Applying The Jacobi-Randers Metric To The Study Of The Gravitational-Magnetic Deflection

This thesis investigates the deflection effects of charged particles in curved space-time with electromagnetic field.To this end,we employ the Jacobi metric,one of the fundamental tools in geometrodynamics,and the Gauss-Bonnet theorem,one of the most important theorems in differential geometry.The role of the Jacobi metric is to convert the non-geodesic motion of charged particles in the 4-dimensional curved space-time+electromagnetic field into geodesic motion in 3-dimensional space.Specifically,in this case,the Jacobi metric is a special class of Finsler metric,the so-called the Randers metric.The application of the Gauss-Bonnet theorem embodies the geometric spirit:curvature determines the deflection angle.This work presents several methods for calculating the deflection angle and examines some typical lensing models.From a theoretical perspective,this study contributes to the application of Finsler geometry in physics and expands upon the research scope of geometrodynamics.From an ob-servational standpoint,this research aids in understanding the origin and propagation of high-energy particles,as well as the electromagnetic field structure surrounding the lens.The main text begins in Chapter 2,with specific contents as follows.Chapter 2 introduces the basic method of studying gravitational deflection using the Gauss-Bonnet theorem and Jacobi metric.The Jacobi metric for a charged particle in a curved spacetime and electromagnetic field background is a Randers type Finsler metric（Jacobi-Randers metric）.Therefore,this chapter also introduces basic concepts such as Finsler geometry,Randers metric and Zermelo navigation problem.In addi-tion,it is still a open question to study gravitational deflection using the Gauss-Bonnet theorem in Finsler geometry.Thus we introduces two Riemannian methods,including the osculating Riemannian manifold method and the generalized Jacobi metric method,which have been applied in previous studies on the deflection of neutral particles（in-cluding photons）.According to Maupertuis principle,we extends these two methods to the case of charged particles.Chapter 3 uses the Jacobi-Randers metric to study the motion of charged particles.These include the derivation of the equations of motion from the geodesic equations of Finsler geometry,as well as a discussion on the gauge invariance of the equations of motion.By drawing an analogy between the equations of motion and the Lorentz equations,the gravitational-magnetic deflection is defined.In light of the asymmetry of Finsler geometry,it is noted that the gravitational-magnetic deflection is a Finsler effect.This chapter introduces two methods for calculating the deflection angle of particles.The first utilizes Zermelo navigation,a fundamental tool of Finsler geometry,while the second employs the F³⁺¹stationary spacetime of the Jacobi-Randers metric and calculates the deflection angle of charged particles through null geodesics.Additionally,we derive the velocity and trajectory equations for particles moving in the equatorial plane based on the Jacobi-Randers metric.In this context,we outlines a weak-field iterative method for determining particle trajectories,as well as an iterative rule for calculating the deflection angle utilizing the Gauss-Bonnet theorem.In Chapter 4,we propose a method for studying weak-field deflection angles us-ing geodesic circular orbits and the Gauss-Bonnet theorem,resulting in a new formula for calculating the deflection angle.This formula is applicable to both asymptotic deflection angles and finite-distance deflection angles.Furthermore,it is suitable for both asymptotically flat spacetimes and asymptotically non-flat spacetimes,thus of-fering a solution to the divergence issues encountered when employing the conven-tional Gauss-Bonnet theorem method in certain asymptotically non-flat spacetimes,such as Schwarzschild-de Sitter spacetime.Moreover,this chapter introduces a method for investigating geodesic circular orbits using the Jacobi metric（including Jacobi-Riemannian and Jacobi-Randers metrics）and derives the equations for geodesic circu-lar orbits.The existence of geodesic circular orbits can be assessed through these orbit equations.Combining these approaches,we apply the new formula to several typical asymptotically non-flat spacetimes,including Schwarzschild-de Sitter spacetime,spher-ically symmetric spacetime in conformal Weyl gravity,Schwarzschild-like spacetime in Bumblebee gravity,and Kerr-de Sitter spacetime.In Chapter 5,we investigate the weak-field deflection of charged particles in Kerr-Newman spacetime.This chapter examines the motion of charged particles from three perspectives:the Jacobi-Randers metric,its Zermelo navigation representation,and the F³⁺¹stationary spacetime representation,thereby employing the osculating Riemannian manifold method,Zermelo navigation method,and geodesic method to calculate the deflection angle.The results obtained from the three methods are consistent.Compared to the deflection of photons or neutral massive particles,the gravitational-magnetic field of charged particles includes an additional spin-charge coupling term.The presence of this term leads to intriguing characteristics regarding the influence of spacetime spin on the deflection angle of charged particles.In particular cases,the deflection angles of charged particles in both positive and negative directions are the same.Chapter 6 explores the deflection of charged particles in a Schwarzschild spacetime+dipole magnetic field background.In this context,the dipole magnetic field is gen-erated by a toroidal current encircling the black hole in the equatorial plane,and its impact on spacetime curvature can be disregarded.This chapter employs the osculat-ing Riemannian manifold method,the Zermelo navigation method,and the generalized Jacobi metric method to calculate the weak-field deflection angle.In particular,by analogy with the influence of Kerr space-time spin on the deflection angle of neutral particles,we investigate the gravitational lensing of charged particles and apply it to constrain the magnetic fields of M87^*and Sgr A^*supermassive black holes.Chapter 7 serves as a summary of the work presented in this thesis and offers an outlook on potential future research.Appendix A introduces the concept of an optical metric for massive particles,which can be employed to calculate the travel time of particles and,consequently,investigate time-delay effects.Appendix B demonstrates that by selecting an appropriate coordinate system（e.g.,harmonic coordinates）,one can circumvent the computational complexities associated with the osculating Riemannian manifold approach.