http://arxiv.org/abs/1708.04011
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 $\alpha$ of the light ray by constructing a quadrangle $\Sigma^4$ on the optical reference geometry ${\cal M}^{\rm opt}$ determined by the optical metric $\bar{g}{ij}$. 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 ${\cal M}^{\rm opt}$ or the curved $(r, \phi)$ subspace ${\cal M}^{\rm sub}$ being a slice of constant time $t$ and (2) the integral formula on the optical reference geometry ${\cal M}^{\rm opt}$ which is the areal integral of the Gaussian curvature $K$ in the area of a quadrangle $\Sigma^4$ and the line integral of the geodesic curvature $\kappa_g$ along the curve $C{\Gamma}$. The curve $C_{\Gamma}$ is the unperturbed reference straight line that is the geodesic $\Gamma$ on the background spacetime such as the Minkowski or the de Sitter spacetime, and therefore $C_{\Gamma}$ is a straight line in the Euclidean sense. 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 ${\cal O}(\Lambda m)$ terms in addition to the Schwarzschild-like part, while order ${\cal O}(\Lambda)$ terms disappear.
H. Arakida
Wed, 16 Aug 17
34/46
Comments: 34 pages, 6 figures. Submitted to PRD