We discuss a possible extension of calculations of the bending angle of light in a static, spherically symmetric and asymptotically flat spacetime to a non-asymptotically flat case. We examine a relation between the bending angle of light and the Gauss-Bonnet theorem by using the optical metric. A correspondence between the deflection angle of light and the surface integral of the Gaussian curvature may allow us to take account of the finite distance from a lens object to a light source and a receiver. Using this relation, we propose a method for calculating the bending angle of light for such cases. Finally, this method is applied to two examples of the non-asymptotically flat spacetimes to suggest finite-distance corrections: Kottler (Schwarzschild-de Sitter) solution to the Einstein equation and an exact solution in Weyl conformal gravity. 95.30.Sf, 98.62.Sb

Continuing work initiated in an earlier publication [Ishihara, Suzuki, Ono, Kitamura, Asada, Phys. Rev. D 94, 084015 (2016) ], we discuss a method of calculating the bending angle of light in a static, spherically symmetric and asymptotically flat spacetime, especially by taking account of the finite distance from a lens object to a light source and a receiver. For this purpose, we use the Gauss-Bonnet theorem to define the bending angle of light, such that the definition can be valid also in the strong deflection limit. Finally, this method is applied to Schwarzschild spacetime in order to discuss also possible observational implications. The proposed corrections for Sgr A * for instance are able to amount to ∼ 10 −5 arcseconds for some parameter range, which may be within the capability of near-future astronomy, while also the correction for the Sun in the weak field limit is ∼ 10 −5 arcseconds. 95.30.Sf, 98.62.Sb