hole surrounded by a cloud of strings in general relativity.

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arXiv2021

Kerr-Newman-Jacobi geometry and the deflection of charged massive particles

In this paper, we investigate the deflection of a charged particle moving in the equatorial plane of Kerr-Newman spacetime, focusing on weak field limit. To this end, we use the Jacobi geometry, which can be described in three equivalent forms, namely Randers-Finsler metric, Zermelo navigation problem, and $(n+1)$-dimensional stationtary spacetime picture. Based on Randers data and Gauss-Bonnet theorem, we utilize osculating Riemannian manifold method and the generalized Jacobi metric method to study the deflection angle, respectively. In the $(n+1)$-dimensional spacetime picture, the motion of charged particle follows the null geodesic, and thus we use the standard geodesic method to calculate the deflection angle. Three methods lead to the same second-order deflection angle, which is obtained for the first time. The result shows that the black hole spin $a$ affects the deflection of charged particles both gravitationally and magnetically at the leading order (order $\mathcal{O}([M]^2/b^2)$). When $qQ/E<2M$, $a$ will decrease (or increase) the deflection of prograde (or retrograde) charged signal. If $qQ/E> 2M$, the opposite happens, and the ray is divergently deflected by the lens. We also showed that the effect of the magnetic charge of the dyonic Kerr-Newman black hole on the deflection angle is independent of the particle's charge.

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arXiv2020

Deflection of charged massive particles by a four-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 four-dimensional charged Einstein-Gauss-Bonnet black hole. We focus on the weak field approximation and consider the deflection angle with finite distance effects. To this end, we use a geometric and topological method, which is to apply the Gauss-Bonnet theorem to the Jacobi space to calculate the deflection angle. We find that the deflection angle contains a pure gravitational contribution $\delta_g$, a pure electrostatic $\delta_c$ 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.

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arXiv2020

Circular orbit of a particle and weak gravitational lensing

The purpose of this paper is twofold. First, we introduce a geometric approach to study the circular orbit of a particle in static and spherically symmetric spacetime based on Jacobi metric. Second, we apply the circular orbit to study the weak gravitational deflection of null and time-like particles based on Gauss-Bonnet theorem. By this way, we obtain an expression of deflection angle and extend the study of deflection angle to asymptotically non-flat black hole spacetimes. Some black holes as lens are considered such as a static and spherically symmetric black hole in the conformal Weyl gravity and a Schwarzschild-like black hole in bumblebee gravity. Our results are consistent with the previous literature. In particular, we find that the connection between Gaussian curvature and the radius of a circular orbit greatly simplifies the calculation.

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arXiv2020

Equivalence of Gibbons-Werner method to geodesics method in the study of gravitational lensing

The Gibbons-Werner method where the Gauss-Bonnet theorem is applied to study the gravitational deflection angle has received much attention recently. In this paper, we study the equivalence of the Gibbons-Werner method to the standard geodesics method, and it is shown that the geodesics method can be derived with the Gibbons-Werner method, for asymptotically flat case. In the geodesics method, the gravitational deflection angle of particle depends entirely on the geodesic curvature of the particle ray in the Euclidean space. The gravitational deflection of light in Kerr-Newman spacetime is calculated by different technologies under the Gibbons-Werner framework, as an intuitive example to show the equivalence.

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arXiv2020

Gravitational deflection of relativistic massive particles by wormholes

In this paper, the gravitational deflection of relativistic massive particles up to the second post-Minkowskian order by static and spherically symmetric wormholes is investigated in the weak-field limit. These wormholes include the Janis-Newman-Winicour wormhole, a class of zero Ricci scalar scalar-tensor wormholes, and a class of charged Einstein-Maxwell-dilaton wormholes. With the Jacobi metric approach, the Gauss-Bonnet theorem is employed to study the gravitational deflection. In this scheme, the deflection angle as a topological effect is considered. Moreover, we analyze the influence of the spacetime parameters on the results.

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arXiv2020

Finite-distance gravitational deflection of massive particles by the Kerr-like black hole in the bumblebee gravity model

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. In particular, we correct a mistake in the previous literature. Furthermore, we consider the finite-distance correction for the deflection angle of massive particles.

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arXiv2019

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.

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arXiv2019

Gravitational Deflection of Massive Particles by a Schwarzschild Black Hole in Radiation Gauge

The exact metric of a Schwarzschild black hole in the true radiation gauge was recently reported. In this work, we base on this gravity and calculate the gravitational deflection of relativistic massive particles up to the fourth post-Minkowskian order. It is found that the result is consistent with the previous formulations for both the case of dropping the fourth-order contribution and the case of light deflection. Our result might be helpful for future high-accuracy observations.

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