Study On Gravitational Lensing Using The Gauss-Bonnet Theorem

Recently,Gibbons and Werner pioneered an elegant geometric approach to studying the gravitational lensing.They applied the Gauss-Bonnet theorem to the corresponding twodimensional optical geometry of static spacetime,and found that the deflection angle is only related to the intrinsic curvature of the space(Gauss curvature),and the formula for calculat ing the gravitational deflection is obtained.Werner then used Finsler geometry to generalize their methods to stationary spacetime.Based on the work of Gibbons and Werner,this paper studies the gravitational lensing of photon and relativistic massive particles for different gravitational sources.Firstly,using Gibbons-Werner method,we study the light deflection of two kinds of static spherical symmetry spacetime,including Elis wormhole spacetime and Janis-NewmanWinicour wormhole spacetime.Using the coordinate transformation method,we obtained the harmonic gauge solutions for these spacetimes.The Gibbons-Werner framework is then substituted into the harmonic coordinate system.We use the post-Minkowski iterative technique to find the photon orbit,and apply the Gauss-Bonnet theorem to the twodimensional optical geometry corresponding to the background spacetime to obtain the gravitational deflection angle.Secondly,we study the light deflection in stationary spacetime taking Kerr-Newman black hole as an example.Since the optical geometry corresponding to the stationary spacetime is the Randers type Finsler geometry,the direct application of the Gauss-Bonnet theorem will be very complicated.To this end,we use two methods: Werner's osculating Riemannian manifold method and the Ono-Ishihara-Asada geodesic curvature method.We calculate the deflection angle to the third order in the harmonic coordinate system,demonstrating the equivalence of this new geometric method applying the Gauss-Bonnet theorem with the standard geodesic method.Thirdly,based on the Jacobi metric,we extend the Gibbons-Werner method for the relativistic massive particle in static spacetime.We investigate in detail three kinds of static spherical wormhole spacetime: Janis-Newman-Winicour wormhole spacetime,a class of scalar-tensor wormhole spacetime and Einstein-Maxwell-dilaton wormhole spacetime.The perturbation method is used to find the first-order particle orbit.By accumulating the Gauss curvature,we obtain the second-order gravitational deflection angle.Finally,we use the Jacobi-Maupertuis Randers-Finsler metric to generalize Werner's method to the relativistic massive particles for stationary spacetime.We study two types of stationary spacetime: rotating black hole(Kerr black hole)and rotating wormhole(Teo wormhole).First,we construct the Jacobi-Maupertuis Randers-Finsler metric corresponding to the stationary spacetime,and then construct the corresponding Riemannian manifold.Based on this,the Gauss-Bonnet theorem is used to calculate the deflection angle up to leading terms.