Generalized Gibbons-Werner method for deflection angle

Abstract

The Gibbons-Werner method has been widely used in the gravitational deflection problem of particles moving in the curved spacetime. By applying the Gauss-Bonnet theorem (GBT) to regions constructed in a two-dimensional manifold, the deflection angle can be expressed and calculated in a geometric manner. For asymptotically flat spacetimes, an infinite region is usually chosen. While for some asymptotically nonflat spacetimes, the infinite region is ill defined due to the singular behavior of the metric, thus some special regions were constructed to solve the ill-defined problem. We propose a generalized method in which the choice freedom of the region where the GBT is applied is significantly enlarged. Our generalized method not only describes the conventional methods uniformly, but also lead to a formula which greatly simplifies the calculation. Additionally, we work out the deflection angle of the massive particle in the static spherically symmetric spacetime in conformal Weyl gravity based on the generalized method and the new formula.