LAUSR

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 $$\alpha $$α of the light ray by constructing a quadrilateral $$\varSigma ^4$$Σ4 on the optical reference geometry $${\mathscr {M}}^\mathrm{opt}$$Mopt determined by the optical metric $$\bar{g}_{ij}$$g¯ij. On the basis of the definition of the total deflection angle $$\alpha $$α and the Gauss–Bonnet theorem, we derive two formulas to calculate the total deflection angle $$\alpha $$α; (1) the angular formula that uses four angles determined on the optical reference geometry $${\mathscr {M}}^\mathrm{opt}$$Mopt or the curved $$(r, \phi )$$(r,ϕ) subspace $${\mathscr {M}}^\mathrm{sub}$$Msub being a slice of constant time t and (2) the integral formula on the optical reference geometry $${\mathscr {M}}^\mathrm{opt}$$Mopt which is the areal integral of the Gaussian curvature K in the area of a quadrilateral $$\varSigma ^4$$Σ4 and the line integral of the geodesic curvature $$\kappa _g$$κg along the curve $$C_{\varGamma }$$CΓ. As the curve $$C_{\varGamma }$$CΓ, we introduce the unperturbed reference line that is the null geodesic $$\varGamma $$Γ on the background spacetime such as the Minkowski or the de Sitter spacetime, and is obtained by projecting $$\varGamma $$Γ vertically onto the curved $$(r, \phi )$$(r,ϕ) subspace $${\mathscr {M}}^\mathrm{sub}$$Msub. We demonstrate that the two formulas give the same total deflection angle $$\alpha $$α 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 $${\mathscr {O}}(\varLambda m)$$O(Λm) terms in addition to the Schwarzschild-like part, while order $${\mathscr {O}}(\varLambda )$$O(Λ) terms disappear.