Light deflection and Gauss-Bonnet theorem: definition of total deflection angle and its applications

In this paper, we re-examine the light deflection in the Schwarzschild and the Schwarzschild-de Sitter spacetime. First, supposing a static and spherically symmetric spacetime, we propose the definition of the total deflection angle a of the light ray by constructing a quadrilateral Sigma(4) on the optical reference geometry M-opt determined by the optical metric (g)over-bar(ij). On the basis of the definition of the total deflection angle alpha and the Gauss-Bonnet theorem, we derive two formulas to calculate the total deflection angle alpha; (1) the angular formula that uses four angles determined on the optical reference geometry M-opt or the curved (r, phi) subspace M-sub being a slice of constant time t and (2) the integral formula on the optical reference geometry M-opt which is the areal integral of the Gaussian curvature K in the area of a quadrilateral Sigma(4) and the line integral of the geodesic curvature K-g along the curve C-Gamma. As the curve C-Gamma, we introduce the unperturbed reference line that is the null geodesic Gamma on the background spacetime such as the Minkowski or the de Sitter spacetime, and is obtained by projecting Gamma vertically onto the curved (r, phi) subspace M-sub. We demonstrate that the two formulas give the same total deflection angle alpha for the Schwarzschild and the Schwarzschild-de Sitter spacetime. In particular, in the Schwarzschild case, the result coincides with Epstein-Shapiro's formula when the source S and the receiver R of the light ray are located at infinity. In addition, in the Schwarzschild-de Sitter case, there appear order O(Lambda m) terms in addition to the Schwarzschild-like part, while order O(Lambda) terms disappear.